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Student Achievement Model
Finding Focus for
Mathematics Instruction
Grade 6
Huron Intermediate School District
March 8, 2010
Finding Focus for Mathematics Instruction – Grade 6
Huron Intermediate School District
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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.
references to research are cited throughout the document.
Finding Focus for Mathematics Instruction – Grade 6
Huron Intermediate School District
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Huron Intermediate School District
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Focusing on Mathematics at Grade 6
Grade
Grade
6 #1
Michigan Focal Point
Developing an
understanding of
operations on all rational
numbers
Related GLCE Topics






Grade
6 #2
Writing, interpreting and
using mathematical
expressions and
equations and solving
linear equations



Grade
6 #3
Describing threedimensional shapes and
analyzing their properties,
including volume and
surface area


Finding Focus for Mathematics Instruction – Grade 6
Huron Intermediate School District
Multiply and divide
fractions
Represent rational
numbers as
fractions or
decimals
Add and subtract
integers and
rational numbers
Find equivalent
ratios
Solve decimal,
percentage and
rational number
problems
Calculate rates
Use variables, write
expressions and
equations, and
combine like terms
Represent linear
functions using
tables, equations,
and graphs
Solve equations
Convert within
measurement
systems
Find volume and
surface area
Targeted Vocabulary





rational number
percentage
rate, ratio
scale up, scale down
integer, positive, negative






equation
expression
coefficient
variable / unknown /
constant
ordered pair
like terms


convert
volume, surface area
Formulas:
 volume of a rectangular
prism
 surface area of a
rectangular prism
 volume of a cube (special
case)
 surface area of a cube
(special case)
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National Math Panel Benchmarks for Grades 5, 6, and 7
Geometry and
Measurement
Fluency with Fractions and Fluency with
Decimals
Whole Numbers
By the end of Grade 5,
students should
By the end of Grade 6, students
should
By the end of Grade 7,
students should
be proficient with
comparing fractions and
decimals and common
percents, and with the
addition and subtraction of
fractions and decimals.
be proficient with multiplication
and division of fractions and
decimals.
be proficient with all
operations involving
positive and negative
fractions.
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).
be able to analyze the properties
of two-dimensional shapes and
solve problems involving
perimeter and area and analyze
the properties of threedimensional shapes and solve
problems involving surface area
and volume.
be proficient with
multiplication and division
of whole numbers.
be proficient with all operations
involving positive and negative
integers.
be able to solve
problems involving
percent, ratio, and rate
and extend this work
to proportionality.
be familiar with the
relationship between
similar triangles and
the concept of the
slope of a line.
Taken from The National Mathematics Advisory Panel Final Report, 2008, p.20
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Grade 6 Focal Point #1: Developing an understanding of operations on all rational numbers
Grade 5
Grade 6
Grade 7
Developing an understanding
of and fluency with division of
whole numbers (NCTM-5th)
Convert within measurement systems
M.UN.06.01
Recognize irrational numbers
N.MR.07.06
National Math Panel Benchmark:
By the end of Grade 5, students
should be proficient with the
multiplication and division of
whole numbers
Understand division of whole numbers
N.X.05.01 - .03
Find prime factorizations of whole
numbers
N.MR.05.07
Multiply and divide whole numbers
N.X.05,04 - .06
Multiply and divide by powers of ten
N.X.05.15 - .17
Find and interpret mean and mode for a
given set of data
D.X.05.03 - .04
Know, and convert among, measurement
units within a given system
M.X.05.01 – .04
Developing an understanding
of and fluency with addition
and subtraction of fractions
and decimals (NCTM-5th)
National Math Panel Benchmark:
By the end of Grade 5, students
should be proficient with
comparing fractions and
decimals and common percent,
and with the addition and
subtraction of fractions and
decimals
Add and subtract fractions using common
denominators
N.FL.05.14
Developing an understanding
of operations on all rational
numbers (NCTM-7th, Michigan
6th)
Developing an understanding
of and applying
proportionality, including
similarity (NCTM-7th)
National Math Panel Benchmark:
By the end of Grade 6, students
should be proficient with all
operations involving positive and
negative integers
National Math Panel Benchmark:
By the end of Grade 6, students
should be proficient multiplication
and division of fractions and
decimals
National Math Panel Benchmark:
By the end of Grade 6, students
should be proficient with all
operations involving positive and
negative integers
Add and subtract integers and rational
numbers
N.X.06.08 - .10
Multiply and divide fractions
N.X.06.01 - .04
Represent rational numbers as fractions
or decimals
N.X.06.05 - .07
Find equivalent ratios by scaling up or
down
N.ME.06.11
Solve decimal, percentage, and rational
number problems
N.X.06.12 - .15
Calculate rates
A.PA.06.01
By the end of Grade 7, students
should be able to solve problems
involving percent, ratio, and rate
and extend this work to
proportionality
National Math Panel Benchmark:
By the end of Grade 7, students
should be familiar with the
relationship between similar
triangles and the concept of the
slope of a line
National Math Panel Benchmark:
By the end of Grade 7, students
should be proficient with all
operations involving positive and
negative fractions
Solve problems involving derived
quantities such s density, velocity, and
weighted averages
N.MR.07.02
Understand and solve problems involving
rates, ratios, and proportions
N.X.07.03 - .05
Understand and apply directly
proportional relationships and relate to
linear relationships
A.PA.07.01 - .05
Understand and solve problems about
inversely proportional relationships
A.X.07.09 - .10
Understand the concept of similar
polygons, and solve related problems
G.X.07.03 - .06
Solve applied problems with fractions
involving addition, subtraction, equivalent
fractions, and rounding
N.X.05.18 - .21
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Huron Intermediate School District
National Math Panel Benchmark:
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Grade 6 Focal Point #1: Developing an understanding of operations on all rational numbers
BIG MATHEMATICAL IDEAS AND UNDERSTANDINGS
FRACTIONS, DECIMALS, AND INTEGERS
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 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 not a 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.

