Thumb-Area
Student Achievement Model
Finding Focus for
Mathematics Instruction –
Grade 3
Huron Intermediate School District
February 8, 2010
Finding Focus for Elementary and Middle School Mathematics Instruction – Grade 3 February 8, 2010
Huron Intermediate School District 2 of 45 printed 12/15/2011 at 6:31:18 AM
Introduction
When teachers plan instruction, they draw on many sources such as state assessment
standards, local curriculum guides, textbook materials, and supplemental assessment
resources. These documents serve as useful sources of information, and it is neither necessary
nor desirable to replace them.
Michigan’s Grade-Level Content Expectations (GLCEs) describe in detail many ways in which
students can demonstrate their mastery of the mathematics curriculum. The GLCEs do not,
however, describe the big ideas and enduring understandings that students must develop in
order to achieve these expectations. The GLCEs describe products of student learning, but
they do not describe the thinking that must take place within the minds of students as they learn.
It is the purpose of this document to focus on the fundamental mathematical ideas that form the
basis of elementary and middle school instruction. Although a variety of research materials
were used in the development of this document, several sources were relied on quite heavily.
In 2006, the National Council of Teachers of Mathematics (NCTM) released Curriculum Focal
Points for Prekindergarten through Grade 8 Mathematics. The Focal Points describe big topics,
or focus areas, for each grade level.
In May, 2009, the Michigan Department of Education published the Michigan Focal Points Core
and Extended Designations. In that document, the NCTM Focal Points were adjusted to align
with Michigan’s GLCEs. The new core and extended designations for the MEAP reflect
Michigan’s Focal Points.
This document is structured around Michigan’s Focal Points and supporting documents, with
significant content included from two other documents:
Charles, Randall I. “Big Ideas and Understandings as the Foundation for Elementary
and Middle School Mathematics.” NCSM Journal of Mathematics Education Leadership.
Spring-Summer, 2005. vol. 8, no. 1, pp. 9 – 24.
“Chapter 4: Curricular Content.” Foundations for Success: The Final Report of the
National Mathematics Advisory Panel. U.S. Department of Education, 2008. pp. 15 – 25.
Particular thanks go to Ruth Anne Hodges for her contributions to this project. Additional
references to research are cited throughout the document.
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Finding Focus for Elementary and Middle School Mathematics Instruction – Grade 3 February 8, 2010
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Focusing on Mathematics at Grade 3
Grade 3 Focal Points
Grade Michigan Focal Point Related GLCE Topics Targeted Vocabulary
Developing Count in steps, and even, odd
Grade 3
understandings of understand even fact family
#1
multiplication and and odd numbers inverse operations
division and strategies Multiply and divide multiplication, product,
for basic multiplication whole numbers division, quotient, remainder
facts and related Problem-solving
division facts with whole
Models of Multiplication
numbers
repeated addition
area / array
Developing an Understand area, perimeter, length,
Grade 3 region
understanding of area meaning of area
#2
and perimeter and and apply in square unit, square inch,
determining the area problems square centimeter
and perimeters of two- Estimate perimeter inch, centimeter
dimensional shapes and area
Solve
measurement
problems
Describing properties Recognize the point, line, line segment,
Grade 3 distance, parallel lines,
of two-dimensional basic elements of
#3 perpendicular lines
shapes and classifying geometric objects
three-dimensional Name and explore parallelogram, trapezoid,
shapes properties of circle, rhombus, cube,
shapes sphere, rectangular prism,
Explore and name pyramid, cone
three-dimensional angle, side, vertex, face,
solids surface, base, edge
compose, decompose
Developing an Understand simple whole / whole unit / one
Grade 3
understanding of fractions, relation to whole number
#4
fractions and fraction the whole, and fraction, numerator,
equivalence addition and denominator
subtraction of equal / equivalent
fractions
number line (on a ruler)
Understand simple
decimal fractions in
relation to money
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National Math Panel Benchmarks for Grades 3, 4, and 5
By the end of Grade 3, By the end of Grade 4, By the end of Grade 5,
students should . . . students should . . . students should . . .
be proficient with the addition be proficient with
and subtraction of whole multiplication and division of
numbers. whole numbers.
be able to identify and be proficient with comparing
represent fractions and fractions and decimals and
decimals, and compare them common percents, and with
on a number line or with other the addition and subtraction of
common representations of fractions and decimals.
fractions and decimals.
be able to solve problems
involving perimeter and area
of triangles and all
quadrilaterals having at least
one pair of parallel sides (i.e.,
trapezoids).
The National Mathematics Advisory Panel Final Report, 2008, p.20
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Grade 3 Focal Point #1: Developing understandings of multiplication and division and
strategies for basic multiplication facts and related division facts
Grade 2 Grade 3 Grade 4
Developing quick recall of Represent whole-number Developing fluency with
addition facts and related problems with objects, words, multiplication of whole
subtraction facts and and mathematical statements numbers (NCTM-4th)
fluency with multi-digit and solve
addition and subtraction Use factors and multiples
(NCTM-2nd) Solve measurement problems
Multiply and divide whole
Count, write, and order whole National Math Panel Benchmark: numbers
numbers By the end of Grade 3, students
should be proficient with the
Add and subtract whole addition and subtraction of whole
numbers numbers
Measure, add, and subtract
length Developing understandings
of multiplication and
Solve measurement problems division and strategies for
involving length, money, and basic multiplication facts
perimeter (M.PS.02.10, and related division facts
M.TE.02.11) (NCTM-3rd
Tell time and solve time Count in steps, and
problems understand even and odd
numbers
Record, add, and subtract
money Multiply and divide whole
numbers
Represent whole-number
problems with objects, words,
and mathematical statements
and solve
National Math Panel Benchmark:
By the end of Grade 5, students
should be proficient with the
multiplication and division of
whole numbers
Key:
bold, non-italic = Michigan Curriculum Focal Points
non-bold, non-italic = GLCE topics associated with that focal point
non-bold, italic = Cross-over GLCE topics associated with another focal point
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Grade 3 Focal Point #1: Developing understandings of multiplication and division and
strategies for basic multiplication facts and related division facts
BIG MATHEMATICAL IDEAS AND UNDERSTANDINGS
Big Idea #5 (Operation Meanings & Relationships)
The same number sentence . . . [e.g. 12 4 = 3] can be associated with different concrete
or real-world situations, AND different number sentences can be associated with the
same concrete or real-world situation.
The real-world actions for multiplication and division of whole numbers are the same for
operations with fractions, decimals, and integers.
Any division calculation can be solved by using multiplication, and multiplication can be used
to check division.
Any two numbers can be multiplied or divided. The product or quotient has a new unit.
o 12 cookies 3 people = 4 cookies per person
o 4 pages per person x 6 people = 24 total pages
The same division problem can be represented with different models, depending on what
information is known.
o Partitive division: 12 cookies are shared equally among 3 children. How many
cookies does each child receive? (12 3 = 4)
o Measurement division: 12 cookies are bundled into groups of 3. How many
bundles can be made? (12 3 = 4)
o Product-factor division: There are 12 cookies on a tray today. That is 3 times
the number of cookies on the tray yesterday. How many cookies were on the
tray yesterday? (12 3 asks the same question as “3 x what = 12?”)
