# MATHEMATICS

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```					                                  MATHEMATICS
Activities

GRADE LEVEL:          Sixth                  COURSE:
Framework Strand:     Probability and Discrete Mathematics
Grade Level Standard:      6-15 Develop, analyze, and solve problems.

Grade Level Benchmark:     1. Count permutations and combinations using
manipulatives, diagrams, and theorems of counting. (VI.2.MS.1)

LEARNING ACTIVITY/FACTS/INFORMATION
CONCRETE (conceptualizing):

   From a group of four students, two students are to be selected. In how many
ways is it possible to select two students? (Use students to model the
combinations – students could be identified as A, B, C, D).

Answer: AB, AC, AD, BC, BD, CD

Vocabulary: permutations

PICTORIAL (symbolic):

   WOL is just one of the order or permutations of the letters O, W, L. See how
many permutations you can find.

Answer: WOL, OWL, OLW, WLO, LOW, LWO

Vocabulary:

ABSTRACT (computational):

   How many permutations of WALRUS have R as the first letter?

Vocabulary:

196
LEARNING ACTIVITY/FACTS/INFORMATION
PROBLEM SOLVING:

   Complete the chart:

Name     # of letters   Answer   # of Permutations   Answer
OWL                     3                            6
HAWK                    4                            24
TIGER                   5                            120
LEOPARD                 7                            5040
FLAMINGO                8                            40,320

Vocabulary:

197
MATHEMATICS
Activities

GRADE LEVEL:               Sixth                     COURSE:
Framework Strand:          Probability and Discrete Mathematics
Grade Level Standard:          6-15 Develop, analyze, and solve problems.

Grade Level Benchmark:         2. Use sets and set relationships to explore and solve
simple algebraic and geometric problems using Venn diagrams. (VI.2.MS.2)

LEARNING ACTIVITY/FACTS/INFORMATION
CONCRETE (conceptualizing):

   Decide on an attribute of your students, such as “wearing blue jeans” or “has
stripes on clothing,” but do not tell the rule to the class. Silently look at one child
at a time and move him or her to the left or right according to this attribute rule.
After a number of students have been sorted, have the next student come up and
ask students to predict which group he or she belongs in. Before the rule is
articulated, continue the activity for a while so that others in the class will have an
opportunity to determine the rule.

Vocabulary: attribute

PICTORIAL (symbolic):

   Identify the following sets as: disjointed, intersecting, or included.

Dogs         Cats                 Fruit
Plants            Food                                                            orange
yellow
fruit

Flower    Tomato
Fish

A                                  B                          C

Answer: a. intersecting                b. disjointed       c. included

Vocabulary: disjointed, intersecting, included

198
LEARNING ACTIVITY/FACTS/INFORMATION
ABSTRACT (computational):

   Find the sum of the fraction in the intersection of the triangle and the circle.

Answer: 5/8 + 1/2 = 5/8 + 4/8 = 9/8 or 1 1/8

1/6                 3/8
2/3
1/3    3/4
5/8

5/6           1/2
2/5
Vocabulary:

PROBLEM SOLVING:

Ice Cream
Cones

   Attributes (values): cone (square, pointed)
flavor (vanilla, chocolate, strawberry)
scoops (one, two, three)

   Construct a Venn diagram using three attributes.

Vocabulary: Venn diagram

199
MATHEMATICS
Activities

GRADE LEVEL:           Sixth                   COURSE:
Framework Strand:      Probability and Discrete Mathematics
Grade Level Standard:       6-15 Develop, analyze, and solve problems.

Grade Level Benchmark:      3. Solve problems using various delivery points, paths
between points, recurrence, and interactions. (VI.2.MS.3)

LEARNING ACTIVITY/FACTS/INFORMATION
CONCRETE (conceptualizing):

   If you are at one corner of a block (point A), and you        A
wish to walk to the opposite corner (point B), the
shortest path is, of course, across the diagonal. But,
let’s suppose that because of houses, mud, and
other obstacles, you must walk only along the street.
In this case, there are two shortest paths; one is east
from A, then south to B; the other is south from A,
then east to B. Therefore, we have: ____ number of                    B
shortest paths.