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.
o
Add fractions with common denominators:
2 3 5
 
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.
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.
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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 wholenumber 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:
o 1/4 + 2/4 = 3/4
o 3/4 = 6/8
o 2/2 = 3/3 = 92/92 = 1
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)

Ratios can be scaled (e.g., 3:5 is equivalent to 6:10). It is sometimes helpful to scale a ratio
up or down in order to solve a problem.

There are different ways to represent equivalent ratios, such as a table of values, a linear
graph, or a constant multiplier (y = kx).
Big Idea #4 (Comparison)
Numbers, expressions, and measures can be compared by their relative values.

A ratio is a multiplicative comparison of quantities; there are different types of comparisons
that can be represented as ratios. Since division is the inverse of multiplication, a ratio is
often written as a division statement.

There are multiplicative relationships within and between ratios.
o Multiplicative relationships within ratios:
o
Multiplicative relationships between ratios:
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
Ratios give the relative sizes of the quantities being compared, not necessarily the actual
sizes.

Rates are special types of ratios where unlike quantities are being compared, such as miles
per hour.

A percent is a special type of ratio where a part is compared to a whole and the whole is
100.

The ratio of two whole number quantities a and b (written a/b) is a multiplicative comparison
telling how much of one quantity there is for a given amount of the other, or how many times
as much one is than the other.
Big Idea #5 (Operation Meanings & Relationships)
The same number sentence (e.g. 12 – 4 = 8) 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 addition and subtraction of whole numbers are the same for
operations with fractions, decimals, and integers.

The real-world actions for multiplication and division of whole numbers are the same for
operations with fractions, decimals, and integers.
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.

Properties of numbers apply to certain operations but not others (e.g., the commutative
property applies to addition and multiplication but not subtraction and division.)

The sum of a number and zero is the number; the product of any non-zero number and one
is the number.

Three or more numbers can be grouped and added (or multiplied) in any order.
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.

Subtraction is the inverse of addition. Any addition problem has related subtraction
problems, and any subtraction problem has related addition problems.

Any subtraction calculation can be solved by adding up, and addition can be used to check
subtraction.
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
Multiplication and division are the inverse 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
Big Idea #11 (Proportionality)
If two quantities vary proportionally, that relationship can be represented as a linear
function.

A proportion is a statement that equates two ratios (a/b = c/d).

If the ratio between two quantities remains constant, the two quantities have a proportional
relationship.

All proportional relationships are linear; not all linear relationships are proportional. The graph of a
proportional relationship is linear and passes through the origin.