The standard multiplication algorithm works because of the distributive property and powers
of 10.
o 32 x 23 = (2 x 23) + (30 x 23) = (2 x 23) + (3 x 23 x 10) = 46 + 690
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Big Idea #6 (Properties)
For a given set of numbers there are relationships that are always true, and these are the
rules that govern arithmetic and algebra.
One way to represent multiplication is as the area of a rectangle. Because the area does
not change when the rectangle is rotated, multiplication is commutative.
o 3 x 5 = 5 x 3 = 15
Division is not commutative, but each division fact has a related division fact:
o 12 4 4 12, but 12 4 = 3, and 12 3 = 4.
Patterns in the multiplication table can be explained by the properties of arithmetic. For
example, the distributive property explains why, for any row, the entries in the 7 column are
the sums of the entries in the 5 and 2 columns (CCSI1).
Big Idea #7 (Basic Facts & Algorithms)
Basic facts and algorithms for operations with rational numbers use notions of
equivalence to transform calculations into simpler ones.
Any unknown multiplication fact can be found by using known facts.
o 6 x 8 is equivalent to 5 x 8 plus one more 8. (6 x 8 = 40 + 8)
o 9 x 3 is 3 less than 10 x 3. (9 x 3 = 30 – 3)
o I don’t know how much 8 groups of 2 is, but I know that 2 groups of 8 is 16.
Since multiplication is commutative, 8 x 2 = 16.
o 3 x 5 x 2 = 5 x 2 x 3 = 10 x 3 = 30 (associative property applies because
multiplication is commutative)
Multiplication and division are inverses of one another and can be used to check each other.
Any multiplication problem has related division problems, and any division problem has
related multiplication problems.
o 6 x 3 = 18 AND 3 x 6 = 18 AND 18 6 = 3 AND 18 3 = 6
1
Common Core Standards Initiative DRAFT 1/13/2010
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Grade 3 Focal Point #1: Developing understandings of multiplication and division and
strategies for basic multiplication facts and related division facts
INSTRUCTIONAL IMPLICATIONS
MODELS OF MULTIPLICATION
Multiplication can be represented as repeated addition or as the area of a rectangle. Because
“real-world” multiplication problems can be found in different contexts, it is important that
students understand and be able to use both models.
Multiplication as repeated addition
Multiplication can be represented as repeated addition of equal-sized groups. For example, if
four students each have three pieces of candy, this could be represented as “four groups of
three”.
As an addition problem, this story situation would be represented as 3 + 3 + 3 + 3, which is the
same as 4 x 3.
Multiplication as the area of a rectangle
The area or array model of multiplication represents multiplication as the area of a rectangle.
Initially, this can be introduced as an extension of equal-sized groups. Consider this example:
Use the square tiles to make 3 groups of 4.
Now, arrange each group in a row. Stack the rows vertically.
How many groups (3)? How many in each group (4)?
How many all together (12)?
Push the rows together so that all the tiles are touching.
How many rows (3)? How many in each row (4)?
How many all together (12)?
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In Grade 3, the area of a region is found by covering the shape with square tiles and counting
the tiles. In Grade 4, students use the area formulas for square and rectangles, with leads to a
more abstract application of the area model. The product of two numbers can be represented
as the area of a rectangle with those dimensions. Area models can represent fractions and
decimals as well as multiplication of whole numbers, fractions, or polynomials. Area models can
be represented visually, such as with grid paper, or tactilely, using tools such as Base 10 blocks
or Algebra Tiles.
Using area models at a variety of levels
2 x 3 = 6 6 ÷ 2 = 3 1/4
1/3 x 1/4 = 1/12
3 x 2 = 6 6 ÷ 3 = 2
21 x 12 = 200 + 10 + 40 + 2 = 252 (2x + 1) (x + 2) = 2x2 + 5x + 2
The example in the lower left (above) illustrates that the standard multiplication algorithm works
because of the distributive property and powers of 10. The rectangle representation of 21 x 12
can be divided into two parts {(20 x 12) + (1 x 12)} or into four parts {(20 x 10) + (1 x 10) + (20 x
2) + (1 x 2)}. For more information about how to use the distribute property to explain the
standard multiplication algorithm, see the Michigan Mathematics Program Improvement Project
(MMPI) (www.michiganmathematics.org, Chapter 4, pp. 9-10)
Assessing the Meanings of Multiplication
One way to assess if students understand the meaning of multiplication is to ask students to
draw pictures to represent story problems, making sure to include both repeated addition and
rectangular area problems. The diagnostic assessments from the Michigan Mathematics
Program Improvement Project (MMPI) contain good examples (www.michiganmathematics.org).
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COMMUTATIVITY OF MULTIPLICATION
Like addition, multiplication is commutative, which means that multiplication facts can be turned
around: 3 x 4 = 4 x 3. One way to illustrate this for students is by using the area or array model
of multiplication. Since the area of a shape does not change when the shape is turned or
moved, a rectangle built with three rows of four tiles each has the same area as a rectangle built
with four rows of three tiles each.
3 x 4 = 12 4 x 3 = 12
DEVELOPING FLUENCY WITH MULTIPLICATION AND DIVISION FACTS
When learning basic facts, students should first understand the meaning of multiplication and
division and represent each with models. For example
Show five groups of six using a picture and a number sentence.
If 36 candies are shared equally among four children, how many candies does
each child receive? Draw a picture and write a number sentence.
When a student understands what multiplication means, he can begin to develop strategies for
finding unknown facts based on known facts. Here are some examples:
Identify facts related to special numbers like 1 or 0. For example, use pictures and
models to help students understand that one group of any number is that number: 1
x 3 = 3; 1 x 4 = 4; 1 x 92 = 92; etc. and that zero groups is always 0: 0 x 5 = 0; 0 x
1,965 = 0; etc.
Use the commutative property of multiplication. When faced with “5 x 1,” a student
should think “5 x 1 is the same as 1 x 5, and one group of five is five.”
Use fact families. When over exposed to flash cards or math fact sheets, students
might see 3 x 4, 4 x 3, 124, and 123 as four unrelated problems rather than a
single fact family. This can also be taught as related facts: When asked “What is
123?” a student might think “what times three equals 12?”
Students often find doubles the easiest to learn, both in addition and subtraction.
Explicit discussion of the relationship between adding a number to itself and
multiplying that number by two can help students to learn the multiples of two.
Multiplying a number by three is the same as doubling the number then adding the
number again.
Multiplying a number by four is the same as doubling the double.
To multiply by five, multiply by 10, then cut it in half.
Multiplying a number by six is the same as multiplying by five then adding the
number again.
Many students also find the square products (2 x 2 = 4; 3 x 3 = 9; 4 x 4 = 16; etc.)
easy to learn.
Students will find some strategies easier than others, but regardless of the strategies used, the
most successful students are those who use numeric reasoning to find unknown facts.