Vocabulary:
W             E
PICTORIAL (symbolic):
S
   We are interested in the number of shortest
paths from P to Q. If we code the path                        P
indicated in our figure with ESSE, meaning
East, South, South, East; list all the shortest
paths this way.

Answer: ESSE, ESES, EESS, SEES, SESE, SSEE
Q
Vocabulary:

200
LEARNING ACTIVITY/FACTS/INFORMATION
ABSTRACT (computational):
A
   How many shortest paths are there from A to B?

Vocabulary:                                                                      B

PROBLEM SOLVING:

   If there are three routes from X to Y and five routes from Y to Z, how many routes
are there from X to Z by way of Y?

U              X

V

W

Vocabulary:

201
MATHEMATICS
Activities

GRADE LEVEL:            Sixth                      COURSE:
Framework Strand:       Probability and Discrete Mathematics
Grade Level Standard:           6-15 Develop, analyze, and solve problems.

Grade Level Benchmark:          4. Explore recurrence relations and iteration through
triangular numbers, square numbers, and patterns. (VI.2.MS.4)

LEARNING ACTIVITY/FACTS/INFORMATION
CONCRETE (conceptualizing):

   Because one, three, six, and ten dots can be arranged in triangles, they are
called triangular numbers.

1                3                  6                   10

   One, four, nine, and sixteen dots can be arranged in squares, so they are called
square numbers

1                4                  9                   16

   Name and sketch the next two square numbers.

Vocabulary: triangular numbers, square numbers

PICTORIAL (symbolic):

   Consider this pattern of blocks.

Building 1         Building 2         Building 3          Building 4           Building 5

202
LEARNING ACTIVITY/FACTS/INFORMATION
PICTORIAL (symbolic): (cont.)

   How many blocks would be in Building 6? Building 7? Building 10?

   Give the next two elements in each pattern:
1. 1, 4, 9, 16
2. AB, DE, GH, JK
3. 2, 5, 8, 11
4.

Vocabulary: element

ABSTRACT (computational):

   Describe a pattern for determining the first ten triangular numbers without using a
drawing.

   Describe a pattern for determining the first ten square numbers without using a
drawing.

Vocabulary: triangular numbers, square numbers

PROBLEM SOLVING:

Problem 1:

Gearing Up for Patterns – Think of how the five gears shown would turn each other –
clockwise and counterclockwise.

1. In what direction would the 2nd gear turn? The 3rd? The 4th? The 5th? The 6th?

5th           3rd
1st

2nd
4th

203
LEARNING ACTIVITY/FACTS/INFORMATION
PROBLEM SOLVING: (cont)

Problem 2:

Use a calculator, a ruler, and the pages of a book to help you compute the thickness of
a single sheet of paper. Use the measure that you get to answer the questions below.

1.   If you fold a sheet of paper one time, how thick is the stack?
2.   If you fold a sheet of paper two times, how thick is the stack?
3.   If you fold a sheet of paper three times, how thick is the stack?
4.   If you fold a sheet of paper four times, how thick is the stack?

Vocabulary: clockwise, counterclockwise

204
MATHEMATICS
Activities

GRADE LEVEL:          Fifth                    COURSE:
Framework Strand:     Probability and Discrete Mathematics
Grade Level Standard:         6-15 Develop, analyze, and solve problems.

Grade Level Benchmark:        5. Model arithmetic algorithms. (VI.2.MS.5)

LEARNING ACTIVITY/FACTS/INFORMATION
CONCRETE (conceptualizing):

   Each student needs four pennies, tails down as shown.

1                  2                   3                      4

   Turn over any three coins for each move. How many moves does it take to make
all the coins show tails?
Answer: Four moves – 1, 2, 3; 2, 3, 4; 1, 2, 4; 1, 3, 4

Vocabulary:

PICTORIAL (symbolic):

   You are helping a family plan a trip
through southern New Hampshire.
Use the road map to write directions
to get from Claremont to
Manchester.

Vocabulary:

205
LEARNING ACTIVITY/FACTS/INFORMATION
ABSTRACT (computational):

Look out! Each person tells the truth in two statements but does not tell the truth in the
third.

Georgia:     I have seven cats.
I have two fewer cats than Amanda has.
I have one more cat than Maggie does.

