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# MATHEMATICS BENCHMARKS

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```									Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)     2000

MATHEMATICS BENCHMARKS

O – means teacher should be able to observe throughout the day – possibly use anecdotal records.
I – Informal Assessment—those marked ―I‖ have an assessment task attached.

Uses properties to create and simplify algebraic expressions and solves linear
equations and inequalities

1)        I -   Solves equations in one variable using addition and subtraction
2)        I -   Models simple addition and subtraction problems using integers on a
number line
3)        I -   Recognizes and continues a number pattern and/or geometric representation
(e.g., triangular numbers)
4)        I -   States a rule to explain a number pattern
5)        I -   Uses whole numbers to complete a function table based on a given rule
6)        I -   Creates and solves proportional equations using one variable

Interprets, organizes, and makes predictions using appropriate probability and
statistics techniques

7)    I     -   Reads and constructs line, bar, and pictographs
8)    I     -   Reads and interprets circle graphs using percents
9)    I     -   Constructs and explains a frequency table
10)   I     -   Uses probability to predict the outcome of a single event and expresses the
result as a fraction or decimal
11) I       -   Estimates and compares data to include mean, median, and mode
12) I       -   Solves problems involving combinations

Writes and solves problems involving standard units of measurement

13) I       -   Measures length to the nearest one-sixteenth inch
14) I       -   Identifies appropriate units for measuring length, weight, volume, and
temperature in the standard (English and metric) systems
15) I       -   Uses appropriate mathematical tools for determining length, weight, volume,
and temperature in the standard (English and metric) systems
16) I       -   Uses estimation to solve problems in the standard (English and metric)
systems
17) I       -   Converts units within a measurement system

(136)
Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

Determines the relationships and properties of two and three-dimensional
geometric figures and the application of properties and formulas of coordinate
geometry

18) I     -   Locates points in all four quadrants of the coordinate plane
19) I     -   Draws points, lines (parallel, perpendicular, intersecting), line segments, and
rays
20) I     -   Identifies, classifies, and measures right, acute, obtuse, and straight angles
21) I     -   Creates tessellations with polygons
22) I     -   Explores the relationships of three-dimensional figures, including vertices,
faces, and edges using manipulative materials
23) I     -   Describes, compares, constructs, classifies, and identifies flips, slides, and,
turns (reflections, translations, and rotations)
24) I     -   Calculates the area of parallelograms (squares and rectangles) without using
calculators
25) I     -   Finds the circumference of a circle with and without the use of manipulative
materials
26) I     -   Determines the area of a circle with and without the use of a calculator
27) I     -   Finds the volume of cubes and rectangular prisms with and without the use
of calculators

Uses basic concepts of number sense and performs operations involving
exponents, scientific notation, and order of operations

28)   I   -   Reads, writes, and rounds twelve-digit whole numbers
29)   I   -   Compares and orders whole numbers using <, >, and =
30)   I   -   Writes twelve-digit whole numbers using expanded notation
31)   I   -   Reads, writes, and rounds decimal numbers to the nearest ten-thousandth
32)   I   -   Compares and orders decimal numbers using <, >, and =
33)   I   -   Writes decimal numbers through the nearest ten-thousandth using expanded
notation
34)   I   -   Uses estimation to determine accuracy of solutions
35)   I   -   Multiplies a three-digit decimal number by a two-digit decimal number
36)   I   -   Divides a five-digit decimal number by a two-digit decimal number
37)   I   -   Rounds decimal quotients to the nearest whole number, tenths, and
hundredth
38) I     -   Estimates and solves one and two-step problems involving addition,
subtraction, multiplication and division of decimals, with and without
calculators
39) I     -   Uses the rules of divisibility to determine factors and multiples of a given
number
40) I     -   Explores the relationships among integers
41) I     -   Models and writes the prime factorization of a number using exponential
notation
42) I    -   Distinguishes between prime and composite numbers with and without the
use of calculators

(137)
Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

43) I     -   Uses the greatest common factor (GCF) to simplify fractions
44) I     -   Uses the least common multiple (LCM) to find common denominators

Determines relationships among real numbers to include fractions, decimals,
percents, ratios, and proportions in real life problems

45)   I   -   Converts among fractions, decimals, and percents
46)   I   -   Finds the percent of a number
47)   I   -   Estimates and calculates sale price and/or original price using discount rates
48)   I   -   Compares and orders fractions as well as mixed numerals
49)   I   -   Determines equivalent forms of fractions
50)   I   -   Uses a variety of techniques to express a fraction in simplest form
51)   I   -   Locates fractions, decimals, and mixed numerals on a number line
52)   I   -   Adds and subtracts mixed numerals with and without regrouping, expressing
the answer in simplest form using like and unlike denominators
53) I     -   Multiplies and divides proper fractions as well as mixed numerals expressing
54) I     -   Estimates, solves, and compares solutions to one and two-step problems
involving addition, subtraction, multiplication, and division of proper fractions
and mixed numerals
55) I     -   Demonstrates different ways to express ratios

(138)
Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

1) Can the student solve equations in                   Provide the student a problem with one
one variable using addition and                      number missing. Tell the student to let ―n‖
subtraction?                                         represent the missing number. Provide
counters as manipulatives so the student
can model equations.

n+7=9                     7–n=2
3 + n = 12                n–8=3

Have the student solve for n with and/or
without the manipulatives.

2) Can the student model simple                         Provide the students with manipulatives
addition and subtraction problems                    and integer problems such as:
using integers on a number line?                      -12 + -2
 -5 + 8
 -4 – 9
 7 – (-3)
 -2 – (-8)

3) Can the student recognize and                        Provide the student with a pattern such as
continue a number pattern and/or                     towering cans.
geometric representation?                                                              * * *
* * *         * * *
* * *        * * *         * * *
Tower 1      Tower 2       Tower 3

Have the student draw tower 4, 5 and 6.
Have the student determine how many
cans are in towers 10 and 15.

4) Can the student state a rule to                      Provide the student with a pattern such as
explain a number pattern?                            8, 4, 0, -4, and -8. Have the student
describe the pattern in three ways.

(139)
Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)        2000

5) Can the student using whole                          Provide the student with a rule and a
numbers complete a function table                    function such as y = x + 5. Have the
based on a given rule?                               student complete the function chart below:

Y

X    0    1    2     3    4    10 100

6) Can the student create and solve                     Provide the student with the following
proportional equations using one                     recipe. Create and solve proportional
variable?                                            equations using one variable.

Sausage Cheese Balls (Serves 20)

1 lb. Sharp cheddar cheese, grated
1 lb. hot sausage
1
2 c Bisquick
2

Mix together. Bake at 325 degrees for 20 minutes.
The student will determine the amount of
ingredients needed to serve 30 people.

7) Can the student read and construct                   Provide the student with a graph and set
line, bar, and pictographs?                          of questions that may be answered using
the graph. Have the student determine:
 The number of items represented by
the graph.
 The variable with the greatest number
of items represented.
 The variable with the least number of
items represented.
 The variable with a specific number of
items represented.

Provide the student with survey data in an
organized frequency table. Have the
student make a graph to display the data.

(140)
Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

8) Can the student read and interpret                   Provide the student with a circle graph
circle graphs using percents?                        (e.g., pie chart) and a set of questions that
can be answered by interpreting data
using percents. Have the student
determine:
 The percentage of occurrences of a
given variable (e.g., What percent
represents yellow?).
 The number of occurrences of a given
variable (e.g., If 20% were yellow, how
many out of 80 were yellow?).

9) Can the student construct and                        Have the student construct a T-Chart
explain a frequency table?                           (frequency table) to record responses for a
given survey question. Have the student
explain the data recorded in the frequency
table.

10) Can the student use probability to                  Have the student solve problems using
predict the outcome of a single event               probability to predict outcomes of a single
and express the result as a fraction                event occurrence (e.g., When you toss a
and/or decimal?                                     coin, what is the probability of it landing on

Have the student express the probability
as a fraction. Have the student express
the probability as a decimal.

8) Can the student estimate and                         Provide a given set of numbers such as
compare data to include mean,                        {15, 19, 14, 16, 15}. Have the student
median, and mode?                                    determine:
 The mean of the set (an average of the
numbers).
 The median (the center location when
the numbers are ordered least to
greatest).
 The mode (the number that occurs
most often in the set).

(141)
Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)         2000

12) Can the student solve problems                      Provide manipulatives like snack photos
involving combinations?                             and drink photos. Have the student model
and list the possible combinations
involving one snack and one drink.

13) Can the student measure length to                   Provide the student with a paper ruler
the nearest one-sixteenth inch?                     marked to sixteenths. Have a variety of
objects to be measured in class like a
paperclip, penny, pencil, and book. Have
the student measure these to the nearest
sixteenth in lowest terms.

14) Can the student identify appropriate                Provide the students with two sets of
units for measuring length, weight,                 cards. The first set names objects to be
volume, and temperature in standard                 measured, and the second set names the
(English and metric) systems?                       measurements. Match the appropriate
cards. Example:
st                                        nd
1 Set                                     2 set
Length of book                            10 in
Length of room                            10 ft
Length of school hallway                  10 yd

Example:
st                                        nd
1 set                                     2 set
weight of paperclip                       5 gm
3
volume of box                             120 cm
length of eraser                          7 cm

15) Can the student use appropriate                     Provide the student with customary
mathematical tools for determining                  measuring devices: ruler, yard stick, tape
length, weight, volume, and                         measure, and metric rulers. Have the
temperature in standard (English and                student measure the following :
metric) systems?                                     Length of pencil
 Length and width of book
 Height of chalk board

(142)
Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

Provide the student with thermometers
(Celsius and Fahrenheit) and warm and
cold water. Have the students find the
temperatures of the water in Celsius and
Fahrenheit.

Provide the student with beakers (metric
and English) and containers of various
amounts of water. Have the student find
the amount of water each container holds.

Provide the student with a balance and/or
scales (metric and English). Have the
student find the weight/mass of:
 bag of sand
 5 paper clips
 eraser

16) Can the student use estimation to                   Provide the student with various objects in
solve problems in standard (English                 the classroom. Let the student estimate
and metric) systems?                                the length in inches and centimeters, then
measure using appropriate tool to find
actual length.

Provide the student with a piece of grid
paper (1 cm). Draw the outline of their
foot. Estimate the number of square
centimeters then count the squares.

Provide the student with a balance. Have
the student estimate the mass, then mass
the following items to find the actual mass.
 cotton balls
 erasers
 quarters

Provide the student with marked beakers.
Have the student estimate then measure
the number of milliliters and fluid ounces in
a jar or cup.

(143)
Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)        2000

17) Can the student convert units within                Provide the student with a chart of equal
a measurement system?                               measures. Have the student convert the
following measurements:
 yards to feet
 quarts to gallons
 100 inches to feet

Provide the student with the following line
chart.

kg   hg     dag    gram    dg      cg   mg

Have the student convert the following
measurements:
 5g =_____ mg
 7cg =____ g
 90g = ___ cg
 3kg = ___ g

18) Can the student locate points in all                Provide the student with a coordinate
four quadrants of the coordinate                    plane. Ask the student to plot and label
plane?                                              the following points on the graph:

(1,2) (‾3, 4) (‾6, ‾6)
(2, ‾5) (0,4) (‾2, 0)

Determine the quadrant or axis of each
point.

19) Can the student draw points, lines                  Given dot paper the student will draw a
(parallel, perpendicular, intersecting),            point, parallel lines, perpendicular lines,
line segments, and rays?                            intersecting lines, line segments, and rays.

(144)
Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

20) Can the student identify, classify, and Provide the student with the following
measure right, acute, obtuse, and       diagram:
straight angles?                                     E   D
    
C

                     
A         F           B

Ask the student to name and/or identify a
right angle, obtuse angle, acute angle,
and straight angle.

Provide the student with a protractor and a
set of angles. Have the student measure
the angles.

21) Can the student create a tessellation               Provide the student with a set of shapes.
using polygons?                                     Have the student create a tessellation
from the shapes.

22) Can the student identify the vertices,              Display an example of a pyramid. Ask the
edges, and faces of three-                          student to identify the vertices, edges, and
dimensional figures?                                faces.

23) Can the student identify and                        Provide the student with a set of pattern
construct flips, slides, and turns?                 blocks and have the student model a flip,
slide, and turn.

24) Can the student calculate the area of               Have the student build a square and a
a square and a rectangle without                    rectangle on the geoboard and find the
using a calculator?                                 areas of each.

(145)
Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

25) Can the student find the                            Give the student a can, a string, and a
circumference of a circle?                          ruler. Using these tools, ask the student
to find the circumference of the can.

26) Can the student find area of a circle               Provide the student with a calculator and
using a calculator?                                 the formula to find the area of a circle
(A=r2). Have the student find the area of
a circle with a radius of 5.

27) Can the student find the volume of a                Provide the student with a base 10 cube.
cube or rectangular prism?                          Have the student measure and find the
volume of the cube.

28) Can the student read, write, and                    Provide the student a card with a number
round twelve-digit whole numbers?                   written on it. The student will read the
number orally, write the number in word
form, and round the number to a given
place value.

29) Can the student compare and order                   Provide the student with a set of numbers.
whole numbers using <, >, and =?                    Ask him or her to compare pairs of
numbers using the symbols. Then place
the set of numbers in order from least to
greatest.

30) Can the student write twelve-digit                  Provide the student with a place value
whole numbers using expanded                        chart. Have the student identify the place
notation?                                           value of digits in a given number. The
student will write the original number in
expanded notation.

31) Can the student read, write, and                    Provide the student with a place value
round decimals to the nearest ten-                  chart. Give the student a number to read
thousandth?                                         orally, to write in word form, and to round
to the ten-thousandths place.

(146)
Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)       2000

32) Can the student compare and order                   Provide the student with decimal
decimal numbers using <, >, and =?                  numbers which they have to compare
and order.
 Example: 4.24       4.024
 Place the following in ascending
order: 0.141, 0.0141, 0.1043

33) Can the student write decimal                       Provide the student with a place value
numbers through the nearest ten-                    chart. Have the student identify the
thousandth using expanded                           place value of digits in a given number.
notation?                                           The student will write the number in
expanded notation.

34) Can the student use estimation to                   Have the student estimate a problem.
determine the accuracy of solutions?                Then have the student solve the same
problem and compare the 2 solutions to
check for accuracy.

35) Can the student multiply a three-digit              Provide the student with a problem such
decimal number by a two-digit                       as:
decimal number?                                           0.718         or      9.23
x 0.45               x 1.6

Have the student estimate, then perform
the operation.

36) Can the student divide a five-digit                 Provide the student with problems such
decimal number by a two-digit                       as:
decimal number?
9.6075 ÷ 0.25 or 0.23193 ÷ 0.03

Have the student estimate, then perform
the operation.

37) Can the student round decimal                       Provide the student with a division
quotients to the nearest whole                      problem containing decimals. Have the
number, tenth, and hundredth?                       student find the quotient rounded to the
nearest whole number, tenth, or
hundredth. Example:

7 65.83           or       0.48 7.1

(147)
Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

38) Can the student estimate and solve                  Provide the student with one and two-
one and two-step problems involving                 step problems involving addition,
addition, subtraction, multiplication               subtraction, multiplication and division.
and division of decimals with and                   Have the student perform the following
 Estimate problems with and without
calculators.
 Solve problems with and without
calculators.

Luke made \$4.95 per hour. He worked 4
hours Monday, 7.5 hours Tuesday, and
much did he make in all?

39) Can the student use the rules of                    Give the student a number such as 4500.
divisibility to determine factors and               Example: Is 4500 a multiple of 3? Have
multiples of a given number?                        the student use the rules of divisibility to
determine factors.

40) Can the student explore the                         Have the student draw and label a
relationship among integers?                        number line. Have the student order the
numbers from least to greatest.

4, -4, 2, 1

Write an integer for a gain of 5 yards, a
loss of 10 lb., deposit of \$100, and
withdrawal of \$8.

41) Can the student model and write the                 Provide the student with a number. Have
prime factorization of a number using               the student model and write the prime
exponential notation?                               factorization for the number. Give the
solution in exponential notation form.
Example:

90 = 21  3 2  5 1

(148)
Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

42) Can the student distinguish between                 Provide the student with a variety of
prime and composite numbers, with                   whole numbers. Have the student
and without the use of calculators?                 determine if the numbers are prime or
composite without using calculators.
Check accuracy of answers using a
calculator. Example:

91(Composite), 200 (Composite),
97 (Prime)

43) Can the student use the greatest                    Give the student a fraction. Have the
common factor (GCF) to simplify                     student determine the greatest common
fractions?                                          factor of the numerator and the
denominator. Have the student simplify
the fraction using the greatest common
factor. Example:
9 3 3
 
12 3 4

44) Can the student use the least                       Give the student two or more fractions
common multiple (LCM) to find                       with unlike denominators. Have the
common denominators?                                student find the LCM of the
denominators to determine the least
common denominator.

45) Can the student convert among                       Have the student perform the following
 Convert 3 to a decimal and then a
4
percent.
 Convert 60% to a fraction and a
decimal.

46) Can the student find the percent of a               Provide the student with a picture of a
number?                                             CD player that indicates the price of the
CD. Have the student determine the
sales tax (7%) of the CD player, which
costs \$129.00.

(149)
Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)           2000

47) Can the student estimate and                        Provide the student with a sale page.
calculate sale price and/or original                Have the student estimate then find the
price using discount rates?                         sale price of a given item using the
discount rate. Then, have the student
determine the original price if given the
sale price and the discount.

48) Can the student compare and order                   Provide the student with sets of fractions
fractions and mixed numbers?                        and mixed numbers. Have him/her
compare sets with two fractions and order
from least to greatest sets of three or more
1 3         1       3      1 1 3
fractions (e.g.,    ,              >     ,     , ,
2 8         2       8      2 3 8
1 3 1
ordered from least to greatest                , , ).
3 8 2

49) Can the student determine equivalent Provide the student with fractions. Have
forms of fractions?                  the student match each fraction in Set A
with its equivalent fraction in Set B.

1 3 1
a)     , ,
2 4 8
5 9   2
b)      , ,
10 12 16

50) Can the student use a variety of                    Provide the student with a fraction. Have
techniques to express a fraction in                 the student express the fraction in
simplest form?                                      simplest form by:
 Finding the greatest common factor
Example: 10  2  5 .
12    2   6
    Using prime factorization.
Example: 10  2  5  5
12    2 23          6

51) Can the student locate fractions,                   Provide the student with a number line.
decimals and mixed numbers on a                     Write fractions, decimals and mixed
number line?                                        numbers on cards. Have the students
locate these places on the number line.

(150)
Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)                   2000

52) Can the student add and subtract                    Provide the student with addition and
mixed numbers, with and without                     subtraction problems with mixed
regrouping, expressing the answer in                numbers. Example: 8 1  1 1
2          4
simplest form using like and unlike
Have the student work problems without
denominators?
the student work problems with
Example: 3 1  1 5
8        6

53) Can the student multiply and divide                 Provide the student with problems using
proper fractions as well as mixed                   multiplication and division of fractions
numbers expressing the answer in                    and mixed numbers. Have the student
simplest form?                                      solve the problems and simplify the
Ex.     3 x 11
4           2
1
Ex.      12  1
3

54) Can the student estimate, solve, and                Provide the student with several one and
compare solutions to one and two-                   two-step problems that involve addition,
step problems involving addition,                   subtraction, multiplication, and division of
subtraction, multiplication, and                    proper fractions and mixed numbers in
division of proper fractions and mixed              which they have to estimate, solve, and
numbers?                                            compare the solutions. For example:

Sal and Felicia went fishing. Felicia caught 3
fish whose total weight was 6 1 lb. Sal caught
2
2 fish. One fish weighed            3   lb. and the other
2
4
weighed    3
1   lb. Estimate the total weight of the
2
fish Sal caught. Whose catch weighed the
most?

55) Can the student demonstrate                         Have the student express the ratio of
different ways to express ratios?                   boys to girls in the class in three different
ways. Example: There are 5 boys and
8 girls in the class.
5:8
5 to 8
5
8

(151)
Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)        2000

Suggested Teaching Strategies for
Uses properties to create and simplify algebraic expressions and solves linear
equations and inequalities

1)      Solve equations in one variable using addition and subtraction
 Use stories to give meaning to equations. For example there are N number
of bears in a boat. Eight jump out and three were left. How many were in the
boat? Have the students act out the story by pretending to be the bears.
 Give the student a set of numbers such as 3, 7, and 10. Student will
construct a fact family. Example:    7 + 3 = 10
3 + 7 = 10
10 – 3 =7
10 – 7 = 3
 Have students use Algebra tiles to model addition and subtraction equations.
Example: x + 2 = ‾1

Step 1                         Step 2                             Step 3           Answer

x     +     + =      -        x     +      + = -                 x =      -       x = ‾3
-      - = --                         - -

Example: x – 4 = -6

Step 1                                                   Step 2

x - - - - = - - - - - -                                 x - - - - = - - - - - -
++++ ++++

x = - -                                                 x = -2

Ask the student to explain the process being used to solve each of the
equations.

   Compare solving an equation to a balance to help students understand the
property of equality. When you add or remove a quantity from one side of a
balance, you must add or remove the same quantity from the other side in
order to keep it balanced. The same is true for an equation.

(152)
Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

2)      Models simple addition and subtraction problems using integers on a number line
 Tape a number line on the floor. The students will explore integers by
walking the number line. To walk the number line, always start facing the
positive numbers. When adding two integers, move forward for positive
numbers and backwards for negative numbers.
If subtracting, the student must turn around and face the opposite direction for
the subtraction sign. This strategy may be used as a game by dividing the
class into two groups. Give a problem to one person in each group. Have
both students walk the number line. The student that is the greatest distance
from zero earns a point for the group. Repeat this process allowing other
students to participate.
 Provide two sided counters to model addition and subtraction. Let red be
negative and yellow be positive. Pull out any zeros. Zeros are one red and
one yellow. Whatever is left is the answer.

Example: 3 + ‾5                        Y        Y        Y
R        R        R       R        R

Two reds are left; therefore, the answer is -2.