Students should not rely solely on inefficient strategies such as repeated addition.
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When a student thoroughly understands the meanings of multiplication and division and is able
to use a variety of strategies to determine unknown facts, he is ready for repeated practice to
develop fluency. Practice should be distributed over time in short pieces. The time allotted for a
given set of problems should be long enough to discourage wild guessing but not long enough
to allow the student to resort to inefficient means.
MULTIPLICATION, DIVISION, AND FRACTIONS
Multiplication, division, and fractions all depend on equal-sized groups. Many student
misconceptions can be traced back to this critical concept.
Before becoming fluent with “naked” numbers, it is important that students understand the
meanings of multiplication, division, and fractions through context. Consider this basic situation:
A total number of objects (t) are shared equally among a number of groups
(g); each group has the same number of objects (n).
The operation this situation represents depends on the information that is known or unknown:
Known Unknown Operation
number of groups total number of objects (t) Multiplication
number of objects in each n x g = t; g x n = t
group
total number of objects number of objects in each Division
number of groups group (g) (partitive / fair shares)
tn=g
total number of objects number of groups (n) Division
number of objects in each (quota or measurement)
group tg=n
total number of objects number of objects in each Division
number of objects in each group OR number of groups (product-factor)
group OR number of n x g = t; g x n = t
groups
For examples of each problem type, see “Big Mathematical Ideas” previously in this section.
While it is not critical that students be able to name the different types of division problem, they
need to be able to recognize them in context. A variety of activities with concrete manipulatives
and pictorial representations is essential to building strong mental images. Students will later be
able to draw on these images when faced with abstract problems such as this one:
Write a story problem for 1¾ ½.
Many teachers struggle with this problem, writing scenarios that actually represent 1¾ 2 or
1¾ x 2. For many people, this problem becomes meaningful in the context of quota division:
I am making bows to decorate packages. I have 1¾ yards of ribbon. Each
bow needs ½ yard of ribbon. How many bows can I make?
Unfortunately, quota or measurement situations like the one above are not presented nearly as
often as partitive or fair share situations. As a result, many students have a limited
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understanding of division. For concrete manipulative activities designed to help students
deepen their understanding, see the Michigan Mathematics Program Improvement Project
(MMPI) at www.michiganmathematics.org (Chapter 4, Pages 5-6).
To complete the cycle of concrete manipulatives, pictorial representations, and abstract
computation, consider having students engage in the following types of activities:
given a story problem, draw a picture
given a number sentence, write a story problem
given a picture, write a number sentence
When students can express equivalent problems in different ways as shown below, they truly
understand the operation.
story problem picture
number sentence
The MMPI Diagnostic Inventories include story problems, pictures, and number sentences that
can be used to assess students’ understanding of multiplication and division and pinpoint
instructional needs: www.michiganmathematics.org (Chapter 4).
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Grade 3 Focal Point #1: Developing understandings of multiplication and division and
strategies for basic multiplication facts and related division facts
RELATED GLCES WITH CORE AND EXTENDED DESIGNATIONS
Number and Operations
Count in steps, and understand even and odd numbers
N.ME.03.04 Count orally by 6’s, 7’s, 8’s, and 9’s starting with 0, making the connection
between repeated addition and multiplication. [Ext-NC]
N.ME.03.05 Know that even numbers end in 0, 2, 4, 6, or 8; name a whole number
quantity that can be shared in two equal groups or grouped into pairs with no
remainders; recognize even numbers as multiples of 2. Know that odd
numbers end in 1, 3, 5, 7, or 9, and work with patterns involving even and
odd numbers. [Ext-NC]
N.ME.03.03 Compare and order numbers up to 10,000 [Ext-NC]
Multiply and divide whole numbers
N.MR.03.09 Use multiplication and division fact families to understand the inverse
relationship of these two operations, e.g., because 3x8=24, we know that
24÷8=3 or 24÷3=8; express a multiplication statement as an equivalent
division statement. [Core-NC]
N.MR.03.10 Recognize situations that can be solved using multiplication and division
including finding “How many groups?” and “How many in a group?” and write
mathematical statements to represent those situations. [Core-NC]
N.FL.03.11 Find products fluently up to 10 x 10; find related quotients using multiplication
and division relationships. [Core-NC]
N.MR.03.12 Find solutions to open sentences, such as 7x □ = 42 or 12÷□ = 4, using the
inverse relationship between multiplication and division. [Ext]
N.FL.03.13 Mentally calculate simple products and quotients up to a three-digit number
by a one-digit number involving multiples of 10, e.g., 500x6, or 400÷8.
[NASL]
N.MR.03.14 Solve division problems involving remainders, viewing the remainder as the
“number left over”; interpret based on problem context, e.g., when we have
25 children with 4 children per group then there are 6 groups with 1 child left
over. [Core]
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Problem solving with whole numbers
N.MR.03.15 Given problems that use any one of the four operations with appropriate
numbers, represent with objects, words, (including “product” and “quotient”),
and mathematical statements; solve. [Core]
Key:
Core – expectation will be assessed with two items on the MEAP
Ext – extended core; expectation will be assessed with no more than one item.
NC – no calculator
NASL – not assessed at the state level; will not be tested on the MEAP
FROM THE 1/13/2010 DRAFT OF THE COMMON CORE STANDARDS INITIATIVE
Operations and the Problems They Solve
Students can and do:
a. Use representations (objects, pictures, story contexts) to describe and justify properties of
multiplication and division.
b. Solve simple multiplication and division word problems involving equal groups, length and
area.
c. Solve up to two-step word problems involving the four operations with whole numbers and
whole number quantities. (Whole number quotients only)
d. Solve multiplicative comparison problems with whole numbers (problems involving the notion
of “times as much”).
e. Draw a scaled bar graph to represent a data set with several categories. Solve “how many
more”/”how many less” problems (two-step problems) using information presented in scaled
bar graphs.
Base Ten Computation
Students can and do:
a. Explain strategies for multiplying and dividing that use the Properties of Arithmetic and
properties of the base ten systems.
b. Rapidly multiply and divide within 100. (A variety of mental strategies are acceptable,
including derived fact strategies and producing products or quotients from memory).
c. Produce full sets of fact families for multiplication and division, as in the set 6 × 7 = 42, 7 × 6
= 42, 42 = 7 × 6, 42 = 6 × 7, 42 ÷ 7 = 6, 42 ÷ 6 = 7, 6 = 42 ÷ 7, 7 = 42 ÷ 6.
d. Find the factor pairs for a given number, as in the factor pairs for the number 42: {42, 1},
{21, 2}, {14, 3}, {7, 6}.
