Helping Your Children
Learn in the
Middle School Years
MATH
Grade 7
A G U I D E T O T H E M AT H C O M M O N C O R E S TAT E S TA N D A R D S
F O R PA R E N T S A N D S T U D E N T S
This brochure is a product of the Tennessee State Personnel Development Grant (SPDG).
It was researched and compiled by Dr. Reggie Curran, University of Tennessee, Knoxville, and reviewed
by Ryan Mathis, Tennessee Department of Education, through a partnership between SPDG and the
University of Tennessee’s Center for Literacy, Education and Employment (CLEE).
Nathan Travis, Grant Director
Tennessee Department of Education
710 James Robertson Parkway
Nashville, TN 37243
615-532-6194
[email protected]
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Donna Parker, Project Manager
Ryan Mathis, Content Reviewer
Margy Ragsdale, Copyeditor
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For more information about the Common Core State Standards and Child Development,
check out this website:
Tennessee Common Core at
www.TNCORE.org
This project is supported by the U.S. Department of Education, Office of Special Education
Programs (OSEP). Opinions expressed herein are those of the authors and do not necessarily
represent the position of the U.S. Department of Education.
UT PUBLICATION NO.: R01-1704-113-002-14
Helping Your Children
Learn in the
Middle School Years
Grade 7
CONTENTS
2 Purpose of this Booklet
3 What are the Common Core State Standards?
4 Overview of Goals for Standards for Mathematical Practice
5 What are the Tennessee Focus Clusters?
Math Standards by Grade
6 Seventh Grade
Math Standards Descriptions
Student Checklist
29Helping Students Succeed in Math
32Tennessee RTI2 Model
33RTI2 Decision-Making Process
36 Additional RTI Resources for Parents
37 Strategies for Families to Help Struggling Learners
40Citations and Acknowledgements
A G U I D E T O T H E M AT H C O M M O N C O R E S TAT E S TA N D A R D S
F O R PA R E N T S A N D S T U D E N T S
1
Purpose of This Booklet
This booklet has two goals:
n to help parents understand more about what their children are learning in school, and
n to help students know if they have mastered the skills their teachers expect them to
know in each grade
T
eachers work from a set of standards
that tell them what to teach. Each state
has created its own standards, and those
standards have not been the same across our
country. However, most states have recently agreed
to use the same set of standards — the Common
Core State Standards. More information is
included about the Common Core State Standards
in the following pages.
This booklet will explain what the Common Core
State Standards are, and about the skills on which
Tennessee teachers will focus math instruction
while transitioning to the Common Core State
Standards. You will find general information that
will give you an overview of what the standards
are and why states are using them.
mastered by the end of the year. Ask your children
to look at them to see if they feel they have
mastered those skills, or if they need some extra
help in specific areas.
At the end of the list of standards and
explanations, you will find a box with an “I know
this!” checklist. These are short statements about
the skills your children will be expected to have
We hope you will find this booklet helpful in your
effort to be a partner in your child’s education and
development.
If you come across a math term and
don’t remember what it is or what it means,
check out the Math is Fun dictionary at
www.mathisfun.com/definitions
2
What are the Common Core State Standards?
A
cademic standards are
statements that describe
the goals of schooling
— what children should know
or be able to do at the end of the
school year. For example, the second grade math
standards state that by the end of the school year,
a second grader should be able to count to 120
and understand what each digit in a three-digit
number represents.
procedural skills with understanding
by finding ways to engage students
in good practices that will help them
understand the math content as they
grow in math maturity and expertise
throughout the elementary, middle, and high
school years.
The Common Core State Standards will provide
students, teachers, and parents with a shared
understanding of what students are learning.
With students, parents, and teachers all on the
same page and working together for shared goals,
we can increase the likelihood that students will
make progress each year and will graduate from
school prepared to succeed and to build a strong
future for themselves and the country.
However, standards have not been the same
across the United States. In the past, states have
had their own sets of standards. This means that
children in one state may be learning different
things at different times (and in different years)
than children in another state. Many states have
recently agreed to use a common set of standards
for learning that takes place in their classrooms;
these are the Common Core State Standards
(CCSS).
Parents: In this booklet, you will find an overview
of the standards for the seventh grade, showing
you what your children should be able to do by the
end of the school year. At the end of the section,
you will find a box with this “I can do it!” symbol.
Discuss these items with your child to see if he/she
is able to complete these tasks.
One major benefit of having common standards
across states is that children are being required
to learn the same information in the same years
in each of those states, so that a child moving
from one state to another will not be behind
the children in the new location. A common
set of standards ensures that all students, no
matter where they live, are focused on graduating
from high school prepared for postsecondary
education and careers.
Students: Find the “I know
this!” box at the end of each
section and check yourself
to see if you can do all those
things.
The Common Core State Standards for Math
have two components: Standards for Mathematical
Practice and Standards for Mathematical Content.
The Practice Standards describe the kind of math
teaching and learning that will produce the most
successful learning and that will help students
dig deeper and better understand math. The
Content Standards outline the concepts and skills
to be learned in each grade; teachers will balance
3
I know
this!
Overview of Goals for Standards for Mathematical Practice
The Standards for Mathematical
Practice describe skills and
behaviors that all students should
be developing in their particular
grades. These practices include
important processes (ways of
doing things) and proficiencies
(how well we do things), including
problem solving, reasoning
and proof, communication,
representation, and making
connections. These practices
will allow students to understand
and use math with confidence.
Following is what children will
be working to be able to do with
increasing ease:
Make sense of problems and
persevere in solving them
•Find the meaning in problems
•Analyze, predict, and plan the
path to solve a problem
•Verify answers
•Ask themselves the question:
“Does this make sense?”
Reason abstractly and
quantitatively
•Be able to translate the meaning
of each math term in any
equation
•Interpret results in the context
(setting) of the situation
Construct arguments and
evaluate the reasoning of
others
•Understand and use information
to build arguments
•Make and explore the truth of
estimates and guesses
•Justify conclusions and respond
to arguments of others
Model with mathematics
•Apply math to problems in
everyday life
•Identify quantities (amounts,
numbers) in a practical situation
•Present, show, or explain the
problem and solution in an
understandable way
Use appropriate tools
strategically
•Consider the available tools
when solving problems, and
know which tool is most
appropriate in the situation
•Be familiar with tools
appropriate for their grade
level or course (pencil and
paper, concrete models,
ruler, protractor, calculator,
spreadsheet, computer
programs, digital content on a
website, and other technological
tools)
Be precise
•Be able to communicate
accurately with others
•Use clear definitions, state the
meaning of symbols, and be
careful when specifying units of
measure and labeling axes (the
“x” and “y” lines that cross at
right angles to make a graph) in
math figures
•Calculate accurately and
efficiently Look for and make use of
structure
•Recognize patterns and
structures
•Step back to find the big picture
and be able to shift perspective
•See complicated things as
single objects, or as being made
up of several objects
Look for and identify ways to
create shortcuts when doing
problems
•When calculations are repeated,
look for general methods,
patterns, and shortcuts
•Be able to evaluate whether an
answer makes sense
The major domains included in the math standards for grades 6-8 are listed below. In each grade, students
build on what they learned previously to form a progression of increasing knowledge, skill, or sophistication.
GRADE
MAJOR DOMAINS FOR
MATH STANDARDS
Ratios and Proportional Relationships
The Number System
Expressions and Equations
6
7
3
3
3
3
3
3
3
3
3
3
Functions
Geometry
Statistics and Probability
4
8
3
3
3
3
3
Tennessee Focus Clusters
T
he Common Core State
Standards in Mathematics
present an opportunity
to engage Tennessee students in
deeper problem solving and critical thinking that
will build the math and reading skills students
will need for success. The new core standards will
allow teachers to provide focus, coherence, and
rigor (difficulty and thoroughness). Students will
think more deeply and know more than how to
just get the answer or read the words on the page
— they will understand! Teachers will link major
topics within grades — math includes reading and
reading includes math (and other subjects as well).
Finally, teachers will provide more challenge to
students so they will understand how to apply
what they are learning to the real world.
While teachers will teach all of
the standards, they will focus
instruction on specific areas that
will build stronger understanding.
To help teachers ease into the move from the
Tennessee State Standards to the Common
Core State Standards, educators in the state of
Tennessee have created a list of clusters (TNCore
Focus Clusters) on which teachers will focus
instruction in the next two years. Clusters are
groups of standards that connect needed concepts
and skills. The table below shows the focus areas
for each grade for school years 2012-2013 and
2013-2014. In addition, teachers will still be
teaching some of the information outlined in the
Tennessee State Standards. Eventually, Tennessee
teachers will be moving fully to the Common
Core State Standards.