   Give a deck of cards to a pair of students. Let the black cards be positive and
the red cards negative. Place eight cards face up on the desk. The objective
of the game is to make zero using as many cards as possible. For instance,
a red 5 and a black 5 represent a ‾5 and a +5 which cancel each other out
and equal zero. The following cards would also equal zero: ‾4 + ‾2 + ‾10 + 10
+ 6 = 0 or red 4, red 2, red 10, black 10, black 6

3)      Recognizes and continues a number pattern and/or geometric representation
(e.g., triangular numbers)
 Use unifix cubes or graph paper and have the students create the following
buildings and extend the pattern.

Building 1                         Building 2                        Building 3

Building #                 # of Windows
1                            3
2                            6
3                            9
4
5
10

(153)
Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)      2000

Students will then complete a function table for 1, 2, 3, 4, 5, and 10. Have the
students extend the pattern for the 100th building and write a rule for this
pattern. Students can also create and extend their own pattern using the
cubes.

   Have the students study the pattern below:

999 x 2 = 1998                                      999 x 4 = 3996
999 x 3 = 2997                                      999 x 5 = 4995

Have the students explain the patterns they see. Have the students extend
the pattern to find 999 x 9. An extension to this activity would be for the
students to use the pattern to find 9,999 x 7 and 99,999 x 8.

   Find the product for 15 x 15, 25 x 25, 35 x 35, and 45 x 45. Have the
students look at the products and the factors to determine the pattern. Have
the students use the pattern to find 85 x 85, 115 x 115, and 135 x 135.
Students can check answer with a calculator.

4)      States a rule to explain a number pattern
 Have the student create a function table for the following example.

The average American family wastes approximately 30,000 gallons of water a
year. F is the number of families and W is the amount of water wasted in a
year.

Down the Drain

F           1           2           3            4             5          10      100
W         30,000      60,000      90,000      120,000        ------      ------   ------

The students will find the pattern and write a rule. Before using the rule to
extend the pattern, have the students check the rule with the values that they
already know. An extension to this assignment would be for the students to
determine how much water is wasted in their own town and in their state.

   The teacher will think of a rule. One student at a time will call out a number.
The teacher applies the rule to the number and tells the class the new
number. Repeat the process until someone determines the rule the teacher
is applying.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)          2000

5)      Using whole numbers to complete a function table based on a given rule
 Give students the rule, then sing ―The Twelve Days of Christmas‖. Discuss
how the rule could be used to determine the total number of gifts given.

Days         1       2      3     4       5      6      7     8       9     10    11    12

Use a scientific calculator to complete a function for the rule 0.5(n + 8)

n              1           2            5           10         15         100
0.5(n+8)          4.5

6)      Creates and solves proportional equations using one variable
 Provide each student with several different colored cubes, such as green and
red. Students will model a ratio of 1 (red) to 3 (green) using the colored
cubes. Students will determine an equivalent ratio for the number of red
cubes to 12 green cubes.
 Use the steps for solving a proportion using cross products, to solve the
following:
n  12
5 6
6n  60
n  10

Interprets, organizes, and makes predictions using appropriate probability and
statistics techniques

7)      Reads and constructs line, bar, and pictographs
 Have students cut out examples of line, bar and pictographs from
newspapers and magazines. The students will answer questions pertaining
to these graphs.
 Divide the class into groups. Each group will collect data on a topic decided
by the group. They will decide which type of graph would best display their
data. They would then construct either a line, bar, or pictograph. If
computers are available, the graphs could be created using a spreadsheet.

8)      Reads and interprets circle graphs using percents
 Have the students to predict how many hours per day they spend doing
various activities. They will create a circle graph from their data.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)    2000

   3-D Circle Graph: Cut the margin off a piece of notebook paper. Make sure
there is a space for each person in the class. Take a survey and make a
Tally Chart. Example: Colors (10 people: 5 blue, 3 red, 2 yellow)

B    B      B      B      B         R     R       R    Y     Y

At this point, when spaces are colored, tape ends of paper together and make
a point on a sheet of paper for the center of the graph. Place taped paper
around point and trace to a circle. Mark on the outside of the circle where
each color ends. Draw a line from the center dot to each mark on the outside
of the circle. Shade in appropriate colors. Change to percents.

Blue 5

Yellow           Red
2               3

5                            3                            2
= 50%                         = 30%                        = 20%
10                           10                            10

9)      Constructs and explains a frequency table
 Give students post-it notes to write down the month they were born. The
students will place their post-it notes by month to make a frequency table and
a bar graph.
 Collect data on the color of students’ math notebooks (binders, folders, etc.).
Have the student make a frequency table from the data. Volunteers will
explain the information in the table.

10)     Uses probability to predict the outcome of a single event and expresses the
result as a fraction or decimal
 Tell students that there are 3 white marbles and 1 blue marble in the bag.
The students will predict the outcome of drawing one marble from the bag
1               3
and test their theory. Example: blue =               ;   white =
4               4
   Have each student write the type of animal they like best on an index card.
Collect cards, make a frequency table, and discuss the data. Have students
predict the probability of choosing their favorite animal if the cards are mixed.
Express as a fraction. Use a calculator to change fractions to decimals.

11)     Estimates and compares data to include mean, median, and mode
 Give students a small bag of candy (skittles, M&M’s). The students will count
the number of candy pieces in each bag. The data will be collected and the
students can find the mean, median and mode. This could be done for
individual colors or for total pieces per bag.

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   Have each student call out his/her shoe size. Make a tally sheet of sizes.
From this data, find the mean, median, and mode. Discuss results.

12)     Solves problems involving combinations
 Give students pictures of 3 different color shirts, 4 different color pants, and 2
different types of shoes. (Pictures may be from catalogs, magazines, or have
students use construction paper to draw each item of clothing.)
 The students will manipulate the different combinations and write each
combination. The class will discuss the results.
 Give students a list of five pizza toppings (cheese, sausage, pepperoni,
bacon, hamburger) and three types of crusts (pan, thin, hand-tossed).
Students will draw or use manipulatives to model the possible combinations.

Writes and solves problems involving standard units of measurement

13)     Measures length to the nearest one-sixteenth inch
1
   Instruct students on how to count by                   increments. Students will simplify
16
fractions to lowest terms. Example: Provide the students with a line that
4
measures 2           in. After students place their ruler on the line, they locate the
16
1     2     3     4
2‖ mark and count by sixteen’s such as                                      ,   etc. Four
16,   16,   16,   16
1                                   1
sixteenths will then be simplified to             . The final answer is 2 .
4                                   4

    Have students collect items such as pine needles, blades of grass, acorns,
leaves, sticks, etc. and measure to the nearest one-sixteenth of an inch.

14)    Identifies appropriate units for measuring length, weight, volume, and temperature
in the standard (English and metric) systems
 Write the words length, weight, volume, and temperature on the board. Give
students a list of choices like 7 feet, 90˚ F, 3 pounds, 5 ounces, 16 fluid
ounces and have them sort into the correct categories. More choices could
be metric like 6 cm, 58 grams, 500 milliliters.
 Have one half of the students give examples of length like 5 inches and the
other half give an example of something that has that length or have one
group name an object and the other group guess a weight, length,
temperature or whatever is being emphasized.
 Have students bring labels of canned goods, liquid measure, lengths of
objects like buttons, ribbon, etc., and categorize them according to length,
weight, and volume.

15)     Uses appropriate mathematical tools for determining, length, weight, volume, and
temperature in the standard (English and metric) systems
 Bring cup, pint, quart, gallon containers, and water to class. Have students
find the number of cups in a pint and so on recording their findings. Have the

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

students create a conversion chart. Then show containers of different sizes
and let the student tell the most appropriate measure to use.
   Bring rulers, meter or yardsticks, and tape measures to class. Have various
objects in the classroom for students to measure. Students measure, record
their measurements and tell why they chose that measuring tool.

16)     Uses estimation to solve problems in the standard (English and metric) system
 Have a Metric Olympic.
a) Hand out graph paper that is marked in 1 centimeter units. Let students
guess how many squares on the paper, then count the squares.
b) Let students draw their hand outline on the paper, guess the number, and
then count the squares.
c) Have students throw a paper plate from some point and guess the
distance in centimeters, then measure.
d) Have students guess the mass of a handful of cotton balls or paperclips,
then find the actual mass. These same activities could be done using the
English measurement system.
 Have students estimate their heights, the heights of their desks, the length of
their books, then find the actual measurements using appropriate tools.
 Keep a thermometer in the classroom and have someone guess the
temperature, then check it. Use Celsius as well as Fahrenheit.

17)     Converts units within a measurement system
 Provide the student with a chart of equal measures. Show students how to

Example: Change 3 pints to cups
2c    6c
3pt             6c
1pt    1

Example: Change 500 feet to yards
1yd 500  1      500         2
500ft               yd      yd  166 yd
3ft   3           3          3

Example: Change 7.5 feet to inches
12 in 7.5  12      90
7.5 ft                  in     in  90 in
1ft     1           1

Explain that whatever unit they begin with must be the unit in the denominator
of the second fraction and the numerator is its equal measure. Multiply
numerators. If a number is greater than 1 in the denominator, divide by that
number.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)       2000

   Provide a metric line chart for each student and one on the chalkboard.
Divide students into groups and give each group enough problems for each
person to have a problem.
   Give 2 M&M’s to each student. These are their decimal points. Place several
white beans on each group’s table. These are the zeros they will add to
which ever side they need. Demonstrate moving the decimal point according
to the number of places to the new measurement.

Metric Chart

Kilo      Hecto      Deka           M          Deci         Centi        Milli

Examples:

34 km = 34000 m                                 34_______ = 34000m
5 cm = 50 mm                                    5_______ = 50 mm
17 mm = 0.017 m                                 17_______ = 0.017 m

The saying ―King Hector Died Monday Don’t Call Me‖ may help students
remember the chart.―

   Convert using multiplication or division. From small unit to large unit, divide;
from large unit to small unit, multiply.

Example: 3 gal. = 12 qt.                                 120 in. = 10 ft.
Multiply by 4                                   Divide by 12

   Draw a very large capital G on the chalkboard/overhead. Explain to the
students that the G stands for one gallon. Within the G write four capital letter
Q’s. This stands for 4 Quarts. Inside each of the 4 Q’s, write two capital
letter P’s. This stands for 2 pints in 1 quart. Within each of the P’s write two
capital letter C’s. This stands for 2 cups in 1 pint. Within each of the C’s
place 8 dots. These dots represent fluid ounces. There are 8 fluid ounces in
1 cup.

Determines the relationships and properties of two and three-dimensional
geometric figures and the application of properties and formulas of coordinate
geometry

18)     Locates points in all four quadrants of the coordinate plane
 Assign students an ordered pair. Using a large floor coordinate plane, have
students locate their point by standing in its location.
 Given the following set of ordered pairs, plot the points on grid paper and
connect the line segments between each of the coordinates given to create a
picture. (-1,3) (8,3) (8, -2) (-1,3)

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19)     Draws points, lines (parallel, perpendicular, intersecting), line segments, and rays
 Using a city street map have students identify streets that are parallel,
perpendicular and intersecting.
 Also have the students to name a point of origin and continue indefinitely
North or South, East or West to represent a ray or corner to corner to
represent a line segment.
 Ask the students where else can they find examples of parallel,
perpendicular, and intersecting lines, segments and rays.

20)     Identifies, classifies, and measures right, acute, obtuse, and straight angles
 Give students a picture of the outside of a house from a magazine. Students
will identify angles in the picture and trace them on a sheet of paper. They
will then classify the angles as right, acute, obtuse or straight. Then they will
use a protractor and measure each angle. Use a 3-D architecture program
on a computer to obtain the original picture.

A                                B                     C
D

Name                        Name                     Name                      Name__________
Estimate                    Estimate                 Estimate                  Estimate
Actual                      Actual                   Actual                    Actual

21)     Creates tessellations with polygons
 Discuss tessellations. Provide examples of tessellations (tile on floor, neck
ties, fabrics, quilts. Give students several geometric figures to discover which
combinations of figures tessellate. (Pattern Blocks may be used)

22)     Explores the relationships of three-dimensional figures including vertices, faces,
and edges using manipulative materials
 Using polyhedrons or straw polyhedrons, have students build a net
and form the following polyhedrons. Answer the following questions.

Polyhedron                                Net

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

How many faces are on a cube?
What is the shape of each face?
How many vertices are on the cube?
How many edges are on a cube?

       Have students build the following polyhedrons and complete the chart.

Polyhedron          Faces           Vertices           Edges          Shapes of
Faces
Tetrahedron
Octahedron
Hepahedron

23)       Describes, compares, constructs, classifies, and identifies flips, slides, turns
(reflections, translations, and rotations)
 Using pattern blocks ask students to take each shape and to produce a
design using slides, turns, and flips.
 Model for students translations (slides), rotations (turns), and reflections
(flips) using pattern blocks, or other designs (tiles).

slide                              rotation                  flip

Be sure to model both vertical and horizontal flips.

   Have students create a tile design and color it. Example:

   Using the tile they have created, have students translate and rotate their tile
design and record the pattern to form a border design.

Slide:

Rotate:

Using another tile, have students create a flip (horizontal) border by
reflections.

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24)      Calculates the area of parallelograms (squares and rectangles) without using a
calculator
 Use a ruler to draw various squares and rectangles on grid paper. Students
will count the number of squares used to form the figure to find the area of
each figure.
 Have students work with a number and partners. Use snap cubes to model
all possible rectangles with areas of 24 squares and 12 squares.
 After students have modeled with snap cubes, have the student use
centimeter grid paper and draw all possible rectangles with area of 10, 16,
and 36. Students complete a chart displaying the length, width, and area of
the rectangles they drew and use this information to develop a formula for
the area of a rectangle.

25)     Finds the circumference of a circle with and without the use of manipulative
materials
 Have students collect various size cans (Pringles, coffee, peanuts, etc.).
Have students estimate the distance around each can before they measure
and record. Compare the estimate to the actual results. Then using a metric
tape, measure around the can and record results.
 Introduce students to the formula for finding the circumference of a circle
 x diameter or C=d, where =3.14.
 Have students find the diameter of the cans with the metric tape and
substitute that measurement with the formula. After the students have
calculated the circumference of the cans using the formula, compare the
answers to those in activity 1.

Example:              Can

Diameter measures (13cm) (3.14) = 40.8cm

26)     Determines the area of a circle with and without the use of a calculator
 Using centimeter grid paper, draw a circle. Count the number of squares
inside the circle, estimating any sections inside the circle that are not
complete squares. (Student may overestimate or under estimate)
 Using grid paper, draw a circle with a diameter of 4 cm. Divide the circle into
eight equal parts. Cut the circle into wedges. Color half of the wedges and
cut the wedges apart. Fit the wedges together to form a shape that looks like
a parallelogram.

Example:

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

The height of the parallelogram is the radius of the circle. The base of the
1
parallelogram is          the circumference of the circle. So, the area of the circle is
2
(height)            (base)
1
r                        (2r)
2
1
 2   r2
2
A= r2

27)     Finds the volume of cubes and rectangular prisms with and without the use of
calculators
 Investigate volume by making containers of different shapes and comparing
how much each container holds.
 Divide the students into small groups. Using 5‖ x 8‖ cards the students will
make 3 containers: one circular, one square, and one triangular. Use tape to
hold cards together. Then tape the ends of each container to another card so
the container has a bottom. The base of each figure has a perimeter of 8
cubes. Each figure is 5 cubes high. Estimate which container will hold the
most, which will hold the least? Fill one container with filler (rice, etc.) Does
the rice fill this container? Is there too much rice? Continue this activity until
you can determine which container holds the most, the least.
 Divide the class into groups of four. Provide each group with models of
prisms and cubes, centimeter cubes, and filler (rice, cereal, sand, etc.). Have
students fill the rectangular prisms with centimeter cubes, counting the
number needed to fill it.
 Have students use the appropriate formula to find the area of the base of a
rectangle or square and then use the height to compute the volume.

Uses basic concepts of number sense and performs operations involving
exponents scientific notation, and order of operations

28)     Reads, writes, and rounds twelve-digit whole numbers
 Hang a clothesline. Label place value through 100 billion above it. Give each
student an index card with a digit, 0-9, written on it. Call students randomly to
bring their number to the place value line. Students attach their card to the
line with a clothespin. Students will read the number, write the word name,
round it to a given number, and expand it.
 Give the students a map of the solar system and have students determine the
planets that are within 999,999,999,999 miles from the Earth. (Make sure
numbers are not in word form on the map.) Have students choose the planet
that is farthest from the Earth within the range and allow them to write the
number in word form.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

29)     Compares and orders whole numbers using <, >, and =
 Give students a deck of cards with face cards and the 10’s removed. The
students will place a given number of cards face up in a row. They will turn
up a second row of cards.

5                4                2          9               3

4                5                9          6               1
The students write a statement using the 2 numbers created. They can use all
operations, compare, round, expand, read, and write the numbers.

    Bring 5 different brands of popcorn to class. Pop the popcorn and have the
students count the number of popped or unpopped kernels from each bag
and then compare and order the different brands.

30)      Writes twelve-digit whole numbers using expanded notation
 Cut out squares from colored construction paper. Students work with a
partner. The teacher will give each pair an index card with a number on it
and provide each pair with a handful of cut out squares. The teacher will
model the process first on the board/overhead using the number 435. The
teacher will point to the digit 5 and ask in what place is this digit?

4                3               5
Ask how many ones are sitting in this place? (5). Then, draw 5 squares
underneath the number 5. Do the same for 3 and 4. (See below)

4                 3                  5

100               10                  1

100               10                  1

100               10                  1

100                                   1

1

400         +      30       +          5 =       435

You can substitute candy by assigning place values to colors.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

    Give each student a whole number with up to twelve digits. Have the
student write the number in expanded notation using parentheses. Then
have students check their own or others using a scientific calculator to
compare with the original number. Example: 431,175,903

(4 x 100,000,000) + (3 x 10,000,000) + (1 x 1,000,000) + (1x 100,000) +
(7 x 10,000) + (5 x 1,000) + (9 x 100) + (0 + 10) x (3 x 1)

Make sure students enclose numbers in parentheses on calculators, if
necessary.

31)     Reads, writes, and rounds decimal numbers to the nearest ten-thousandth
 Have students use base-ten blocks and place value charts to model decimal
numbers, such as 2.24. See example below. The teacher can either put
decimal numbers, one at a time, on the board for students to model or can
hand each student an index card that has a decimal number written on it to
model. The students will read the number to a partner, write the word name
of the decimal number, and round it to the nearest __________.

2 ONES                         2 TENTHS                     4 HUNDREDTHS

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Copies of centimeter graph paper can be made, laminated, and cut out so
that each student has 4 wholes (4 10-by-10 cm squares), 2 tenths
(2 1-by-10 cm squares), and 20 hundredths (20 1-by-1 cm squares).

   Students play the ―Memory Game‖ individually, or with a partner. Each group
will use a deck of 16 cards. Eight of the cards should have a decimal number
written on it. The remaining 8 should have the word name match for the
decimal number. The cards should be placed face down in rows of 4. Player
#1 will turn up 2 cards. If a match is not made, Player #2 will turn up 2 cards.
If there is a match the player continues until no match is made. The winner is
the player who gets the most matches.

Two and one
hundred forty-
2.141                   one thousandths
0.6                    six tenths

Cards can also be constructed for rounding. See example below.

4.24                                   4.2

   Students use a place-value chart and blocks to do this activity. Ahead of time
the teacher will take 12 index cards (1 set of 12 for each pair of students) and
write a one-digit number on each of 11 cards and a decimal point on one
card. These 12 cards will be kept in a baggie. One student will construct a
number using all 12 cards on the place value chart. The other student will
read the number, write the word names for the number and round it to the
nearest tenth, hundredth, thousandth and/or ten-thousandth. Player #2 will
then do the same thing.
   Place students in pairs. Each pair will receive 1 sheet of newspaper and 2
different colored markers. Each pair will circle as many decimals as they can
find on their newspaper within 5 minutes. The pair who finds the most will
place their newspaper on the bulletin board. Students will be called at
random to go to the bulletin board to read one of the circled decimal numbers.
   Have the teacher hold up a sheet of construction paper that has a decimal
number written on it. Students can write the word name for that number and
round it to a given place.
   Set up an interactive bulletin board in which students can use as a center to
practice reading, writing, and rounding decimal numbers.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

32)     Compares and orders decimal numbers using <, >, and =
 Students are provided a stack of index cards that have decimal numbers
written on them. The students will place these cards in order from least to
greatest. These can also be placed on a number line. Write a less than,
greater than, or = statement using appropriate symbols.
 Play the game of ―Battle‖. Students work in groups of 2-4. Each group is
provided a stack of decimal cards in which a decimal has been written on it.
The stack of cards is dealt among the players. Each player places a card
face-up. The player that has greatest number gets the cards. The person
with the most cards at the end of the game is the winner.
 Prepare a paper bag containing 11 cards-one each for the numbers 0-9 plus
a card with a decimal point. Each student will make an answer board by
drawing a row of five boxes in which to place digit choices. (See below.) The
teacher will pull one card from the bag, show it to the students, then return the
card to the bag. Students will write the digit in one of their boxes.

If the teacher draws the decimal point from the bag, it will go before the next
digit is drawn. After the decimal point is drawn, leave it out so it cannot be
drawn again. The teacher will draw from the bag until all boxes have been
filled in. The student that has the largest number will draw from the sack next
game. This is a good activity for small groups.

33)     Writes decimal numbers through the ten-thousandths’ place using expanded
notation
 Have students play the ―Memory Game‖ individually, or with a partner. Each
group will use a deck of 16 cards. Eight of the cards should have a decimal
number written on it. The remaining eight should have the expanded form
written on it. (See example below.) The cards should be placed face down in
rows of 4. Player #1 will turn up 2 cards. If a match is not made, Player #2
will turn up 2 cards. If there is a match the player continues until no match is
made. The winner is the player who gets the most matches.