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Grade 3 Focal Point #2: Developing an understanding of area and perimeter and determining
the areas and perimeters of two-dimensional shapes
Grade 2 Grade 3 Grade 4
Developing an
understanding of area and
perimeter and determining
the areas and perimeters of
two-dimensional shapes
(NCTM-4th)
Understand meaning of area
and perimeter and apply in
problems
Solve measurement problems Estimate perimeter and area
involving length, money, and for squares and rectangles
perimeter (M.PS.02.10,
M.TE.02.11) Solve measurement problems
National Math Panel Benchmark:
By the end of Grade 5, students
should be able to solve
problems involving perimeter
and area of triangles and all
quadrilaterals having at least
one pair of parallel sides (i.e.,
trapezoids)
Key:
bold, non-italic = Michigan Curriculum Focal Points
non-bold, non-italic = GLCE topics associated with that focal point
non-bold, italic = Cross-over GLCE topics associated with another focal point
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Grade 3 Focal Point #2: Developing an understanding of area and perimeter and determining
the areas and perimeters of two-dimensional shapes
BIG MATHEMATICAL IDEAS AND UNDERSTANDINGS
Big Idea #17 (Measurement)
Some attributes of objects are measurable and can be quantified using unit amounts.
Measurement involves a selected attribute of an object (length, area, mass, volume,
capacity) and a comparison of the object being measured against a unit of the same
attribute.
The larger the unit of measure, the fewer units it takes to measure the object.
A unit of measure can be partitioned into equal-sized parts, whose sizes can be represented
as fractions of the unit.
A given measurement can be expressed in many equivalent forms of different units of the
same attribute or dimension:
o 2 feet = 24 inches
o 1 cubic yard = 27 cubic feet
The magnitude of the attribute to be measured and the accuracy needed determines the
appropriate measurement unit.
The unit used to measure an object’s attribute depends on the dimension of the attribute:
o Length is measured in linear units like inch, centimeter, meter, etc. This includes
height, width, distance, perimeter, and circumference.
o Area is measured in square units like square meter, square yard, acre, etc. This
includes the area of two-dimensional figures and the surface area of three-
dimensional shapes.
o Volume is measured in cubic units like cm3, in3, etc.
A square with side length 1 unit is said to enclose “one square unit” of area. The area of a
closed plane figure can be measured by the number of square units that fit inside it with no
gaps or overlaps (CCSI2).
The perimeter, circumference, area, surface area, or volume of an object depends on the
object’s linear dimensions, interior angles, and curves. For many common shapes, formulas
can be used to calculate the perimeter, area, volume, surface area, or circumference. (At
Grade 3, students calculate perimeter by adding the lengths of the sides and find area by
covering the region with squares.)
A figure or object can be constructed from or decomposed into figures of the same
dimension. The measurement of a given attribute of the object is equal to the sums of the
measurements of the components of the object for that attribute:
2
Common Core Standards Initiative DRAFT 1/13/2010
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o if a polygon is decomposed into other polygons, the area of the original polygon
is equal to the sum of the areas of the component polygons
o the perimeter of a polygon can be found by adding together the lengths of the
sides
o if an angle is composed from smaller angles, the measure of the total angle is
equal to the sums of the measures of the component angles
o if a box is composed from smaller boxes, the total volume of the box is equal to
the sum of the volumes of the component boxes
When two polygons or circles are similar by a factor of r, their perimeters or circumferences
are similar by a factor of r.
When two polygons or circles are similar by a factor of r, their areas are similar by a factor of
r2 (e.g., if you triple the length of each side of a triangle, the area increases to nine times that
of the original area).
For a given perimeter there can be a shape with area close to zero.
The maximum area for a given perimeter and a given number of sides is a regular polygon
with that number of sides. (In a regular polygon, all sides are congruent and all angles are
congruent).
Given a regular polygon with fixed perimeter, the more sides there are, the larger the area
will be.
The maximum area for a given perimeter is a circle with that circumference. (Think of a
circle as a regular polygon with an infinite number of sides).
[ The light gray points are related to the same Big Idea and topic, but are addressed at a later
grade level. ]
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Grade 3 Focal Point #2: Developing an understanding of area and perimeter and determining
the areas and perimeters of two-dimensional shapes
INSTRUCTIONAL IMPLICATIONS
Concrete-Representational-Abstract is an instructional strategy with a strong research base.
Following the steps of “Build It – Draw It – Write It,” students first explore a concept with
concrete manipulatives or kinesthetic activities. Students then draw pictures to represent the
actions that were performed, writing symbolism as appropriate. The concrete phase is dropped
when students are ready, but students may continue to draw pictures or sketches for quite a
while, including abstract symbolism whenever possible. This process forms strong mental
images in students’ minds so that even when solving problems with abstract symbolism alone,
students retain a strong understanding of concept.
Concrete-Representational-Abstract with Area and Perimeter
Begin with a concrete representation. For example, ask students to count how many steps it
takes to walk around the classroom along the walls or use popsicle sticks to measure the
perimeter of a student desktop. A concrete example of area would be to count the floor tiles in
the cafeteria or cover a math book with square tiles. Students record their work by drawing a
picture (e.g., a picture of a desk with the correct number of popsicle sticks around the edge).
After some basic concrete experiences, students explore area and perimeter through story
problems and sketches:
Mr. Smith’s garden is a rectangle 8 feet long and 12 feet wide. If he wants to put
up a fence that goes completely around the garden, how many feet of fence does
he need?
Trace your foot onto a piece of grid paper. How many squares does it take to
cover your footprint?
When transitioning from manipulatives and pictures to formulas, it is important that students
understand this big idea:
A figure or object can be constructed from or decomposed into figures of the
same dimension. The measurement of a given attribute of the object is equal to
the sums of the measurements of the components of the object for that attribute:
if a polygon is decomposed into other polygons, the area of
the original polygon is equal to the sum of the areas of the
component polygons (if I count the squares it takes to cover
my footprint, that is the area of my footprint)
the perimeter of a polygon can be found by adding together
the lengths of the sides
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Grade 3 Focal Point #2: Developing an understanding of area and perimeter and determining
the areas and perimeters of two-dimensional shapes
RELATED GLCES WITH CORE AND EXTENDED DESIGNATIONS
Measurement
Understand meaning of area and perimeter and apply in problems
M.UN.03.05 Know the definition of area and perimeter and calculate the perimeter of a
square and rectangle given whole number side lengths. [Core]
M.UN.03.06 Use square units in calculating area by covering the region and counting
the number of square units. [Core-NC]
M.UN.03.07 Distinguish between units of length and area, and choose a unit
appropriate in the context. [Core-NC]
M.UN.03.08 Visualize and describe the relative size of a square inch and one square
centimeter. [Ext-NC]
Estimate perimeter and area
M.TE.03.09 Estimate the perimeter of a square and rectangle in inches and
centimeters; estimate the area of a square and rectangle in square inches
and square centimeters. [Core-NC]
Solve measurement problems
M.PS.03.10 Add and subtract lengths, weights, and times using mixed units, within the
same measurement system. [Ext]
M.PS.03.11 Add and subtract money in dollars and cents. [Ext]
M.PS.03.12 Solve applied problems involving money, length, and time. [Core]
M.PS.03.13 Solve contextual problems about perimeters of rectangles and areas of
rectangular regions. [Core]
Key:
Core – expectation will be assessed with two items on the MEAP
Ext – extended core; expectation will be assessed with no more than one item.