TNCore FOCUS CLUSTERS FOR MATH – 2012-2013 and 2013-2014
6th Grade
• Understand ratio concepts and use ratio reasoning to solve problems.
• Apply and extend previous understandings of arithmetic to algebraic expressions.
• Apply and extend previous understandings of numbers to the system of rational numbers.
• Reason about and solve one-variable equations and inequalities.
7th Grade
• Analyze proportional relationships and use them to solve real-world and mathematical problems.
• Solve real-life and mathematical problems using numerical and algebraic expressions and
equations.
• Apply and extend previous understandings of operations with fractions to add, subtract, multiply,
and divide rational numbers.
• Use properties of operations to generate equivalent expressions.
8th Grade
• Understand the connections between proportional relationships, lines, and linear equations.
• Define, evaluate, and compare functions.
• Analyze and solve linear equations and pairs of simultaneous linear equations.
• Use functions to model relationships between quantities.
+ –
x÷
Mathematics
5
Seventh Grade Math
Focus Standards for Seventh Grade – Teachers will focus on specific skills in each
grade. For seventh grade, the instructional focus will center on these skills:
• Analyze proportional relationships and use them to solve real-world and
mathematical problems.
• Solve real-life and mathematical problems using numerical and algebraic
expressions and equations.
• Apply and extend previous understandings of
operations with fractions to add, subtract, multiply,
and divide rational numbers.
• Use properties of operations to generate
equivalent expressions.
Skills that focus on these areas appear in the shaded
box below. While these skills are priority areas,
students will be learning all of the skills listed in the
following sections.
Ratios and Proportional Relationships
For seventh graders, the math standards outline the
skills that should be developing, so that a student
can say, “I can … (insert math goal).” For example, a
student might say, “I can add, subtract, multiply, and
divide rational numbers.” Your child will be working on
the following skills this year.
Ratios and Proportional Relationships
Analyze proportional relationships and use them to solve real-world and
mathematical problems.
1.Compute unit rates associated with ratios of fractions, including ratios of lengths,
areas and other quantities measured in like or different units.
For example: If 1/2 gallon of paint covers 1/6 of a wall, then how much paint
is needed to cover the entire wall? The amount of paint needed for the entire
wall can be computed by 1/2 gallon divided by 1/6 wall. This calculation
gives 3 gallons.
6
2.Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship, e.g., by testing
for equivalent ratios in a table or graphing on a coordinate plane and observing
whether the graph is a straight line through the origin. For example: The table
below gives the price for different numbers of books. Do the numbers in
the table represent a proportional relationship?
Equivalent Ratios Versus
Equivalent Fractions
If the amounts from the table below are graphed (number
of books, cost), the pairs (1, 3), (2, 6), and (3, 9) will form
a straight line through the origin (0 books cost 0 dollars),
indicating that these pairs are in a proportional relationship.
The ordered pair (4, 12) means that 4 books cost $12.
However, the ordered pair (5, 13) would not be on the line,
indicating that it is not proportional to the other pairs.
1
3
2
6
3
9
4
12
5
13
COST
NumberCost
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Equivalent Ratios
Equivalent Fractions
5
4
3
2
1
0
1
2
3
4
5
NUMBER OF BOOKS
blue
cups
2
4
6
total
cups
3
6
9
2
_
3
=
4
_
6
=
more parts,
same size parts
more parts,
smaller parts
more total cups,
more blue cups
same whole amount,
same portion
b.Identify the constant of proportionality (unit rate) in tables, graphs, equations,
diagrams, and verbal descriptions of proportional relationships.
Cost of Bananas
BANANAS
160
140
PRICE (cents)
This graph represents the price of the bananas at
one store. What is the constant of proportionality?
From the graph, it can be determined that 4 pounds
of bananas is $1.00; therefore, 1 pound of bananas
is $0.25, which is the constant of proportionality for
the graph. The cost of bananas at a store can be
determined by the equation: P = $0.25n, where P is
the price and n is the number of pounds.
120
100
80
60
40
20
0
0
2
4
6
8
POUNDS
c. Represent proportional relationships by equations. For example: If total cost
t is proportional to the number n of items purchased at a constant price
p, the relationship between the total cost and the number of items can
7
6
_
9
be expressed as t = pn. A student is making trail mix. Create a graph
to determine if the quantities of nuts and fruit are proportional for
each serving size listed in the table. If the quantities are proportional,
what is the constant of proportionality or unit rate that defines
the relationship? Explain how you determined the constant of
proportionality and how it relates to both the table and graph.
y
1234
8
Cups of Nuts (x) 1234
7
Cups of Fruits (y)2468
6
The relationship is proportional.
For each of the other serving sizes
there are 2 cups of fruit for every 1
cup of nuts (2:1). The constant of
proportionality is shown in the first
column of the table and by the slope
of the line on the graph.
FRUIT (cups)
Serving Size
5
4
3
2
1
0
1
2
3
4
5
6
7
8
x
NUTS (cups)
d.Explain what a point (x, y) on the graph of a proportional relationship
means in terms of the situation, with special attention to the points (0, 0)
and (1, r) where r is the unit rate.
y
Packs of Gum (g)
0
1
2
3
4
Cost in Dollars (d)
0
2
4
6
8
8
7
6
COST (dollars)
Number of
5
4
3
2
The graph to the right represents the
cost of gum packs as a unit rate of
$2 dollars for every pack of gum. The
unit rate is represented as $2/pack.
Represent the relationship using a
table and an equation.
1
0
1
2
3
4
5
6
7
8
x
PACKS OF GUM
Equation: d = 2g, d is the cost in dollars
and g is the number of packs of gum.
3.Use proportional relationships to solve multistep ratio and percent problems.
Examples are simple interest, tax, markups and markdowns, gratuities and
commissions, fees, percent increase and decrease, percent error.
For example: After eating at a restaurant, your bill before tax is $52.50.
The sales tax rate is 8%. You decide to leave a 20% tip for the waiter
8
based on the pre-tax amount. How much is the tip you leave for the waiter?
How much will the total bill be, including tax and tip? Total bill will be food
+ tax + tip, or $52.50 + (.20 x $52.50) + (.08 x $52.50).
The amount paid = Tip = (0.20 x $52.50) = $10.50
Tax = (0.08 x $52.50) = $4.20
Total Bill = $52.50 + $10.50 + $4.20 = $67.20
The Number System
Apply and extend previous understandings of operations with fractions to
add, subtract, multiply, and divide rational numbers.
1.Represent addition and subtraction
on a horizontal or vertical number
line diagram.
2+3=5
2
3
01234567 8910
a. Describe situations in which
opposite quantities combine to
make 0. On this number line, the
numbers a and b are the same
distance from 0. What is the sum
of a + b?
b0 a
b.Understand p + q as the number located a distance | q | from p, in the positive
or negative direction depending on whether q is positive or negative. Show
that a number and its opposite have a sum of 0 (are additive inverses).
Interpret sums of rational numbers by describing real-world contexts.
For example: John earns $3 for raking the leaves, but he owes his
brother $3. How much money will John have after he pays his brother?
(- 3 + 3) = 0. -3 and 3 are shown
to be opposites on the number
-303
line because they are equal
distance from zero and therefore
have the same absolute value. The sum of the number (3) and its
opposite (-3) is zero.
c. Understand subtraction of rational numbers as adding the additive inverse,
p - q = p + (-q). Show that the distance between two rational numbers on the
number line is the absolute value of their difference, and apply this principle
in real-world contexts. For example: If one of the integers is negative,
9
subtract the absolute value of it from the other number.
Example: 14 + (-6) = 14 – 6 = 8
If both of the integers are negative, add their absolute values and prefix
the number with a negative sign.
Example: (-14) + (-6) = |14| + |6| = (- 20)
Rules for Multiplying and Dividing
d.Apply properties of operations as
strategies to add and subtract rational
numbers. (See Properties Charts in the
following section.)
2.Apply and extend previous understandings of
multiplication and division and of fractions to
multiply and divide rational numbers.
Signed Numbers
MULTIPLYING
• The product of two numbers with the
same signs is positive.
• The product of two numbers with
different signs is negative.
DIVIDING
• The quotient of two numbers with the
same signs is positive.
• The quotient of two numbers with
different signs is negative.
a. Understand that multiplication is extended
from fractions to rational numbers by requiring that operations continue
to satisfy the properties of operations (see charts below), particularly the
distributive property, leading to products such as (-1)(-1) = 1 and the rules
for multiplying signed numbers. Interpret products of rational numbers by
describing real-world contexts.
b.Understand that integers can be divided, provided that the divisor is not
zero, and every quotient of integers (with a non-zero divisor) is a rational
number. If p and q are integers, then -(p/q) = (-p)/q = p/(-q). Interpret
quotients of rational numbers by describing real world contexts.