6.25                                  (6  1) + (2  0.1) + (5  0.01)

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

   Have students use a place-value chart and blocks to do this activity. Ahead
of time the teacher will take 12 index cards (1 set of 12 for each pair of
students) and write a one-digit number on each of eleven cards and a decimal
on one card. These 12 cards will be kept in a baggie. One student will
construct a number using 12 decimal cards on the place value chart. The
other student will write the expanded form of the number. If correct, the
students swap turns.
   The teacher will hold up a sheet of construction paper that has a decimal
number written on it. Have students write the expanded form for that number.
The opposite can also be done. (An expanded notation may be held up and
students may write the decimal number.)

34)     Uses estimation to determine accuracy of solutions
 Bring an apple to class and the students estimate the weight of the apple.
The students discuss their conclusion. The students weigh the apple and
determine the accuracy of their solution.
 Have the students estimate the answer to a given problem:
Example: John wants to buy 3 shirts that cost \$8.65 each and 2 pair of pants
at \$9.39 each. If he hands the cashier three \$20.00 bills, how much change
will he receive? First estimate the change, then solve and find actual change

35)     Multiplies a three-digit decimal number by a two-digit decimal number
 Write several 2 and 3 digit decimal numbers on the board. Have students
select one three-digit decimal number and one two-digit decimal number to
multiply and solve the problem. Use a calculator to check for accuracy.
 Give the students several multiplication problems and allow them to use a
calculator to determine the answer. (Do not allow the answer to end in zero.)
The students will write down the answer. Tell the students to focus on why
the decimal appears where it does. The students will discover the rule for
multiplying decimals.

Example:                         4. 26               3 numbers behind the decimal
x 5 .2

22 .152                   3 numbers behind the decimal

36)     Divides a five-digit decimal number by a two-digit decimal number
 Have students use the calculator to divide a 5-digit decimal number by a 2-
digit decimal number.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

   Give students several problems with the same numbers but with the decimal
point in different places (up to five digits). These numbers will be divided by
two 2-digit decimal numbers (same numbers, decimal point in different
places). Possible quotients will be given. Students will match the problems
with the correct quotients. Calculators can be used to check results.

Example:              6.75  0.25 .675  0.25 67.5  0.25
6.75  2.5  .675  2.5  67.5  2.5

Possible Quotients: 27, 2.7, 270, 0.27, 0.027

37)     Round decimal quotients to the nearest whole number, tenth, and hundredth
 Give the students a problem such as:

Four people go to the store and purchase candy for \$9.61. How much does
each have to pay? Round the solution to the nearest hundredth (cent).

   Give students 10 proper or improper fractions with denominators, which will
not terminate when changed to a decimal. Use calculators to convert each
fraction to decimal form. Round each to the nearest whole number, tenth,
and hundredth.

38)     Estimates and solves one and two step problems involving addition, subtraction,
multiplication, and division of decimals, with and without calculators
 Have the students create a menu for a concession stand for their school.
They decide what will be sold and how much each item will cost. The
students display the menu on a sheet of poster board. The teacher selects a
specific amount of money and displays it on the board/overhead. Student
places an order and the class will decide if the amount displayed will be
enough to pay for the order or not enough to pay. The students then use a
calculator to compute the exact amount needed to purchase the order to
determine whether or not the estimation was adequate.
 Have the students work in pairs or small groups. Give each group a map and
have the students plan a trip. They choose the kind of car to be used. The
teacher will list several makes and models along with a list of miles per gallon
of gas each car will get. The students will estimate the total number of miles
they will travel and estimate the cost of the gas. The students then find the
exact distance they will travel and calculate the exact cost of gas (a calculator
may be used).

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

39)     Uses the rules of divisibility to determine factors and multiples of a given number
 Give each student a hundreds chart and a ziploc bag containing 50 beans. (A
hundreds chart is a 10 x 10 square with the numbers marked on it.)

1     2      3      4      5      6     7      8       9     10
11    12     13     14     15     16     17     18     19     20
21    22     23     24     25     26     27     28     29     30
31    32     33     34     35     36     37     38     39     40
41    42     43     44     45     46     47     48     49     50
51    52     53     54     55     56     57     58     59     60
61    62     63     64     65     66     67     68     69     70
71    72     73     74     75     76     77     78     79     80
81    82     83     84     85     86     87     88     89     90
91    92     93     94     95     96     97     98     99    100

Tell students to cover every number that 2 will divide into evenly. These
numbers are divisible by 2 and are multiples of 2. Remove the beans and do
the same for 3, 5, 6, 9, and 10. Discuss divisibility rules for that number each
time.

   Provide the rules of divisibility to each student. Have each students write an
example of each rule with all examples being greater than 100. Students will
swap with a partner and check each other’s work.

40)     Explores the relationship among integers
 Make a large thermometer on the board/overhead. Ask the students to locate
various pairs of temperatures and write an inequality to compare them.
 Put students in groups and distribute scissors, rulers, three magic markers,
and a foot of string to each student. Fold strings in half and color this point
with one of the markers. This represents 0. Measure 1 inch on each side of
this point and tie a knot. These knots represent +1 and –1. Color the right
side one color (positive) and the left side another color (negative). Measure 1
inch from each of these points and tie 2 more knots. These represent +2 and
–2. Color accordingly. Continue doing this as far as they can go adding a
knot each time.
 Make a set of index cards with one number on each card using positive and
negative numbers including zero. Give each student a number and have
them form a line from least value to greatest value.
 Have a number line marked on the floor. Give each student a problem and
have a student walk a problem on the number line, such as 3 + 2, 5 + -3, -6
+ˉ1, 8 – 4, ˉ7 – 2. Be sure to explain that subtract means to add its inverse.
This means 8 – 4 is really 8+ ˉ4 and ˉ7 – 2 is ˉ7 +ˉ2.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)      2000

41)     Models and writes the prime factorization of a number using exponential notation
 Use a factor tree to find the prime factors of a number.

Example:

110                        81

11 x 10                  9 x       9

11 x     5 x 2           3x3x3x3

Answers:              110 = 2 x 5 x 11                 81 = 34

   Make a prime factor column. Always use the smallest prime number that will
go into the dividend. Factors go on the outside of the upside down division
sign with the quotient on the inside of the division sign. Keep factoring with
primes until the last quotient is 1.

Example: 2         18                                               Answer: 2 x 32 = 18
3 9
3 3
1

42)     Distinguishes between prime and composite numbers, with and without the use
of calculators
 Give each student a ziploc bag of 100 cubes or tile squares. Begin with 1,
then 2, 3 through 20 and arrange each number of tiles in rectangles. If the
cubes can be arranged in only one way, the number is prime. If the cubes
can be arranged in more than one rectangle the number is composite.

Example:

3                    is the same as         Both are 1 x 3 so 3 is prime.

6
But 6 = 1 x 6 or 3 x 2, so 6 is composite.
3
2

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

   Give each student a large 10 x 10 square. Have them number each square
from 1 – 100. Have them cross out every second number after two, every
third number after 3, every fifth number after 5, every seventh number after 7.
The numbers that are left are prime. This is called the Sieve of Eratosthenes.
Using a calculator, let the students find a factor of the numbers they thought
were prime.

43)     Use the greatest common factor (GCF) to simplify fractions
 Write the numerator and the denominator of a fraction as prime factors. Draw
a line through the factors that are the same in both. What is left is the
simplified fraction. The factors that are the same in numerator and
denominator determine the GCF.

Ex. 9            3x 3         3 ; GCF = 3
12           2 x 2 x3        4

   Use Venn diagrams to simplify fractions. First factor the numerator and
denominator into primes. The number contained by both circles is the GCF.
What is not in both circles is the simplified fraction.
Ex. 6          2x3                                        GCF = 2
16        2x2x2x2                                    2
3         2
3                                  2   2
Simplified fraction             3
2x2x2 8

44)     Uses the least common multiple (LCM) to find common denominators
 Use factor trees to find the prime factors. Put the factors in a Venn diagram.
Multiply all the factors together.
Example:            4  20               12                  15
12 60
 7  28
15 60
3 x 4             3     x 5

3    x     2 x 2

2
3      5             2  2  3  5 = 60
2

   Multiply the two denominators together. Divide by the highest number the
denominators have in common. The results is the least common multiple of
the two numbers.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)         2000

3 and 7 4 x 12  48  4  12
4     12
2 and 7   5 x 8  40 (They have no factor in common)
5      8
9 and 7 10 x 15  150  5  30
10     15

Determines relationships among real numbers to include fractions, decimals,
percents, ratios, and proportions in real life problems

45)     Converts among fractions, decimals, and percents
 Provide students with fictitious sports data or data that has been collected
from the school’s teams. Students can determine shooting percentages
during basketball season and batting averages during baseball season.

Example:              Number Shots Made                  or       Number of Hits
Total Shots Attempted                   Total Number of at Bats

   Have students work in groups of 4. Each group is given 4 index cards.
Students will write their first name on an index card and determine the
number of consonants or vowels on their first name. These will be written in
ratio, decimal, and percent form. These cards can be placed in order from
least to greatest.
   Play the ―Memory Game‖. Students work with a partner. Each group is given
a stack of 24 cards. Eight of the cards will have a ratio written on them, eight
will have the decimal form, and the remaining eight will have the percent form.
See example below. Shuffle the cards and place face down in rows of four.
One player turns up 3 cards to see if he/she gets a 3-way match of a ratio,
decimal, and percent. He/she continues until a 3-way match is not made.
When this happens, the next player takes a turn. The student with the most
matches at the end of the game is the winner.

3
4                    0.75                 75%

   Students work in groups of 2-4. Each group receives a deck of 42 cards.
(These cards have fraction, decimals, and/or percents on them.) Each
student is dealt 7 cards. The remainder of the cards are placed in the center
of the table. Students play according to ―Gin Rummy‖ rules.
   Give each student an index card that has either a ratio, decimal, or percent
written on it. Students are instructed to get out of their seats and locate their
3
match. For instance, if a student is holding the fraction                 ,   he/she will look for
4
the 0.75 and the 75%.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)       2000

   Have students look in reading books or textbooks and locate 3 lines. They can
do any of the following to determine fraction, decimal, and percent form:
Make a line plot depicting the letters of the alphabet. For instance,

X                                         X
X      X             X                    X
X      X      X      X                    X            X
A      B      C      D      E      F      G      H     I      J      K      L       M

   Repeat the above example and compare the number of 3-letter words vs. the
number of 4-letter words. Results will be written in fraction, decimal, and
percent form. Students determine the fraction, decimal, and percent form of
common nouns vs. adjectives. Graphs can be constructed to display the data.
   Collect grocery store receipts. Try and get receipts from at least 2 different
grocery stores for at least 2 weeks before this activity. Divide the class into 2
groups. One group will get the receipts from one store, while the other group
gets the receipts from the 2nd store. These 2 groups can be subdivided when
the activity begins. Students will predict which digit they think will appear most
frequently as the last digit in a price from the receipts. Each group will
construct a line plot for each digit 0-9 using the data from the receipts.

See example below.

0      1      2      3      4      5      6     7      8      9
X             X                    X                   X
X      X             X             X      X                   X

The results can be expressed in decimal, fraction, or percent form. Graphs can
also be constructed to display data. The results can be compared as to which
grocery store had the number ____ to show up the most.

    Students work with a partner. Each group will be given a page from the local
newspaper and a transparent centimeter grid sheet to be used as an overlay.
Students will estimate the total area of the newspaper page, excluding the
margins, and determine the area of each category. The categories could
photographs, weather, obituaries, etc. Students then express the area of the
article to the area of the page as a fraction, decimal, and percent. The class
then records all findings. The total area of each of the categories is calculated.
These totals are compared to the total number of pages of the newspaper.
Using this data, students decide how much of the newspaper is really news.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

46)     Finds the percent of a number
    Sample Problem: If my class has 30 students and 60% are girls, how many
girls are in the class?
    Give the students 10 M & M’s. They must give their neighbor 20%. How
many M & M’s will they give away?

47)     Estimates and calculates sale price and/or original price using discount rates
 Set up a class store that contains pictures of items found in a department
store. Have different departments set up, such as a furniture department, a
clothing department, a shoe department, etc. Have prices on the items.
Display a percent off sign in each department. Students rotate centers and
estimate first what the sale price will be, then calculate to find the exact
answer. Calculators can be used to check work.
 Provide students a menu, or have them make up a menu of their own. Each
student orders a meal from the menu. The total cost of the meal will be
determined. Students can determine the sales tax (7%) and gratuity (15%).
 Each student gets a flier from J.C. Penney, Wal-Mart, or K-Mart, etc. that
shows the regular price and sale price, a pair scissors, glue, and a sheet of
construction paper. Have students cut out 3 ads and glue to the construction
paper. Students then find the discount. Swap and solve.
 Calculate the original price if the sale price is \$60 after a 25% discount has
been deducted.

48)     Compares and orders fractions as well as mixed numerals
 Give the student a combination of two proper fractions, two mixed numbers,
or one from each set. Have the student draw rectangles of similar size to
illustrate each fraction. Have the student compare the two illustrations to
determine if the fractions are greater than, less than, or equal to each other.
3                            2
4                            3

3 2
Example:       >
4 3

   Have students use cross-products to compare fractions.
5  4
Example:        < because 5 x 7 is less than 9 x 4.
9  7

   Work in groups of 5 or 6. Have the students measure the length of their
shoes to the nearest 1 of an inch. Let one student record the information on
16
an index card. Have the group arrange the measures in order from least to
greatest. Have students compare their data with the other groups to
determine the smallest measure and the greatest measure.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

49)     Determines equivalent forms of fractions
 Have students work in pairs. Make a gameboard by using a 3 by 3 grid. Put
a fraction in each of the 9 squares. Each player is given either nine 0’s or
nine’s. Player 1 should put a marker over one of the fractions and call out an
equivalent fraction. Player 2 will decide if the two fractions are equal. If an
equivalent fraction is given, the first player claims the box. If an equivalent
fraction is not given, the player must give the box to his opponent. The 1st
player to get 3 in a row vertically, diagonally, or horizontally wins.
 On a sheet of paper, list several fractions. On the opposite side, in random
order, give equivalent fractions. Have the students connect a fraction from
the left to a fraction on the right. Give more fractions on the right side so that
some will be left over.

50)     Uses a variety of techniques to express a fraction in simplest form
 Give each student a fraction to simplify using two methods, such as dividing
numerator and denominator by their GCF, and prime factorization of
numerator and denominator. (See Strategy #43) Students will solve and
swap with a partner to check.
 Make two lists of equivalent fractions (Set A will not be simplified. Set B will
be simplified). Use a calculator that will simplify fractions to match each
fraction in Set A with its equivalent form in Set B.

51)     Locates fractions, decimals, and mixed numerals on a number line
 Have students work in groups of 4. Each group is given a roll of adding
machine tape and 4 index cards with each card containing a fraction, decimal,
or mixed number on it. Students will tear off a piece of tape and display their
number on it. These can be put in order from least to greatest.
 String plastic clothesline wire across the room to serve as a number line.
Clothespins can be used to clip index cards that have fraction numbers,
decimal numbers, or mixed numbers on them.

52)     Adds and subtracts mixed numerals, with and without regrouping, expressing the
answer in simplest form using like and unlike denominators
 Have the students work in pairs. Give each student a stack of number cards
with the digits 1 – 9 written on them. Place the cards face down and spread
them out. Have the students draw 3 cards from their own pile. The students
each will form one mixed number from the 3 cards.

   Example: If the student draws the number 2, 3, and 4 some possible
2     3           2
combinations would be 2 , 4 , 3 ). Have the students add the 2 mixed
4     3     4
numbers and simplify the answer, then have them repeat the process using
subtraction.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

   Assign each letter in the alphabet a point value (e.g., vowels are 2 1 points,
4
capitals are 2 3 ; consonants are 1 1 points).
8                    2
   Have the students determine the value of each of their spellings words,
names, vocabulary words, etc. Have the students add and/or subtract these
values.

53)     Multiplies and divides proper fractions as well as mixed numerals expressing the
 Write fractions on index cards in blue marker. On the opposite side put the
reciprocal of the fraction in red marker. Give a stack of cards to each student.
On another card, put a division sign on one side and a multiplication sign on
the other side. Students can work alone or with a partner. Have the student
draw one card and lay it face up with the blue side showing. The student will
choose to multiply or divide by using the appropriate side of the operation
card. The student then chooses a 2nd fraction card and lays it face up with
blue showing. If the operation sign is multiplication, the student then solves
the problem. If the operation sign is division, the student turns the second
card over to the red side (reciprocal), exchanges the division sign for the
multiplication sign, and solves the problem.
 Students will choose a problem from Set A (division problems) and a problem
from Set B (multiplication problems) that should give the same answer. They
will write reasons for their choices. Students will then exchange papers and
work out each pair the other student chose. A point will be awarded for each
correct match.

Example:              Set A                                      Set B

1 1                                        1 1
                                          
2 3                                        2 3

1
3                                       3 2
2

1                                      1
3                                            3
2                                      2

1 1                                        1
                                           2
3 2                                        3

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

54)     Estimates, solves, and compares solutions to one and two-step problems
involving addition, subtraction, multiplication, and division of proper fractions and
mixed numerals.
 Each student is given a grease pencil and a baggie. Students are to draw a
picture on the baggie using the grease pencil. Students are selected at
random to place their drawing on the overhead projector. Students will write
a fraction word problem about the picture and solve. These can be swapped.
 Students work in pairs. Each pair will get a number cube that has numbers
written on it. Spinners can also be used. Student #1 will draw on his/her
paper the following: ─ + ─ =. Student #1 will roll the number cube or spin the
spinner 4 times. The first number will be placed in the numerator spot of the
fraction. The second number will be placed in the denominator of the fraction.
The third number will be placed in the numerator spot of the second fraction,
and the fourth number will place in the denominator of the second fraction. A
word problem will be written, estimated, and solved for this problem. Player
#2 will do the same.

55)     Demonstrates different ways to express ratios
 Use circle graphs that contain percents and have students express these
percents as ratios.
 Collect data such as birthday months, favorite ice cream, TV shows, etc.
Students determine the ratio, decimal, and/or percent from the data. An
extension of this would be to have students construct a graph to display this
data.
 Have students work in groups of 4 and determine the ratio, decimal, and
percent for the different colors of M&M’s that are in a small bag. Estimation
can be connected to this activity as well as graphing the collected data.
 Draw a rectangle on the overhead/chalkboard and label the sides 6-ft. and 4-
ft. Ask students to give the ratio of the length of the longer side to the length
of the shorter side.
 Give each pair of students a baggie that contains 100 dry butter beans. One
of the students will reach in the bag and pull out a handful of beans. The
student will first estimate the number of beans he/she has in their hand.
The student will count the beans in his/her hand. This will be written as a ratio
out of 100. This ratio can also be written in decimal and percent form.
 Have students work with a partner. Each group will need a measuring tape.
They are to measure* each other to determine each of the following listed
below. Each ratio is to be expressed in simplest form.
 arm to foot
 foot to height
 around the neck to arm
 foot to arm

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

Example: A person has a wrist measurement of 5 in. and a height of 50 in.
The ratio of wrist to height is 5:50 or 1:10 in simplest form. *Measurements
should be given to the nearest inch.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)     2000

MATHEMATICS BENCHMARKS

O – means teacher should be able to observe throughout the day – possibly use anecdotal records.
I – Informal Assessment—those marked ―I‖ have an assessment task attached.

Uses properties to create and simplify algebraic expressions and solves linear
equations and inequalities

1) I       -       Describes and extends patterns in sequences
2) I       -       Identifies and uses the commutative, associative, distributive, and identity
properties
3) I       -       Translates between simple algebraic expressions and verbal phrases
4) I       -       Solves linear equations using the addition, subtraction, multiplication, and
division properties of equality with integer solutions
5) I       -       Writes a real world situation from a given equation
6) I       -       Writes and solves equations that represent problem-solving situations

Interprets, organizes, and makes predictions using appropriate probability and
statistics techniques

7)     I   -       Organizes data in a frequency table
8)     I   -       Interprets and constructs histograms, line graphs, and bar graphs
9)     I   -       Interprets and constructs circle graphs when given degrees
10)    I   -       Interprets and constructs stem-and-leaf plots and line plots from data
11)    I   -       Estimates and compares data including mean, median, mode, and range of a
set of data
12) I -            Predicts and recognizes data from statistical graphs
13) I -            Determines probability of a single event
14) I -            Uses simple permutations and combinations

Writes and solves problems involving standard units of measurement

15) I          - Converts within a standard measurement system (English and metric)
16) I          - Converts temperature using the Fahrenheit and Celsius formulas
17) I          - Uses standard units of measurement to solve application problems

Determines the relationships and properties of two and three-dimensional
geometric figures and the application of properties and formulas of coordinate
geometry

18)    I       -   Identifies polygons with up to twelve sides
19)    I       -   Classifies and compares the properties of quadrilaterals
20)    I       -   Classifies and measures angles
21)    I       -   Classifies triangles by sides and angles
22)    I       -   Finds the perimeter of polygons

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

23)   I   -   Finds the area of triangles and quadrilaterals
24)   I   -   Finds the circumference and area of a circle
25)   I   -   Identifies congruent segments, angles, and polygons
26)   I   -   Develops relationships of faces, vertices, and edges of three-dimensional
figures
27) I     -   Performs transformations (rotations, reflections, translations) on plane figures
using physical models and graph paper
28) I     -   Investigates symmetry of polygons
29) I     -   Develops and applies the Pythagorean Theorem to find missing sides of right
triangles
30) I     -   Graphs ordered pairs on a coordinate plane

Uses basic concepts of number sense and performs operations involving
exponents, scientific notation, and order of operations

31)   I   -   Uses powers of ten to multiply and divide decimals
32)   I   -   Uses patterns to develop the concept of exponents
33)   I   -   Writes numbers in standard and exponential form
34)   I   -   Converts between standard form and scientific notation
35)   I   -   Finds and uses prime factorization with exponents to obtain the greatest
common factor (GCF) and least common multiple (LCM)
36) I     -   Uses patterns to develop the concept of roots of perfect squares with and
without calculators
37)   I   -   Recognizes and writes integers including opposites and absolute value
38)   I   -   Compares and orders integers
39)   I   -   Adds, subtracts, multiplies, and divides integers with and without calculators
40)   I   -   Uses the order of operations to simplify and/or evaluate numerical and
algebraic expressions with and without calculators

Determines the relationships among real numbers to include fractions, decimals,
percents, ratios, and proportions in real-life problems

41) I     -   Compares, orders, rounds, and estimates decimals
42) I     -   Adds, subtracts, multiplies, and divides decimals in real-life situations with
and without calculators
43)   I   -   Converts among decimals, fractions, and mixed numbers
44)   I   -   Expresses ratios as fractions
45)   I   -   Adds, subtracts, multiplies, and divides fractions and mixed numbers
46)   I   -   Uses estimation to add, subtract, multiply, and divide fractions
47)   I   -   Explores equivalent ratios and expresses them in simplest form
48)   I   -   Solves problems involving proportions
49)   I   -   Determines unit rates
50)   I   -   Uses models to illustrate the meaning of percent
51)   I   -   Converts among decimals, fractions, mixed numbers, and percents
52)   I   -   Determines the percent of a number
53)   I   -   Estimates decimals, fractions, and percents

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

54) I    -    Uses proportions and equations to solve problems with rate, base, and part
with and without calculators
55) I    -    Finds the percent of increase and decrease
56) I    -    Solves problems involving sales tax, discount, and simple interest with and
without calculators

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

1) Can the student describe and extend                  Provide the student with arithmetic (+ and
a pattern in a sequence?                             −) and geometric (x and ÷) sequences.
a) 2, 5, 8, 11, –, –, –
b) 2, 2, 4, 8, 16, –, –, –

Have the student extend patterns such as:
c) 1, 1.25, 1.5, 1.75, 2, ___, ___, ___, ___
d) 256, 128, 64, 32, ___, ___, ___.