Amanda:      There is a difference between the number of cats Maggie has and the
number of cats I have.
I do not have the smallest number of cats.
Maggie has 10 cats.

Maggie:      I have fewer cats than Georgia has.
Amanda has 3 more cats than Georgia has.
Georgia has 8 cats.

How many cats does each person have?

Assume that Georgia’s first statement is true. Could the other people have made two
true statements and one statement that is not true? Pick another statement and
assume that it is true.
Answer: Georgia – eight cats; Amanda – ten cats; Maggie – seven cats
Vocabulary:

PROBLEM SOLVING:

At 2:00AM on July 4, Overbyte Computers in Dallas was broken into, and their Wonder-
Y Computer was smashed. You have been hired to find the culprit. The police tell you
the following:
The guilty person is definitely one of five suspects: Brainy, Gizmo, Shifty, Trouble, or
Zapper. Each suspect has made four statements, three of which are true and one of
which is false. The statements were these.
Brainy:    a.   I did not smash the computer.
b.   Gizmo has never been in Dallas.
c.   I never saw Shifty before.
d.   Trouble was with me in Waco the night of July 4.

Shifty:    a.   I was in Reno when the computer was smashed.
b.   I have never touched a computer.
c.   Gizmo is the guilty man.
d.   Brainy and I are friends.

206
LEARNING ACTIVITY/FACTS/INFORMATION
PROBLEM SOLVING: (cont)

Zapper:    a.   Trouble lied about never working on a computer.
b.   The computer was smashed on Independence Day.
c.   Shifty was in Reno at the time.
d.   One of us is guilty.

Gizmo:     a.   I did not smash the computer.
b.   I have never been in Dallas.
c.   I never saw Trouble before now.
d.   Shifty lied when he said I was guilty.

Trouble: a.     I did not smash the computer.
b.     I never worked on a computer in my life.
c.     Gizmo knows me.
d.     I was in Waco on July 4.

Find two statements of Zapper’s you already know are true. Two of Gizmo’s statements
say the same thing. Remember that each suspect makes only one false statement.
You can also find Shifty’s false statement.

Use the table to keep track of what you deduce by writing true of false for each
statement.
a.     b.     c.    d.           a.      b.     c.      d.
Brainy                                       T       T       F      T
Gizmo                                        T       T       F      T
Shifty                                      F       T       T      T
Trouble                                      T       T       F      T
Zapper                                       F       T       T      T

Who was the guilty subject?___________________

Vocabulary:

207
MATHEMATICS
Activities

GRADE LEVEL:            Sixth                  COURSE:
Framework Strand:       Probability and Discrete Mathematics
Grade Level Standard:        6-15 Develop, analyze, and solve problems.

Grade Level Benchmark:       6. Solve problems by determining the best solutions using
various strategies. (VI.2.MS.6)

LEARNING ACTIVITY/FACTS/INFORMATION
PROBLEM SOLVING:

   Below are the work schedules of 4 employees of a frozen yogurt store. All of
them are working on October 1, and all four are trying to figure out who works
with whom and when for the rest of the month. Help them solve their dilemma.
Be sure to check for more than one solution.

EMPLOYEE                   DAYS                TIME
Joey                    Every other day    1:00-8:00 p.m.
Debbie                  Every third day    12:00-7:00 p.m.
Eddie                   Every fourth day   2:00-9:00 p.m.
Melanie                 Every sixth day    3:00-10:00 p.m.

1.   On what day, other than October 1, will all four of them be working together?
2.   Who will be working together on October 5?
3.   On what days will only Eddie and Joey be working together?
4.   On what days will only Joey, Debbie, and Melanie be working together?
5.   Why is it that every day that Melanie works, so do Joey and Debbie?
6.   On October 13 at 7:30 p.m., who will be working?
7.   On October 13 at 2:25 p.m., who will be working?

Answers: 1. October 13, 25                  2. Eddie and Joey
3. Oct. 5, 9, 17, 21, 29           4. Oct 7, 19, 31
5. Joey works every 2 days, Debbie every 3 days. Both 2 and 3 are
factors of 6.
6. Joey, Eddie, and Melanie        7. Joey, Debbie, and Eddie

Vocabulary:

208
MATHEMATICS
Assessment

GRADE LEVEL:             Sixth                        COURSE:
Framework Strand:        Probability and Discrete Mathematics
Grade Level Standard:            6-15 Develop, analyze, and solve problems.