NC – no calculator
NASL – not assessed at the state level; will not be tested on the MEAP
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FROM THE 1/13/2010 DRAFT OF THE COMMON CORE STANDARDS INITIATIVE
Quantity and Measurement
Students can and do:
a. Measure lengths using rulers marked with halves and fourths of inches. Make a dot plot
to show repeated measurements.
b. Convert compound units to a smaller or a larger unit, and solve problems involving
mixed units (feet and inches, yards and feet).
c. Using customary units, demonstrate and justify correct processes for measuring,
comparing, and estimating length, mass, capacity, and durations of time, including unit
selection, partitioning and iterating units, and transitivity.
d. Compute perimeters of polygons by adding given side lengths, and find an unknown
length in a polygon given the perimeter and all other side lengths. Represent these
problems with equations involving a symbol for the unknown quantity.
2 2 2
e. Determine and compare areas by counting square units (improvised units, cm , m , in ,
2
ft ).
f. Compute elapsed time and solve problems involving elapsed time (to the nearest
minute).
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Grade 3 Focal Point #3: Describing properties of two-dimensional shapes and classifying three-
dimensional shapes
Grade 2 Grade 3 Grade 4
Composing and Describing properties of
decomposing geometric two-dimensional shapes
shapes (NCTM-1st) (part of NCTM-3rd – see also
5th grade) and classifying
Identify and describe shapes 3D shapes (MI language)
Work with unit fractions Recognize the basic elements
of geometric objects
Name and explore properties
of shapes
Explore and name three-
dimensional solids
Key:
bold, non-italic = Michigan Curriculum Focal Points
non-bold, non-italic = GLCE topics associated with that focal point
non-bold, italic = Cross over GLCE topics associated with another focal point
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Huron Intermediate School District 23 of 45 printed 12/15/2011 at 6:31:18 AM
Grade 3 Focal Point #3: Describing properties of two-dimensional shapes and classifying three-
dimensional shapes
BIG MATHEMATICAL IDEAS AND UNDERSTANDINGS
Big Idea #14 (Shapes & Solids)
Two- and three-dimensional objects with or without curved surfaces can be described,
classified, and analyzed by their attributes.
Two-dimensional figures are composed of lines, line segments, and curves.
Three-dimensional objects are composed of flat surfaces and curved surfaces.
Point, line, line segment, distance, and plane are the core attributes of flat shapes.
Polygons are two-dimensional shapes composed of straight line segments.
Polygons can be described uniquely by their sides and angles.
Polygons can be constructed from or decomposed into other polygons.
When a polygon is composed of other polygons, the area of the final polygon is equal to the
sum of the areas of the component polygons.
Triangles and quadrilaterals can be described, categorized, and named based on the
relative lengths of their sides and the sizes of their angles.
Polyhedra are three-dimensional solids composed of flat surfaces. All polyhedra can be
described completely by their faces, edges, and vertices.
Some shapes or combinations of shapes can be put together without overlapping to
completely cover the plane.
There is more than one way to classify most shapes and solids.
Big Idea #15 (Orientation & Location)
Objects in space can be oriented in an infinite number of ways, and an object’s location
in space can be described quantitatively.
Two distinct lines in the plane are either parallel or intersecting; two distinct lines in space
are parallel, intersecting, or skew.
Perpendicular lines intersect at a right angle (90).
The orientation of an object does not change the other attributes of the object.
A number of degrees can be used to describe the size of an angle’s opening.
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A right angle is a special kind of angle measuring 90. There are 90 degrees in a quarter
turn.
Angles that form a straight line add up to 180. There are 180 degrees in a half turn.
Angles that surround a point add up to 360. There are 360 degrees in a full turn.
Angles can be classified as acute, right, obtuse, or straight by comparing their measures to
right angles (90) and straight lines (180).
Unknown angles can be determined by looking at their relationships to known angles:
o complementary angles form a right angle and add up to 90
o supplementary angles form a straight line and add up to 180
o vertical angles have the same measure, which can be shown by their
relationships to a common supplementary angle
o the angles formed when a line intersects two parallel lines in a plane also have
special relationships and may be congruent (equal) or supplementary (add up to
180)
In some polygons, the angles and sides have special relationships:
o the sum of the interior angles of a triangle is 180
o the sum of the interior angles of a quadrilateral is 360.
o in an equilateral triangle, all of the sides are congruent (equal), and each angle is
60
o in isosceles triangles, the angles that are opposite the two congruent sides are
congruent (equal)
o in a parallelogram, opposite angles are congruent (equal)
o in a parallelogram, opposite sides are congruent (equal)
o in a right triangle, the relationships among the lengths of the sides is shown in
the Pythagorean Theorem: a2 + b2 = c2
Every point in the plane can be described uniquely by an ordered pair of numbers based on
two intersecting number lines.
o The two number lines are perpendicular and intersect at 0 on each. This point
(0,0) is called the origin.
o The first number in the ordered pair tells the distance to the left (-) or right (+) of
zero on the horizontal number line.
o The second number in the ordered pair tells the distance above (+) or below (-)
zero on the vertical number line.
o This scheme is called the Cartesian Coordinate System (the Cartesian plane).
The system can be extended to name points in space.
The Pythagorean Theorem can be used to find the distance between two points in the
Cartesian plane. This is expressed as the distance formula, where the two points are (x1, y1)
and (x2, y2): D = D ( x2 x1)2 ( y 2 y1)2
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Big Idea #16 (Transformations)
Objects in space can be transformed in an infinite number of ways, and those
transformations can be described and analyzed mathematically.
Congruent figures remain congruent (the same shape and size) through translations,
rotations, and reflections.
Shapes can be transformed to similar shapes (but larger or smaller) with proportional
corresponding sides and congruent corresponding angles.
Triangles have unique properties which allow us to show that two triangles are similar
without knowing all of the angles and sides:
o if all corresponding angles are congruent (AAA), the two triangles are similar
o if the ratios of all pairs of corresponding sides are equal (SSS), the two triangles
are similar
o if the ratios of two corresponding sides are equal and the included angles are
congruent (SAS), the two triangles are similar
Algebraic expressions can be used to generalize transformations for objects in the plane.
Some shapes can be divided in half where one half folds exactly on top of the other (line
symmetry).
Some shapes can be rotated around a point in less than one complete turn and land exactly
on top of themselves (rotational symmetry).
Big Idea #17 (Measurement)
Some attributes of objects are measurable and can be quantified using unit amounts.
Measurement involves a selected attribute of an object (length, area, mass, volume,
capacity) and a comparison of the object being measured against a unit of the same
attribute.
The larger the unit of measure, the fewer units it takes to measure the object.
The unit used to measure an object’s attribute depends on the dimension of the attribute:
o Length is measured in linear units like inch, centimeter, meter, etc. This includes
height, width, distance, perimeter, and circumference.
o Area is measured in square units like square meter, square yard, acre, etc. This
includes the area of two-dimensional figures and the surface area of three-
dimensional shapes.
o Volume is measured in cubic units like cm3, in3, etc.