Have the student describe the pattern and
label it as arithmetic or geometric
sequence.

2) Can the student identify and use the                 Provide the student with the following
commutiative, associative,                           properties:
distributive, and identity properties?
a+b=b+a                               a+0=a
(a  b)  c = a  (b  c)             ax1=a
a(b + c) = ab + ac

Have the student identify the properties as
the commutative, associative, distributive
or identity properties.

Simplify the following using the common
associative or distributive properties: 25 x
168 x 4 = (25 x 4) x 168

3) Can the student translate between                    Read the following verbal phrases:
simple algebraic expressions and                      three times a number
verbal phrases?                                       a number increased by five
 the difference of a number and six

Have the student write an algebraic
expression for each.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

4) Can the student solve linear                         Provide the student with equations such
equations using the addition,                        as the following:
subtraction, multiplication, and                      x+4 = -6
division properties of equality with                  m-13 = -5
integer solutions?                                    -3x = 21
a
        24
12

The student may use manipulatives, if
needed.

5) Can the student write a real world                   Give the student the equation: x - 5 = 17
situation from a given equation?                     Have the student write a word problem to
Five years ago Mary was 17. How old is
she now?

6) Can the student write and solve                      Give the student examples of real life
equations that represent problem-                    problems such as, ―Bob is three years
solving situations?                                  older than twice his brother’s age. If Bob
is thirteen, how old is his brother?‖
Have the student write an equation and
solve. Example: 2a + 3 = 13, a = 5

7) Can the student organize data in a                   Provide students with given data such as
frequency table?                                     temperatures for the week and have them
create a frequency table.

8) Can the student interpret and                        Provide students with data and have them
construct histograms, line graphs,                   construct histograms, line graphs, and bar
and bar graphs?                                      graphs. After graphs have been
constructed, each group will interpret the
different graphs.

9) Can the student interpret and                        Provide students with given degrees such
construct circle graphs when given                   as 90, 45, 45, 75, 105. Have them
degrees?                                             construct a circle graph using a compass
and a protractor. Each student must be
able to interpret the circle graph.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)    2000

11) Can the student estimate and                        Provide students with the following data:
compare data including mean,                        23, 85, 76, 63, 76, 94. Students are to
median, mode, and range of a set of                 estimate and then find actual results of
data?                                               mean, median, mode, and range.

Compare actual results with the estimate.

12) Can the student predict and                         Provide students with a statistical graph.
recognize data from statistical                     For example, a sports graph comparing
graphs?                                             previous year’s averages to present
averages. Students should predict the
following year’s average after interpreting
data from graph.

13) Can the student determine the                       Provide the student with a spinner that
probability of a single event?                      has been divided into a specific number of
sections such as eighths in which the
numbers 1 through 8 will be written.
Students will determine the probability of
spinning a composite/prime number.

14) Can the student use simple                          Have students solve problem such as:
permutations and combinations?                      Given a group of 4 people, how many
different ways can they line up to go to
lunch?

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)       2000

15) Can the student convert within a                    Provide the student with a chart of equal
standard measurement system                         measures. Have the student complete the
(English and metric)?                               following conversions and explain why
one would multiply or divide.

9 gal.       =        ____ qt.
17 ft.       =        ____ in.
23 c.        =        ____ pt.

Provide the student with the following
metric system line:

k      h       da    meter     d          c
m                    gram
liter

7g    =                   kg
37cm =                    m
5KL =                     L
6.9cm =                   mm

16) Can the student convert temperature                 Provide the student with the Fahrenheit
using the Fahrenheit and Celsius                    and Celsius formulas. Have the student
formulas?                                           convert the following temperatures.

86ºF =                    ˚C
59º F =                   ˚C
35º C =                   ˚F
120º C=                   ˚F

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

17) Can the student use standard units of Provide the student with standard English
measurements to solve application     units of measure. Have the student
problems?                             answer the following problems showing all
his work.
 Ann needed 5 cups of milk to make a
recipe. How many pints does she need
to make this recipe?
 Tom rode his bike 2.5 miles. How
many feet did he ride his bike?
 Janet has 3 pounds and 4 ounces of
grapes to divide into 4 baskets. How
many ounces are to be put into each

Provide the student with metric units of
measure. Have the student answer the
following questions:
 Tammie’s pencil is 13 cm long. How
many millimeters is her pencil?
 John ran the 400-meter dash. How
many kilometers did he run?

18) Can the student identify polygons                   Provide the student with a set of polygons.
with up to twelve sides?                            Have the student name each polygon in
the set.

19) Can the student classify and                        Give the student a set of quadrilaterals.
compare the properties of                           Have the student identify which ones are
trapezoids, and rhombi and explain why.

20) Can the student classify and                        Provide the student with a set of pre-
measure angles?                                     drawn angles in various positions. Have
the student determine whether the angles
are right, acute, obtuse, or straight and
define each.

Provide the student a protractor and a set
of angles. Have the student measure
each angle and classify each angle.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

21) Can the student classify triangles                  Provide the student with a set of triangles.
according to their sides and angles?                Have the student name the triangles
according to their sides and angles.

22) Can the student find the perimeter of               Give the student a meter stick or trundle
polygons?                                           wheel. Have the student find the
perimeter of the classroom, hallway,
shapes on the floor, etc.

23) Can the student find the area of                    Give the student grid paper and a ruler.
triangles and quadrilaterals?                       Have the student draw a triangle, a
rectangle, and a parallelogram on the grid
paper and find the area of each figure.

24) Can the student find the                            Give the student a circle with a diameter of
circumference and area of a circle?                 5 cm. Have the student find the
circumference and area of the circle using
the appropriate formulas.

25) Can the student identify congruent                  Provide the student with the following sets:
segments, angles, and polygons?                     angles, segments, and polygons. Have
the student identify the congruent shapes
in each set.

26) Can the student develop                             Provide the student a set of three-
relationships of faces, vertices, and               dimensional figures. Ask the student to
edges of three-dimensional figures?                 identify all of the faces, edges, and
vertices.

27) Can the student perform                             Provide the student with a geoboard. Ask
transformations on plane figures                    the student to model/demonstrate a
using physical models and graph                     transformation of a given design.
paper?

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)    2000

28) Can the student identify lines of                   Provide the student with a triangle,
symmetry in polygons?                               square, rectangle, octagon, parallelogram,
etc. Ask the student to draw lines of
symmetry for each figure.

29) Can the student develop and apply                   Use the Pythagorean Theorem formula to
the Pythogorean Theorem to find the                 find the longer leg of a right triangle, if the
missing side of a right triangle?                   hypotenuse is 25 units and the shorter leg
is 7 units.

30) Can the student graph ordered pairs                 Provide the student with a pegboard with
on a coordinate plane?                              a coordinate plane sketched on it and a
handful of golf tees. Have the student plot
several ordered pairs on the pegboard
from a given set of ordered pairs.

31) Can the student use powers of ten to                Provide the student with a decimal
multiply and divide decimals?                       number. Have the student:
 Multiply the decimal number by a
power of ten.
 Divide the decimal number by a power
of ten.

Example:
a) 0.125 x 10
b) 3.5 x 100
c) 152 ÷ 1000
d) 0.39 ÷ 100

32) Can the student use patterns to                     Give the student a number. Have the
develop the concept of exponents?                   student find a geometric pattern using
multiplication. Example:

1, 4, 9, 16 _, _, _
(12, 22, 32, 42…)

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)       2000

33)      Can the student write numbers in               Give the student a number. Have the
standard and exponential form?                 student write the number in exponential
form. For example: 154 = (1 x 102) +
(5 x 10) + (4 x 1)

Give the student a number written in
exponential form. Have the student write
the number in standard form. For
example: (3 x 103) + (5 x 102) +
(8 x 1) = 3, 508

34) Can the student convert between        Give the student a number written in
standard form and scientific notation? scientific notation and have him/her
convert the number to standard form.

Example: Have the student convert a
number such as 6,200,000,000 to
scientific notation. (6.2 x 109)

35) Can the student find and use prime                  Provide the student with two or more
factorization with exponents to obtain              whole numbers. Example: 20 = 22 x 51
the greatest common factor (GCF)                                                 14 = 21 x 71
and least common multiple (LCM)?                    Have the student:
 Find the prime factorization of each
number.
 Find the GCF of any 2 numbers of the
given set using the factorizations.
GCF = 2 1 = 2
 Find the LCM of any 2 numbers of the
given set using the prime
factorizations. LCM = 2 2  5 1  7 1 = 140

36) Can the student use patterns to                     Have the student draw squares with
develop the concepts of roots of                    lengths of 1 cm to 10 cm. Have the
perfect squares with and without                    student calculate the area of each square.
calculators?                                        Have the student complete the following
pattern:

12 = 1  1 =1             1  1 1  1
22 = 2  2 =4             4    22  2
32 = 3  3 =9             9  33  3

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)      2000

37) Can the student recognize and write                 Provide the student a set of integers.
integers including opposites and                    Have the student write the opposite of the
absolute value?                                     given integer. For example:

Integer                       Opposite
a) 8                             -8
b) -24                           24
c) 0                              0

Have the student write the absolute value
of the given integer. For example:

Integer                       Absolute
Value
a) 4                            4
b) 8                            8
c) 0                            0

38) Can the student compare and order                   Provide the student with a number line
integers?                                           and a given set of integers. Have the
student:
 Compare sets of integers.
 Order from least to greatest.

39) Can the student add, subtract,                      Provide the student with a variety of
multiply, and divide integers with and              addition subtraction, multiplication, and
without calculators?                                division problems using integers. Have
the student solve the problems without a
calculator and then check each problem
using a calculator. For example:

a) -8 + 12
a) –55 ÷ 11
b) –15 – 14
c)   4 x 16

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)    2000

40) Can the student use the order of                    Have the student explain the correct order
operations to simplify and/or evaluate              in which to simplify numerical expressions.
numerical and algebraic expressions                 Have the student simplify the following
with and without calculators?                       numerical expressions.

15  10  5
2 6  4 2
30  5 22  2
6  4(3  2)

Provide the student with a calculator to

41) Can the student compare, order,                     Provide the student with a set of decimals.
round, and estimate decimals?                       Have the student:
a) Compare two or more decimals
b) Order the set from least to greatest,
c) Round the decimal to a given value
d) Estimate the decimals.

Example:
a) 0.56____ .05
b) 0.56, 4.5, 5.6, .05
c) Round 0.56 to the nearest tenth
d) Estimate: 5.6 + 0.56 = ____

42) Can the student add, subtract,                      Provide the student with several decimal
multiply, and divide decimals in real-              numbers that have been written on the
life situations with and without                    board or overhead and ask them to write
multiplication, and/or division word
problems using the numbers. The student
will then solve their problems.
Example:      28.7     2.5      426

\$1.31   7.236

Shelley’s gas tank holds 28.7 gallons of
gas. If she can travel 426 miles on a tank
of gas, how many miles per gallon does
her car get?

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)        2000

43) Can the student convert among                       Provide the student with a chart in which
decimals, fractions, and mixed                      they have to fill in the correct form, such
numbers?                                            as the following:

Decimal        Fraction          Mixed Number
2.5
804
100

3
14

44) Can the student express ratios as                   Provide the student with 4 yellow squares
fractions?                                          and 8 brown squares that have been cut
from construction paper. Have the student
 What is the ratio of yellow squares to
brown squares?
 What is the ratio of brown squares to
yellow squares?
 What is the ratio of yellow squares to
all squares?
Answers are to be written in ratio and
fraction form.

45) Can the student add, subtract,                      Provide the student with problems that
multiply, and divide fractions and                  involve addition, subtraction, division, and
mixed numbers?                                      multiplication. Use denominators that are
found in everyday life, such as
2,3,4,8,12,16.

Example:

At the family reunion Kerra saw 60 of her
relatives. She remembered the names of
3
of them. How many names did she
4
remember?

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

46) Can the student use estimation to                   Have the student estimate the following:
add, subtract, multiply, and divide                       1     4
fractions?                                          a) 3 8  6 5  3  7  10
5        2
b) 2 6 x 7 9  3 x 7  21

47) Can the student explore equivalent                  Provide the students with cards with ratios
ratios and express them in simplest                 written on them. Have the student match
form?                                               a ratio card with another card that is an
equivalent ratio in simplest forms.

48) Can the student solve problems                      Provide the student with proportion
involving proportions?                              problems to solve such as the following:
 9 to 15 is the same as 12 to_______
8    12
 Solve :      =
12     n

49) Can the student determine unit                      Provide the student with an index card
rates?                                              containing information such as the
following on it.

Apples
\$1.60
5

Have the student will determine the cost of
1 apple.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)       2000

50) Can the student use models to                       Provide the student with graph paper and
illustrate the meaning of percent?                  a ruler in which he/she is to model
percents such as the following on the
graph paper. Using 10-by-10 grids, have
the student model his/her answers to each
of the problems.

3
a)
10
85
b) 4
100
c) 23%

51) Can the student convert among                       Provide the students with a chart in which
decimals, fractions, mixed numbers,                 they have to fill in the correct form, such
and percents?                                       as the following.

Decimal      Fraction   Mixed Number     Percent

0.25

3
1
5

6                       33%
10

52) Can the student determine the                       Provide the student with a piece of graph
percent of a number?                                paper and have him/her model 25% of 80.
The student will then calculate the answer.

53) Can the student estimate decimals ,                 Provide the student with several word
fractions, and percents?                            problems to estimate using decimals,
fractions, and/or percents, such as the
following:

A Big Mac at McDonalds’s has 940 calories. Jesse
1
3
did he eat?

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

54) Can the student use proportions and                 Have the students count the number of
equations to solve problems with                    girls and boys in the classroom. Knowing
rate, base, and part with and without               this information, have the students predict
calculators?                                        how many boys and how many girls are in
the seventh grade class with a total of 750
students. Find the percent of the class that
is boys.

55) Can the student find the percent                    Provide the student with several
of increase and percent of decrease?                basketball players’ shooting averages
from last year and this year. Have the
student calculate the percent of increase
or percent decrease of last year’s average
and this year’s average.

56) Can the student solve problems                      Provide the student with a restaurant
involving sales tax, discount, and                  menu. Have the student order a meal.
simple interest with and wihtout                    Have the student calculate the total cost of
calculators?                                        the meal, including 7% sales tax and a
15% tip.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

Suggested Teaching Strategies for
Uses properties to create and simplify algebraic expressions and solves linear
equations and inequalities

1)      Describes and extends patterns in sequences
 Use pattern blocks to develop a tessellation.
 Use pencil and paper to use patterns to develop string art.

2)      Identifies and uses the commutative, associative, distributive, and identity properties
 Divide the class into two groups. Give each student a card with a variable, an
operation, parentheses, or an equal sign written on each card. Have the
students to work cooperatively to create and identify a property using the
given cards.
 Create one set of cards with the property names written on them and another
set with an example of the property written on them. Have the students play
―Memory‖ by matching the property name with the property example.

3)      Translates between simple algebraic expressions and verbal phrases
 Create an expression search puzzle using algebraic expressions. Give the
students verbal phrases as clues. Have the student find and circle the

A number (x) increased by five                      x+5
Twice a number (n)                                  2n
Six minutes less than Bob’s time (t)                t–6
3 years younger than Seth (s)                       s–3

   Give the students a number. Have the students perform the instructed
operations on that number.

Example:
Twice that number                  12  2 = 24
Less 8                             24 - 8 = 16
Divided by 4                       16 ÷ 4 = 4
Increased by 2                     4+2=6
Squared                            62 = 36

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

4)      Solves linear equations using the addition, subtraction, multiplication, and
division properties of equality with integer solutions
 Have the students solve equations using a flow chart. On the top of the flow
chart create the equation and solve the equation on the bottom.

Example:              3a – 4 = 11

+4       +4               3a = 15
3 3                     a=5

   Give the students ten equations that are already solved. Have the students
mark the equations that are worked correctly and rework the equations
worked incorrectly.

5)      Writes a real world situation from a given equation
 Give the students the equations such as 3 + (2.50)x = 11.50. Have the
students write a story problem based on the equation.

Sample Questions: The Rollo Skating Rental charges an entrance fee of \$3
and an additional \$2.50 per hour. If Sam paid \$11.50 to skate, how many
hours did he skate?

   Have the students create a small business that will sell a product or service.
Have the students create an advertisement for the product or service that
includes a description of what they are selling and the cost. Based on this
information, have the students write an equation describing the relationship.

6)      Writes and solves equations that represent problem-solving situations
 Have the students write an equation and solve problems.

Examples:

    The sum of a number and 8 is 21. What is the number?
    The sum of two consecutive numbers is 23? What are the numbers?
    Mike’s test scores are 75, 93, 88, and 82. What must he make on his next
test to have an average of 85?

John made 14 points in the basketball game on Tuesday. By the end of the
second game on Friday, his two game total was 30 points. How many points
did John make in the second game? Solution: Tuesday’s game + Friday’s
game = total points of 14 + f = 30

   Choose several occupations and write problem-solving situations for each
occupation on index cards. Divide the class into groups according to the
occupations chosen and distribute the cards for that occupation to the
appropriate group. Each group will write and explain equations that could be

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

used to solve each problem. Group presentations can be made
demonstrating how various occupations use the equations in the business.

Example: Carpenters

One sheet of 4x8 plywood will cover 32 sq. ft. How many sheets will be
needed to cover 640 sq. ft.? 32 p = 640

Interprets, organizes, and makes predictions using appropriate probability and
statistics techniques

7)      Organizes data in a frequency table
 Provide the students different types of balls and collect data on how many
times the balls bounce. Transfer information to a frequency table.
 Give students a list of their grades. Have students organize their grades in a
frequency table.

8)      Interprets and constructs histograms, line graphs, and bar graphs
 Use the computer to show different types of bar graphs and line graphs.
 Take the students outside and choose students to run sprints. Students will
time and record data. Divide the students into groups. Assign each group a
different type of graph to construct using the data collected.

9)      Interprets and constructs circle graphs when given degrees
 Have the students use protractors to construct circle graphs when given
degrees. Have the students use compasses to draw a circle.
 Have the students use the computer to construct circle graphs. Have the
students write two questions about their graphs.

10)     Interprets and constructs stem-and-leaf plots and line plots from data
 Time the students as they toss paper into the trash cans. Count the number
of baskets each student makes. Have the students to create stem-and-leaf
plots and line plots from the collected data.
 Give the student a set of data such as 23, 25, 40, 71, 21, 32, 66, 54, 40, 47
and 78 which could represent the number of points scored in a game.
Have the students put the data in order from least to greatest. Have the
student choose an appropriate stem value such as tens digits and leaf values
such as one digits. Plot the data in a stem-and-leaf plot. Explain the plot
giving an explanation of the stems and leaves. Be sure to give the stem-and-
leaf plot a title.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

Examples:                      Number of Points Scored

Stems         Leaves
2            1,3,5
3              2
4           0, 0, 7
5              4
6              6
7             1, 8

   List the grades made on a particular test in random order. Students will
construct and interpret stem and leaf and/or line plots from this data.

11)     Estimates and compares data including mean, median, mode, and range of a set
of data
 Have students estimate the number of drops of water that will fit on a penny.
Provide students with a dropper, water, and a penny. Have the student find
the mean, median, mode, and range for the class data.
 Have the students use the mean, median, and range feature on the graphing
calculator.

12)     Predicts and recognizes data from statistical graphs
 Have the student go to a web site on the Internet dealing with the stock
market page. Have students predict and recognize data.
 Provide students with several examples of statistical graphs from newspapers
and magazines. Divide the class into groups and let each group choose a
graph to use. Have the students in each group present an explanation of the
data represented by their graph to the class. Then have them use their
graphs to predict what might happen in the future and explain what might
cause these changes.
 Have students collect a variety of graphs from magazines or newspapers.
Have each student write questions about a selected graph. Students
exchange graphs with a partner and answer the questions about the graphs.