PROBLEM SOLVING ASSESSMENT:

     Design an algorithm to accomplish a task:
- Design a flow diagram to “assemble” a sandwich.
- Use a map to write directions to get from point A to point B.

PROBLEM RUBRIC:
BEGINNING              DEVELOPING              ACHIEVING             EXCEEDING
1                      2                       3                     4
Appropriate      Attempts a strategy    Begins with an         Uses an                Carries appropriate
Strategy                                appropriate strategy   appropriate strategy   strategy to correct
solution
Understanding    Shows minimal          Shows some             Shows significant      Understands the
Problem          understanding of       understanding of       understanding of       content of the
the content of the     the content of the     the content of the     problem
problem                problem                problem
Errors           Makes serious          Makes some errors      Makes minor errors     Makes no
errors but shows       (requires some         (could complete the    meaningful errors
reasoning (requires    instruction prior to   task with a non-
significant            being able to          instructional hint)
instruction prior to   complete the task)
being capable of
completing the
Supporting       Provides few           Provides some (but     Provides nearly        Provides clear and
Calculations,    supporting             not complete)          complete               complete
Arguments,       calculations/          supporting             supporting             supporting
and/or           arguments/             calculations/          calculations/          calculations/
Justifications   justifications         arguments/             arguments/             arguments/
justifications         justifications         justifications
Use of Tools     Uses inappropriate     Recognizes an          Uses appropriate       Uses available
tools and/or           appropriate tool or    tools and              tools and
representations        representation but     representations        representations
uses it                correctly but reads    correctly with
inappropriately        the scale              correct results
incorrectly or has
an error in
interpretation

209
MATHEMATICS
Activities

GRADE LEVEL:           Sixth                    COURSE:
Framework Strand:      Technology
Grade Level Standard:        6-16 Use technology tools to solve mathematical problems.

Grade Level Benchmark:       1. Use calculator operation keys such as memory recall,
square root, and order of operation.

LEARNING ACTIVITY/FACTS/INFORMATION
CONCRETE (conceptualizing):

   Define and locate the following memory and function keys on a calculator.

Memory Plus:             M+

Memory Minus:            M-

Clear Memory:            CM

Recall Memory:           RM

Vocabulary: memory plus, memory minus, clear memory, recall memory

PICTORIAL (symbolic):

   See following pages – “Meet Your Calcu-u-vue.”

Vocabulary:

ABSTRACT (computational):

   Solve these problems using M+, M-, and MRC. Remember to press ON/C before
each problem.

(12 x 5) – (48 ÷ 8) =              54
(100 + 60) – (8 x 5) =             120
(3 x 5) + (20 ÷ 4) – (80 ÷ 4) =    0

210
LEARNING ACTIVITY/FACTS/INFORMATION
ABSTRACT (computational): (cont)

(4 x 9) – (7 x 2) + (28 ÷ 4) =      29
(42 – 4) + (5 x 3) – (54 ÷ 9) =     47
(36 ÷ 9) + (6 x 4) – (49 ÷ 7) =     21
(100 ÷ 5) – (8 x 2) + (60 ÷ 6) =    14
(50 – 10) + (45 ÷ 9) – (6 x 3) =    27

Vocabulary:

PROBLEM SOLVING:

   See activity worksheet “Pay Day.”

Answers:           DAY               PAY           CUMULATIVE EARNINGS
1                 3¢                2¢
2                 4¢                6¢
3                 8¢                14¢
4                 16¢               30¢
5                 32¢               62¢
6                 64¢               \$1.26
7                 \$1.28             \$2.54
8                 \$2.56             \$5.10
9                 \$5.12             \$10.22
10                \$10.24            \$20.46
11                \$20.48            \$40.94
12                \$40.96            \$81.90
13                \$81.92            \$163.82
14                \$163.84           \$327.66
15                \$327.68           \$655.34
16                \$655.36           \$1,310.70
17                \$1,310.72         \$2,621.42
18                \$2,621.44         \$5,242.86
19                \$5,242.88         \$10,484.74
20                \$10,485.76        \$20,971.50
Regular Wages:            1st week         \$40
2nd week         \$80     40 x 2
3rd week         \$120    40 x 3
4th week         \$160    40 x 4
The 13th day he earned \$20,811.50.
Vocabulary:

211
212

Percent key. Use Change sign key.
Finds the       together with one Changes a number        Dual function. Not only
square root of      or more of the  from a positive to a    turns the calculator on,
a number         basic operation   negative and vice     but also clears all pending
keys.             versa.             operations. Does not
clear the memory.