Integer – a number with no fractional parts
-10-9-8-7-6-5-4 -3-2-1
01234567 8910
Integers include the counting numbers (1, 2, 3…), zero (0), and the negative of the
counting numbers (-1, -2, -3…).
An rational number is any number that can be made by dividing one
integer by another. The term comes from the word “ratio.”
EXAMPLES:
• 1/2 is a rational number (1 divided by 2, or the ratio of 1 to 2).
• 0.75 is a rational number.
• 1 is a rational number (1/1).
• 2 is a rational number (2/1).
• 2.12 is a rational number (212/100).
• -6.6 is a rational number (-66/10).
Rational
Number
a
b
c. Apply properties of operations as strategies to multiply and divide rational
numbers. (See Properties Charts in the following section.)
10
d.Convert a rational number to a decimal using long division; know that the
decimal form of a rational number eventually terminates in zero or repeats.
Terminating decimal numbers can easily be written in decimal form.
For example: 0.67 is 67/100
For example: 3.40938 = 340938/100000.
Non-terminating decimal numbers can
An irrational number is a real number that
also be rational numbers. For example,
cannot be written as a simple fraction.
Irrational means not rational. For
1/9 converted into a decimal is 0.1111…
example, pi (π) is a famous irrational
(doesn’t end), but since it can be written
number:
as a fraction (1/9), it is a rational number.
3.1415926535897932384626433832795
In fact, every non-terminating decimal
(and more...).
You cannot write down a simple fraction
number that REPEATS a certain pattern of
that equals pi. The popular approximation
digits is a rational number.
of 22/7 = 3.1428571428571...is close but
not accurate.
3.Solve real-world and mathematical problems
involving the four operations with rational numbers. For example: Your cell
phone bill is automatically deducting $32 from your bank account every
month. How much will the deductions total for the year?
-32 + -32 + -32 + -32 + -32 + -32 + -32 + -32 + -32 + -32 + -32 + -32 =
12 (-32) = -$384
Properties of Addition and Subtraction
Commutative Property. The commutative
property says that the positions of the numbers
in a mathematical equation do not affect the
ultimate solution. Five plus three is the same as
three plus five. This applies to addition, regardless
of how many numbers you add together. The
commutative property allows you to add a large
group of numbers together in any order. The
commutative property does not apply to
subtraction. Five minus three is not the same as
three minus five.
Associative Property. The associative property
applies to more complicated equations that use
parentheses or brackets to separate groups
of numbers. The associative property says
that numbers you are adding together can be
grouped in any order. When you are adding
numbers together, you can move the parentheses
around. For example, (3 + 4) + 2 = 3 + (4 + 2).
The associative property also does not apply
to subtraction since (3 - 4) - 2 does not equal
3 - (4 - 2). This means that if you are working
11
on a subtraction equation, you cannot move the
brackets around.
Identity Property. The identity property says
that any number plus zero equals itself. For
example, 3 + 0 = 3. The identity property also
applies to subtraction since 3 - 0 = 3. Zero
is known as the identity number because in
addition and subtraction it does not affect other
numbers. When a child is adding or subtracting
large groups of numbers, remind her that the
number zero does not affect other numbers in the
equation.
Inverse Operations. In addition to the properties
that affect addition and subtraction separately,
addition and subtraction also relate to each
other. They are inverse operations, which is
similar to saying that addition and subtraction
are opposites. For example, five plus three minus
three equals five because adding and then
subtracting the threes cancels both of them out.
Encourage your child to look for numbers that
cancel each other out when he is adding and
subtracting.
Properties of Multiplication
Commutative Property. The order of the
numbers doesn’t change the result (answer to
the problem). pxq=qxp
Associative Property. The grouping of the
factors doesn’t change the answer.
(p x q) x r = p x (q x r)
Distributive Property. Multiplying the sum
(or difference) by a number is the same
as multiplying each number in the sum (or
difference) by the number and adding (or
subtracting) the product.
9 x (20 - 3) = (9 x 20) – (9 x 3)
8 x (40 + 5) = (8 x 40) + (8 x 5)
Zero Property. When any number is multiplied
by zero, the answer is zero.
98,756,432 x 0 = 0
Expressions and Equations
Use properties of operations to generate equivalent expressions.
1.Apply properties of operations as strategies to add, subtract, factor, and expand
linear expressions with rational (whole numbers that may or may not be expressed as
fractions, for example, 4 or 4/1) coefficients. For example:
• We can use the commutative and associative properties to add linear
expressions with rational coefficients
(e.g., -4x + (3 + x) = -4x + (x + 3) = (-4x + x) + 3 = -3x + 3).
• We can use the distributive property to add and/or subtract linear expressions
with rational coefficients (e.g., -1/5x + 3/5x = (-1/5 + 3/5)x = 2/5x).
• We can use the distributive property to factor a linear expression with rational
coefficients (e.g., 6x + 9 = 3(2x + 3).
• We can use the distributive property to expand a linear expression with rational
coefficients (e.g., 2/3(9x + 6) = (2/3 × 9x) + (2/3 × 6) = 6x + 4).
2. Understand that rewriting an expression in different forms in a problem context can
shed light on the problem and how the quantities in it are related.
For example: If you know that your rent is $500 now and is going up 5% next
year, how much will the new rent payment be? 5% is the same as .05, so
a + 0.05a = 1.05a. “Increase by 5%” is the same as “multiply by 1.05”, so $500
multiplied by 1.05 equals $525.00 — your new rent.
12
Solve real-life and mathematical problems using numerical and algebraic
expressions and equations.
1.Solve multi-step real-life and mathematical problems posed with positive and
negative rational numbers in any form (whole numbers, fractions, and decimals),
using tools strategically. Apply properties of operations to calculate with numbers in
any form; convert between forms as appropriate; and assess the reasonableness
of answers using mental computation and estimation strategies. For example: If a
woman making $25 an hour gets a 10% raise, she will make an additional 1/10
of her salary an hour, or $2.50, for a new salary of $27.50.
If you want to place a towel bar 9 3/4 inches long in the center of a door that
is 27 1/2 inches wide, you will need to place the bar about 9 inches from each
edge; this estimate can be used as a check on the exact computation.
27 _1 inches
2
27 _1 – 9 _3 = 17 _3
2
8 _7 inches
4
17 _3 ÷ 2 = 8 _7
4
4
8
9 _3 inches
8
8 _7 inches
4
8
estimate of 9 inches correct
2.Use variables to represent quantities in a real-world or mathematical problem, and
construct simple equations and inequalities to solve problems by reasoning about
the quantities. For example: The youth group is going on a trip to the state fair.
The trip costs $52. Included in the price is $11 for a concert ticket and the
cost of 2 passes, one for the game booths and one for the rides. Each of the
passes cost the same price. Write an equation representing the cost of the
trip and determine the price of each of the passes.
x
x
11
52
x = cost of one pass
2x + 11 = 52
2x = 41
x = $20.50
13
a. Solve word problems leading to equations of the form px + pq = r and
p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of
these forms fluently. Compare an algebraic solution to an arithmetic solution,
identifying the sequence of the operations used in each approach.
For example: The perimeter of a rectangle is 54 cm. Its length is 6 cm. What
is its width?
p=2l+2w
54 = (2 x 6) + (2 x w)
Width = 21 centimeters
54 = 12 + 2w
54 – 12 = 2w
42 = 2w
21 = w
54 = (2 x 6) + (2 x 21), also 54 = 2(6 +21)
b.Solve word problems leading to inequalities of the form px + q ≥ r or px + q ≤ r,
where p, q, and r are specific rational numbers. Graph the solution set of the
inequality and interpret it in the context of the problem.
For example: As a salesperson, you are paid $50 per week plus $3 per sale.
This week you want your pay to be at least $100. Write an inequality for the
number of sales you need to make, and describe the solutions.
Salary for Week
px + q ≥ r
3x + 50 ≥ 100
3x ≥ 50
50 ÷ 3 = 16.66, so if you want
to make at least $100, you
must make at least 17 sales.
50 + (3 x 17) = 50 + 51 = 101
110
110
107
104
101
98
98
95
92
89
86
Salary
86
83
80
77
74
74
71
68
65
62
62
59
53
50
56
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Sales
14
Geometry
Draw, construct, and describe geometrical figures and describe the relationships between them.
1.Solve problems involving scale drawings of geometric figures, including computing
actual lengths and areas from a scale drawing and reproducing a scale drawing at a
different scale. Computations with rational numbers extend the rules for manipulating
fractions to complex fractions.
7 in.
Perimeter is (7 x 2) + (2 x 2) = 18
Area is (7 x 2) 2 = 14 in 2
2 in.