13)     Determines probability of a single event
 Have the students roll a number cube to find the probability of rolling an even
number, odd number, prime number, composite number, and a given number.
 Have the students roll two number cubes, add the numbers together, and find
their sum. Have the students determine the probability of getting the sum of
9, 12, etc.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

14)     Uses simple permutations and combinations
 Announce to the students that they are going to elect a president, vice-
president, and secretary for the class. Have the students nominate five
students to run for the three positions. Have the students predict the possible
outcomes.

Examples: P (5,3) = 5 · 4· 3 = 60

*Order is important in permutations.

Choose a three-member team from a group of five people. How many teams
are possible?

543
C (5,3) =              10 outcomes
3  2 1

*Order is not important in combinations.

Writes and solves problems involving standard units of measurement

15)     Converts within a standard measurement system (English and metric)
 Provide the students with a chart of equal measures. Show the students how
to use cancellation to convert measurements.

Example:

a) Convert 5 gal. to quarts.
4qt
5 gal          5  4qt  20 qt
1gal

b) Change 1000 ft. to yards
1yd 1000  1yd 1000yd      1
1000ft                         333 yd
3ft    3         3         3

c) Change 8.7 ft. to inches
12in 8.7  12
8.7ft                 in  104.4in
1ft    1

Explain that whatever unit the student starts with must be the unit in the
denominator of the second fraction, and the numerator is its equal measure.
Multiply numerators. Divide by denominator.

   Provide each student a metric line chart. Divide the students into groups of
fours. Give each group a ziploc bag of cubes, a ziploc bag of beans, and two
M & M’s for each student. The cubes represent all numbers except 0, the

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)      2000

beans represent zeroes, and the M & M’s represent decimal points. Give
each group four problems. Have each student take a problem and model the
conversion by moving the decimal.

Example: 17 m = 17000 mm                                            0.45 g = 0.00045 kg

Metric Chart

Kilo       Hecto Deka         M       Deci      Centi      Milli

This saying may help students remember the chart ―King Hector Died
Monday Don’t Call Me.‖

   Make a floor chart of the metric system. Make a black circle for the decimal
point. Have students write a conversion problem, swap problems with their
neighbor, and walk the problems off on the floor chart.

16)     Converts temperature using the Fahrenheit and Celsius formulas
 Write the formulas on the board or overhead.
 Demonstrate solving formulas, emphasizing that substitution is used. Have
students use order of operations to convert temperatures.

Example: Change 50˚ F to ˚C
5
˚C=   (˚F – 32)
9
5
C  (50 - 32)
9
5
C  (18)
9
C = 10C

Change 0˚ to ˚F

9
F= C + 32
5
9
F= (0) + 32
5
F=0 + 32
F=32F

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)       2000

   Explain that a formula shows a relationship between quantities. Give the
student the formulas for converting degrees Fahrenheit to degrees Celsius
5
and for converting degrees Celsius to degrees Fahrenheit: C =                       (F - 32) and
9
9
F=     C + 32.
5
   Give a problem such as, the temperature outside is 10C. What is the
temperature measured in degrees Fahrenheit? Have the students explain
which formula should be used. Then have the student write the correct
formula and plug in the appropriate information.

9
Examples:             F=    C +32
5
9
F = (10) +32
5
F = 18 + 32
F = 50

*Note: In order for students to use this formula, they must know how to
multiply fractions. Use a scientific calculator to convert between Fahrenheit
Celsius temperatures.

17)     Uses standard units of measurement to solve application problems
 Provide the students with Standard English and metric units of measure and
calculators. Give a series of questions in different areas and require students
to work in pairs to answer the question.

Examples:

1. A beauty shop has about 66 customers per day, keeping 3 beauticians
busy for most of a 9-hour day. How long does each beautician average
with each customer? Give answer in minutes.
2. John said ―It is 35˚ C outside.‖ Is this a hot, a cold, or a very pleasant
comfortable day?
3. Dominick is recovering from surgery and is not suppose to lift more than
25 pounds. He works in a grocery store as a stocker. A bag of flour
weighs 5 pounds. Each case contains 6 sacks. Can he carry the cases to
the aisle and stock the shelf? Explain.
4. John’s coach has ordered him to drink a liter of Gatorade during practice.
The containers hold 1,000 ml, 500 ml, and 250 ml. Which size should he
get? Explain.
5. Find the amount of carpet needed to cover the floor, the paint to paint the
walls, and the border to go around a room with certain dimensions.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

   Use plastic straws, a hole punch, and brads to construct various
quadrilaterals. (D-Stix or other commercial sets may be used.) Use different
colored straws for different lengths so they can be compared.

Examples:

4 equal sides, angles not all equal

4 equal sides, no right angles

4 equal sides, all right angles

4 equal sides, square or rhombus

   Have students measure the dimensions of the classroom floor to determine
the area of the floor.

Determines the relationships and properties of two and three-dimensional
geometric figures and the application of properties and formulas of coordinate
geometry

18)     Identifies polygons with up to twelve sides
 Students work with a partner to create shapes of polygons on a geoboard.
Record on dot paper and identify each polygon by name
 On a geoboard construct each of the following figures with one rubber band
and record each result on dot paper.
a. an acute triangle
b. a right triangle
c. a triangle
d. a isosceles triangle
f. an equilateral triangle
g. a rectangle with four congruent sides
h. a parallelogram with four right angles
i. a pentagon
j. a hexagon

19)      Classifies and compares properties of quadrilaterals
 Using a Venn diagram, compare and contrast the following quadrilaterals.
Explain why they were grouped in a particular way. Provide the student with
pictures of the five kinds of quadrilaterals. Have the student explain the
properties of each set that are alike and the properties of each set that are
different (number of sides, types of interior angles, congruent angles,
congruent sides, length of sides).

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

20)     Classifies and measures angles
 Cut out models of rays from tag board or card stock. Connect two rays with a
paper fastener to form a model of an angle. Explore forming angles of
different sizes by rotating the rays around the vertex. Sketch the angles you
modeled. Next to each angle write the name of each. Measure with a
protractor to validate the name of the angle. List examples around the
classroom that are real-world types of angles.
 Let students work in small groups of 2 or 3. Give each student a sheet of
white paper and a protractor. Each student will draw several angles using the
protractor. Each student should draw at least one of each of the following:
acute angle, obtuse angle, right angle, and straight angle. Encourage
students to draw these angles opening in different directions. The students
will exchange papers and use the protractor to measure and classify each
angle.

21)     Classifies triangles by sides and angles
 Use a circular geobard template to connect the points. Then measure the
length of each side of the figure with a ruler. What type of triangle is
represented?

Correct Points               Type of Triangle
1.   P7 P15      P23
2.   P1 P6       P11
3.   P11 P15     P20
4.   P20 P6      P10

Using your circular template, connect the points. Then measure each angle
and classify the triangle.

Correct Points              Type of Triangle
1.   P5 P14      P21
2.   P4 P10      P16
3.   P24 P16     P20
4.   P5 P13      P21

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

   Use the circular template or geoboard to investigate which of the following
triangles are possible
1. an acute scalene
2. an obtuse scalene
3. an obtuse isosceles
4. an obtuse equilateral
5. an acute equilateral
6. a scalene right
7. an isosceles right

22)     Finds the perimeter of polygons
 Give students geoboards, rubberbands, and dot paper. Students will model
polygons with perimeter of 20 units each. Each polygon will be drawn on dot
paper.
 Give students drawings of polygons and the measurement of the sides.
Students will find the perimeters.

23)     Finds the area of triangles and quadrilaterals
 Discuss the materials that are necessary to paint the interior of a house.
Discuss the specific materials needed to paint each room. Determine how
much of each material will be needed to paint a seven room house.
 Divide the student into teams. Have each team to choose an area of the
school to paint a new and better color. Each team will determine perimeter
and area of the floor, the walls, and ceiling. Then they are to determine the
amount of paint needed for their chosen area.

24)     Finds the circumference and area of a circle
 Give students grid paper. Trace a small and large circle on the grid paper.
Have the students count the squares and approximate the area of each circle.
Wrap a string around the outside of each circle. Measure the string to
determine the perimeter.
 Use a calculator to find the area and circumference of various circles given
the formulas A=r2 and C=d. Discuss the  key on a calculator and the
22
common values used for  are                or 3.14.
7
   Have several circular items in the room. Let the student choose one or more
of the circular items. Have the student measure the circumference and the
diameter of each item. This information should be recorded. Be sure
students know that pi () is approximately 3.14. Students may use a
calculator to solve the problem.

25)     Identifies congruent segments, angles, and polygons
 Using a set of tangrams, the student will find all tangram shapes that are the
same size and shape. Ask the students to explain how they know that they
are the same size and shape. Tangrams that have the same size and shape

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)       2000

are congruent. Also have students create congruent shapes using two or
more tangrams.

Note: After activities read the ―Grandfather Story,‖ and play Dominoes by
matching congruent segments, angles, and polygons.

Examples:

Define the term congruent (the same size and shape) and give the symbol used
to show two items are congruent (). On a sheet of paper, draw several line
segments, angles, and polygons. Be sure that some or all have a match. Have
the student match the items that are congruent.

26)     Develops relationship of faces, vertices, and edges of three-dimensional figures
 Use one or more of these strategies: Make a table, use a formula, draw a
diagram or guess and check to complete the following table.
 N - The number of sides in the base
 F - The number of faces
 V - The number of vertices
 E - The number of edges

Complete the table and identify the relationship between the number of vertices
and edges.

Figures                  N                  F                  V                  E
Triangular Prism
Rectangular Prism
Pentagonal Prism

   Construct three-dimensional figures from straws or commercial sets letting
each length be a different color. Describe the number of faces, the shape of
each face and the number of vertices and edges. Compare with other three-
dimensional figures.

Example: Triangular prism and rectangular pyramid.
Discuss the similarities and differences.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

27)    Performs transformations (rotation, reflections, translations) on plane figures
using physical models and graph paper
 Use a geoboard to create a quadrilateral and demonstrate slides, flips, and
turns.

Slide

    Use grid paper to form a coordinate plane. Plot the following points, and then
translate (slide) the point and make the new coordinates. Use colored pencils
to work new points.

Example:         (-2, 4) Slide 4 units right (2, 4)
(4, 0) Slide 3 units down (4, -3)

    Working in pairs, reflect the following shapes across the line of symmetry.

A                                                         B

C                                      D

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

28)     Investigates symmetry of polygons
 Have each student take a shape (made from cardstock) and one sheet of
tracing paper. Trace the cardstock shape on the tracing paper. Then have
each student see if the traced shape could be folded so that the 2 halves will
match exactly. If the traced shape can be folded exactly in half, then the fold
line is the line of symmetry. Try to see if the traced shape can be folded
another way to match exactly. There may be more than one line of
symmetry.
 Have each student use another sheet of tracing paper and trace the same
shape or different shapes. Each student then takes a mirror or mira and
places it on the figure so that half the figure together with the reflection of that
half forms the entire figure. The line along which the mirror is placed is the
line of symmetry.
 Have each student use another sheet of tracing paper and trace a shape.
Students will cut out the traced shape, then try to fold the shape so that each
half is the same along the folded line of symmetry. *Note: Look for more
then one line of symmetry in some shapes.

29)     Develops and applies the Pythagorean Theorem to find missing sides of right
triangles
 Using a geoboard construct a right triangle that has legs each 3 units + 4
units long. Form two squares using the length of each leg as the side of each
square. Also form a square using the hypotenuse as one side. The area of
square ―a‖ plus square ―b‖ = the area of square ―c‖.

   Using the Pythagorean Theorem a2 + b2 = c2, find the missing side (Use
calculator if needed)

Leg 1                  Leg 2                   Hypotenuse
3                      4                         5
5                     12                         13
8                     15                         17

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

30)     Graphs ordered pairs on a coordinate plane
 Use dot paper to form a coordinate grid. Give each student a grid. Call out
ordered pairs, and have students plot them on the grids.
 Track a real or imaginary hurricane on a hurricane tracking map. Discuss
latitude as the x-axis and longitude as the y-axis.
 Create a design by connecting points represented by x, y coordinates.

(2, 1)          (8,1)
(4, -1)          (2, 1)
(6, -1)

   Provide the student with a coordinate plane. Discuss the x-axis and y-axis
and the ordered pair of numbers which indicate the coordinates of a point (x,
y). Have the student plot several coordinates on the graph. Examples: (5, 2),
(3, 4), (0-03), (-4, -2).
   If there is a floor with square tiles, have students determine an origin and use
tape to make the x-axis and the y-axis. Call out an ordered pair and have the
student walk out the correct number of squares to get to the given point.

Uses basic concepts of number sense and performs operations involving
exponents, scientific notation and order of operations

31)     Uses powers of ten to multiply and divide decimals
 Make a set of index cards with problems such as 2.756 x 300 on them. Make
another set of index cards with the answer 826.8 on them. As students enter
the classroom they are given an index card. Students are instructed to find
their match.

32)     Uses patterns to develop the concept of exponents
 Have the students use grid paper to explore the patterns of exponents.

Example:
20     21    22            23

   Have the students multiply or divide decimals by powers of ten using long
multiplication or division. Students will then look for patterns and develop
shortcuts.

33)     Write numbers in standard and exponential form
 Have the students use dominos to write numbers in standard and exponential
form.

= 42 = 44                          = 34 = 3333

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)         2000

   Students find the prime factorization of a number and write the answer in
standard and exponential form 810 = 2  3  3  3  3  5 = 21  34  51.

34)     Converts between standard form and scientific notation
 Have the students use a map of the solar system to convert the distances into
scientific notation. Earth to sun: 93,000,000 miles = 9.3 x 10 7
 Have the students brainstorm on when they would use scientific notation.

Example: Thickness of a strand of hair in inches
The length of a bedroom wall in millimeters

35)     Finds and uses prime factorization with exponents to obtain the greatest common
factor (GCF) and least common multiple (LCM)
 Create a Vienn Diagram. First allow the students to create a factor tree.

GCF =                12                                             20

2 6                                         2        10

22  3             list common multiples   2        2  5
12           20

3      22        5

GCF is 2 x 2 = 4

3    2 2     5
LCM

LCM          3  4  5 = 60

   Write a prime factorization for a set of numbers such as 18 and 30. Use
common bases and lowest exponents for GCF. Use all bases and largest
exponents for LCM.

Example:                       18 = 2  32                           30 = 2  3  5
GCF = 2  3 = 6                       LCM = 2  32  5 = 90

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)       2000

36)     Uses patterns to develop the concept of roots of perfect squares with and without
calculators
 Provide the students with cm squares. Have the students arrange the pieces
into squares and tell how many pieces it takes to make the squares

Example:

1                    4                   9                 16

1x1                2x2                   3x3
4x4
   Provide the students with a set of numbers. Have the students predict which
number are perfect squares. Have the students use the calculator to check

Number                        Prediction                  Calculator Check
1                              yes                           1  1 yes
10                             no                            10  3.2 no
18                             yes                          18  4.2 no

37)     Recognizes and writes integers including opposites and absolute value
 Have the students make a human number line. Have one student walk this
number line and turn every time opposite is mentioned.

Opposite of –3
-3           0             3

   Use this same number line to demonstrate absolute value and discuss the
definition of absolute value.

38)     Compares and orders integers
 Place students into groups of four or less. Provide the students with a set of
cards with integers and >, <, =, and absolute value signs written on them.
Have the students arrange the cards from least to greatest.

Example:

-96   -50       -31    0        30       48      101       250

Ask the students comparison questions. Have the students in each group
use the cards to answer the questions.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)            2000

   Provide the students with a number line made from a piece of rope and the
rope knotted at equal intervals. Have the students go to a knot on the
number line and move forward as a problem is read out loud. Have the
students make comparisons.

39)     Adds, subtracts, multiplies, and divides integers with and without calculators
 Have the students use a number line rope. Call out addition and subtraction
problems. Have the students move up and down the rope and tell their new
position.
 Have the students play ―around the world.‖ Provide each student with one
integer card. Have each student stand by another student. Have the two
students hold up their cards. Call out an operation. The student who answers
first exchanges cards and moves around to the next person. The person who
goes around the room first wins.
 Have the students play domino chase in pairs. Provide the students with a
game board, two tokens, and a + - card.

start
finish

                    -2 + 4 = 2 move forward 2 places
     

-                  +         -5 + 1 = -4 move backward 4 places

      


40)     Uses the order of operations to simplify and/or evaluate numerical and algebraic
expressions with and without calculators
 Have students simplify problems using the correct order of operations:
Example: 3 + 2 – 4  3 = ‾ 7
 Have the students solve problems and expressions. Have the students check
their work with calculators.
 Students may learn the phrase ―Please Excuse My Dear Aunt Sally‖, to assist
them in learning order of operations. (Parentheses, Exponents, Multiplication,
and Division left to right, Addition and Subtraction left to right)

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

Determines the relationships among real numbers to include fractions, decimals,
percents, ratios, and proportions in real life problems

41)     Compares, orders, rounds, and estimates decimals
 Provide students cards containing decimal numbers. Have the class arrange
itself in order from least to greatest. Have each student remain in line and tell
what his/her number is, when rounded to the nearest given place value.
 Have two students show their card numbers containing a decimal. Have the
other students estimate the sum of these two numbers. A statement will also
be written to compare the two numbers such as 42.347 > 41.38.

42)     Adds, subtracts, multiplies, and divides decimals in real-life situations with and
without calculators
 Use place value blocks and charts to model all four operations. Have
students construct word problems that fit the problems modeled using the four
operations.
 Place value blocks can be made by taking a grid sheet of 1 cm squares and
dividing it into several 10-by -10 cm squares. Make and laminate enough
copies for each student to have at least six 10-by-10 cm squares, 15 1-by-10
cm squares, and 15 1-by-1 cm squares. These can be stored in baggies so
that each student will have his/her own squares.
 Write several problems with addition and subtraction of money on the
overhead/board, for example, \$2.96 + \$0.63 and \$0.73 – \$0.20. Have the
students solve the problems and discuss the similarity between decimals and
money amounts. Explain that adding and subtracting money is similar to
 Have the students estimate the amount their family spends on food each
month. Have the students bring to class their cash register receipts for food
for the month. Have the students add the cash register receipts and compare
them with original estimates.

43)     Converts among decimals, fractions, and mixed numbers
 Have students play the ―Memory Game ― individually or in groups of up to four
players. Make twenty-four square-sized cards. Write different fractions on
eight squares, the decimal form of the fraction on eight squares, and the
percent form on eight squares. (See below) The 24 cards are placed
facedown with 6 rows of 4. One player turns up 3 cards trying to get a 3-way
match of a fraction, decimal, and percent card. If a correct match is not
made, the next player turns up three cards. If a correct match is made, that
player continues until a match is not made. The player with the most matches
at the end of the game is the winner.
FRACTION                      DECIMAL                     PERCENT

3
0.75                       75%
4

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

   Provide each student an index card on which either a decimal, fraction, or a
1
percent has been written, such as              , 50%, or 0.5. Have the students locate
2
their matches.
   Have students determine what letters of their first name are vowels, and have
them record this information on an index card. Convert the number to
decimal and percent forms. A large piece of string can be strung across the
room for a number line for students to tape their index cards. Have the
students repeat procedure for their last name. Have the students develop
graphs from the class data.

44)     Expresses ratios as fractions
 Have the student write the following as a fraction.
a) 3:5
b) 6 to 9
c) 10 out of 20
d) John made 4 free throws when he shot 7 times.

   Give the student a problem and have the student express the ratio as a
fraction. Example: There are 14 girls and 17 boys. What is the ratio of girls
14                                                17
to boys?         What is the ratio of boys to girls and boys?      What is the
17                                                31
31
ratio of girls and boys to girls?
14
   Collect data from the class such as number of boys and girls or number of
red, blue, white, and mixed shirts. For example: 12B, 14G, 8r, 4b, 42, 10m.
Write fractions (in lowest terms) to show comparisons such as:
14     7
 the fractional part of the class that is girls.      =
26    13
4     2
 the fractional part of the class wearing a blue shirt.       =
26   13

45)     Adds, subtracts, multiplies, and divides fractions and mixed numbers
 Have students roll fraction number cubes twice. Have the students performs
all four operations using the two fractions rolled.
 Have the students work with a partner. Provide each group with a baggie and
grease pencil. The groups are to draw a picture on the baggie with the
grease pencil. Have each group individually place its baggie on the overhead
projector. Have the class to write an addition, subtraction, multiplication, and
division word problem that fits the picture. Have the students share the
problem with the class.
 Have the students work with a partner to construct a spinner containing the
digits 1-9. Have a student write on a sheet of paper the following:


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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

Have the same student spin the spinner four times. (The first number will be
placed in the numerator spot of the first fraction. The second number will be
placed in the denominator spot of the second fraction. The third number will
be placed in the numerator spot of the second fraction, and the fourth number
will be placed in the denominator spot of the second fraction.) Have another
student spin four times and do the same thing as the other student. Have the
students then add their fractions. The student with the larger fraction is the
winner. This activity can also be done with subtraction, multiplication, and
division. In subtraction, if the first fraction is smaller than the second fraction,
the student must swap them.

46)     Uses estimation to add, subtract, multiply, and divide fractions
 Have the student use Cuisennaire Rods to estimate adding, subtracting,
multiplying and dividing fractions.
 Have the students locate fractions on a ruler and round to the nearest whole
number.

Example:

0                    x     1                      x    2
3                         13
         1
4                         16
1 + 2 = 3 estimation
   Have the students use the stock quotes in a newspaper to estimate gains or
losses for a stock on a particular day. Add or subtract to compare with their
estimates.

47)     Explores equivalent ratios and expresses them in simplest form
 Write a ratio on one side of fifteen index cards and its equivalent (simplest
form) on the other side. Cut the cards in various ways in the middle. Give
each student 1 of a card. Do not let the students see each other’s cards.
2
One student stands and calls out his ratio and the student who thinks he has
a match stands. If they are correct, their cards will fit together.
 Give students several pairs of equivalent ratios. Have students draw a
rectangle to show the ratio draw a rectangle the same size showing the ratio
with the smaller numbers. Compare the shaded part of the two rectangles. It
1 2
should be the same. Example:                ,
2 4

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

48)     Solves problems involving proportions
 Model the procedure for solving a proportion using cross products. Point out
to the students that the position of the terms in each ratio MUST correspond.
 Have the students set up a proportion. Example: ―Sheila wants to buy a CD
player that costs \$240. She earns \$25 in 5 hours babysitting. How many
hours will Sheila have to work in order to be able to buy the CD player?