Corresponding parts of similar figures are proportional.
[ 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 6 Focal Point #1: Developing an understanding of operations on all rational numbers
INSTRUCTIONAL IMPLICATIONS
REPRESENTATIONS OF FRACTIONS
When developing understanding of fractions, concrete models are critical. Fractions can be
modeled in many ways, but the Rational Number Project (RNP) has found that the fraction circle
model is the most effective representation for building mental image for fractions.1 Blackline
masters can be found in the RNP materials
(http://www.cehd.umn.edu/rationalnumberproject/rnp2.html).
For a virtual representation, see “Fraction Model 1” at illuminations.nctm.org
(http://illuminations.nctm.org/ActivityDetail.aspx?ID=11).
In addition to concrete models, it is important that students have opportunities to translate
among pictures, contexts, verbal representations, and symbols. By drawing a picture to
represent a fraction, or writing an equation to accompany a story problem, students deepen
their understanding of fractions.
IDENITIFICATION OF THE WHOLE
As part of developing deep understanding of fractions, students should experience working with
different wholes. Consider these examples:
If this $10 bill represents one, show me one-tenth. Show me one-hundredth.
Show me 10. What if the $1 bill represents one? What if a $5 bill represents
one?
Using pattern blocks, ask, If the yellow hexagon represents one, what is onehalf? (the red trapezoid) What is one-sixth? (the green triangle).” Then change
the whole and ask, If the red trapezoid is one, how much is the green triangle?”
(one-third)
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 create strong
mental images showing that the more equal parts an object is divided into, the smaller each part
is. Students should 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 makes 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:
Cramer, Kathleen; Wyberg, Terry; and Leavitt, Seth (2008). “The Role of Representations
in Fraction Addition and Subtraction.” Mathematics Teaching in the Middle School (13,8)
pp. 490-496.
1
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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.
ADDITION AND SUBTRACTION WITH FRACTIONS
Before performing operations with fractions, students must first develop a strong mental model
of fractions and use a variety of reasoning strategies to compare fractions. The ability to
compare fractions to benchmarks supports students in making reasonable estimates to fraction
addition and subtraction.
In addition to the need for visualization and estimation, the Rational Number Project (RNP) has
drawn several conclusions:2
Students need to experience acting out addition and subtraction concretely with
an appropriate model before operating with symbols. (p. 494)
Researchers found that most students needed extended periods of time with the fraction circles
before formalizing the algorithms for addition and subtraction. Students should discover for
themselves the need to exchange pieces for a different color before making the connection to
common denominators in a symbolic representation.
Making connections between concrete actions and symbols is an important part
of understanding. Students should be encouraged to find their own way of
recording with symbols. (p.494)
The transition from concrete model to symbolic representation is not automatic. As students act
out addition and subtraction with fraction models, they use pictures and written explanations to
describe what they did. By developing their own record keeping systems, students develop a
deeper understanding of the relationship between actions on fraction manipulatives and
symbolic representation of fraction operations.
Students need easy recall of their multiplication and division facts. (p. 495)
While concrete models are necessary for building understanding, students must eventually
compute symbolically without the model. Most students understand what to do with the
symbols, but students with poor fact recall struggle to find the equivalent fractions necessary for
addition and subtraction.
Connecting the procedure to a new representation may be an effective strategy
to reinforce the procedure. (p. 496)
Just as whole number addition and subtraction can be represented with equal-sized “jumps” on
a number line, addition and subtraction of fractions with like denominators can also be
represented on a number line. However, this is best used as reinforcement after the idea of
common denominators has already been developed with fraction circles.
Cramer, Kathleen; Wyberg, Terry; and Leavitt, Seth (2008). “The Role of Representations
in Fraction Addition and Subtraction.” Mathematics Teaching in the Middle School (13,8)
pp. 490-496.
2
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Understanding – and finding – common denominators is critical to students’ ability to add and
subtract fractions. However, it is not always necessary to find the least common denominator.
In fact, Michigan’s GLCEs for Grade 5 specify that students should be able to “add and subtract
fractions with unlike denominators . . . using the common denominator that is the product of the
denominators of the 2 fractions” (N.FL.05.14).
MULTIPLICATION AND DIVISION WITH FRACTIONS AND DECIMALS
The models of multiplication and division that are used with whole numbers also apply to
fractions and decimals.
Two models of multiplication are repeated addition and the area model. When multiplying two
non-integer quantities, it may be easiest to use an area model. Finding the area of a rectangle
that is 1.3 x 1.4 may be easier to model for some students than counting out 1.3 groups of 1.4.
For more information on this strategy, see the “Rectangular Multiplication” Power Point from the
Michigan Mathematics Improvement Project (MMPI) (www.michiganmathematics.org, Chapters
4,5, and 6).
The two models of division that are described in MMPI are partitive division (fair shares) and
quota or measurement division. When dividing with fractions or decimals, students may find the
quota model easier to begin with: how many groups of 1.4 are in 4.9? This question can be
answered with manipulatives or pictures and can be applied to fractions as well.
Division can also be modeled as the missing factor in a multiplication problem using an area
model. To use the rectangular multiplication model to solve the division problem 4.9  1.4, first
gather Base 10 blocks to represent 4.9 (use the “100 square” as one, the “ten rod” as 0.1, and
the “one cube” as 0.01). Using the “Decimal Multiplication Mat” from MMPI
(www.michiganmathematics.org, Chapter 5), build a rectangle with a height of 1.4 and an area
of 4.9. Exchanges will need to be made. The quotient is the missing dimension of the
rectangle: if 1.4 x 3.5 = 4.9, then 4.9  1.4 = 3.5. This activity is time-consuming and may lead
to some complex discussion, but it can also deepen students’ understanding of division.
REPRESENTATION OF INTEGERS AND ADDITION AND SUBTRACTION WITH INTEGERS
Perhaps the most straightforward way to introduce integers is through the concept of “opposite.”
On a number line, you can move forward (right or up) or backward (left or down). When you
reach zero, you keep going, but numbers get a new name: -1, -2, -3, etc. One visual model
that might make sense to students is a vertical number line in the form of a ladder that goes
infinitely above ground (positive) and also down into a whole in the ground (negative). (For a
literature connection, try Papa, Get the Moon for Me by Eric Carle.)
Once students understand the concept of an integer, they should know that the four operations
(addition, subtraction, multiplication, division) work the same on integers as on other rational
numbers.
Addition and subtraction can be modeled on a number line as described in the Michigan
Mathematics Improvement Project (MMPI) (www.michiganmathematics.org, Chapter 3, pp. 5, 9,
10). For example, to model 4 + (-8), use two number lines. One number line is mounted on a
desk or wall; the other number line is free. Find 4 on the fixed number line. Place the second
number line above so that the 0 is above the 4. The -8 of the free number line matches to -4 on
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the fixed number line, showing that if you start at 4 and add 8 units in the negative direction, you
will end up at -4 (4 + -8 = -4). To model subtraction such as 4 – (-8), turn the top number line
over, because subtraction is the opposite of addition.
Another powerful model for integer operations are integer chips, where positive and negative
chips are represented by different colors. Addition and subtraction are modeled by putting on
and taking away chips. “Zero pairs” can also be added or removed as needed. The National
Library of Virtual Manipulatives has two good virtual models of number chips: Color Chips –
Addition and Color Chips – Subtraction (www.nlvm.usu.edu  Number and Operations,
Grades 3-5).
MULTIPLICATION AND DIVISION WITH INTEGERS
While integer multiplication can be modeled using an area model, it requires the use of four
quadrants and can be confusing. A repeated addition model using number lines or integer chips