For example: if the rectangle above is enlarged using a scale factor of 1.5, what
will be the perimeter and area of the new rectangle?
The perimeter is linear or one-dimensional. Multiply the perimeter of the given
rectangle (18 in.) by the scale factor (1.5) to get an answer of 27 in. We could
also increase the length and width by the scale factor of 1.5 to get 10.5 inches
for the length (7 x 1.5) and 3 in. for the width (2 x 1.5). The perimeter could be
found by adding 10.5 + 10.5 + 3 + 3 to get 27 in.
The area is two-dimensional so the scale factor must be squared. The area
of the old rectangle is 14 but the scaled up rectangle would be 14 x 1.52, or
14 x 2.25, which is 31.5.
2.Draw (freehand, with ruler and protractor, and with technology) geometric shapes
with given conditions. Focus on constructing triangles from three measures of angles
or sides, noticing when the conditions determine a unique triangle, more than one
triangle, or no triangle.
Example 1: Which of these are quadrilaterals with at least one set of parallel
sides and no right angles?
xx – has parallel
sides and right
angles
√√ – has parallel
sides and
no right angles
√√ – has parallel
sides and
no right angles
xx – has parallel
sides and
right angles
√√√ – has no
parallel sides
and no right
angles
Example 2: Will three sides of any length create a triangle?
Possibilities to examine are:
Answer: “A” will not work; “B” and “C” will
a. 13 cm, 5 cm, and 6 cm
work. Students should recognize that the sum
b. 3 cm, 3 cm, and 3 cm
of the two smaller sides must be larger than
c. 2 cm, 7 cm, 6 cm
the third side.
15
3.Describe the two-dimensional figures that result from slicing three-dimensional
figures, as in plane sections of right rectangular prisms and right rectangular
pyramids. Cuts made parallel will take the shape of the base; cuts made
perpendicular will take the shape of the lateral (side) face.
a. If a pyramid is cut with a plane (green) parallel to the base,
the intersection of the pyramid and the plane is a cross
section (red).
b.If the pyramid is cut with a plane (green) passing through
the top vertex and perpendicular to the base, the
intersection of the pyramid and the plane is a triangular
cross section.
c. If the pyramid is cut with a plane (green) perpendicular to
the base but not through the top vertex, the intersection
of the pyramid and the plane is a trapezoidal cross section
(red).
Solve real-life and mathematical problems involving
angle measure, area, surface area, and volume.
1.Know the formulas for the area and circumference of a circle and use them
to solve problems; give an informal derivation of the relationship between the
circumference and area of a circle.
Students understand the relationship between radius and diameter. Students
also understand the ratio of circumference to diameter can be expressed as
pi (π). Building on these understandings, students generate the formulas for
circumference and area.
The circumference of a circle (the distance around the circle) is 2 times the radius
(the distance from the center to the edge of a circle) times π (3.14). ( c = 2πr).
mference
rcu
Ci
Rad
ius
Dia
The ratio of a circle’s circumference
(distance around the circle) to the
diameter is π, or 3.14159… (the digits
go on forever without repeating, so we
use 3.14 as an approximate number in
calculations).
met
er
The area of a circle is circle is
π times the radius squared.
r
16
CIRCLE
Area = π x r 2
Circumference = 2 x π x r
r = radius
The illustration shows the relationship between the circumference and area. If
a circle is cut into wedges and laid out as shown, a parallelogram results. Half
of an end wedge can be moved to the other end and a rectangle results. The
height of the rectangle is the same as the radius of the circle. The base length is
½ of the circumference, which is (2 π r). The area of the rectangle (and therefore
the circle) is found by the following calculations:
Area = Base x Height
Area = ½ (2 π r) x r
Area = π r x r
Area = π r 2
πr
The base of the
rectangle is made of
half the outer wedges
of the circle. The other
half forms the top
edge of the rectangle.
r
Real life example:
If a circle is cut from
a square piece of plywood,
how much plywood would
be left over?
The area of the square is 28 x 28 or 784 in2.
The diameter of the circle is equal to the
length of the side of the square, or 28'',
so the radius would be 14''. The area of the
circle would be approximately 615.44 in2.
The difference in the amounts (plywood left
over) would be (784 – 615.44) or 168.56 in2.
28''
2.Use facts about supplementary, complementary, vertical, and adjacent angles in
a multi-step problem to write and solve simple equations for an unknown angle
40º
in a figure.
Complementary angles: Two angles are complementary
50º if they add up to
90 degrees (right angle).
These two angles (40º and
50º) are complementary
angles because they add
up to 90º.
40º
50º
However, the angles
don’t have to be together.
27º
63º
Notice that together they
make a right angle.
27º
63º
17
These two are
complementary because
27º and 63º = 90º.
Supplementary angles: Two angles are supplementary if they
add up to 180 degrees.
These two angles (40º and 140º
are supplementary angles
because they add up to 180º.
140º
40º
Notice that together they make a
straight angle.
However, the angles don’t have
to be together.
120º
60º
These two angles are
supplementary because
60º + 120º = 180º.
Vertical angles: Angles that are opposite each other when
two lines cross.
In this example, aº and bº
are vertical angles.
aº
The interesting thing here
is that vertical angles are
equal:
aº = bº
bº
Adjacent angles: Two angles are adjacent if they have a common
side and a common vertex (corner point) and don’t overlap.
What is and isn’t an adjacent angle?
aº aº
bº bº
aº
aº aº
aº
b ºb º
bº
These are adjacent angles.
They share a vertex
and a side.
aº aº
bº
These are not adjacent angles.
They only share a vertex,
not a side.
18
aº
bº bº
bº
These are not adjacent angles.
They only share a side,
not a vertex.
A
26º
Angle ABC is adjacent
to angle CBD because
they have a common side
(line CB) and they have a
common vertex (point B).
C
33º
B
D
Example: Write and solve an equation
to find the measure of angle x.
40º
Solution: The right angle at the bottom
xº
is 90°, and the top angle is 40°. Since
the angles of a triangle add up to 180°,
the equation would be (180 - 90 - 40), or 50°. The missing angle is 50°. The
measure of angle x is supplementary to 50°, so subtract 50 from 180 to get
a measure of 130° for x.
3.Solve real-world and mathematical problems
involving area, volume and surface area of
two- and three-dimensional objects composed
of triangles, quadrilaterals, polygons, cubes,
and right prisms. Students will know some
formulas for finding area, volume, and
surface areas, but “knowing the formula”
does not mean memorization of the formula;
it means to have an understanding of why the formula works and how the
formula relates to the measure (area and volume). For example: Students
can build on their work with nets in the 6th grade, recognizing that finding
the area of each face of a three-dimensional figure and adding the areas will
give the surface area.
In addition, students’ understanding of volume can be supported
by focusing on the area of base times the height to calculate
volume, and understanding of surface area can be supported by
focusing on the sum of the area of the faces. For example: The
surface area of a cube is 96 in2. What is the volume of the
cube?
Solution: The area of each face of the cube is equal. Dividing
96 by 6 gives an area of 16 in2 for each face. Because each face is a square,
the length of the edge would be 4 in. The volume could then be found by
multiplying 4 x 4 x 4 or 64 in3.
19
Statistics and Probability
Use random sampling to draw inferences about a population.
1.Understand that statistics can be used to gain information about a population
by examining a sample of the population; generalizations about a population
from a sample are valid only if the sample is representative of that population.
Understand that random sampling tends to produce representative samples
and support valid inferences.
Random sample: A sample in which each individual or object in the
population has an equal chance of being selected.
For example: The school food service wants to increase the number of
students who eat hot lunch in the cafeteria. The student council was
asked to conduct a survey of the student body to determine the students’
preferences for hot lunch. They have determined three ways to do the
survey. The three methods are listed below. Determine if each survey
option would produce a random sample. Which survey option should the
student council use and why?
1.Write all of the students’ names on cards and pull them out in a draw to
determine who will complete the survey.
2.Survey the first 20 students that enter the lunchroom.
3.Survey every 3rd student who gets off a bus.
The student council should use solution #1; it would be a true random sample
because every student has the same chance to have his/her name pulled to
complete the survey. In #2, only those students who get to lunch early have a
chance, and in #3, only those students who ride the bus have a chance.
2.Use data from a random sample to draw inferences about a population with
an unknown characteristic of interest. Generate multiple samples (or simulated
samples) of the same size to gauge the variation in estimates or predictions.
Examples: Estimate the mean word length in a book by randomly
sampling words from the book; predict the winner of a school election
based on randomly sampled survey data. Gauge how far off the estimate
or prediction might be.
For example: Below is the data collected from two random samples of
100 students regarding students’ school lunch preference. Make at least
two inferences based on the results.