(hours worked)5   n(hours worked)

(earnings)    25      (earnings)

   Have the students draw a rectangle that measures 1 cm by 3 cm on graph
paper. Have the students draw another rectangle whose sides are 3 times
the length of the first rectangle. Have the students label the lengths. Have the
students write a proportion whose ratios are the widths over the length of
each rectangle.
   Provide each student with centimeter squares. Have students construct
rectangles with one side twice as long as the other. Have students create
their own proportions using the pieces and share these with the class.
   Utilize the Russian dolls that come in proportional sizes for students to
actually see how the dolls are proportionally made. Sand is a good material
for students to experiment with.

49)     Determines unit rates
 Set up a class store. Have at least a produce section, a meat section, and a
canned good section. Have students bring pictures of items that are found in
these sections. The students will cut out the picture and glue them to
construction paper. Provide prices by each item, such as, green beans-
4/\$1.00, apples-3/\$0.89, etc. Set up the items in centers. Have the students
rotate from one item to the next to determine the cost of one of each item.
Have the students then work with a partner to see if they have the same
 Have the student bring in grocery ads from newspapers. Have the students
figure and compare unit prices on given items at two different stores.

50)     Uses models to illustrate the meaning of percent
 Allow the student time to experiment with place value mats and blocks to
build decimal numbers in relation to percents. Write percent and decimal
numbers on index cards. Give each student an index card that has a decimal
number on it. Have the students model their number as a percent on the
place value chart.
 Have the students play the ―Memory Game‖. Make index cards for students
to match. Have the students work with a partner or individually. Give each
group a deck of 16 cards. Write a percent on eight of the cards. Provide a
picture of a 10-by-10 cm grid that has been shaded on the other eight cards.
(See example below.) The 16 cards are placed face down in rows of four.
One player turns up 2 cards trying to and get a match, such as 16% is the

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)       2000

same as a 10-by-10 cm grid that has 16 squares colored. If a correct match
is not made, the other player turns up 2 cards. If a correct match is made,
that player continues until a match is not made. The player with the most
matches at the end of the game is the winner.

16%

   Provide each student with a laminated 10-by10 cm grid and an index card
that has a percent written on it. Have the students use a watercolor marker to
model their number on the grid.

51)     Converts among decimals, fractions, mixed numbers, and percents
 Place the students in groups of 2-4. Provide each group with a deck of 43
cards. Write a decimal number on fifteen of the cards, a fraction or mixed
number on fourteen cards, and a percent on the remaining fourteen. Deal
each student 7 cards. Place the remainder of the cards in the center.
Students will play according to the rules of ―Gin Rummy‖.
 Model for students that division is another way to convert fractions to percents,
1                      1                         1
such as:        1  4  25% and that 1  8  0.125 12.5% 12 % .
4                      8                         2
   Have the students work in groups of 4. Give each group 4 index cards. Have
the students write their first name on their index card and determine the part
of their first name that is vowels. These will be written in fraction, decimal,
and percent form. Students will then do the same for their last name. These
cards can be ordered from least to greatest within the individual groups or for
the whole class. The fraction forms can be hung on a number line.
   Students are asked to look in any of their textbooks and locate 3 lines. They
can do any of the following to determine fraction, decimal, and percent form.
Make a line plot depicting the letters of the alphabet. For instance:

A      B      C      D      E      F      G      H     I      J      K      L       M
X      X      X             X             X      X     X      X      X
X                           X             X      X     X                    X

Students can then determine fraction, decimal, and percent forms for what
part of the 3 lines of print each letter represents. Use the same 3 lines and
compare the number of 3-letter words vs. the 4-letter words. These will be
written in fraction, decimal, and percent form.

   Using a paragraph from the newspaper, students will determine the fraction,
decimal, and percent form of common nouns vs. adjectives. Graphs can be
constructed to display the data in these activities.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)      2000

52)     Determines the percent of a number
 Students work with a partner. Percent problems will be written on the back of
index cards such as:

25% of 200                         37.5% of 95
19% of 125                         50% of 315

The cards will be divided evenly between the two. Each student will place a
card face up. The student with the greater card takes both cards. This will
continue until all cards are played. The student with the most cards is the
winner.

   Have students practice using mental math with easier numbers, such as
1         1        1
of 16,   of 10,   of a dozen.
8         2        3
   Have the students work with a partner. Provide each group with a newspaper
that shows regular price and rate of discount of items, scissors, glue, and
construction paper. Have students cut and paste ads on the construction
paper. Have each group compute the sale prices of the items.
   Use 10-by-10 cm grids for students to model such problems as:
 10% of 100
 30% of 60
 50% of 75

53)     Estimates decimals, fractions, and percents
3        1
questions to help determine the estimate. Example: 7        + 2
4        3
―Is the sum more or less than        ?‖ (Fill in the blank with various
numbers.)
 Have the student use calculators to check for exact answer after estimation
has been done.
3   1
   Hold up cards that have problems on them, such as 46.68 – 14.66, 2                       +7 ,
4   8
or 28%  54%. Have the students write the estimated answer. The first
student to calculate the correct answer will hold up the next card.

54)     Uses proportions and equations to solve problems with rate, base, and part with
and without calculators
 Provide each student with a triangular card. *See below. Display the
following problem: 15 is what percent of 60?

Identify P = 15
b = 60

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

Have the students cover the unknown part to determine the appropriate
operation. (If ―r‖ or ―b‖ is the missing part, the student is to divide; if ―p‖ is the
missing part, the student is to multiply.

p




r    x          b

Use the proportion to set up and solve percent problems. For example: 7% of
7      n
80 is _______.            ; Cross multiply and divide. (7x80)  100 = 5.60
100 80
60 2 2 1
Example: 60 is _____% of 90 (Simplify first           );       ; Cross multiply
90 3 3 100
2
and divide. (2 x 100)  3 = 66 % . Use a calculator for multiplication and
3
division.

56)     Finds the percent of increase and decrease
 Have students use base ten blocks to model percent increase. For example,
students put a hundred square on their desk and identify it as 1 whole. Ask
the students what fraction is 1 one-square (1/100). Then ask, ―What percent
is this?‖ (1%). Next have the students put 1-square next to the hundred
squares. By doing this hundred square is increased by 1%. Have the
students use the blocks to increase the hundred-square by 10%. Students
will add a ten-strip. Ask what fraction is 1 ten-strip. (10/100) and what
percent is this? (10%). Inform students that by adding a ten-strip they
increase the hundred-square by 10%. Have students use the blocks to
increase the hundred-square by 20% (2 ten-strips added), 6% (6 one-squares
added), and 14% (1 ten-strip and 4 one-squares added). In finding percent
decrease, students use their hands to cover the amount decreased. Having
worked students through this strategy, provide them with this information.

Unknown           x        Original         =      Subtracted
Percent                   Amount                   Amount

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

   Provide the students with given data from line graphs to plot. Have the students
determine the percent of increase and decrease based on the data. For
example:
Unemployment Rate

7.5
7.0
6.5
6.0
5.5
5.0
Jan.                 Feb.                       Mar.              Apr.

Ask the question, ―What was the percent increase from January to February?‖

57)       Solves problems involving sales tax, discount, and simple interest with and
without calculators
 Have student get a sales advertisement from a department store. Pretend
the student has \$100 to spend. Have the student make a purchase and
figure the total cost including a 7% sales tax without overspending. Have the
student find the amount of discount, by purchasing the item or items on sale.
 Have the students look through the newspaper car sales. The students
should select a car, and then figure the total cost with the given rate of
interest and the selected time period.
 Students work in groups of 2-4. Provide each group with two triangular
cards* as shown below. Provide one problem at a time for the groups to
determine the missing part. Have the students cover the missing part to
determine the appropriate operation. If ―r‖ or the other item in the lower right
hand corner is the missing part; the student will divide to find the answer.

If the C and D located at the top of the triangle are missing; the student will

*               C                                        D
                                        

                                        
original
r      x      amount                     r      x     price

     Have the students make up their own menu or provide one. Have each
student order a meal from the menu. Have the students determine the total
cost of the meal. Have the students calculate the amount of sales tax with
and without calculators.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

   Have the students determine the gratuity (tip) of 15% and add to the final
cost. *The triangular cards can be used to assist the student. Calculators
can be used.
   Have the students work with a partner. Provide each group newspaper ads
showing regular price and sale price, a pair of scissors, glue, and a sheet of
construction paper. Have the students cut out the ads, glue them to the sheet
of construction paper, and calculate the discount and rate of discount.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)     2000

MATHEMATICS BENCHMARKS

O – means teacher should be able to observe throughout the day – possibly use anecdotal records.
I – Informal Assessment—those marked ―I‖ have an assessment task attached.

Uses properties to create and simplify algebraic expressions and solves linear
equations and inequalities

1) I - Identifies and applies the commutative, associative, and distributive
properties
2) I - Distinguishes between numerical and algebraic expressions, equations, and
inequalities
3) I - Converts between word phrases or sentences and algebraic expressions,
equations, or inequalities
4) I - Simplifies and evaluates numerical and algebraic expressions
5) I - Solves and checks one and two-step linear equations and inequalities
6) I - Solves and checks multi-step linear equations using the distributive property
7) I - Graphs solutions to inequalities on a number line
8) I - Writes a corresponding real life situation from an algebraic expression

Interprets, organizes, and makes predictions using appropriate probability and
statistics techniques

9) I - Interprets and constructs frequency tables and charts
10) I - Finds the mean, median, mode, and range of a given set of data
11) I - Interprets and constructs bar graphs, line graphs, circle graphs, and
pictographs from given data
12) I   Interprets and constructs stem and leaf, box and whisker, and scatter plots
from given data
13) I - Predicts patterns or trends based on given data
14) I - Uses combinations and permutations in application problems
15) I - Calculates and applies basic probability

Writes and solves problems involving standard units of measurement

16) I - Converts, performs basic operations, and solves word problems using
standard measurements
17) I - Measures line segments and finds dimensions of given figures using standard
measurements
18) I - Writes and solves real life problems involving standard measurements
19) I - Selects appropriate units of measurement for real life problems

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

Determines the relationships and properties of two and three-dimensional
geometric figures and the application of properties and formulas of coordinate
geometry

20) I - Identifies parallel, perpendicular, intersecting, and skew lines
21) I - Identifies and describes characteristics of polygons
22) I - Finds the perimeter and area of polygons and circumference and area of
circles
23) I - Classifies, draws, and measures acute, obtuse, right, and straight angles
24) I - Identifies and finds the missing angle measure for adjacent, vertical,
complementary and supplementary angles
25) I - Locates and identifies angles formed by parallel lines cut by a transversal
(e.g., corresponding, alternate interior, and alternate exterior)
26) I - Classifies triangles by sides and angles and finds the missing angle measure
27) I - Identifies three-dimensional figures and describes their faces, vertices, and
edges
28) I - Uses the Pythagorean Theorem to solve problems with and without a
calculator
29) I - Identifies the x and y-axis, the origin, and the quadrants of a coordinate
plane
30) I - Plots ordered pairs
31) I - Labels the x and y coordinates for a given point
32) I - Uses tables and graphs simple linear equations

Uses basic concepts of number sense and performs operations involving
exponents, scientific notation, and order of operations

33) I - Simplifies expressions using order of operations
34) I - Uses the rules of exponents when multiplying or dividing like bases and
when raising a power to a power
35) I - Multiplies and divides numbers by powers of ten
36) I - Converts between standard form and scientific notation
37) I - Multiplies and divides numbers written in scientific notation
38) I - Evaluates and estimates powers, squares, and square roots with and without
calculators

Determines relationships among real numbers to include fractions, decimals,
percents, ratios, and proportions in real life problems

39) I - Classifies and gives examples of real numbers such as natural, whole,
integers, rational, and irrational
40) I - Identifies, compares, and orders fractions and decimals
41) I - Rounds and estimates using fractions and decimals
42) I - Solves real life problems involving addition, subtraction, multiplication, and
division of fractions, decimals, and mixed numbers
43) I - Determines the absolute value and additive inverse of real numbers

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

44) I - Classifies, compares, and orders integers and rational numbers
45) I - Adds, subtracts, multiplies, and divides integers and rational numbers with and
without calculators
46) I - Writes ratios comparing given data
47) I - Converts among ratios, decimals, and percents
48) I - Solves proportions
49) I - Solves for part, rate, or base
50) I - Finds commissions and rates of commission, discounts, sale prices, sales
tax, and simple interest
51) I - Finds percent of increase and decrease
52) I - Writes and solves real life word problems using percents with and without
calculators

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)      2000

1) Can the student identify and apply                   Provide the student with the following
the commutative, associative, and                    statements:
distributive properties?                              (2 + 5) + 3 = 2 + (5 + 3)
    (6a)b = 6(ab)
    7 x 32 = 32 x 7
    8(6 + 7) = 8(6) + 8(7)

Have the student label each as the
commutative, associative, or distributive
property.

2) Can the student distinguish between                  Provide the student with cards containing
numerical and algebraic expressions,                 the following 1, 2, 3, 4, 5, 6, 7, 8, 9, x, y, z,
equations, and inequalities?                         +, -, =, <, and > written on them.

Using the cards have the student create
numerical and algebraic expressions,
equations, and inequalities using the
cards.

3) Can the student convert between                      Provide the student with a word phrase or
word phrases or sentences and                        sentence similar to the following:
algebraic expressions, equations, or                 1) Three times a number less than
inequalities?                                           sixteen
2) Twenty more than twice is a negative
thirty.
3) Three times a number increased by 4
is at least 16.

Have the student write expressions,
equations, or inequalities for each.
Provide the student with the following
expressions, equations, and inequalities:
1) 10x – 7
2) 6x + 2 = 20
3) 4x – 7 < 17

Have the student write a word phrase for
each.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

4) Can the student simplify and                         Provide the student with algebraic
evaluate numerical and algebraic                     expressions. Have the student roll a
expressions?                                         number cube. Have the student evaluate
the expression by substituting the number
rolled for the variable.

5) Can the student solve and check one                  Provide the student with two-step linear
and two- step linear equations and                   equations and inequalities. For example:
inequalities?                                        a) 2x + 5 = 9               b) ‾5a – 2 < 8

Have the student solve the equations and
inequalities with manipulatives. Have the
student check the solution of the equation
by substituting it for the variable.

6) Can the student solve and check                      Provide the student with a set of multi-step
multi-step linear equations using the                linear equations. Have the student solve
distributive property?                               and check the equation with or without the
use of manipulatives. For example:
2(3x – 4) = 10

7) Can the student graph solutions to                   Provide the student with inequalities and
inequalities on a number line?                       number lines. Have the student solve and
graph the inequality on the number line.
Example:      2x – 5 > 15

8) Can the student write a real-life                    Provide the student with an equation such
situation from an algebraic                          as 3x – 5 = 15. Have the student write a
expression?                                          real-life situation that is represented by
this equation.

9) Can the student interpret and                        Have the student gather responses to a
construct frequency tables and                       survey question and construct a frequency
charts?                                              table and/or chart. Have the student
concerning the table and/or chart.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

10) Can the student find the mean,                      Provide the student with a given set of
median, mode, and range of a given                  data such as grades.
set of data?                                         Have the students find the mean.
 Have the students find the mode.
 Have the students find the median.
 Have the students find the range.

11) Can the student interpret and                       Divide the class into groups. Have each
construct bar and line graphs, circle               group take a survey of favorite television
graphs, and pictographs from given                  programs. Assign each group a graph to
data?                                               construct. Have each group interpret a
different group’s graph.

12) Can the student interpret and                       Provide the student with a set of data
construct stem-and-leaf, box-and-                   gathered by measuring the circumference
whisker, and scatterplots from                      of the students’ heads. Have the student
given data?                                         construct stem-and-leaf and box-and-
whisker graphs.

13) Can the student predict patterns or                 Provide the student with a graph showing
trends based on given data?                         minimum wages for a 10-year period.
Have the student predict what the
minimum wage will be in five years.

14) Can the student use combinations                    Provide the student with the following
and permutations in application                     examples:
problems?                                                                        654
Combinations     C (6, 3) =            20
3  2 1
Permutations     P(6, 3) = 6  5  4 =20

Have students find how many ways a first,
second, and third place winner can be
selected from 20 people.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)            2000

15) Can the student calculate and apply                 Have the student find the number of
basic probability?                                  phone numbers that can be assigned if
the three-digit prefix remains the same
and the last four numbers can be
rearranged.

Example :           736 – N N N N
10 · 10 · 10 · 10
10,000

16a) Can the student convert from one                   Provide the student with a chart of equal
unit to another using standard                     measures. Have the student convert the
measurements?                                      following measurements showing his/her
work.
16b) Can the student perform basic                      1. 10 quarts to gallons.
operations using standard                          2. 4 feet 7 inches to inches
measurements and solve word                        3. 34 ounces to pounds and ounces
problems using standard
measurements?                                      Provide the student with the following line
chart.

k    h            da      meter d     c            m

Have the student convert the following
conversions:
1. 9 g = ______ g
2. 14.5 ml = ______ L
3. 25 m = _____ km

Provide the student with a chart of equal
measures. Have the student perform the
following.

3 qt 2 pt                  12 lb
+ 4 qt 3 pt                 - 3 lb 4 oz

5 ft 3 in
x        5 in

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)     2000

Provide the student with standard English
and metric units of measure. Have the
and show all work.
1. Janet’s room is 12 feet by 11 feet.
How many square feet of carpet will
she need?
2. If Mary rode her bike 3.2 miles, how
many yards did she ride the bike?
3. Tyler has a board 24 cm long. He
needs 3 boards, 7 cm long. Can he
get all three from the board?
4. If a nickel weighs 5 grams, how many
nickels weigh 1 kilogram?

17) Can the student measure line                        Provide the student with some pre-drawn
segments and find dimensions of                     line segments. Have the student measure
given figures using standard                                                   1
these to the nearest         of an inch or to
measurements?                                                                 16
the nearest cm.
Provide the student with an English and
metric ruler. Have the student measure a
side of a square on the floor, the diameter
of a penny, the length of book, etc., in
English and metric measures.

18) Can the student write and solve real-               Provide the student with several vegetable
life problems involving standard                    can labels. Ask the student to find the
measurements?                                       number of cans needed to feed 3, 10, or
25 people.
Provide the student with a grocery ad and
ask the student to find the cost of a certain
number of pounds of some food.
Use a grocery ad and have the student
write a problem involving the purchase of
2 or 3 items such as fruit or deli (per
pound) items.
Provide the student with information about
races or some event involving metric
measurement. Have the student write a
problem using this information.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)    2000

19) Can the student select appropriate                  Present the student with a list of objects
units of measurement for real-life                  such as length of a pencil, volume of a
problems?                                           tank, or area of a room, and have the
student tell if it should be measured in
inches, feet, square inches, square feet,
cubic inches, or cubic feet.

Provide the student with the following
items to match with the appropriate
measurement.

a) Distance from Memphis to Columbus 1. 270 km
b) Length of shoe                    2. 270 m
c) Diameter of quarter               3. 19 cm
4. 19 mm
5. 3 cm
6. 3 mm

20) Can the student identify parallel,                  Ask the student to use the shape below to
perpendicular, intersecting, and skew               identify a pair of parallel, perpendicular,
lines?                                              intersecting, and skew lines.
A                   B
E
F

D               C
G
H

21) Can the student identify and describe               Have the student draw and name a
characteristics of polygons?                        polygon. Have them describe the
characteristics of the polygon drawn.

22) Can the student find the perimeter                  Using a map of Mississippi, have the
and area of polygons and                            student find the perimeter of the polygon
circumference and area of                           formed by joining Jackson, Meridian,
circles?                                            Laurel, and Collins.
Ask the student to build a five-sided, four-
sided, and six-sided polygon on the
geoboard and find the area of each.
Ask the student to find the circumference
of a circle with a diameter of 8 cm using
the formula c=  d.

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23) Can the student classify, draw, and                 Ask the student to name an acute, obtuse,
measure acute, obtuse, right, and                   right, and straight angle using the figure
straight angles?                                    below:
A                  C          D

B
E         F                   G

Have the student use a protractor to find
the actual measure of ABC, BFG, and
EFG in the figure above.

24) Can the student identify and find the               Ask the student to draw and label a
missing angle measure for adjacent,                 diagram to show each of the following:
vertical, complementary, and                        1) Angles ABC and CBD are adjacent
supplementary angles?                                  angles
2) Angles XYZ and AYC are vertical
angles
3) Angles ABC and CBD are
complementary

Have the student find the measure for the
missing angle in each figure.

a)                             b)
75
?                               110
?

c)

45
?

25) Can the student locate and identify                 Provide the students with two parallel lines
angles formed by parallel lines cut by              cut by a transversal. Have student identify
a transversal (e.g., corresponding,                 the corresponding, alternate interior, and
alternate interior and alternate                    alternate exterior angles.
exterior)?                                                                              1 2
3 4
5 6
7 8

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)         2000

26a) Can the student classify triangles                 Have the student draw and label an
by sides and angles?                               isosceles triangle, a scalene triangle, and
an equilateral triangle. Have the student
draw a sketch of an acute, obtuse, and
right triangle.

26b) Can the student find a missing                     Have the student find the value of the
angle measure?                                     missing angle(s). Then classify each
triangle as acute, right, and obtuse.

a)                          b)
45
60    60

c)
120
30

27) Can the student identify three-                     Provide the students the following figures:
dimensional figures and describe
their faces, vertices, and edges?                   a)                               b)

c)

Have the student name the type of
polygon that forms the faces and edges.
Have the student list the number of faces,
vertices and edges of each solid.