may be easier for students to grasp. Consider these examples:
3  -5 means “three groups of negative 5”
3  5 means “three groups of (positive) 5”
+
+
+
+
+
+
+
+
+
+
+
+
-
+
+
-
-
+
-
-
-
-
-
-
-
-
-
-3  5 means “the opposite of three groups of positive 5”
-
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
-3  -5 means “the opposite of three groups of negative 5”
-
-
-
-
-
-
-
-
-
-
-
-
-
Division can also be modeled with integer chips using a quota model: How many groups of -5
can be made from 15? Since I can form 3 groups of +5, I would need the opposite of each
group to have -5, so 15  -5 = -3. This can be checked with multiplication. It is essential that
students have a variety of experiences working with number chips, number lines, and other
models before learning shortcuts such as “two negatives makes a positive.” Isolated “tricks”
such as this lead to misunderstandings and errors.
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Grade 6 Focal Point #1: Developing an understanding of operations on all rational numbers
RELATED GLCES WITH CORE AND EXTENDED DESIGNATIONS
Number and Operations
Multiply and divide fractions
N.MR.06.01 Understand division of fractions as the inverse of multiplication e.g.,
if 4/5 ÷ 2/3 =□ , then 2/3• □ = 4/5, so □ = 4/5 • 3/2 = 12/10. [Core-NC]
N.FL.06.02 Given an applied situation involving dividing fractions, write a mathematical
statement to represent the situation. [Core-NC]
N.MR.06.03 Solve for the unknown in equations such as: ¼ ÷ □ =1, ¾ ÷ □ = ¼, and
½=1•□. [Core-NC]
N.FL.06.04 Multiply and divide any two fractions, including mixed numbers, fluently.
[Core-NC]
Represent rational numbers as fractions, or decimals
N.ME.L06.05 Order rational numbers and place them on the number line. [Ext]
N.ME.06.06 Represent rational numbers as fractions or terminating decimals when
possible, and translate between these representations. [Ext]
N.ME.06.07 Understand that a fraction or a negative fraction is a quotient of two
integers, e.g., -8/3 is -8 divided by 3. [Ext-NC]
Add and subtract integers and rational numbers
N.MR.06.08 Understand integer subtraction as the inverse of integer addition.
Understand integer division as the inverse of integer multiplication. [Ext-NC]
N.FL.06.09 Add and multiply integers between –10 and 10; subtract and divide integers
using the related facts. Use the number line and chip models for addition and
subtraction. [Core-NC]
N.FL.06.10 Add, subtract, multiply and divide positive rational numbers fluently. [CoreNC]
Find equivalent ratios
N.ME.06.11 Find equivalent ratios by scaling up or scaling down. [Core]
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Solve decimal, percentage and rational number problems
N.FL.06.12 Calculate part of a number given the percentage and the number. [Core-NC]
N.MR.06.13 Solve contextual problems involving percentages such as sales taxes and
tips. [Ext]
N.FL.06.14 For applied situations, estimate the answers to calculations involving
operations with rational numbers. [Core]
N.FL.06.15 Solve applied problems that use the four operations with appropriate decimal
numbers. [Core]
Algebra
Calculate rates
A.PA.06.01 Solve applied problems involving rates, including speed, e.g., if a car is
going 50 mph, how far will it go in 3 ½ hours? [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
Ratios and Proportional Relationships
Students understand that:
1. Multiplicative comparisons can be extended from whole numbers to fractions and
decimals. When the ratio q/m is formed, or when q is r times as much as m, the numbers
q, r and m can be fractions or decimals.
2. p% of a quantity means p/100 times as much as the quantity. The number p can be a
fraction or decimal, as in 3.75%.
3. A unit rate is the multiplicative factor relating the two quantities in a ratio. Two quantities
q and m can be compared by q = r × m, where the unit rate r tells how much q per m.
4. Given two quantities in a ratio (e.g. distance and time), finding the unit rate produces a
new type of quantity (e.g. speed).
Students can and do:
a. Solve for an unknown quantity in a problem involving two equal ratios.
b. Find a percentage of a quantity; solve problems involving finding the whole given a part
and the percentage.
c. Solve unit rate problems including unit pricing and constant speed. (See table.)
D=s×T
A car driving at a speed
of 30 miles per hour for
6 hours travels a
distance of 180 miles.
D÷T=s
If a car drives 180 miles
for 6 hours at a constant
speed, that speed is 30
miles per hour.
D÷s=T
When a car drives 180
miles at a speed of 30
miles per hour, the trip
takes 6 hours.
d. Represent unit rate problems on a coordinate plane where each axis represents one of
the two quantities involved, and find unit rates from a graph. Explain what a point (x, y)
means in terms of the situation, with special attention to the points (0, 0) and (1, r) where
r is the unit rate.
The Number System
Students understand that:
1. The Properties of Arithmetic govern operations on all numbers.
2. Division of fractions follows the “invert and multiply” rule because multiplication and
division are inverse operations. For example, (2/3) ÷ (5/7) = 14/15 because (14/15) ×
(5/7) = 2/3.
3. Every nonzero fraction has a unique multiplicative inverse, namely its reciprocal. Division
can be defined as “multiplying by the multiplicative inverse.” Then (2/3) ÷ (5/7) = 14/15
because the division symbol indicates multiplication by the multiplicative inverse.
4. A two-sided number line can be created by reflecting the fractions across zero. Numbers
located to the left of zero on the number line are called negative numbers and are
labeled with a negative sign.
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5. Two different numbers, such as 7 and –7, that are equal distant from zero are said to be
opposites of one another. The opposite of 7 is –7 and the opposite of –7 is 7. The
opposite of the opposite of a number is the number itself. The opposite of 0 is 0. The
operation of attaching a negative sign to a number can be interpreted as reflecting the
number across zero on the number line.
6. The absolute value of a number is its distance from zero on the number line. For any
positive number q, there are two numbers whose absolute value is q, namely q and –q.
7. The absolute value of a signed quantity (e.g. account balance, elevation) tells the size of
the quantity irrespective of its sense (debit or credit; above or below sea level).
8. Comparison of numbers can be extended to the full number system. The statement p > q
means that p is located to the right of q on the number line, while p < q means that p is
located to the left of q on the number line. The statement p > q does not mean |p| > |q|.
Students can and do:
a. Divide fractions, and divide finite decimals by expressing them as fractions.
b. Solve problems requiring arithmetic with fractions presented in various forms, converting
between forms as appropriate and estimating to check reasonableness of answers.
c. Find and position rational numbers on the number line.
d. Use rational numbers to describe quantities such as elevation, temperature, account
balance and so on. Compare these quantities using > and < symbols and also in terms
of absolute value.
e. Graph points and identify coordinates of points on the Cartesian coordinate plane in all
four quadrants.
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Grade 6 Focal Point #2: Writing, interpreting, and using mathematical expressions and
equations, and solving linear equations
Grade 5
Grade 6
Grade 7
Writing, interpreting, and using
mathematical expressions and
equations (NCTM-6th) and
solving linear equations(from
NCTM-7th)
Analyzing and representing
linear functions and solving
linear equations and systems
of linear equations (NCTM-8th,
Michigan 7th )
Represent linear functions using tables,
equations, and graphs
A.X.06.08 - .10
Use variables, write expressions and
equations, and combine like terms
A.X.06.03 - .07
Solve equations
A.X.06.11 - .14
Understand and represent linear
functions
A.X.07.06 - .08
Apply basic properties of real numbers in
algebraic contexts
A.PA.07.11
Combine algebraic expressions and solve
equations
A.X.07.12 - .13
Recognize irrational numbers
N.MR.07.06
Compute with rational numbers
N.X.07.07 - .09
Represent and interpret data
D.X.07.01 - .02
National Math Panel Benchmark:
By the end of Grade 7, students
should be familiar with the
relationship between similar
triangles and the concept of the
slope of a line
Note:
Solving simultaneous linear
equations and inequalities in two
variables are Grade 8 GLCEs (A.
X. 8.11 - .13) but are not related
to a focal point at Grade 8
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 6 Focal Point #2: Writing, interpreting, and using mathematical expressions and
equations, and solving linear equations
BIG MATHEMATICAL IDEAS AND UNDERSTANDINGS:
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.
 Equivalent expressions can be expressed in an infinite number of ways:
o 5=3+2
o 3+2=8–3
o 8 – 3 = 15  3
o 15  3 = 3(1 + 1 + 1 + 2)  (4 – 1)
o 3x = 2x + x
o 2x + x = 5x – 2x
o 5x – 2x = 6x  2
Big Idea #5 (Operation Meanings & Relationships)
The same number sentence (e.g. 12-4 = 8) 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.