Student Sample
#1
#2
Hamburgers
Tacos
Pizza
TOTAL
12
14 74100
12
11 77100
Solution: • Most students prefer pizza.
• More people prefer pizza than hamburgers and tacos combined.
20
Draw informal comparative inferences about two populations.
1.Informally assess the degree of visual overlap of two numerical data distributions
with similar variability, measuring the difference between the centers by expressing
it as a multiple of a measure of variability.
For example: Jason wanted to compare the mean height of the players on
his favorite basketball and soccer teams. He thinks the mean height of the
players on the basketball team will be greater but doesn’t know how much
greater. He also wonders if the variability of heights of the athletes is related
to the sport they play. He thinks that there will be a greater variability in
the heights of soccer players as compared to basketball players. He used
the rosters and player statistics from the team websites to generate the
following lists.
Basketball Team – Heights of Players in inches for 2010-2011 Season
75, 73, 76, 78, 79, 78, 79, 81, 80, 82, 81, 84, 82, 84, 80, 84
Soccer Team – Heights of Players in inches for 2010 Season
73, 73, 73, 72, 69, 76, 72, 73, 74, 70, 65, 71, 74, 76, 70, 72, 71, 74, 71, 74, 73,
67, 70, 72, 69, 78, 73, 76, 69
To compare the data sets, Jason
creates two dot plots on the same
scale. The shortest player is 65 inches
and the tallest players are 84 inches.
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84
Height of Soccer Players (in.)
In looking at the distribution of the
data, Jason observes that there is
x
x x x x x
x
some overlap between the two data
x
x x
x x x x x
x
sets. Both teams have some players
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84
between 73 and 78 inches tall.
Height of Basketball Players (in.)
Jason decides to use the mean and
mean absolute deviation to compare
the data sets. Jason sets up a table for each data set to help him with the
calculations. The mean (the number found by the dividing the sum of a set of
numbers by the number of addends, sometimes called the average) height
of the basketball players is 79.75 inches as compared to the mean height of the
soccer players at 72.07 inches, a difference of 7.68 inches. The mean absolute
deviation (MAD) is calculated by taking the mean of the absolute deviations
(Deviation just means how far from the normal, and absolute deviation is a
positive number) for each data point. The difference between each data point
and the mean is recorded in the second column of the table. Jason used rounded
values (80 inches for the mean height of basketball players and 72 inches for the
mean height of soccer players) to find the differences. The absolute deviation,
the absolute value (positive number) of the deviation, is recorded in the third
21
Heights of Players on Soccer and Basketball Teams
Soccer Players (n = 29)
Height (in.)
65
67
69
69
69
70
70
70
71
71
71
72
72
72
72
73
73
73
73
73
73
74
74
74
74
76
76
76
78
∑ = 2090
Deviation from
Mean (in.)
-7
-5
-3
-3
-3
-2
-2
-2
-1
-1
-1
0
0
0
0
+1
+1
+1
+1
+1
+1
+2
+2
+2
+2
+4
+4
+4
+6
Basketball Players (n = 16)
Deviation from
Height (in.)
Mean (in.)
73
-7
75
-5
76
-4
78
-2
78
-2
79
-1
79
-1
80
0
80
0
81
1
81
1
82
2
82
2
84
4
84
4
84
4
Absolute
Deviation (in.)
7
5
3
3
3
2
2
2
1
1
1
0
0
0
0
1
1
1
1
1
1
2
2
2
2
4
4
4
6
∑ = 62
∑ = 1276
Mean = 2090 ÷ 29 = 72 in.
MAD = 62 ÷ 29 = 2.13 in.
Mean = 1276 ÷ 16 = 80 in.
MAD = 40 ÷ 16 = 2.5 in.
column. The absolute deviations are summed and divided by the number
of data points in the set. The mean absolute deviation is 2.14 inches for the
basketball players and 2.53 for the soccer players. These values indicate
moderate variation in both data sets. There is slightly more variability in the
height of the soccer players. The difference between the heights of the teams
is approximately 3 times the variability of the data sets (7.68 ÷ 2.53 = 3.04).
22
Absolute
Deviation (in.)
7
5
4
2
2
1
1
0
0
1
1
2
2
4
4
4
∑ = 40
2.Use measures of center and measures of variability
x
x
for numerical data from random samples to
x
x
x
x x
draw informal comparative inferences about two
x x x x
x x
x x x x
x x x
x x x x x
x x x x x
populations.
x x x x x
x x x x x x x
For example: The seventh grade’s essays were
0 1 2 3 4 5 6
0 1 2 3 4 5 6
scored on specific traits. The scores for ideas
Scores
Scores
were 2, 3, 6, 5, 3, 5, 2, 4, 6, 5, 5, 6, 4, 3, 4, 3, 4, 5,
for Ideas
for Organization
4, 6, 5, and the scores for organization were 2, 6,
3, 5, 4, 0, 3, 1, 3, 4, 3, 6, 5, 3, 5, 4, 4, 3. Which trait got higher scores?
Showing the two graphs side by side helps students make comparisons. For example: Students would be able to see from the display of the two
graphs that the ideas scores are generally higher than the organization
scores. One observation students might make is that the scores for
organization are clustered around a score of 3 whereas the scores for
ideas are clustered around a score of 5.
Investigate chance processes and develop, use, and evaluate probability
models.
1.Understand that the probability of a chance
event is a number between 0 and 1 that
_1
expresses the likelihood of the event
0
1
2
occurring. Larger numbers indicate greater
The closer the fraction is to 1, the greater the
likelihood. A probability near 0 indicates
probability the event will occur. Larger numbers
indicate greater likelihood. Students also recognize
an unlikely event, a probability around 1/2
that the sum of all possible outcomes is 1.
indicates an event that is neither unlikely nor
likely, and a probability near 1 indicates a likely
event.
Example 1: There are three choices of jellybeans — grape, cherry and
orange. If the probability of getting grape is 3/10 and the probability of
getting cherry is 1/5, what is the probability of getting orange?
3/10 + 1/5 + x = 1
x = 1 – 5/10
3/10 + 2/10 + x = 1
x = 5/10, or 1/2
There is a 1/2 chance you will get orange.
23
5/10 + x = 1
Example 2: The container at the right
contains 2 gray, 1 white, and 4 black
marbles. Without looking, if Eric chooses
a marble from the container, will the
probability be closer to 0 or to 1 that Eric
will select a white marble? A gray marble?
A black
marble?
Solution: You have a better chance at
choosing a black marble
• Gray has 2/7 chance which is closer to 0
• White has a 1/7 chance, which is closer to 0
• Black has a 4/7 chance, which is closer to 1
2.Approximate the probability of a chance event
by collecting data on the chance process that
produces it and observing its long-run relative frequency, and predict the
approximate relative frequency given the probability.
For example: A baseball player comes up to bat. He has hit the ball 118
times in his last 342 times at bat. What is the probability that he will hit
the ball this time? One way to interpret what this probability number means
is what people call long run relative frequency. The phrase “long run” is an
important part of this interpretation. Let’s take the example of the baseball
player. We’re partway into the season and he has batted 342 times. He’s
gotten a hit 118 times. Another way of saying this is that the frequency of at
bats is 342 and the frequency of successful at bats, that is the frequency of
hits, is 118. Relative frequency is defined by taking the total hits and dividing
them by the total at bats 118/342 (which doesn’t reduce evenly below 59/171,
but is a little more than 1/3. The relative frequency is .345 minus the proportion
of total at bats that resulted in a hit. People who are involved with playing
baseball tend to multiply that relative frequency or proportion by a thousand
and say the batting average is 345.
EXAMPLE: A
baseball player
gets 118 hits in
342 times at bats.
EXAMPLE: A
baseball player
gets 118 hits in 342
times at bats.
A common and useful
interpretation of
probability is long run
relative frequency.
The frequency of
opportunities to
get a hit (at bats)
is 342.
The frequency of
(at bats) is 342.
The baseball player’s
relative frequency of
hits is .345.
The frequency of
successes (hits)
is 118.
The frequency of
(hits) is 118.
The RELATIVE FREQUENCY of hits
per at bats is ____________.
118
RELATIVE FREQUENCY: ___
= .345
342
24
We could think of the
player’s probability
of getting a hit as
the player’s long run
relative frequency.
3.Develop a probability model and use it to find probabilities of events. Compare
probabilities from a model to observed frequencies; if the agreement is not good,
explain possible sources of the discrepancy. Probability is such a natural part
of your life that you rarely think about it. However, every time you use a
word like “might,” “may,” “undoubtedly,” “without fail,” or “maybe,” you are
voicing a probability that an event will occur. Scientists and mathematicians
like to express probability more accurately.
a. Develop a uniform probability model by assigning equal probability to all
outcomes, and use the model to determine probabilities of events. For
example: If a student is selected at random from a class, find the
probability that Jane will be selected and the probability that a girl will be
selected.