Can the student use the Pythagorean                     Have the student use the Pythagorean
Theorem to solve problems with and                    Theorem (a2 + b2 = c2) to find the length of
without a calculator?                                the missing leg.
5

3

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)          2000

29) Can the student identify the x-axis,                Provide the student a coordinate plane
the y-axis, the origin, and the                     and ask the student to label the x-axis, y-

30) Can the student plot ordered pairs?                 Ask the student to plot these points on the
coordinate plane:

(0,0) (-2,3) (4, -2) (-3, -3) (0, 4) (-5, 0).

Provide the student a negative x
coordinate of an ordered pair. Ask the
student ―In which quadrants will the point
be plotted?‖ Have the student give two
examples of an ordered point using the
given x coordinate.

Provide the student a positive y-coordinate
of an ordered pair. Ask the student ―In
which quadrants will the point be plotted?‖
Have the student give two examples of an
ordered pair using the given y coordinate.

31) Can the student label the x and y                   Provide the student with a coordinate
coordinates for a given point?                      plane on which several points have been
drawn. Have the student name the x-
coordinate and the y-coordinate of each.
Have the student tell which quadrant
contains each ordered pair.

32) Can the student use tables to graph                 Provide the student with a pegboard, golf
simple linear equations?                            tees, and a simple linear equation.
 Have the student make a table of
values.
 Have the student graph the equation
by placing the pegs in the correct
holes.
 Have the student complete the table
and graph the solution to the equation:
y = 2x + 1.
x       2x + 1    y           x, y

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

33) Can the student simplify expressions                Provide the student with several
using order of operations?                          expressions containing more than one
operation. Have the student simplify each
expression by using order of operations.
Example: 4 + 3  5 = 4 + 15 = 19

34) Can the student use the rules of                    Provide the student with multiplication and
exponents to multiply or divide like                division problems with like bases and
bases and to raise a power to a                     exponents. Provide the student with
power?                                              problems raising a power to a power.
Have the student use the rule of
exponents to solve the problems.
Example:
35  32 = 37, 98  9 = 97, (24)5 = 220

35) Can the student multiply and divide                 Provide the student with a set of problems
numbers by powers of ten?                           requiring multiplication and division by
powers of ten.
Example: 2.1 x 104 = 21,000
2.1 ÷ 104 = 0.00021

36) Can the student convert between        Provide the student with a set of numbers
standard form and scientific notation? in standard form. Have the student
convert to scientific notation.
Example: 3,500= 3. 5 x 103
Provide the student with a set of numbers
written in scientific notation. The student
must convert to standard form.
Example: 7.4 x 10-2 = 0.074

37) Can the student multiply and divide                 Provide the student with a set of problems
numbers written in scientific                       in scientific notation. Have the student
notation?                                           multiply and divide the numbers.
Example: 2.4 x 105 = 1.2 x 10 2
2 x 10 3

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)                   2000

38) Can the student evaluate and                        Provide the student with a set of numbers
estimate powers, squares, and                       written as powers, squares and square
square roots with and without                       roots. Have the student estimate and
calculators?                                        evaluate the numbers with calculators?
Have the student estimate and evaluate
the numbers without calculators?
Example: Estimate then evaluate.
38 21 2

39) Can the student classify and give                   Provide the student with a set of numbers.
examples of real numbers (natural,                  Have the student classify each number as
whole, integers, rational and                       natural, whole, integer, rational or
irrational)?                                                                     1                    7
irrational. Ex: ( -2,            , 0, 5, 0.6,         )
3                    2
Have the student give examples of natural
numbers, whole numbers, integers,
rational numbers and irrational numbers.

40) Can the student identify, compare                   Provide the student with a group of
and order fractions and decimals?                   fractions and decimals. Have the student
compare fractions and decimals
3        3
Example: ( , 0.6) (  0.6)
4        4
Order the group of numbers from least to
1       1   2
greatest.       (0.7 ,       , ,         , 0.5)
3       9   5
1 1 2
( , , , 0.5, 0.7)
9 3 5

41) Can the student round and estimate                  Give the student a variety of fractions and
fractions and decimals?                             decimals. Have the student round each
fraction to the nearest whole number. Ex:
3
1       = 2. Round each decimal to a given
4
place value. Ex: Round 7.0236 to the

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)             2000

Have student answer questions, such as
3                       7
―If I have 2 pizzas, and Mary has 1
4                       8
pizzas, do we have as much as 4 whole
pizzas together?‖

43) Can the student determine absolute                  Ask the student to state the absolute value
value and additive inverse of real                  for each of the following:
numbers?                                                           
 3,   0,         2
,     4 .5
3

1
inverse of 6, -5, 0, 2 , 0.45.

44) Can the student classify, compare,                  Given the following numbers:
and order integers and rational                                         2
ˉ5,      ,      7 , 0, ˉ2.7, 45, 0.4
numbers?                                                                3

Ask the student to tell whether the number
is an integer, a rational number, neither, or
both. Ask the student to arrange the
following numbers in order from least to
greatest.

5             2
, -4, 0, -1, , 2
4             3

45) Can the student add, subtract,                      Provide the student with a variety of math
multiply, and divide integers and                   problems using the four basic operations.
rational numbers with and without                   Include integers and rational numbers in
calculators?                                        the problems.

Have the student solve some of the
problems without calculators. Have the
student solve some of the problems using
a calculator. Ex: -3 + 24 = 21

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

47) Can the student convert among                       Have the student perform the following
 Convert a ratio to a decimal and to a
percent.
Example: 3:5 = 0.6 = 60%
 Convert a decimal to a ratio and a
percent.
6 3
Example: 0.6 =        60%
10 5
 Convert a percent to a ratio and to a
decimal.
60 3
Example: 60% =          0 .6
100 5

48) Can the student solve proportions?                  Ask the student to explain how to solve a
proportion. Provide the student with a set
of dominoes. Have the student draw two
dominoes and determine if they form a
proportion. (If a blank domino is drawn,
the student must determine the number
needed to form a proportion.)

49) Can the student solve for part, rate or             Have the student count the number of girls
base?                                               and boys in the classroom. Have the
student use this information to predict how
many boys and how many girls are in the
eighth grade class with a total of 1000
students.

50) Can the student find commissions                    Provide the student with a list or pictures
and rates of commission, discounts,                 of prices of specific items (e.g., homes,
sale prices, sales tax, and simple                  CD players, food, cars). Provide the
interest?                                           student with interest rates or sales tax
rates. Have them determine discount,
sale price, sales tax, and/or simple
interest.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

51) Can the student find percent of                     Provide the student with several
increase and decrease?                              basketball players shooting averages from
last year. Have them find the percent of
increase or decrease as compared to this
year’s averages.

52) Can the student write and solve real-               Write several percents on the overhead
life word problems using percents                   board for the student to select at least two.
with and without calculators?                       Have the student write and solve a
problem involving the two percents
chosen. Example:
60% 40% 33 1 %
3
25% 17%

Janie got 40% of her questions correct on
Maurice got 25% of his questions correct
on a spelling test that had 80 questions.
How many questions did each student
miss?

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)        2000

Suggested Teaching Strategies for
Uses the properties to create and simplify algebraic expressions and solves
linear equations and inequalities

1)      Identifies and applies the commutative, associative, and distributive properties.
 Make a deck of cards with the name, the algebraic notation, and a numerical
example of the commutative, associative, and distributive properties. Include
cards with incorrect properties written on them. Have the students play
Rummy. An example of a set would be

Distributive                  a(b + c) = ab + ac                   6(2 + 3)= 6(2) + 6(3)

   Give the student expressions already simplified. Have the students
determine which expressions are equivalent and which ones are not. Have
the students correct the mistakes made and identify the name of the property.

Examples: ˉ2(x -5) = ˉ2x – 5 incorrect                   ˉ2(x – 5) = ˉ2x + 10 Distributive

3 + (x + 5) = (3 + x) + 5 correct               Associative for addition

   Use boxes to help students remember to distribute the number on the outside
of the parentheses correctly.

Examples:             ˉ6(x – 8)

ˉ6x      48      =        ˉ6x + 48

2)      Distinguishes between numerical and algebraic expressions, equations, and
inequalities
 Divide the class into three teams. Hold up a card with an algebraic
expression, equation, or inequality written on it. Give one point to the team
who can correctly identify the card as an expression, equation, or an
inequality. Give bonus points to the team that can simplify the expression or
solve the equation or inequality. Provide bonus points or a small prize to the
team who earns the most points.
 Discuss the difference between operation and relation signs. Provide
students with various examples of numerical and algebraic expressions,
equations and inequalities. Have students identify the type example it is.
They must explain ―why‖ in order to receive full credit.
Example: 4a – 2 is an algebraic expression - Why: variable, no relation sign
Example: y < 6 inequality - Why: relation sign, less than

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

   Put several examples of numerical and algebraic expressions, equations and
inequalities on the board. Have the student classify each example.
k
3 + a; 5 x 7 + 12 ÷ 4; a +16 = 72; 5b = 75;                   <5
13

3)      Converts between word phrases or sentences and algebraic expressions,
equations, or inequalities
 Divide the students into teams. Play Jeopardy with the categories being
expressions, equations, and inequalities. Have the team choose the category
and the point value for the question. If the team can simplify the expression
or solve the equation or inequality, they earn an additional 100 points.

Example: Team A picks equations for 200 points. The teacher would read an
equation problem such as three times a number increased by 7 is 52. Team
A would respond, ―What is 3x + 7 = 52?‖ Team A would earn 200 points for
the correct equation and 100 points for the correct solution.

   Provide students with algebraic phrases or sentences written on one side and
word phrases or sentences on the other side. Orally discuss the odds on
each side together. Students complete the evens individually or in groups.
   Have students come to the overhead to work and explain their answers.
Discuss multiple ways to explain the algebraic phrases or sentences.
Example: 2 + 5a, two plus five times ―a,‖ the sum of 2 and five times ―a.‖

4)      Simplifies and evaluates numerical and algebraic expressions
 Give students an expression such as 3 52  2  4 . Have students write the
problem several times on a piece of paper and insert parentheses to create
as many expressions as possible. Have the student evaluate each
expression. Have the class make a list of all the possible answers.
 Give the class four numbers and the desired answer. Have the students add,
subtract, multiply, or divide the four numbers to get the desired answer.

Example: 4, 6, 5, 2            desired answer is 9
(6 – 4)  2 + 5                6–4=2x2=4+5=9

   Give each group a deck of cards with algebraic expressions written on them
and a number cube. Give each person in the group one card. Have a
student roll the number cube and have the other students evaluate the
expressions with the number generated. (The person with the highest answer
earns a point. After using all cards, the person with the most points is the
winner.) Another option for scoring is to keep a sum of all possible answers.
The person with the largest sum is the winner.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

5)      Solves and checks one and two-step linear equations and inequalities
 Have the student solve equations and inequalities using a flow chart.

Example: ˉ5x + 7 > ˉ38

Start:

X        ˉ5      ˉ5x     +7        ˉ5x + 7

Answer      <         9        ÷-5      ˉ45     ˉ7        ˉ38         >
x<9

When solving inequalities using a flow chart, remind students to look at the
number with the variable. If the number is negative the inequality must reverse.

   Discuss opposites, zero values, equations, and inequalities. Demonstrate
how to use a balance to solve equations. Have students come to the
overhead to solve, then check equations. Example: 3a – 2 = 10
aaa –2 = 10. Replace each original with 4. Extend this to solving inequalities.
+2          +2
aaa  2             10

aaa                 12

a                   4

 Give the student one and two-step equations and inequalities. Have the
student solve. Let the student use a calculator to check the solution.

4a + 2 = 22
4a + 2 – 2 = 22 – 2                subtract 2 from both sides
4a = 20                            divide both sides by 4
4    4
a=5

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

6)      Solves and checks multi-step linear equations using the distributive property
 Have the students use the flow chart, algebra tiles, etc., to solve the equation.

Example:             -4(2x + 11) = 92

2x – 4 -8x + 11  -4 -44 -8x + -44 = 92

x  -8 -8x – 44 – 8x – 44 =

-17  -8 136 + 44 92

   Use parentheses and arrows to model the solution of equations and
inequalities using the distributive property.

Example:                -2(4a – 5)    = -22
-8a + 10     = -22
-10     = -10
 8a         32
=
8          8
a     =4

   Have the student use the distributive property to solve the equation. Use a
calculator to check solution.

5(2 x + 3)       = 65
(5  2x) + (5  3)      = 65
10 x + 15        = 65
10 x +15 – 15        = 65 - 15
10 x       = 50
10          10
x       =5

7)      Graphs solutions to inequalities on a number line
 Create a number line on the floor using masking tape. Give each student a
card with a number written on it. Write this number on the number line.
Provide an example such as x < 2. Have any student who has a number less
than two stand on the number line. Have a class discussion about the spaces
in between the numbers now marked on the number line. Have the student
on the number line hold hands to represent that the correct graph would
include all numbers less than 2, even the fractions and decimals in between
each number.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

8)      Writes a corresponding real life situation from an algebraic expression
 Have students create a small business that will sell a product or service.
Have the students create an advertisement for the product or service that
states what they are selling and the cost. Have the students write an
expression showing the relationship.
 Divide the class into 4 or 5 groups. Have each group write an algebraic
expression. Groups then pass their expression to the next group who will
write a real-life situation from the expression. Continue passing around until
each group receives their original expression back. Each group will present
to the class all the real life situations given for their expression.
Example: 2a – 5. My sister is 5 years less than twice my age.
 Have student write an expression for each of the following:
a. Tom has two times as many marbles as John. Let m stand for John’s
marbles. Write an expression for Tom’s marbles, such as 2m.
b. Denise ran 5 more laps today than she did yesterday. Let y stand for the
laps run yesterday and write an expression for the number of laps she ran
today.      Example: 5 + y

Interprets, organizes, and makes predictions using appropriate probability and
statistics techniques

9)      Interprets and constructs frequency tables and charts
 Have students make paper airplanes and go outside to fly them. Have the
students record the distance in feet and make a frequency table using the
distances the airplanes flew.
 Punch out holes from several colors of construction paper. Provide each
student with a small amount of different color holes. Have the students glue
dots on graph paper to create a frequency table. Have the student use this
information to create a story.

10)     Finds the mean, median, mode, and range of a given set of data
 Have the student count the number of each color of M&M’s. Have the student
find the mean, median, mode, and range for each color of M&M. Discuss the
results.
 Have students measure their height in cm. Make a frequency table. Then
find the mean, median, mode and range of their heights.
 Give the student the following test scores: 100, 80, 93, 85, 80, 90, 65, 50, 93,
98, 57. Have the student place the test scores in order from least to greatest
and find the mean, mode, median and range.

Solution:    50, 57, 65, 80, 80, 85, 90, 93, 93, 98, 100
Mode:        80 and 93
Median:      85
Range:       100 – 50 = 50
Mean:        891 ÷ 11 = 81

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

11)     Interprets and constructs bar graphs, line graphs, circle graphs, and pictographs
from given data
 Construct a frequency table from the class’ favorite football team. Divide the
class into 3 groups. Assign each group a different type of graph to construct
from the data (circle, bar, pictograph). Have the groups present their graphs
to the class. Conduct a class discussion to help decide which type of graph
best represents the data.
 Provide the class with several examples of bar graphs, circle graphs, and
pictographs. Working in pairs or small groups, have the students select a
graph. The students will interpret the data given in their graphs. They should
be able to discuss the key, the intervals used, the type of graph used and
what the graph represents.

12)     Interprets and constructs stem-and-leaf, box and whisker, and scatter plots from
given data
 Have the students measure their height and armspan to the nearest
centimeter. Construct two frequency tables. Divide the class into 3 groups.
Have one group construct a stem-and-leaf plot, another group a box and
whisker plot and the last group a scatterplot from both sets of data. Have
each group present and discuss the graphs. Use the list function on a
graphing calculator to create the above graphs.

13)     Predicts patterns or trends based on given data
 Provide a chart showing the number of cell phones per family since 1990.
Have the students predict the number of cell phones per family in 2025.
 Use the Internet to track the progress of a given stock for the past year.
Predict patterns or trends for the next month.

14)     Uses combinations and permutations in application problems
 Introduce the factorial key on the calculator. Provide students with
permutation and combination problems to solve using the calculator.
 Have the student determine how many ways a president, vice president, and
secretary can be selected from a group of 10 people. Example: C (10, 3) =
10  9  8
3  2 1

15)     Calculates and applies basic probability
 Have the students find the number of possible car tags that can be assigned
in a state if the tag must have 3 letters and 3 numbers.
 Discuss why a town of 20,000 households with one phone each must have
multiple prefixes in addition to the 4 digit phone numbers.

Example:
10 x 10 x 10 x 10
10,000 possible numbers = 10,000 possible numbers < 20,000 households

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

   Have students show how prefixes could determine 400,000 different phone
numbers.
   Use only the face cards from a deck of cards. Lay the cards face-up in a
random order. State this rule: A card is chosen at random. What is the
probability of each of the following events:
4       1
a. Choosing a queen?           or
12       3
6      1
b. Choosing a red card         or
12      2
8     2
c. Choosing jack or king?         or
12     3

Writes and solves problems involving standard units of measurement

16)     Converts, performs basic operations, and solves word problems using standard
measurements
 Provide students with calculators, standard English and metric units of
measures, and a series of questions to be answered in class

Examples:
1. Your new hot water heater is calibrated in degrees Celsius. What setting
should you use if you want a hot water temperature of 149˚ F?
2. How many pounds of ground beef should you buy to make 140
hamburgers, if each hamburger patty weighs 6 ounces before cooking?
1
3. You want to increase a recipe by               . Your measuring cup is marked with
4
thirds of a cup and milliliters. The recipe calls for 2 cups of sugar.
a. If each cup is equivalent to 8 fl. oz., how many ounces of sugar do you
need?
b. If a cup is about 250 milliliters, how many milliliters do you need?
c. Can you express the amount of sugar in ―thirds of a cup‖? If so, how
many thirds do you need?

   Give the student the following problems to solve.
1. A carpenter uses 4 pieces of wood to make a frame. The pieces of wood
total 8ft 2in. What is the average length of each piece of wood? Simplify

17)     Measures line segments and finds dimensions of given figures using standard
measurements
 Let the students make figures with tangrams, trace around their figures, and
write measurements in English and metric measures above each line.
 Have a scavenger hunt for two or three days. Ask students to bring in objects
that are specified lengths.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

   Measure temperature, length, liquid measure, and weight/mass of given
objects.
   Measure line segments and find dimensions of given figures using standard
measurements.

18)     Writes and solves real life problems involving standard measurements
 Provide the students a recipe of something they can make in class like ―nuts
and bolts‖ or Cheks Party mix. Have them double, triple, half, 1 1 times the
2
recipe. Have the students make the recipe and enjoy. Use an easy cookie
recipe and do the same thing. They love to cook. Let them bring cookies or
verify that they made the recipe for extra credit.
 Divide class into groups. Have each group build a swing out of straws, string,
cardboard and glue. Final presentation must list lengths and heights of each
object like swings, play area, and seesaw.

19)     Selects appropriate units of measurements for real life problems
 Have students take 4 index cards each. Write an object to be measured on
two cards. Write the appropriate measure of the objects on the other two
cards. Have the students gather the information from local newspapers,
magazines, etc. Take up cards every day for several days. Display about 4
or 5 objects and the correct answers in random order. Have the students
match the object with the correct measure.
 Have the student choose the most appropriate unit of measurement:
a. The length of a football field.
b. The volume of a fish tank.
c. The height of a door.
d. The weight of a book.
e. The amount of water a glass can hold.
f. The distance from Jackson to the Gulf Coast.

Determines the relationships and properties of two and three-dimensional
geometric figures and the application of properties and formulas of coordinate
geometry

20)     Identifies parallel, perpendicular, intersecting, and skew lines
 Use the classroom as the model and have students identify parallel,
perpendicular, intersecting and skew lines.
 Use a city street map. Have students name streets that are parallel, streets
that intersect, and streets that are perpendicular.
 Use given examples and have the students identify skew lines, parallel lines,
and perpendicular lines.

21)     Identifies and describes characteristics of polygons
 Arrange the students in groups of four. Ask the students to discuss the set
of shapes, then begin sharing sentences that involve ALL of the shapes,

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

SOME of the shapes, and NONE of the shapes. The recorder will post on
chart paper and groups will report out.

ALL of the shapes

SOME of the shapes

NONE of the shapes

   Give students sets of pictures of polygons with several characteristics in
common. Have students give at least one similarity and one difference in
each set.
Example:        Rectangle                   Square

Same: Right angle             Different: Length of sides
(Note: A square is a rectangle, but not all rectangles are squares.)

22)    Finds the perimeter and area of polygons and circumference and area of circles

Have students make shapes of various polygons on the geoboard. Count the
number of squares to find the area. Count the border edges to find the
perimeter.
 Give a shape and ask students to add tiles until the perimeter of the figure is
16. (Squares that are added must meet so that they are touching on at least
one side) What is the area of the original figure? What is the area since the
addition of the tiles? Where would you place a tile to increase the perimeter
by 1? By 2? By 3? What happened to the area as the perimeter increased?
How could you increase the area by 3 and not increase the perimeter? What
is the fewest number of tiles that can be added to increase perimeter to 16
units? The greatest number? Describe this new shape. What is the area of
each shape?

FINDING THE AREA OF A CIRCLE

Method 1:             Counting Squares

1. Draw a circle on the grid paper. Count all the whole centimeter squares
that lie completely inside the circle. (This underestimates the area of the
circle).

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)    2000

2. Now make an overestimate of the area of the circle by taking the number
of whole squares that lie inside the circle (the same number you got for
step 1 above) and add to it the number of squares that touch the circle
and lie partly inside and partly outside the circle.

3. Find the average of your two counts.

Method 2:             Inscribing and Circumscribing Squares

1. Circumscribe a square about the circle. Find its area.
2. Inscribe a square inside the circle. Find its area.
3. Find the average of the two areas.