When simplifying expressions, only quantities with the same unit (like terms) may be added
or subtracted.
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.

Two quantities equal to the same third quantity are equal to each other.
Big Idea #10 (Variable)
Mathematical situations and structures can be translated and represented abstractly
using variables, expressions, and equations.

Numerical variables represent numbers and follow the same rules as numbers. They are
used to represent generalized properties, unknowns in equations, and relationships between
quantities.

An expression is a combination of numerals, variables, and operations:
o 5+3
o x–4
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
Some mathematical phrases can be represented as algebraic expressions (e.g., “five less
than a number” can be written as n – 5).

Some problem situations can be represented as algebraic expressions (e.g. “Susan is twice
as tall as Tom;” If T = Tom’s height, then 2T = Susan’s height).
Big Idea #13 (Equations & Inequalities)
Rules of arithmetic and algebra can be used together with notions of equivalence to
transform equations and inequalities so solutions can be found.

An equation equates two expressions (an equation is a true statement and includes an
equal sign):
o 5+3=4x2
o x–4=8

For a given algebraic equation, there may be zero, one, or more values for a variable that
make the equation true. These values are solutions to the equation.

If two equations have the same solution, the graphs of the equations will intersect at that
corresponding point.

If the same real number is added or subtracted to both sides of an equation, equality is
maintained.

If both sides of an equation are multiplied or divided by the same real number (not dividing
by 0), equality is maintained.

In order to maintain equality when multiplying both sides of an equation by the same real
number, the entire expression on each side of the equation must be multiplied by that
number (the distributive property applies).

Adding or subtracting 0 to one side of an equation does not change the equation. “0” can
take any form (i.e., + 5 – 2 – 3 = 0).

Multiplying or dividing one side of an equation by 1 does not change the equation. “1” can
take any form (i.e., x 4/4 = x1).
[ 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 6 Focal Point #2: Writing, interpreting, and using mathematical expressions and
equations, and solving linear equations
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, recording 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 leading to deeper understanding.
Concrete-Representational-Abstract and Linear Equations
A strong visual model for solving linear equations is a balance beam based on the premise that
an equation in balance must remain in balance. There are several manipulatives that represent
this model:



Hands-On Equations uses “pawns” and number cubes to represent linear equations with
integer coefficients and variables. This system of manipulatives is designed for use at
Grades 4-6.
Algebra Tiles, Algebra Lab Gear, and AlgeBlocks all use manipulatives to represent
variables and constants and can be used to solve linear equations. However, these models
are more often used to model the factoring of polynomials.
“Algebra Balance Scales – Negatives” is a free virtual balance beam from the National
Library of Virtual Manipulatives (www.nlvm.usus.edu  Algebra Grades 6-8).
As with any manipulative, it is important that students have a variety of experiences with the
model in order to build a strong mental image. It is equally critical that the movements that
students make with the model should also be represented with pictures and symbols in order to
build a connection to the abstract symbolism. The goal is for students to be able to solve
equations symbolically, where the picture or manipulative becomes a mental image to be drawn
on when needed.
Certain critical skills such as combining like terms and solving linear equations should be
practiced to fluency. As with any procedural skill, there are critical factors to be considered
when building fluency3:
3
Sarama, Julie and Clements, Douglas H. (2009) Early Childhood Mathematics and
Education Research. New York, NY: Routledge. pp. 139-140.
Marzano, Robert J., Pickering, Debra J., and Pollock, Jane E. (2001) Classroom
Instruction that Works. Alexandria, VA: Association for Curriculum and Development. pp.
66-69.
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1. Focus on essential core skills only. Not every skill needs to be practiced for both speed
and accuracy.
2. Develop concepts and strategies first. Before becoming fast at solving equations,
students must first understand what it means to maintain a balanced equation, why that
is important, and what it looks like symbolically. Students must have a variety of
strategies based in mathematical reasoning that allow them to solve a problem without
resorting to an inefficient method or rote procedure.
3. Provide distributed practice. Students should practice small sets of problems spread out
over time. If practice is timed, each session should provide enough time to discourage
wild guessing but not enough time to resort to inefficient strategies. Practice sessions
should be challenging but doable.
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Grade 6 Focal Point #2: Writing, interpreting, and using mathematical expressions and
equations, and solving linear equations
RELATED GLCES WITH CORE AND EXTENDED DESIGNATIONS
Algebra
Use variables, write expressions and equations, and combine like terms
A.FO.06.03 Use letters, with units, to represent quantities in a variety of contexts,
e.g., y lbs., k minutes, x cookies. [Core-NC]
A.FO.06.04 Distinguish between an algebraic expression and an equation.
[Core-NC]
A.FO.06.05 Use standard conventions for writing algebraic expression e.g., 2x+1
means “two times x, plus 1” and 2(x+1) means “two times the quantity (x+1)”.
[Ext-NC]
Represent linear functions using tables, equations, and graphs
A.RP.06.08 Understand that relationships between quantities can be suggested
by graphs and tables. [Ext]
A.PA.06.09 Solve problems involving linear functions whose input values are
integers; write the equation; graph the resulting ordered pairs of integers, e.g.,
given c chairs, the “leg function” is 4c; if you have 5 chairs, how many legs?: if
you have 12 legs how many chairs? [Ext]
A.RP.06.10 Represent simple relationships between quantities using verbal
descriptions, formulas or equations, tables, and graphs, e.g., perimeter- side
relationship for a square, distance-time graphs, and conversions such as feet to
inches. [Ext]
Solve equations
A.FO.06.11 Relate simple linear equations with integer coefficients, e.g., 3 x = 8
or
x +5 = 10, to particular contexts and solve. [Core-NC]
A.FO.06.12 Understand that adding or subtracting the same number to both
sides of an equation creates a new equation that has the same solution. [CoreNC]
A.FO.06.13 Understand that multiplying or dividing both sides of an equation by
the same non-zero number creates a new equation that has the same solutions.
[Core-NC]
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A.FO.06.14 Solve equations of the form ax + b = c, e.g., 3x + 8 =15, by hand for
positive integer coefficients less than 20, using calculators otherwise, and
interpret the results. [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 THE COMMON CORE STANDARDS
Expressions and Equations
Students understand that:
1. A number that is the result of a sequence of operations with other numbers can be
expressed in different ways using conventions about order of operations and
parentheses, rules for working with fractions, and the Properties of Arithmetic. All such
expressions are equivalent.
2. A letter is used to stand for a number in an expression in cases where one doesn't know
what the number is, or where, for the purpose at hand, it can be any number in the
domain of interest. Such a letter is called a variable.
3. An equation is a statement that two expressions are equal, and a solution to an equation
is a value of the variable (or a set of values for each variable if there is more than one
variable) that makes the equation true.
Students can and do:
a. Represent an unknown number using a letter in simple expressions such as y + 2, y – 3,
6 + y, 5 – y, 3y, y/2, and (3±y)/5.
b. Interpret 3y as y + y + y or 3 × y, y/2 as y ÷ 2 or 1/2 × y, (3±y)/5 as (3 ± y) ÷ 5 or 1/5 × (3
± y).
c. Evaluate simple expressions when values for the variables in them are specified
(exclude expressions with a variable in denominator).
d. Choose variables to represent quantities in a word problem and construct simple
equations to solve the problem by reasoning about the quantities.
e. Solve equations of the form x + p = q (for p < q) and px = q where p and q are fractions.
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Grade 6 Focal Point #3: Describing three-dimensional shapes and analyzing their properties,
including volume and surface area
Grade 5
Grade 6
Grade 7
Analyzing properties of twodimensional shapes, including
angles (includes NCTM-3rd,
Michigan 5th)
Describing three-dimensional
shapes and analyzing their
properties, including volume
and surface area (NCTM – 5th,
Michigan 6th)
National Math Panel Benchmark:
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)
National Math Panel Benchmark:
Find areas of geometric shapes using
formulas
M.X.05.05 - .07
By the end of Grade 6, students
should be able to analyze the
properties of two-dimensional
shapes and solve problems
involving perimeter and area,
and analyze the properties of
three-dimensional shapes and
solve problems involving surface
area and volume
Know the meaning of angles, and solve
problems
G.X.05.01 - .06
Find volume and surface area
M.X.06.02 - .03
Solve problems about geometric shapes
(find unknown angles and sides using
properties of shapes)
G.GS.05.07
Convert within measurement systems
M.UN.06.01
By the end of Grade 7, students
should be able to solve problems
involving percent, ratio, and rate
and extend this work to
proportionality
Understand and solve problems involving
rates, ratios, and proportions
N.X.07.03 - .05
Understand the concept of similar
polygons, and solve related problems
G.X.07.03 - .06
National Math Panel Benchmark:
By the end of Grade 7, students
should be familiar with the
relationship between similar
triangles and the concept of the
slope of a line
Know, and convert among, measurement
units within a given system
M.X.05.01 – .04
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 6 Focal Point #3: Describing three-dimensional shapes and analyzing their properties,
including volume and surface area
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 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 threedimensional 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.