Jane has a 1/20 chance of being selected.
A female has an 8/20 chance of being selected.
Students
in class
20
Male
Students
12
Female
Students
8
b.Develop a probability model (which may not be uniform) by observing
frequencies in data generated from a chance process. For example: Find
the approximate probability that a spinning penny will land heads up or
that a tossed paper cup will land open-end down. Do the outcomes for
the spinning penny appear to be equally likely, based on the observed
frequencies?
If you toss a penny in the air, the
probability (P) that it will land heads
up can be expressed: P = # of
times it lands heads/total number
of coin tosses.
The number will be 5/10, which
reduces to ½. This means that you
have a 1 out of 2 chance that the
penny will land heads.
The graph (at right) shows the results of an experiment in which a
coin was tossed thirty times. Find the experimental probability of
tossing tails for this experiment.
18
3
__
__
= 30 = 5
3
The experimental probability of tossing tails is __
.
5
Compare the experimental probability you found in example 2 to
its theoretical probability.
1
__
The theoretical probability of tossing tails on a coin is 2 , so the
experimental probability is close to the theoretical probability.
25
Number of Tosses
number of times tails occurs
P
(tails) = number of possible outcomes
You call it!
20
18
16
14
12
10
8
6
4
2
0
Heads
Tails
4.Find probabilities of compound events (an event made of two or more simple
events) using organized lists, tables, tree diagrams, and simulation.
a. Understand that, just as with simple events, the probability of a compound
event is the fraction of outcomes in the sample space for which the
compound event occurs.
b.Represent for compound events using methods such as organized lists,
tables and tree diagrams.
Example 1: How many ways could the three students, Amy, Brenda
and Carla, come in 1st, 2nd and 3rd place?
Solution: Making an organized list will identify that there are six ways
for the students to win a race:
A, B, C A, C, B B, C, A B, A, C C, A, B C, B, A
Example 2: Students conduct a bag pull experiment. A bag contains
five marbles. There is one red marble, two blue marbles and two
purple marbles. Students will draw one marble without replacement
and then draw another one. What is the sample space for this
situation? Explain how the sample space was determined and how it
is used to find the probability of drawing one blue marble followed by
another blue marble.
Example 3: A fair coin will be tossed three times.
What is the probability that two heads and one
tail in any order will result? (Adapted from SREB
publication Getting Ready for Algebra I: What Middle
Grades Students Need to Know and Be Able to Do)
Solution: HHT, HTH, and THH, so the probability
would be 3/8.
Example 4: Show all possible arrangements of
the letters in the word FRED using a tree diagram.
If each of the letters is on a tile and drawn at
random, what is the probability of drawing
the letters F-R-E-D in that order? What is the
probability that a “word” will have an F as the first
letter?
Solution: There are 24 possible arrangements
(4 choices • 3 choices • 2 choices • 1 choice).
The probability of drawing F-R-E-D in that order is
1/24. The probability that a “word” will have an F
as the first letter is 6/24 or 1/4.
26
R
F
E
D
F
R
E
D
Start
F
E
R
D
F
D
R
E
E
D
R
D
R
E
D
E
D
R
E
R
E
D
F
D
F
E
R
D
F
D
R
F
R
E
F
E
F
R
D
R
D
F
E
F
D
R
D
F
F
R
E
R
E
F
R
F
c. Design and use a simulation (an experiment that models a real-life
situation) to generate frequencies for compound events. Simulations
require the use of random numbers. Random numbers have no pattern;
they cannot be predicted in any way. Knowing a random number in no
way allows you to predict the next random number. An example of a
random number generator is a standard number cube or dice, which will
randomly generate a number from 1 to 6 when you roll it, or a scientific
calculator that may or may not generate a larger range of numbers.
COMPOUND EVENTS
The combination of two or more events
EXAMPLES
Roll two dice
Deal two cards
When the mathematics becomes too complicated to figure out the
theoretical probability of certain events, statisticians often use computer
simulations instead.
Complex simulations like modeling the weather, traffic patterns, cancer
radiation therapy, or stock market swings require tens of billions of
random numbers. For those simulations a large computer running for
hours or even days is needed.
27
Seventh Grade Student Self-Check List
I know
this!
Students: You have been working on learning these things this year. Are you able
to do these things? Check the box next to the skill if you can do it.
Seventh grade students deepen their understanding of proportional relationships
to solve complicated problems. They extend their understanding of rational
numbers to include computation (add, subtract, multiply, and divide). Irrational
numbers are introduced in seventh grade. Algebraic foundations are practiced
and extended. Students continue to extend their understanding of probability and
statistics by describing populations based on sampling, and investigate chance to
develop, use, and evaluate probability models.
Use proportional relationships to solve multistep operation and percent problems.
Add, subtract, factor, and expand linear
expressions.
If a person walks ½ mile in each ¼ hour, what
is her speed per hour?
Construct simple equations and inequalities
to solve problems.
Compute unit rates.
Draw, construct, and describe geometrical
figures and describe the relationships
between them.
Add, subtract, multiply, and divide rational
numbers.
Solve problems involving angle measure,
area, surface area, and volume (cylinders,
cones, and spheres).
Know irrational numbers (numbers that are
not rational) and approximate them with
rational numbers. [Writing √2 (an irrational
number) as a decimal is 1.4142435623.
Understand that √2 is between 1 and 2, then
between 1.4 and 1.5. we can approximate it
in a fraction, but it will not be exact.]
Know formulas for volumes of cones,
cylinders, and spheres.
Know the formulas for area and
circumference of a circle.
Use properties of operations to solve
algebraic equations.
Use random sampling to describe and
compare populations.
Use square root and cube root symbols to
represent solutions to equations.
Find, calculate, and explain the probability of
a chance event.
Evaluate square root and cube roots (of small
perfect square roots and cube roots).
For example, if a student is selected from a
class, find the probability that Jane will be
selected and the probability that a girl will be
selected.
Know that √2 is irrational.
Use numbers multiplied by a power of ten
to estimate very large or very small quantities
(the population of the United States is
3 x 108).
Or if 40% of donors have type A blood, what
is the probability that it will take at least 4
donors to find one with type A blood?
28
Helping Students Succeed in Math
Response to Intervention: A New Way of Teaching that Helps All Students
Learn, and a New Way of Identifying Possible Learning Disabilities
Response to Intervention (RTI)
RTI is a tiered process of instruction that allows
schools to identify struggling students early and
provide appropriate instructional interventions.
Early intervention means more chances for success
and less need for special education services. RTI
would also address the needs of children who
previously did not qualify for special education.
provided to a small group of students with similar
skill needs.
Tier 3: At this tier, intensive interventions are
provided for a few students (1-5%) needing the
MOST help. For example, intervention time may
be 60 minutes per day in addition to the Core (Tier
1). These interventions are typically implemented
by other trained and qualified staff that could
include the special education teacher. For some
students, this could also be a replacement
core curriculum. Some students may need a
replacement core when they are significantly
discrepant (more than two years behind) from
typical peers and the general core is not explicit
enough to meet their intense skill needs.
http://www.wrightslaw.com/info/rti.parent.guide.htm
Tiers in RTI
RTI is a delivered to students in tiers or levels.
There is much discussion about how many
tiers should be in RTI models. The three-tiered
model is the most common. This means there
are different levels of intervention, based on the
needs of the student. The level of intervention
increases in intensity if a child does not respond to
instruction.
Key Terms in RTI
Universal Screening is a step taken by school
personnel early in the school year to determine
which students are “at risk” for not meeting
grade level standards. Universal screening can be
accomplished by reviewing recent results of state
tests, or by administering an academic screening
test to all children in a given grade level. Those
students whose test scores fall below a certain
cut-off are identified as needing more specialized
academic interventions.
Tier 1: This is the core curriculum and
instruction that all students receive. A key question
to ask is “Does the core curriculum meet the needs
of most (80-90%) students?” Whole Group/Core
Instruction is differentiated (effective teaching that
involves providing students with different ways to
acquire content, often in the same classroom) to
meet ALL students’ needs and is implemented by
the general education teacher.
Tier 2: This tier involves small group
interventions for some students (5-15%)
needing MORE instruction or supplementary
interventions. For example, intervention time
may be 30 minutes per day in addition to the Core
(Tier 1) and can be implemented by the general
education teacher or other trained and qualified
staff (math specialist teacher, related services staff,
paraprofessional). These interventions are usually
Student Progress Monitoring is a scientifically
based practice that is used to frequently assess
students’ academic performance and evaluate the
effectiveness of instruction. Progress monitoring
procedures can be used with individual students
or an entire class.