Method 3:             The “Curvy Parallelogram” Method

1. The circle has been divided into 8 congruent sectors.
2. Cut out the sectors and arrange them to form a curvy parallelogram.
3. Approximate the area of the curvy parallelogram.

Recording the Results

Record the information from methods 1, 2, & 3 in the table below.

Method No.                     Title                                                   Area

1                              Counting Squares

2                              Inscribing and Circumscribing Squares

3                  The ―Curvy Parallelogram‖
______________________________________________________________

When you and your partner have completed all the methods, answer the
following question in writing: Which method do you ―trust‖ the most? Why?

23) Classifies, draws, and measures acute, obtuse right, and straight angles
 Use a circular geoboard, template and protractor. Using a rubber band
connect P1 to P19. Then use another rubberband to connect P19 to P17
What is the vertex of the angle? Estimate the measure of the angle. Use a
protractor to measure the angle. What type of angle is formed?
 Construct any angle on geoboard; transfer to template. Name, measure, and
classify the angle according to its measure.
 Construct polygons on the circular template. Measure and classify each
angle of the polygon.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

Enrichment: Construct these 3 polygons on the same circle using colored
pencils and color after completion.

Note: Circle may be divided into units other than 24. Students may color in
the designs made and have beautiful art work on display.

24)    Identifies and finds the missing angle measure for adjacent, vertical,
complementary, and supplementary angles
 Draw 90 angles. Draw another ray from the vertex to create two adjacent
angles. Measure the two angles created. Ask students to write a statement
regarding these angles.

The following are 180 angles. Draw another ray from the vertex to create
adjacent angles. Measure the angels created. Write a statement regarding
these angles?

   Use dowel rods to model two intersecting lines. Have the students name the
angles formed and note their findings about the relationships of the angles.
   Identify and find the missing angle measure for adjacent, vertical and
complementary angles. Find the measure of angle X, Y and Z.

Y       X
Z       60

25)    Locates and identifies angles formed by parallel lines cut by a transversal (e.g.,
corresponding, alternate interior, and alternate exterior)
 Give students these materials: grid sheet, colored pencils, and tracing paper.
This grid is formed by sets of parallel lines.
Example:

Use one color and color in one set of corresponding angles.
Use a different color and color in a different set of corresponding angles.

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Use a third color and color in a set of alternate interior angles.
Using a fourth color and color in a set of alternate exterior angles.
Using the tracing paper, trace over the grid marking the corresponding angles
on top of the other. What observations can be made? Do the same with
alternate interior and alternate exterior. Try several to validate the conjecture.

   Use a subway map to locate and identify parallel lines, transversals, interior
and exterior angles.

26) Classifies triangles by sides and angles and finds the missing angle measure
 Have the class construct triangles on the geoboard with bands. Students will
sort triangles according to their sides and angles and name triangles.
 On geoboard construct a triangle. Working in pairs, do not show it to your
partner, but give partner careful instructions on how to create triangle on his
geoboard. When finished, compare your triangles. Are they congruent? If
not why? Reverse role.
 Use a circular geoboard to connect.

1) P1 and P7                           What type of triangle do you have?
P7 and P15
P15 and P1                          Measure each angle to verify the type of
triangle.

2) P10 and P14
P14 and P19
P19 and P10

3) P1 and P5                           Measure the sides to determine the type of
P5 and P13                          triangle.
P13 and P1

27)    Identifies three-dimensional figures and describes their faces, vertices, and
edges
 Use materials such as straws, bobby pins or pipe cleaners to:
1. Make a square with straws
2. At each corner, attach a straw (Straws must be the same length)
3. Join these four straws at a point.
4. What shape was constructed? How many faces? edges? vertices?

Make a triangular pyramid. Count its vertices, edges and faces.
Complete the chart

Vertices           Edges             Faces
Square pyramid
Triangular pyramid

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

   Use three-dimensional figures or pictures to compare faces, vertices, and
edges of these figures.

Example: Rectangular prism and rectangular pyramid
Example: Cube and rectangular prism
Example: Triangular prism and triangular pyramid

28)    Uses the Pythagorean Theorem to solve problems with and without a calculator
 Have student use the Pythagorean Theorem to determine whether the
triangles are right triangles.

3             5                          5         15

4                                    12

   Find the unknown length of given right triangles

25
x
x                                     8

24                              15

When a TV is advertised a having a 19-inch screen, it means that the
diagonal is 19 inches long. If a 19 inch TV screen has a height of 12 inches,
what is the width?

29)    Identifies the x -and y-axis, the origin, and the quadrants of a coordinate plane
 Identify a point on the x-axis, y-axis and identify the origin when given a
coordinate plane with several points on it. Tell in which quadrants the
following points are located.

(-3,2)            (0, 0)        (6,1)
(5,-4)            (-2,-2)

30)    Plots ordered pairs
 Trace a picture of Mississippi on a coordinate plane. Write the ordered pairs
that form the state. Give the set of ordered pairs to a friend to plot. See if
they get Mississippi.

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   Play a game of ―Battleship‖. Try to sink your opponent’s ship by giving the
correct coordinates.

31)     Labels the x and y coordinates for a given point
 The grid shows the location of three sailboats, A, B, and C. Give an ordered
pair for each sailboat. Use the game Battleship to locate coordinates for a
given point.

32)     Uses tables and graphs simple linear equations
 Write and graph an equation that shows the relationship between the amount
of time Susan studies and the amount of time she practices the piano. Susan
spends four more hours each week studying than she does practicing the
piano.

Solution:

Let x = time she practices piano
Let y = time she studies
y=x+4

Make a table of values and graph the ordered pairs on a coordinate plane.

x      x +4          y     x, y
0       0+4          4     0, 4
1       1+4          5     1, 5
2      2+ 4          6     2, 6

Use a graphing calculator to solve linear equations for given values of x.
Then graph.

Uses basic concepts of number sense and performs operations involving
exponents, scientific notation and order of operations

33)     Simplifies expressions using order of operations
 Give students a Bingo Card with expressions written in each space. Allow the
students time to simplify the expressions. Call out numbers, which are
solutions to various expressions. Students mark their cards according to their
solutions. (The winner must show how they simplified each expression used.)
 Give one group of students a scientific calculator and another group a regular
calculator. Give all students 10 expressions to simplify. After they have
worked them with the given calculator, compare answers. Discuss which they
have alike and which are different and why.

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34)     Uses the rules of exponents when multiplying or dividing like bases and when
raising a power to a power
 Have students model/demonstrate the meaning of multiplying the bases. For
example: 35 · 32 =( 3 · 3 · 3 · 3 · 3) · (3 · 3) = 37
 Discuss the shortcut to use when multiplying like bases. (Adding exponents).
Work examples by using the shortcut.
 Have students demonstrate the meaning of dividing like bases. Ex. 35  32
3 3 3 3 3  33
3 3
 Discuss the short cut for dividing like bases – (subtracting exponents). Work
examples using the shortcut.
 Have the students demonstrate the meaning of
(82)3=(8  8)  (8  8)  (8  8) = 86
 Discuss the shortcut of raising powers to powers (multiplying exponents).
Work examples.

35)     Multiplies and divides numbers by powers of ten
 Have the student multiply or divide by powers of ten by using exponents.

3.5  103                 (3 places) = 3500
3.5  104                 (4 places) = .00035

   Have the student count the number of zeroes in order to know how many
places to move.

7.2 x 10000                    (4 zeroes)  (4 places)   72,000
7.22 ÷ 10000                   (4 zeroes)  (4 places) 0.000722

   Compare multiplying by powers of ten with negative exponents (6.8 x 10 -3)
and dividing by powers of ten with positive exponents (6.8 ÷ 10 3). Work
several example. Generalize a rule.

6.8  10-3 = 0.0068
6.8  103 = 0.0068

Multiplying by a negative power of ten and dividing by the same positive
power of ten will give the same answer.

36)     Converts between standard form and scientific notation
 Find examples of large whole numbers or very small decimal numbers in
magazines or newspapers and convert to scientific notation.
 Give each group a chemistry book. Have them write 5 numbers they found
written in scientific notation and what they were describing. Convert to
standard form. Have each group demonstrate the numbers and explain ways
they were used.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)         2000

37)     Multiplies and divides numbers written in scientific notation
 Use properties and powers of ten to regroup and multiply.

(7.1  103)              (0.14  104)
(7.1  0.14)             (103 x 104)
0.994                    107
9,940,000

   Have the student use properties and powers of ten to regroup and divide.

6.8  10 3
(6.8  103)  (0.4  102)                                        17  101  170
0.4  10 2

38)     Evaluates and estimates powers, squares, and square roots with and without
calculators
 Find the missing standard numerals

32 =              , 33 =        , 105 =                      , 9 squared =           ,
8 cubed =                , 26 =

   Have the student write the expanded form and standard form of each
42 =               108 =

   Have the student estimate the square root by determining which perfect
squares the number is between. 40        Estimate between 6 and 7, because
40 is between 36 and 49.
   Estimate which numbers the square root of 85 is closer to and why. 85 Is
closer to 9 or 10? (9, because 85 is closer to 81 than 100)
   Use the x2 , x4 , and keys on a calculator to simplify the given expression.
192 =
37=
22500 =

Determines relationships among real numbers to include fractions, decimals,
percents, ratios and proportions in real life problems

39)     Classifies and gives examples of real numbers such as natural, whole, integers,
rational, and irrational
 Give each student an index card with natural, whole, integer, rational, or
irrational written on it. Tell the students to write on their card any number that
called out that is an example of the word on their card. Call out various
numbers. (The student with the most correct numbers written on their card
for each category wins). Let the class help decide if the numbers are correct
and why.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

   Use Venn Diagrams to show relationships among natural, whole, integers,
rational, and irrational numbers.
R
I
W
N                              IR

40)     Identifies, compares, and orders fractions and decimals
 Have students match the fraction or decimal form and give a word name.

Seven and 8 tenths                     7.8

Nine hundredths                 9
100
   Have the students cross multiply to compare fractions.

3 5
<
8 9

27<40

   Find common denominators to compare fractions.

3 27                      3 5
                         <
8 72                      8 9

5 40

9 72

   Change fractions to decimals, and then compare.
3
 0.375
8
0.375 < . 5
5
 0.5
9

   Use baseball cards to record batting averages. Order from least to greatest.
Discuss what the average means.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)                   2000

41)     Rounds and estimates fractions and decimals
 Have the student round decimals to a given place. Example: Round 7.176 to
the nearest tenth.
7
 Estimate to which whole number of a given fraction is closer. 2 is closer to
8
which whole number ( 2 or 3)?

42)     Solves real life problems involving addition, subtraction, multiplication, and
division of fractions, decimals, and mixed numbers
 Have students select two stocks they would be interested in buying. Students
will work individually or in a group. Provide each student/group with a set
amount of money. Students will use the stock market page to determine
increase and/or decrease of their stock for at least a week. Graphs can be
drawn to model how their stocks did for that period of time. Percentages can
be calculated with this activity also.
 Use the sports page to determine shooting percentages during basketball
season and batting averages during baseball season. Students can also
determine these with their own basketball and baseball teams.
 Have students work with a partner. Each group will be given a catalog, such
as J.C. Penny, Sears, etc. Each group will choose 10 items and record each
item and its price on a sheet of paper. Have the students write a word
problem for each of the operations.

43)     Determines the absolute value and additive inverse of real numbers
 Draw a number line. Have students locate points on the number line such as
6 and -6
 Discuss that they are both 6 units from zero.
 Give students examples of integers and ask them to give the absolute
value and the additive inverse of each. Compare the additive inverse of

Example:              3 absolute value = 3 additive inverse = -3
-2 absolute value = 2 additive inverse = 2

44)     Classifies, compares, and orders integers and rational numbers
1                 1           1
   Give students a set of rational numbers such as (                   , 0, 0.5, 2       , -3,       )
3                 2           4
   Have students identify the integers, place all numbers in ascending order,
and compare each consecutive pair of numbers.
   On a poster board write 10 integers. Write 2 using blue numbers, 2 using red
numbers, 2 using green numbers, 2 using yellow numbers and 2 using
orange numbers.
a. Compare the 2 green integers.
b. Order the green, blue and red integers from least to greatest.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)     2000

45)     Adds, subtracts, multiplies and divides integers and rational numbers with and
without calculators
 Pair students and play a game using dominos. The students can use

Example:
start                             finish
game                                              0
board
0

_
(+)      (-)                                       +
•      • •
•      • •               -2 + 4 = 2 move forward 2 places.

The students will pull a domino and place on the card.
-                +
•
• •               -3 + 2 = ˉ1 move backward 1 place.
• •

The student that reaches the finish line 1st wins.

     When students multiply and divide they use the hand-flipping rule to remind
them of the appropriate rules. Have the student put a positive sign on the top
of one hand and a negative sign on the palm of the other hand. Begin with
the positive sign facing up for each problem. Each time you have a negative
sign appear, turn your hand over. (Which ever sign is facing up at the end of
the problem is the sign of the number in the answer).

Example: 4 x ˉ3 = ˉ12                *Begin with + up
*Flip hand because of the ˉ3.
*Negative ends facing up so the answer is
negative
ˉ8 x ˉ6 = 48              *Begin with + up
*Flip hand because of the ˉ8
*Flip again because of the ˉ6
*Positive ends facing up so the answer is
positive

46)     Writes ratios comparing given data
 Have the students work in groups of 4. Provide each group a small package
of M&M’s or baggie that contains different colored squares. Students will
determine what part of the package/baggie contains each color. These will
be written in ratio form. Students can also change these ratios to percent
form. Data will be shared with other groups. Students could then graph the
data gathered from the entire class.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)     2000

   Provide each student an index card that has a ratio or a fraction on it. Instruct
the students to locate their match. (For instance, if a student is holding the
3
fraction     , he/she will look for the ratio that matches his. These can also be
4
placed in order from least to greatest with the entire class and placed on a
number line that has been hung across the classroom.)
   Have the students work with a partner. Provide each group with a measuring
tape that will be used to determine the measure* of each of the following for
each partner. Write in ratio form:

-arm to foot                   -foot to height                   -around the neck to arm
-foot to arm                   -head to foot

Example: A person has a wrist measurement of 5 in. and a height of 50 in.
The ratio of wrist to height is 5:50 or 1:10 in simplest form. Measurements
should be given to the nearest inch and in simplest form.

47)     Converts among ratios, decimals, and percents
 Provide students with fictitious sports data or data that has been collected
from your school’s teams. Students can determine shooting percentages
during basketball season and batting average during baseball season.
 Play the ―Memory Game‖. Students work with a partner. Each group is given
a stack of 24 cards. Eight of the cards will have a ratio form written on them;
eight will have the decimal form and the remaining eight will have the percent
form. *See example below. Shuffle the cards and place face down in rows of
four. One player turns up 3 cards to see if there is a 3-way match of a ratio,
decimal, and percent. The game continues until a 3-way match
is not made. When this happens, the other player plays. The student with
the most matches at the end of the game is the winner.

*      33 1 %                  1              .3333
3                  3

   Students work in groups of 2-4. Each group receives a deck of 42 digit cards.
These cards have fraction, decimals, and/or percents written on them. Each
student is dealt 7 cards. The remainder of the cards go in the center of the
table. Students play according to ―Gin Rummy‖ rules.
   Provide the student an index card that has a ratio, decimal, or percent form
on it. Students will be instructed to get out of their seats and find their match.
3
(For instance, if a student is holding the fraction              , find the student that has
4
0.75 and the 75%.)

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

   Students can bring in grocery store receipts. Have the student get receipts
from at least two different grocery stores. Collect these for at least two weeks
before this activity is done. Divide the class into two groups. Provide one
group the receipts from one store, and the other gets the receipts from the
store. Subdivide these two groups when the activity begins. Students will
predict which digit they think will appear most frequently as the last digit in a
price from the receipts. Have each group construct a line plot for each digit 0-
9 using the data from the receipts. See example below.

0       1        2        3       4        5        6       7        8        9
x                x                         x                         x
x       x                 x                x        x                         x

Have the students express the results in decimal, fraction, or percent form.
Have them construct a graph to display the data. The results can be
recorded as to which grocery store had the number _____ to show up most
often.

   Students work with a partner. Each group will be given a page from the local
newspaper and a transparent centimeter grid sheet to be used as an overlay.
   Students will estimate the total area of the newspaper page, excluding the
margins, and determine the area of each category. The categories could
photographs, weather, obituaries, etc. Students then express the area
of the article to the area of the page as a fraction, decimal, and percent. The
class then records all findings. The total area of each of the categories is
calculated. These totals are compared to the total number of pages of the
newspaper. Using the data, students decide how much of the newspaper is
really news.

48)     Solves proportions
 Have students conduct a survey of 5 blocks in the area in which they live and
count the number of cats and dogs seen in a given period of time. Students
can set up a proportion and make an estimate of the total number of cats and
dogs in their neighborhood.
 Provide each student with several 2 colored cubes or counters, for instance,
yellow and red. Have the students model a ratio of 1 to 3 using the
cubes/counters. Have them set up a proportion to determine the number of
yellow or red cubes/counters it will take to solve this proportion.
1 n

3 12
The student will also model this proportion using the cubes or counters.

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)       2000

   Utilize the Russian dolls that come in proportional sizes for students to
actually see how the dolls are proportionally made. Sand is a good material
for students to experiment with.
   Show the class small and large photographs of the same picture to review the
concept of ratio. Have students investigate the relationship between their arm
span and their height or the measurement of the length of their foot to the
measurement of the distance around their fist. The ratio is about 1 to 1.
Show a picture of the statue of Liberty to the students. Inform students that
the Statue of Liberty’s nose measures 4 feet 6 inches from the bridge to the
tip. Having this information, groups are to try and determine the length of the
Statue of Liberty’s right arm, the one holding the torch. The actual length of
her arm is 42 feet. Have students then compare the ratios of the
measurements of the lengths of their noses to the lengths of their arms.

49)     Solves for part, rate, or base
 Provide student with a triangular card. *See below. The teacher will display
the following problem: 15 is what percent of 80?
Identify p = 15
b = 80

Students cover the unknown part, which will determine the appropriate
operation. (If ―r‖ or ―b‖ is the missing part, the student is to divide; if ―p‖ is the
missing part, the students is to multiply.)

*        p



r        x         b

Conduct a class survey of favorite things (ice cream, songs, TV shows, etc.).
Have students find the percent of their favorite compared to the whole class.

Example: Ice Cream
10 – chocolate                 5 – vanilla        4- strawberry           1- banana

Student A likes vanilla.

5 1
  25%
20 4

50)    Finds commissions and rates of commission, discounts, sale prices, sales tax,
and simple interest
 Work in groups of 2-4. Each group is provided 3 triangular cards as shown
below. The teacher will provide one problem at a time for the groups to
determine the missing part. Students are to cover the missing part, which will

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)                  2000

determine the appropriate operation. If r and the other item in the lower right
hand angle is the missing part, the student will divide to find the answer. If
the C, T, or D located at the top of the triangle is missing, the student will

COMMISSION                         TAX                   DISCOUNT

C                             T                   D             *

•                             •                        •
•                             •                        •
original
r       x     sales          r     x       amount           r       x    price

   Make up a menu or provide one. Each student is to order a meal from the
menu. The total cost of the meal is to be determined. They are to find the
amount of sales tax. Gratuity (tip) of 15% can be determined and added to
the final cost. *The triangular cards can be used to assist the student.
   Work with a partner. Each group is to be given newspaper ads showing
regular price and sale price, a pair of scissors, glue, and a sheet of
construction paper. Students cut out the ads, glue them to the sheet of
construction paper, and calculate the discount and rate of discount.
   Find the commission a real estate agent makes on the sale of a \$65,000
home. Use a variety of rates. Example: \$65,000 x 3%; \$65,000 x .035.

51)     Finds the percent of increase and decrease
 Students will be given data from line graph to plot. They will determine the
percent of increase and decrease based on the data. For example:

Unemployment Rate
Rate of unemployment

7.5
7.0
6.5
6.0                                                  •
5.5
5.0
Jan.         Feb.              Mar.        April

What was the percent of increase from Jan. to Feb.?

   Have students use base ten blocks to model percent of increase. For
example, students put a hundred squares on their desks and identify this as 1
1
whole. Ask the students what fraction is 1 one-square. (                              ). Then ask
100

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Mathematics Instructional Intervention Supplement (Benchmarks, Informal Assessments, Strategies)   2000

what percent is this (1%). Next have the students put a 1-square next to the
hundred squares. By doing this, the hundred square is increased by 1%.
Have students use the blocks to increase the hundred-square by 10%. They
should add a ten-square ten-strip. Ask what fraction is 1 ten-strip. (10/100)
and what percent is this. (10%). Inform students that by adding a ten-strip
they increase the hundred-square by 10%. Have students use the blocks to
increase the hundred-square by 20% (2-ten strips added), 6% (6 one-squares
students through this strategy, provide them with this information:

Percent                   Amount                    Amount

p                        x 100                     = 10
= 10/100
= 1/10
= 10%

In finding percent of decrease, students use their hands to cover the amount
decrease.

52)     Writes and solves real life word problems using percents with and without
calculators
 Allow students each day for a 2-week (ten-day) period to shoot a wad of
paper from a designated spot into the garbage can. Each student is to keep
up with his or her data (number of hits and misses). This information will be
given to the teacher at the end of the 2-week period. Students will be given
the data, such as, ―Michael shot at the garbage can 10 times. The paper
went into the garbage can 60% of the time. How many times did the paper go
into the can?‖ The teacher will supply the data of several more students.
 Students determine the percent of common nouns vs. the percent of
adjectives in a paragraph from a reading, math, or social studies book.
Newspapers can also be used.
 Conduct a class survey of favorite things, such as, ice cream, song, brand
shoe, TV show, radio station, etc. Students will work in-groups of 2-4 and
create a circle graph displaying the data in percents.
 Two students are given a deck of index cards that contain percents. Each
student draws from the deck and writes a word problem incorporating their
percent. The two students will swap cards* and work the problem. A
calculator will be used to check the answers. *These cards can be put in a
file for students to use throughout the year as enrichment.

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