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.
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
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 6 Focal Point #3: Describing three-dimensional shapes and analyzing their properties,
including volume and surface area
INSTRUCTIONAL IMPLICATIONS
VOLUME, SURFACE AREA, AND MEASUREMENT CONVERSIONS
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 Surface Area, Volume, and Measurement Conversions
In Grade 3, students covered regions with tiles or squares to determine area. In Grade 5,
students used formulas to calculate the areas of triangles and parallelograms. Students may
also have explored the concept of volume although it is not a focal point in Grade 5. In Grade 6,
students build on previous knowledge to compute the volume and surface area of cubes and
rectangular prisms given the lengths of their sides.
Before students can apply formulas, they must have many concrete experiences to understand
what is surface area means. Cutting apart and opening up cereal boxes, drawing the patterns
(nets), and cutting and folding paper to build new solids, for example, can create strong mental
images. When students can see the faces of a rectangular prism as individual rectangles, they
can find surface area even before being introduced to the formula. The bridge between
concrete objects and formulas is pictures with “notes”. Students sketch a solid and find the area
of each face, using prior knowledge. As students take the time to work slowly through a few indepth examples, they deepen their understanding of what surface area means and how to find it
in a variety of situations. When the surface area formula is introduced, it becomes a “shortcut”
for a concept that is thoroughly understood.
A similar strategy can be applied to an understanding of volume. Just as area is first explored
by covering regions with square tiles, students’ first experiences with volume should include
filling shapes with cubes. By looking at the structure of cubes, students build connections and
make and test hypotheses. For example, students might predict how many cubes it will take to
build a rectangular prism that is 3 cubes wide, 4 cubes deep, and 5 cubes tall. After making
their predictions, they would test the hypothesis by building the solid. Pictures and written
explanations help build connections to the formula. For example, a student might write, “Since
each layer is 3 cubes wide and 4 cubes deep, there are 12 cubes in each layer. Since there are
5 layers with 12 cubes in each, there must be 60 cubes all together.” When students are able to
express this type of reasoning, they can begin to move away from concrete models and rely
more and more on sketches, accompanied by calculations and written explanations. Since
each problem is explored in depth, students may only do a few problems in order to build
understanding. Eventually, the formula is introduced, and students can begin to practice
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computing volume from written dimensions. Because of the prior experience with concrete
objects, students who still need to draw pictures will know how to do so.
Concrete experiences and pictorial representations can also build the understanding needed to
perform measurement conversions, especially where square units are concerned. For example,
if a student is asked, “How many square inches are in 3 square feet,” he could draw a sketch of
just one square foot to determine that there are 144 square inches in 1 square foot. From this,
he would determine that there are 432 square inches in a square foot – not 36! When a student
can form a picture in his mind, he will remember the concept for other situations.
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Grade 6 Focal Point #3: Describing three-dimensional shapes and analyzing their properties,
including volume and surface area
RELATED GLCES WITH CORE AND EXTENDED DESIGNATIONS
Measurement
Convert within measurement systems
M.UN.06.01 Convert between basic units for measurement within a single measurement
system, e.g., square inches to square feet. [Ext]
Find volume and surface area
M.PS.06.02 Draw patterns (of faces) for a cube and rectangular prism that, when out,
will cover the solid exactly (nets). [Core-NC]
M.TE.06.03 Compute the volume and surface area of cubes and rectangular prisms
given the lengths of their sides, using formulas. [Core-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
Geometry
Students understand that:
1. Triangles and parallelograms can be dissected and reassembled into rectangles with the
same area; this leads to a formula for area in terms of base and height.
2. Polygons can be dissected into triangles in order to find their area.
Students can and do:
a. Find the area of right triangles, other triangles, special quadrilaterals, and polygons (by
dissection into triangles and other shapes).
b. Find surface area of cubes, prisms and pyramids (include the use of nets to represent
these figures).
c. Solve problems involving area, volume and surface area of objects.
d. Examine the relationship between volume and surface area. Exhibit rectangular prisms
with the same surface area and different volume, and with the same volume and
different surface area.
e. Use exponents and symbols for square roots and cube roots to express the area of a
square and volume of a cube in terms of the side length, and to express the side length
in terms of the area or volume.
Finding Focus for Mathematics Instruction – Grade 6
Huron Intermediate School District
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Grade 6 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 5
Grade 6
Grade 7
Understand meaning of decimal
fractions and percentages
N.X.05.08 - .09
Use exponents
N.ME.06.16
Draw and construct geometric objects
G.X.07.01 - .02
Volume and Surface Area Grade 8
Understand fractions as division
statements; find equivalent fractions
N.X.05.10 - .11
Multiply an divide fractions
N.X.05.12 - .13
Grade 6
Express, interpret, and use ratios; find
equivalences
N.X.05.22 - .23
Grade 6
Understand the concept of volume
M.X.05.08 - .10
Grade 6
Construct and interpret line graphs
D.X.05.01 - .02
Understand rational numbers and their
location on the number line
N.X.06.17 - .20
Compute statistics about data sets
D.X.07.03 - .04
Grade 8
Understand the coordinate plane
A.RP.06.02
Linear Functions Grade 7
Understand and apply basic properties
of lines, angles, and triangles
G.GS.06.01
Understand the concept of congruence
and basic transformations
G.X.06.02 - .04
Similarity Grade 7
Construct geometric shapes
G.SR.06.05
Volume and Surface Area Grade 8
Understand the concept of probability and
solve problems
D.X.06.01 - .02
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 Mathematics Instruction – Grade 6
Huron Intermediate School District
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Grade 6 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 Operations
Use exponents
N.ME.06.16 Understand and use integer exponents, excluding powers of negative
bases; express numbers in scientific notation. [Ext]
Understand rational numbers and their location on the number line
N.ME.06.17 Locate negative rational numbers (including integers) on the number line;
know that numbers and their negatives add to 0, and are on opposite sides and at equal
distance from 0 on a number line. [Ext-NC]
N.ME.06.18 Understand that rational numbers are quotients of integers (non-zero
denominators), e.g., a rational number is either a fraction or negative fraction. [Ext-NC]
N.ME.06.19 Understand that 0 is an integer that is neither negative nor positive. [ExtNC]
N.ME.06.20 Know that the absolute value of a number is the value of the number
ignoring the sign; or is the distance of the number from 0. [Ext-NC]
Algebra
Understand the coordinate plane
A.RP.06.02 Plot ordered pairs of integers and use ordered pairs of integers to identify
points in all four quadrants of the coordinate plane. [Ext-NC]
Geometry
Understand and apply basic properties
G.GS.06.01 Understand and apply basic properties of lines, angles, and triangles,
including