Scientific, Research-Based Instruction refers to
specific curriculum and educational interventions
29
that have been proven to be effective; that is, the
research has been reported in scientific, peerreviewed journals.
special education. Often, these students did not
demonstrate significant discrepancies between
their achievement and intellectual ability until
the third grade; therefore, leaving use of the
discrepancy model coined the “wait to fail
model.” In 2004, the Individuals with Disabilities
Education Act (IDEA) was reauthorized to reflect
an important change in the way schools meet
individual student need(s). An emphasis was
placed on early intervention services for children
who are at risk for academic or behavioral
problems. Schools can no longer wait for students
to fail before providing intervention. Instead,
they should employ a problem-solving model to
identify and remediate areas of academic concern.
It is important to the Tennessee Department of
Education that the RTI² framework represents
a continuum of intervention services in which
general education and special populations work
collaboratively to meet the needs of all students.
This includes shared knowledge and commitment
to the RTI² framework, its function as a process
of improving educational outcomes for ALL
students, and its importance to the department
to meet requirements related to the Individuals
with Disabilities Education Act (IDEA) and the
Elementary and Secondary Education Act (ESEA).
RTI in Tennessee Schools
http://state.tn.us/education/speced/doc/rtimanual.
pdf
Introduction
The role of the public education system is to
prepare ALL students for success after high school.
The Tennessee Department of Education (TDOE)
believes that the framework surrounding positive
outcomes for ALL students in Tennessee is the
Response to Instruction and Intervention (RTI²)
model. This framework integrates Common Core
State Standards, assessment, early intervention,
and accountability for at risk students in the belief
that ALL students can learn.
What is RTI²?
The RTI² framework is aligned with the
department’s beliefs and allows for an integrated,
seamless problem-solving model that addresses
individual student needs. This framework relies
on the premise of high-quality instruction and
interventions tailored to student need where
core instructional and intervention decisions are
guided by student outcome data. In Tennessee,
the education system will be built around a
tiered intervention model that spans from
general education to special education. Tiered
interventions in the areas of reading, math, and/
or writing occur in general education depending
on the needs of the student. If a student fails
to respond to intensive interventions and is
suspected of having a Specific Learning Disability,
then the student may require special education
interventions (i.e. the most intensive interventions
and services). As always, parents reserve the right
to request an evaluation at any time.
Individuals with Disabilities Education
Act (IDEA), as reauthorized in 2004, states
that a process that determines whether the
child responds to scientific, research-based
interventions may be used to determine if a
child has a specific learning disability. IDEA also
requires that an evaluation include a variety of
assessment tools and strategies and cannot rely
on any single procedure as the sole criterion for
determining eligibility.
A Response to Instruction and Intervention (RTI²)
method will now be used (effective July 2014) to
determine whether a child has a specific learning
disability (SLD) in basic reading skills, reading
comprehension, reading fluency, mathematics
calculation, mathematics problem solving,
Historically, the primary option available to
students who were not successful in the general
education classroom was a placement in
30
or written expression for students in grades
K-12. Other areas of SLD including listening
comprehension and oral language, in addition to
behavioral concerns, may be added in the future.
Parent contact is an essential component of
RTI² and reinforces the culture of collaboration.
A variety of means to reach parents may be
used, including: automated phone systems,
electronic mail, US Mail, and student-delivered
communications. Local Education Agencies
(districts and schools) must designate a person to
coordinate and/or make contact with parents at
the school level.
The RTI² Framework is a model that promotes
recommended practices for an integrated system
connecting general and special education by the
use of high-quality, scientifically research-based
instruction and intervention. The RTI² framework
is a 3-Tier model that provides an ongoing
process of instruction and interventions that allow
students to make progress at all levels, particularly
those students who are struggling or advancing.
This person must contact parents for each
of the following reasons: before initiating
or discontinuing tiered interventions, to
communicate progress monitoring data in
writing every 4.5 weeks for students receiving
tiered interventions, in the event there is a referral
to special education, and regarding the dates and
duration of universal screenings.
The Tennessee RTI² Model (on the following
page) is a picture of a well-run RTI² system. It
represents the goal of what an RTI model will
look like. When Tier I instruction is functioning
well, it should meet the needs of 80-85% of the
student population. Only 10-15% of the student
population should need Tier II interventions and
only 3-5% should need Tier III interventions.
31
Tennessee
Guiding Principles
RTI Model
2
TS
80-85%
10-15%
In ADDITION to Tier I, interventions are
provided to students that fall below the
25% percentile on universal screening and are
struggling academically and/or behaviorally.
Research-based interventions will be provided to
students within their specific area(s) of deficit. These
students are progress monitored using a tool that is
sensitive to change in the area of deficit and that
provides a Rate of Improvement (ROI) specific
to the individualdeficit.
TIER III
FEW
FOR
STU
EVALUATE
Data-Based
Decision
Making
ORT
DEFINE
ANALYZE
INC
REA
IMPLEMENT
UPP
TIER II SOME
DEN
All students receive researched-based,
high-quality, general education
instruction using Common Core
Standands in a positive behavior
environment that incorporates ongoing
universal screening and ongoing assessment
to inform instruction.
GS
ALL
SIN
TIER I
• Leadership
• Culture of Colloboration
• Prevention and Early Intervention
3-5%
In ADDITION to Tier I, interventions are provided to
students who have not made significant progress in Tier II,
are 1.5-2.0 grade levels behind or are below the 10th percentile.
Tier III interventions are more explicit and more intensive than
Tier II interventions. Researched-based interventions will be
provided to students within their specific area(s) of deficit.
These students, who are struggling academically and/or behaviorally,
are monitored for progress using a tool that is sensitive to change
in the area of deficit and that provides a Rate of Improvement (ROI)
specific to the individual deficit.
32
RTI2 Decision-Making Process
UNIVERSAL SCREENING
Does not meet
gradel-level expectations
(below 25th percentile)
TIER I
Ready for
grade-level instruction
Exceeds advanced
expectations
Core Instruction 80-85%
• High quality instruction aligned to Common Core Standards
• Instructional decisions driven by ongoing formative assessment
• High quality professional development and support
Ongoing Assessment
Does not meet
grade-level expectations
If student
is more than
1.5-2 years
behind, may
need Tier III
intervention
TIER II
Meets grade-level
expectations
Exceeds grade-level
expectations
Provide
enrichment
Targeted Intervention 10-15%
• Addresses the needs of struggling and advanced students
• Additional time beyond time allotted for the core instruction
• HIgh quality intervention matched to student-targeted area of need
• Provided by highly trained personnel
Progress monitoring required for data-based decision making
Does not meet
grade-level expectations
TIER II
Meets grade-level
expectations
Targeted Intervention 3-5%
• Addresses small percentage of struggling students
• More explicit and more intensive intervention targeting specific
area of need
• Intervention provided by highly trained professionals
Progress monitoring required for data-based decision making
Does not make
significant progress
Makes significant
progress
Consider possible need for Special Education referral
after Tier II and Tier III interventions and fails to
make adequate progress based on gap analysis.
33
Sixth through Twelfth Grade (6-12)
Mathematics
While the Common Core State Standards specifies
the content necessary for all students to become
college and career ready, we recognize that not
every student moves at a uniform pace to meet that
goal.
The following is an example of Tier I
instruction in 6-12 Mathematics:
Tier I instruction in mathematics should be 90
minutes (55 minutes if on traditional schedule) of
uninterrupted instructional time. Students should
receive regular, systematic, direct instruction
from the teacher. The teacher should demonstrate
problem-solving strategies, provide models
for different representations of mathematical
concepts, and develop the students’ mathematical
vocabulary.
In 6-12 Mathematics, the core curriculum
(or Tier I) addresses the needs of all students.
Flexible small groups may be used. Instruction
in 6-12 should be student-focused, with constant
opportunities to engage in mathematical thinking
and reasoning. As teachers shift toward a balance
of conceptual understanding, procedural fluency,
and application, they will engage students in
a variety of tasks and activities that address
specific goals, always embedding the Standards
for Mathematical Practice in all instruction and
assessments. Problem solving will be at the heart
of the mathematics classroom. Students will have
the opportunity to make sense of mathematical
concepts on their own and regularly discuss
their ideas with peers. Teachers will frequently
assess student understanding and press students
toward the mathematical goals and essential
understanding without telling students how to
solve problems. Teachers will orchestrate classroom
discussions that promote connections between
student ideas and multiple representations
for deeper understanding. Students will have
regular practice and support in demonstrating
fluency with both number facts and algebraic
manipulation. Students will have the opportunity
to apply problem-solving skills in new and
unfamiliar contexts and situations.