The perimeter, circumference, area, surface area, or volume of an object depends on the
object’s linear dimensions, interior angles, and curves. For many common shapes, formulas
can be used to calculate the perimeter, area, volume, surface area, or circumference. (At
Grade 3, students calculate perimeter by adding the lengths of the sides and find area by
covering the region with squares.)
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A figure or object can be constructed from or decomposed into figures of the same
dimension. The measurement of a given attribute of the object is equal to the sums of the
measurements of the components of the object for that attribute:
o if a polygon is decomposed into other polygons, the area of the original polygon
is equal to the sum of the areas of the component polygons
o the perimeter of a polygon can be found by adding together the lengths of the
sides
o if an angle is composed from smaller angles, the measure of the total angle is
equal to the sums of the measures of the component angles
o if a box is composed from smaller boxes, the total volume of the box is equal to
the sum of the volumes of the component boxes
When two polygons or circles are similar by a factor of r, their perimeters or circumferences
are similar by a factor of r.
When two polygons or circles are similar by a factor of r, their areas are similar by a factor of
r2 (e.g., if you triple the length of each side of a triangle, the area increases to nine times that
of the original area).
[ The light gray points are related to the same Big Idea and topic, but are addressed at a later
grade level. ]
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Grade 3 Focal Point #3: Describing properties of two-dimensional shapes and classifying three-
dimensional shapes
INSTRUCTIONAL IMPLICATIONS
Explicit vocabulary instruction is important in all mathematics instruction but is particularly
critical with this topic. Students should not simply copy the definitions of important terms.
Rather, they should be able to:
Define the term in their own words
Draw pictures
Give examples and non-examples
Identify similar words
Compare to related words
Sample vocabulary activities
Term Activity
line, line segment compare line to line segment
sketches of lines have arrows on the end;
sketches of line segments do not
parallel lines define in your own words
draw examples and non-examples
perpendicular lines identify objects that do and do not have
right angles
draw examples and non-examples of
perpendicular lines
parallelogram, trapezoid, rhombus create a chart comparing different
attributes of these shapes, i.e.:
o how many sides
o how many pair of parallel sides
o how many equal sides
cube, sphere, cone, pyramid, cylinder sort three-dimensional solids by
o only curved surfaces
o only flat surfaces
o both curved and flat surfaces
It is also critical that students build with and draw concrete objects as they develop
understanding of shape properties. The Concrete-Representational-Abstract strategy is
particularly appropriate (see Grade 3 Focal Point #2- page 20).
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Grade 3 Focal Point #3: Describing properties of two-dimensional shapes and classifying three-
dimensional shapes
RELATED GLCES WITH CORE AND EXTENDED DESIGNATIONS
Geometry
Recognize the basic elements of geometric objects
G.GS.03.01 Identify points, line segments, lines, and distance. [Core-NC]
G.GS.03.02 Identify perpendicular lines and parallel lines in familiar shapes and
in the classroom. [Core-NC]
G.GS.03.03 Identify parallel faces of rectangular prisms, in familiar shapes and in the
classroom. [Core-NC]
Name and explore properties of shapes
G.GS.03.04 Identify, describe compare and classify two-dimensional shapes, e.g.,
parallelogram, trapezoid, circle, rectangle, square, and rhombus, based
on their component parts (angles, sides, vertices, line segment) and on
the number of sides and vertices. [Core-NC]
G.SR.03.05 Compose and decompose triangles and rectangles to form other familiar
two-dimensional shapes; e.g., form a rectangle using two congruent right
triangles, or decompose a parallelogram into a rectangle and two right
triangles. [Core-NC]
Explore and name three-dimensional solids
G.GS.03.06 Identify, describe, build, and classify familiar three-dimensional solids,
e.g., cube, rectangular prism, sphere, pyramid, cone, based on their
component parts (faces, surfaces, bases, edges, vertices). [Core-NC]
G.GS.03.07 Represent front, top, and side views of solids built with cubes. [Ext-NC]
Key:
Core – expectation will be assessed with two items on the MEAP
Ext – extended core; expectation will be assessed with no more than one item.
NC – no calculator
NASL – not assessed at the state level; will not be tested on the MEAP
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FROM THE 1/13/2010 DRAFT OF THE COMMON CORE STANDARDS INITIATIVE
The DRAFT of the Core Standards addresses properties of shapes, parallelism,
perpendicularity, and angle measure at Grade 4:
At GRADE 4, students can and do (Shapes):
a. Draw points, lines, line segments, rays and angles; identify these in geometric figures.
o
b. Associate angles of a quarter turn (subtending ¼ of a circle) with angle measure 90 , a half
o o
turn (½ of a circle) with angle measure 180 , ¾ turn (¾ of a circle) with angle measure 270 ,
o
and a full turn (complete circle) with angle measure 360 .42
c. Draw perpendicular and parallel lines; identify these in geometric figures.
d. Identify right angles and angles smaller than/greater than a right angle in geometric figures;
recognize right triangles.
e. Given a quadrilateral, say whether it is a square, whether it is a rectangle, and whether it is a
parallelogram (with an understanding that a given shape may fit more than one category).
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Grade 3 Focal Point #4: Developing an understanding of fractions and fraction equivalence
Grade 2 Grade 3 Grade 4
Developing an Developing an Developing an
understanding of the base- understanding of fractions understanding of decimals
ten numeration system and and fraction equivalence and fractions, including the
place-value concepts (NCTM-3rd) connections between them
(NCTM-2nd) (NCTM-4th)
Understand simple fractions,
Count, write, and order whole relation to the whole, and Understand fractions
numbers addition and subtraction of
fractions Read, interpret, and compare
Understand place value decimal fractions
Understand simple decimal
Work with unit fractions fractions in relation to money
National Math Panel Benchmark:
By the end of Grade 4, students
should be able to identify and
represent fractions and decimals
and compare them on a number
line or with other common
representations of fractions and
decimals
National Math Panel Benchmark:
By the end of Grade 5, students
should be proficient with
comparing fractions and
decimals and common percents
Key:
bold, non-italic = Michigan Curriculum Focal Points
non-bold, non-italic = GLCE topics associated with that focal point
non-bold, italic = Cross-over GLCE topics associated with another focal point
Finding Focus for Elementary and Middle School Mathematics Instruction – Grade 3 February 8, 2010
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Grade 3 Focal Point #4: Developing an understanding of fractions and fraction equivalence
BIG MATHEMATICAL IDEAS AND UNDERSTANDINGS
Big Idea #1 (Numbers)
The set of real numbers is infinite, and each real number can be associated with a unique
point on the number line.
Fractions and decimals are numbers:
o A fraction describes the division of a whole (area, set, or length) into equal parts.
The more equal pieces a whole is divided into, the smaller each piece is.
o A fraction is relative to the size of the whole or unit.
Would you rather have all of a mini candy bar or ½ of a king-sized candy
bar?
o Each fraction can be associated with a unique point on the number line, but not
all of the points between integers can be named by fractions.
o There is no least or greatest fraction on the number line.
o There are an infinite number of fractions between any two fractions on the
number line.
o A decimal is another name for a fraction and thus can be associated with the
corresponding point on the number line.
o Whole numbers can be written as fractions, as in 3/3 = 1; 4/1 = 4; 15/3 = 5.