 Triangle inequality
 Relationships of vertical angles, complementary angles, supplementary
angles
Finding Focus for Mathematics Instruction – Grade 6
Huron Intermediate School District
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Congruence of corresponding and alternate interior angles when parallel
lines are cut by a transversal, and that such congruencies imply parallel
lines
Locate interior and exterior angles of any triangle and use the property
that an exterior angle of a triangle is equal to the sum of the remote
(opposite) interior angles
Know that the sum of the exterior angles of a convex polygon is 360º
[Ext-NC]
Understand the concept of congruence and basic transformations
G.GS.06.02 Understand that for polygons, congruence means corresponding sides and
angles have equal measures. [Ext-NC]
G.TR.06.03 Understand the basic rigid motions in the plane (reflections, rotations,
translations), relate these to congruence, and apply them to solve problems. [Ext-NC]
G.TR.06.04 Understand and use simple compositions of basic rigid transformations,
e.g., a translation followed by a reflection. [Ext-NC]
Construct geometric shapes
G.SR.06.05 Use paper folding to perform basic geometric constructions of perpendicular
lines, midpoints of line segments and angle bisectors; justify informally. [NASL]
Data and Probability
Understand the concept of probability and solve problems
D.PR.06.01 Express probabilities as fractions, decimals, or percentages between 0 and
1; know that 0 probability means an event will not occur and that probability 1 means an
event will occur. [Ext]
D.PR.06.02 Compute probabilities of events from simple experiments with equally likely
outcomes, e.g., tossing dice, flipping coins, spinning spinners, by listing all possibilities
and finding the fraction that meets given conditions. [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 Mathematics Instruction – Grade 6
Huron Intermediate School District
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FROM THE 1-13-2010 DRAFT OF THE COMMON CORE STANDARDS
Statistics
Students understand that:
1. The mean is a measure of center in the sense that it is the balance point; the mean is
the value each data point would take on if the total value of all the data points were
redistributed fairly.
2. When the mean and median of a data set differ substantially, both measures should be
provided, and the difference explained in terms of the data values.
Students can and do:
a. Collect data to answer a predefined question about a measurement quantity.
Make a dot plot to display the data, and describe the data using measures of
center and measures of variation.
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Huron Intermediate School District
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Suggested Sixth Grade Vocabulary
Taken from Huron County Mathematics Curriculum Framework
January 3, 2006
Number and Operations
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absolute value
additive identity*
ascending
associative
associative property
base
brackets*
calculate
commutative
commutative property
compare
composite*
consecutive integers*
cross multiplication
cross-multiply
cube numbers*
descending
difference
discount
distributive
distributive property
divisibility test
divisible
division
division key*
division symbols
equivalent ratios
estimate*
evaluate*
exponential notation*
exponents
factor tree*
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fraction
identity of 0
identity of 1
integer
integers, fractions
(positive & negative),
decimals (positive)
inverse operations
LCD (least common
denominator)
LCM (least common
multiple)*
mixed fractions
mixed numbers
multiple
multiplication
multiplication symbols
multiplication symbols
x* [ ]( )
multiplicative identity*
negative
negative integers*
number line
operation
order
order of operation*
percent
percentage
perfect square*
positive
positive integers*
positive/negative
fractions*
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powers
powers of 10*
prime factorization*
prime*
quotient
rate
ratio
rational number
rational number line
rational numbers
reciprocal
reduce*
repeating decimals*
roots*
sales tax
scale factor*
scaling up / down
scientific notation
square numbers
square roots*
standard notation*
sum
terminating decimals
tips
tree diagram*
unit
unit rate*
unknown
unlike denominators
values*
word problems*
* Instructional term on which student might not be assessed
Finding Focus for Mathematics Instruction – Grade 6
Huron Intermediate School District
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Algebra
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algebraic equation
algebraic expression
charts
coefficient
constant*
coordinate plane
coordinates
direct*
distance
equation
equivalent equations*
evaluate
expression
formula
function f(x)
geometric shapes*
graph
image*
inequality*
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integer
intersecting*
inverse operation
inverse*
like terms
linear
linear equation
linear function
mph
non-linear*
ordered pair(s)
origin
pattern
plane
point
prediction
predictions
proportional*
quadrant
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quadrants
quantity
rates
ratio*
relationship
simplify
speed
substitute*
table
tables
term*
values*
variable
variable equation*
word problems*
x and y coordinates
x-axis
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right angle
side
solid
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sq. inches, sq. feet, sq.
yards
surface area
surface area
triangles – equilateral,
scalene,
isosceles, right
triangular prism
U.S. customary
vertex* / vertices*
volume
Measurement
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area
base (area of base
face of a prism
and length
of base side of
a triangle)
centi- (0.01)
circumference*
convert / conversion
corner
cube
deci- (0.10)
deka- (10)
face
formula
hecto- (100)
height (of a triangle
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and of a
prism)
inch, foot, yard, mile
kilo- (1,000)
length
mass (weight)
metric
mg, g, kg
milli- (0.001)
ml, l, kl
mm, cm, dm, m, km
net
ounce, cup, pint,
quart, gallon
perimeter
rectangular prism
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* Instructional term on which student might not be assessed
Finding Focus for Mathematics Instruction – Grade 6
Huron Intermediate School District
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Geometry
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alternate interior angles
angle bisector
angle of rotation
center of rotation
complementary angles
concave polygon*
congruent
convex polygon*
corresponding angles
corresponding sides
exterior angles
flip
geometric construction
interior angles
line
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line segment
lines of reflection*
lines of symmetry*
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midpoint
orientation
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parallel lines
perpendicular lines
polygon
quadrilateral
reflection
remote angles
right, scalene,
obtuse, acute
rotation
rotational symmetry*
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similar*
slide
straight angle
supplementary angles
transformation
translation
transversal
triangle inequality
triangle: equilateral,
isosceles,
turn
vertical angles
view: front,
side, top
Data and Probability
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certain events
chances*
cumulative frequency*
equally likely events
equally likely outcomes
event
experimental probability
favorable outcomes*
flow chart
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fractions, decimals,
percents
impossible events
interquartile range*
line of best fit*
mean/average
median*
mode
organized lists*
outcome
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paths
possible outcomes*
probability
quartile*
range
relative frequency*
routes
scatter plot
simulation*
theoretical probability
* Instructional term on which student might not be assessed
Finding Focus for Mathematics Instruction – Grade 6
Huron Intermediate School District
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