Dual function. Pressed
once, this key clears the
last entry only. Pressed
twice, it clears all pending
operations. It does not
clear the memory.

Basic arithmetic
operation keys.

Stores numbers in
numbers directly to
the memory.

Subtracts numbers
directly from the
negative numbers Dual function. Pressed
Decimal point
to the memory.    once, this key brings to
the display any number
stored in the memory.
Pressed twice, it wipes
the memory clear.

213
PAY DAY

Shaun was offered a job by his father.
He would have to clean his father’s
shop every afternoon after school.
Being smart as well as hard-working,
Shaun asked his father for a special
wage. Instead of getting paid the
regular hourly wage, Shaun said he
wanted to get paid only 2¢ for the first
day, 4¢ the second day, 8¢ the third
day, and so on. His father agreed to
this reasonable agreement, and so Shaun began his job.

How much had Shaun earned at the end of:

The first week (five working days)? _______________________________
The second week? ___________________________________________
The third week? _____________________________________________
The fourth week? ____________________________________________

If Shaun had worked for a regular wage, he would have earned \$4.00 per hour
for 2 hours every afternoon. At this rate, how much would he have made at the
end of:

The first week (five working days)? ______________________________
The second week? ___________________________________________
The third week? _____________________________________________
The fourth week? ____________________________________________

On what day did the amount that Shaun earned using his new pay scheme
exceed the amount he would have earned on an hourly rate? ______________

How much more did Shaun earn during the four-week period using his pay
scheme than he would have earned had he been paid an hourly rate?
________________________

214
MATHEMATICS
Activities

GRADE LEVEL:           Sixth                  COURSE:
Framework Strand:      Technology
Grade Level Standard:       6-16 Use technology tools to solve mathematical problems.

Grade Level Benchmark:      2. Solve problems using calculators.

LEARNING ACTIVITY/FACTS/INFORMATION
CONCRETE (conceptualizing):

   Imagine the following situations:

1. You and your friends have lunch at a local restaurant. The cashier gives you a
bill for \$11.93.
2. You receive the bank statement for your savings account. The bank informs
you that you have “close to \$100 in your account.”
3. You are babysitting and ask what time the children need to go to bed. The
parents answer, “At 7:33 and 42 seconds.”

Which of these situations seem strange to you? Why?

Vocabulary: estimate, calculate

PICTORIAL (symbolic):

   NA

Vocabulary:

ABSTRACT (computational):

   Use any operation (+, -, x, ÷) to complete the following sentences:

215
LEARNING ACTIVITY/FACTS/INFORMATION
ABSTRACT (computational): (cont)

1.   5      (18 9) 6 = 1           +÷–
2.   9      5 6 18 = 2             ++–
3.   18      9 6 5=3               ÷+–
4.   18      6 9 5=7               ÷+–
5.   18      9 5 6=8               –+–
6.   6     (18 9) 5 = 9            –÷+
7.   9     5 (18 6) = 11           +–÷
8.   6     5 (18 9) = 28           x–÷
Vocabulary:

PROBLEM SOLVING:

   Do the following worksheet “Broken Calculator Keys.”

Answers will vary. Examples:

1. a.     Multiply 3 x 3 x 3, then subtract 5 four times. 3 x 3 x 3 – 5 – 5 – 5 – 5 = 7
b.     Multiply 7 x 3, then subtract 5 twice. 7 x 3 – 5 – 5 = 11
c.     Multiply 11 x 3, then subtract 5. 11 x 3 – 5 = 28
d.     Use [5] [yx] [3] or 5 x 5 x 5

2. 73 x 20 = 1460 leaves; 292; 73 x 4 – 292; 20 + 4 – 24 or 1752 x 1/73

3. 570 + 570 – 57 – 57

4. 40 x 37 + 4 x 37 + 4 x 37

Vocabulary:

216
BROKEN CALCULATOR KEYS
Describe how you would solve the following problems on your calculator if the
listed keys were broken.