When a whole, 1, is divided into b equal parts, the size of the parts is written 1/b. To show
1/b of something, divide the thing into b equal parts.
The same fraction can describe different situations:
o ¾ describes how much of a candy bar is eaten if a candy bar is divided into 4
equal parts and 3 of the parts are eaten
o ¾ also describes how much of a candy bar one person eats if 3 candy bars are
shared fairly (divided evenly) among 4 people
Just like whole numbers, fractions and decimals of the same unit can be added, subtracted,
or counted:
o Count by tenths: 0.1, 0.2, 0.3, . . .
o Count by fourths: one-fourth, two-fourths, three-fourths, four-fourths (one), etc.
2 3 5
o Add fractions with common denominators:
4 4 4
Any fraction can be written as the sum of unit fractions (e.g., ¾ = ¼ + ¼ + ¼)
Integers are numbers:
o Integers are the whole numbers and their opposites on the number line, where
zero is its own opposite.
o Each integer can be associated with a unique point on the number line, but there
are many points on the number line that cannot be named by integers.
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o An integer and its opposite are the same distance from zero on the number line.
o There is not a greatest or least integer on the number line.
Big Idea #2 (The Base Ten Numeration System)
The base ten numeration system is a scheme for recording numbers using digits 0-9,
groups of ten, and place value.
Decimal notation is an extension of place value based on powers of ten. As with whole-
number place value, each place is ten times the value of the place to the right:
o 3 ones = 30 tenths
o 2 tenths = 20 hundredths
o 4 tenths = 400 thousandths
Teacher note: 10n, . . . 103, 102, 101, 100, 10-1, 10-2, 10-3, . . . 10-n
Big Idea #3 (Equivalence)
Any number, measure, numerical expression, algebraic expression, or equation can be
represented in an infinite number of ways that have the same value.
Fractions and decimals can be expressed in equivalent forms using different units:
1 2 3
o
4 4 4
3 6
o
4 8
2 3 92
o 1
2 3 92
o 0.2 = 0.20 = 0.200
Whole numbers and integers can be written in fraction or decimal form (e.g., 4 = 4/1;
-2 = -8/4; 3 = 3.0)
Two fractions are equal when they represent the same portion of a whole, or when they
have the same length on a number line.
[The light gray points are related to the same Big Idea and topic, but are addressed at a later
grade level.]
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Grade 3 Focal Point #4: Developing an understanding of fractions and fraction equivalence
INSTRUCTIONAL IMPLICATIONS
REPRESENTATIONS OF FRACTIONS
Fractions can be represented using a set model, a linear model, or an area model. Students
who understand one model may become confused when presented with a different model. For
an example of a virtual representation, see the activity “Fraction Model I” at
illuminations.nctm.org (http://illuminations.nctm.org/ActivityDetail.aspx?ID=11)
When creating concrete representations of these visual models, use a model with the
appropriate dimension. For example, fraction strips sometimes confuse students who aren’t
sure if it represents an area model or a linear model. One way around this would be to use
string to measure a particular distance and then fold the string to divide it evenly.
ORDERING AND COMPARING FRACTIONS
A common misunderstanding among students is to think that 1/4 is larger than 1/3 because four
is larger than three. By using concrete manipulatives and drawings, students should create a
strong mental image showing that the more equal parts an object is divided into, the smaller
each part is. It is also important that students be able to explicitly state this relationship and
apply it to unit fractions: 1/5 is larger than 1/8, because dividing an object into five pieces gives
larger pieces than dividing the same object into eight pieces.
When comparing fractions or placing fractions on a number line, it is often useful to compare
fraction quantities to certain benchmark fractions, such as 0, ½, or 1. For example:
1/3 is less than 1/2, and 3/4 is more than 1/2, so I know that 1/3 is less than 3/4.
3/4 is one-fourth less than one, but 5/6 is only one-sixth less than one. So 3/4 is
less than 5/6.
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Grade 3 Focal Point #4: Developing an understanding of fractions and fraction equivalence
RELATED GLCES WITH CORE AND EXTENDED DESIGNATIONS
Number and Operations
Understand simple fractions, relation to the whole, and addition and subtraction of
fractions
N.ME.03.16 Understand that fractions may represent a portion of a whole unit that has
been partitioned into parts of equal area or length; use the terms
“numerator” and “denominator.” [Core-NC]
N.ME.03.17 Recognize, name and use equivalent fractions with denominators 2, 4,
and 8, using strips as models. [Core-NC]
N.ME.03.18 Place fractions with denominators of 2, 4, and 8 on the number line; relate
the number line to a ruler; compare and order up to three fractions with
denominators 2, 4, and 8. [Core-NC]
N.ME.03.19 Understand that any fraction can be written as a sum of unit fractions,
e.g., 3/4 =1/4 + 1/4+1/4. [Ext-NC]
N.MR.03.20 Recognize that addition and subtraction of fractions with equal
denominators can be modeled by joining or taking away segments on the
number line. [Ext-NC]
Understand simple decimal fractions in relation to money
N.ME.03.21 Understand and relate decimal fractions to fractional parts of a dollar,
e.g., 1/2 dollar= $0.50; 1/4 dollar=$0.25. [Ext]
Key:
Core – expectation will be assessed with two items on the MEAP
Ext – extended core; expectation will be assessed with no more than one item.
NC – no calculator
NASL – not assessed at the state level; will not be tested on the MEAP
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FROM THE 1/13/2010 DRAFT OF COMMON CORE STANDARDS INITIATIVE
Fractions
Students can and do:
a. Use fractions to describe quantities and parts of wholes.
b. Compare and order fractions with equal numerators or equal denominators, including in
contextual situations, using the fractions themselves, bar strip drawings, number line
representations, and area models.
c. Reason about fractions to establish equivalences between fractions with unlike
denominators 2, 3, 4 and 6 (e.g. 1/2 = 2/4, 4/6 = 2/3).
d. Add and subtract fractions with like denominators.
e. Solve word problems that involve adding, subtracting, ordering and comparing fractions.
f. Represent fractions of the form a/10 in decimal notation; compare and order to tenths in
decimal notation.
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Grade 3 GLCEs not related to a focal point
Key: Builds on previous grade(s) Related to topics within or beyond mathematics
Later grade at which topic relates to a focal point
Grade 2 Grade 3 Grade 4
Understand the meaning of Understand and use Understand and use number
notation and place value
multiplication and division number notation and place
Grades 3, 4, 5 value Add and subtract whole
numbers
Understand the concept of Add and subtract whole
area numbers Add and subtract fractions
Grade 3 Grade 5
Measure and use units for
Multiply fractions by whole
Read thermometers length, weight, temperature, numbers Grade 6
and time
Use coordinate systems Add and subtract decimal fractions
Use bar graphs Grade 6
Create, interpret, and solve
problems involving Multiply and divide decimal
fractions Grade 6
pictographs
Estimate
Measure using common tools
and appropriate units
Convert measurement units
Grade 5
Use perimeter and area formulas
Grade 5
Understand right angles Grade 5
Solve contextual problems about
surface area Grades 5, 6, 8
Understand perpendicular, parallel,
and intersecting lines
Identify basic geometric shapes
and their components, and solve
problems
Recognize symmetry and
transformations Similarity Gr. 7
Represent and solve problems for
given data
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Grade 3 GLCEs not related to a focal point
Approximately 70% - 80% of Tier 1 instruction should relate to the grade-level Focal Points
identified previously. No more than 20% - 30% of Tier 1 instruction should be devoted to the
following GLCEs, which are not related to a focal point.
Number and Operation
Understand and use number notation and place value
N.ME.03.01 Read and write numbers to 10.000 in both numerals and words, and
relate them to the quantities they represent, e.g., relate numeral or written
word to a display of dots or objects. [Ext-NC]
N.ME.03.02 Identify the place value of a digit in a number, e.g., in 3,241, 2 is in the
hundreds place. Recognize and use expanded notation for numbers
using place value through 9,999, e.g., 2,517 is 2000 + 500 +10+7; 4
hundreds and 2 ones is 402. [Ext-NC]
N.ME.03.03 Compare and order numbers up to 10,000. [Ext-NC]
Add and subtract whole numbers
N.FL.03.06 Add and subtract fluently two numbers through 999 with regrouping and
through 9,999 without regrouping. [Ext-NC]
N.FL.03.07 Estimate the sun and difference of two numbers with three digits (sums
up to 1000), and judge reasonableness of estimates. [Ext-NC]
N.FL.03.08 Use mental strategies to fluently add and subtract two-digit numbers.
[NASL]
Measurement
Measure and use units for length, weight, temperature, and time
M.UN.03.01 Know and use common units of measurements in length, weight,
temperature, and time. [Ext]
M.UN.03.02 Measure in mixed units within the same measurement system for length,
weight and time: feet and inches, meters and centimeters, kilograms and
grams, pounds and ounces, liters and milliters, hours and minutes,
minutes and seconds, years and months. [Ext]
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M.UN.03.03 Understand relationships between sizes of standard units, e.g., feet and
inches, meters and centimeters. [Ext]
M.UN.03.04 Know benchmark temperatures such as freezing (32ºF, 0ºC); boiling
(212ºF 100ºC); and compare temperatures to these, e.g., cooler, warmer.
[Ext-NC]
Data and Probability
Use bar graphs
D.RE.03.01 Read and interpret bar graphs, in both horizontal and vertical forms. [Ext-
NC]
D.RE.03.02 Read scales on the axes and identify the maximum, minimum, and range
of values in a bar graph. [Ext-NC]
D.RE.03.03 Solve problems using information in bar graphs, including comparison of
bar graphs. [Ext]
Key:
Core – expectation will be assessed with two items on the MEAP
Ext – extended core; expectation will be assessed with no more than one item.
NC – no calculator
NASL – not assessed at the state level; will not be tested on the MEAP
Finding Focus for Elementary and Middle School Mathematics Instruction – Grade 3 February 8, 2010
Huron Intermediate School District 39 of 45 printed 12/15/2011 at 6:31:18 AM
Finding Focus for Elementary and Middle School Mathematics Instruction – Grade 3 February 8, 2010
Huron Intermediate School District 40 of 45 printed 12/15/2011 at 6:31:18 AM
Suggested Third Grade Vocabulary
Taken from Huron County Mathematics Curriculum Framework
January 3, 2006
Number and Operations
addend* equation(s)* patterns
addition equivalence* place value
alternately* equivalent fractions* predict*
ascending* estimate product
associative property* even property of zero*
attributes* extend* quantity*
balance* factor quotient
base ten block* fair Share* regroup*
cent(s) fourths relationships*
characteristics* fraction remainder(s)
classify* fraction strip* rounding*
combinations* greater than (>) sequence*
commutative property* half/halves subtraction
compare* hundredths* sum
consecutive numbers* identify* symbols*
decimal fraction* less than (<) symmetry
decimal(s) mathematical sentence* tenths*
denominator mixed number(s)* trade*
descending* multiple values*
difference multiplication variable*
digit* number line whole
distributive property* number sentence whole numbers*
division numerator
dollar(s) odd
double* order
eighths ordinal numbers*
equal part
* instructional term on which students might not be assessed
Finding Focus for Elementary and Middle School Mathematics Instruction – Grade 3 February 8, 2010
Huron Intermediate School District 41 of 45 printed 12/15/2011 at 6:31:18 AM
Measurement
a.m. inch (in.) rectangle
analog clock kilogram (kg) ruler
area kilometer (km)* scales*
Celsius length second
cent(s) linear unit* square
centimeter (cm) liter (l) square centimeter
cubic unit* mass* square inch
cup measurement square unit
customary* meter (m) standard units*
day metric surface area*
decimeter* millimeter (mm)* temperature
degree minute thermometer
digital clock money time
dimension month volume*
distance non-standard units* week
dollar(s) ounces (oz.*) weight
Fahrenheit p.m. width
foot/feet (ft.) perimeter length yard (yd.)
gallon pint* year
gram (g) pounds (lbs.*)
hour quart*
* instructional term on which students might not be assessed
Finding Focus for Elementary and Middle School Mathematics Instruction – Grade 3 February 8, 2010
Huron Intermediate School District 42 of 45 printed 12/15/2011 at 6:31:18 AM
Geometry
angle open/closed same shape
attributes* triangle orientation* same size
base* parallel scale drawing*
circle parallel line segments side
cone parallel lines similar*
congruent* parallel planes* solid
coordinate* parallelogram sphere
corner perpendicular square
cube perpendicular line square two-dimensional
cylinder segments (2-D)
distance perpendicular lines surface
edge point symmetrical
equal sides polygon* tangram*
face polyhedron* tessellate*
flip position* tessellation*
fold prism three-dimensional (3-D
hexagon* pyramid front)
three-dimensional (3-D)
line quadrilateral*
tile two-dimensional (2-D)
line of symmetry rectangle
top
line segment rectangular prism
trapezoid
mirror image rhombus
turn
north, south, east, west right angles
vertex/vertices
N-S-E-W right triangle
octagon* rotate/rotation*
* instructional term on which students might not be assessed
Finding Focus for Elementary and Middle School Mathematics Instruction – Grade 3 February 8, 2010
Huron Intermediate School District 43 of 45 printed 12/15/2011 at 6:31:18 AM
Data and Probability
axis/axes impossible probable*
bar graph less likely range
bias* likely scale
category* maximum survey*
certain median* table
chance middle tally
chart minimum title
data more likely trend*
fair/not fair pictograph* trial*
graph picture graph vertical
grid population*
horizontal possible
hypothesis /hypotheses* predict /prediction*
* instructional term on which students might not be assessed
Finding Focus for Elementary and Middle School Mathematics Instruction – Grade 3 February 8, 2010
Huron Intermediate School District 44 of 45 printed 12/15/2011 at 6:31:18 AM