1. The number keys 0, 1, 2, 4, 6, 7, 8, and 9 are broken.

a. How would you enter the number 7?
______________________________________________________________
______________________________________________________________

b. How would you change the 7 to 11?
______________________________________________________________
______________________________________________________________

c. How would you change the 11 to 28?
______________________________________________________________
______________________________________________________________

d. Clear the calculator. How would you enter the number 125?
______________________________________________________________
______________________________________________________________

2. The + key is broken. Find the missing number in 73 * _______ = 1752.
Explain what you did.
______________________________________________________________
______________________________________________________________

3. The x key is broken. Solve 57 * 18 = _______. Explain what you did.
______________________________________________________________
______________________________________________________________

4. The 8 key is broken. Solve 48 * 37 = _______. Explain what you did.
______________________________________________________________
______________________________________________________________

217
MATHEMATICS
Activities

GRADE LEVEL:           Sixth                  COURSE:
Framework Strand:      Technology
Grade Level Standard:       6-16 Use technology tools to solve mathematical problems.

Grade Level Benchmark:      3. Use appropriate computer skills.

LEARNING ACTIVITY/FACTS/INFORMATION
CONCRETE (conceptualizing):

   Use the computer to locate the calculator and review the skills.

Vocabulary:

PICTORIAL (symbolic):

   Use Excel on the computer to solve basic mathematic equations for home loans,
etc.

Vocabulary:

ABSTRACT (computational):

   Solve addition, subtraction, multiplication, and division problems with Excel.

Vocabulary:

PROBLEM SOLVING:

   NA

Vocabulary:

218
MATHEMATICS
Activities

GRADE LEVEL:            Sixth                   COURSE:
Framework Strand:       Technology
Grade Level Standard:        6-16 Use technology tools to solve mathematical problems.

Grade Level Benchmark:       4. Solve problems using the computer.

LEARNING ACTIVITY/FACTS/INFORMATION
PROBLEM SOLVING:

All of the following programs can be done using the software examples or by using web
sites.

1.   Students learn to solve word problems by selecting data, choosing operations, and
computing using software such as Puzzle Solving software (Addison Wesley).

2.   Students add to or subtract from the amount of liquid in two tanks to fill a third tank
using software such as Puzzle Tanks software (Sunburst).

3.   Students utilize software such as Easy Graph (Grolier) to make pictographs, bar
graphs, or circle graphs.

4.   Students choose reduced and equivalent fractions using software such as Fraction
Crunchers (MECC).

5.   Students make Escher designs and tessellations using software such as
Tessellmania.

Vocabulary:

219
MATHEMATICS
Activities

GRADE LEVEL:           Sixth                    COURSE:
Framework Strand:      Employability/Career Skills
Grade Level Standard:        6-17 Introduce, explore, and study various careers using
mathematics.
Grade Level Benchmark:       1. Understand the connection between mathematics and
career clusters.

LEARNING ACTIVITY/FACTS/INFORMATION
CAREERS:

   Use the computer to look up “job” sites, finding a variety of skills that relate to
mathematics.

Vocabulary:

220
MATHEMATICS
Activities

GRADE LEVEL:           Sixth                 COURSE:
Framework Strand:      Employability/Career Skills
Grade Level Standard:      6-18 Explore various financial management situations in
daily activities.
Grade Level Benchmark:     1. Develop skills for maintaining personal finances.

LEARNING ACTIVITY/FACTS/INFORMATION
PERSONAL MANAGEMENT:

   Use Excel to set up a personal budget.

Vocabulary: budget

221
MATHEMATICS
Activities

GRADE LEVEL:          Sixth                 COURSE:
Framework Strand:     Employability/Career Skills
Grade Level Standard:     6-19 Use cooperation and negotiation between and within
groups.
Grade Level Benchmark:    1. Develop skills for interacting with others.

LEARNING ACTIVITY/FACTS/INFORMATION
TEAMWORK:

   Break up the class into groups and give them on-the-job mathematics questions
to solve.

Vocabulary:

222

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