6-12 Mathematics Minimum
6-8
Recommended Instructional (traditional)
Times: Tier I
Mathematics 55 (daily)
Students should be given time to work individually
to build perseverance in problem solving.
Depending on the students, teachers can work
to develop perseverance by starting with a short
private think time a few times a week and then
increase in frequency and duration. Teachers
should work to develop students who are able to
sustain productive individual engagement in a task
for 6-8 minutes daily.
Students should spend time in small groups of
3-5 students discussing and sharing ideas on a
regular basis. Students can explore mathematical
ideas together and listen to other students’ ideas
as they begin to develop mathematical reasoning
and arguments. Small group time can also be
stations set up for students to work individually
or collectively on specific skills according to the
needs of the students as determined by the teacher
through frequent formative assessment data. It is
recommended that the teacher work to interact
with as many groups as possible daily and have
contact with individual students at least every
other day.
6-8
(block)
9-12
(traditional)
9-12
(block)
90
55 (daily)
90
34
Students should also engage productively in
whole class discussion facilitated by the teacher
where they can share ideas and demonstrate their
reasoning to the class. Students will learn how to
present and defend their ideas as well as listen to
and critique the reasoning of others in a respectful
manner.
or are below the 10th percentile and require the
most intensive interventions immediately. Students
at this level should receive daily, intensive, small
group, or individual interventions targeting
specific area(s) of deficit, which are more intense
than interventions received in Tier II.
Tier II in 6-12 Mathematics:
Tier II addresses the needs of struggling and
advanced students. Advanced students should
receive reinforcement and enrichment. Students
who require assistance beyond the usual time
allotted for Tier I instruction should receive
additional intensive small group attention.
Teachers should use the CCSS to identify standards
from previous grades that might be prohibiting
a student from accessing grade-level standards.
Research indicates that students’ struggles in
mathematics are often attributed to a lack of
conceptual understanding of number sense. It is
important to diagnose specific student deficiencies
through carefully designed assessments in order
for the proper support to be given. Students who
struggle with fluency can oftentimes continue
to learn grade-level concepts. In this case, Tier II
intervention should target the necessary fluencies
to support conceptual understanding.
Tier III in 6-12 Mathematics:
Tier III addresses 3-5 percent of students who have
received Tier I instruction and Tier II intervention
and continue to show marked difficulty in
acquiring necessary mathematics skill(s). It could
also include students who are 1.5 to 2 years behind
Consideration for Special Education
A referral for special education for a specific
learning disability (SLD) in basic reading
skills, reading fluency, reading comprehension,
mathematics calculation, mathematics problem
solving, or written expression will be determined
when the data indicate that Tier III is ineffective.
Information obtained from any screenings
completed during the intervention process may
be used as part of the eligibility determination
following informed written parental consent.
Consent for an evaluation may be requested or
received during Tier III interventions, but evidence
from Tier III must be a part of determination, and
a lack of response to Tier III interventions may not
be pre-determined. An evaluation for SLD may be
in conjunction with the second half of Tier III but
may not be concluded before Tier III interventions
are proven ineffective at the end of Tier III.
Team members involved in making a decision to
refer for special education may include:
• School psychologist
• Principal or other designee
• Intervention/Support team members
Parents must be invited to a meeting to discuss a
referral for special education evaluation.
35
Additional RTI Resources for Parents
A Parent’s Guide to Response to
Intervention
http://www.ncld.org/learning-disability-resources/
ebooks-guides-toolkits/parent-guide-responseintervention
Millions of school-age children experience
difficulties with learning.
Their struggles in
school may be due to factors such as cultural or
language differences, poor attendance or a lack of
appropriate instruction. In some cases, a learning
disability can make learning difficult for a child.
The National Center for Learning Disabilities’
RTI Action Network has developed this guide for
parents and schools involved in implementing
response to intervention (RTI) in the elementary
grades. As schools work to implement this new
approach, some confusion may arise, so parents
should feel free to ask questions and raise concerns
along the way.
The Parent’s Guide to RTI includes Parent
Perspectives, Glossary of terms, Explanations
of the Tiered model and why it works, sample
intervention plans, and checklists and worksheets.
The ABC’s of RTI
http://www.rti4success.org/resources/familyresources
List of resources for families who want to
understand RTI, Progress Monitoring, and how
families can help their children.
Response to Intervention (RTI): A Primer
for Parents
http://www.nasponline.org/resources/factsheets/
rtiprimer.aspx
National Association of School Psychologists
(NASP) Key Terms and Components of RTI
explained to parents.
Tennessee State Personnel Development
Grant (SPDG) Math Resources for RTI2
http://www.tnspdg.com/math/
RTI Action Network: What is RTI?
http://www.rtinetwork.org/learn/what/whatisrti
36
Strategies for Families to Help Struggling Learners
—Compiled by Karen Harrison, Support and Training for Exceptional Parents (STEP)
I
f your children are struggling with math,
following are some ideas about what you can
do to help them to gain understanding and be
more successful. Although the following tips were
written for students with disabilities, many of these
ideas apply to any student who is struggling with
challenging learning.
1. Ask questions of school personnel.
• What is my child’s grade level in math? What
does that mean that he/she can do?
• What areas need improvement?
• Are you using a specific program to teach my
child? If so, what skills does this program focus
on? How did you determine this math approach
would meet my child’s learning style?
• What specific kinds of things are you doing to
help my child succeed in math? (for example,
support by a math specialist, technology,
providing different materials, teaching skills they
can apply independently).
• What can I do at home to help my child with
math mastery?
those accommodations affect students’ learning
and their performance on tests? The challenge
for educators and families is to decide which
accommodations will help students learn new
skills and knowledge—and which will help them
demonstrate what they’ve learned. The Online
Accommodations Bibliography at the National
Center on Educational Outcomes is a good
source of information on the range of possible
accommodations and the effects of various testing
accommodations for students with disabilities.
http://www.cehd.umn.edu/NCEO/OnlinePubs/
AccommBibliography/AccomStudies.htm.
What helps one student may not address another’s
needs at all. Decisions about accommodations
must be made on an individualized basis, student
by student, by the IEP team. Students can help
inform these decisions by talking with the team
about what works best for them. Involving
students and parents in the process of determining
goals and respecting their voices about which
accommodations might best help them achieve
those goals recognizes them as valued participants
and can ultimately lead to feelings of increased
control and responsibility in their education. For
more information on this topic, see http://nichcy.
org/research/ee/assessment-accommodations.
2. Gather specific information on how
your child learns new information. Ask for
a learning style inventory to be completed. Talk to
previous teachers about ways your child learned
new information, retained the information, and
demonstrated what they had learned. Ask that
this information be placed in the Individualized
Education Program (IEP).
4. Consider the need for an Assistive
Technology evaluation. Completing an
Assistive Technology evaluation may provide
critical information on multiple ways to engage
and teach students with disabilities, as well
ways for students to demonstrate competency.
The only way to know what a child needs is to
evaluate what factors are affecting the student. For
example, one method for assessing the need for
assistive technology is the SETT method (Student,
Environment, Tasks, Tools), which includes
questions that address which students need
3. Consider multiple ways for your
child to demonstrate what he or she
knows. A critical part of teaching and assessing
students with disabilities is providing them
with accommodations that support learning
and that support their ability to show what they
know and can do. But what accommodations
are appropriate for which students, and how do
37
assistive technology, which kinds of technology
are needed, and who should make the decisions.
For more information about the SETT framework,
go to http://www.atto.buffalo.edu/registered/
ATBasics/Foundation/Assessment/sett.php.
5. Consider the impact of the student’s
disability and how it will affect mastery of
math. If there could be a direct relationship, talk
about the issues and solutions as part of the IEP
process. It is important to look at the underlying
cause of a student’s difficulties, and then choose
tools, techniques, or technology for intervention.
For example:
• Poor visual processing can affect how students
line up numbers
• Writing difficulties may impact students’ abilities
to write symbols and fractions
• Difficulties with language in determining key
information can impact comprehension of math
problems
• Poor recall abilities, short or long term memory
deficits may affect math success
• Attention issues which impact following
sequential steps may be a barrier
• Struggling readers, those below grade level in
reading, and students with comprehension
deficits will struggle with math. There
is a significant amount of reading and
comprehension required for math mastery.
Address this with the Individualized Education
Planning (IEP) team or your child’s teacher.
• For more information about how you can help
your child in math, go to the STEP (Support
and Training for Exceptional Parents) website at
http://www.tnstep.org/.
6. Learn about evidence-based practices
and ask that they be used to assist your
child in math mastery. When it comes time to
determine how to best teach math to a particular
student, it is important to select an instructional
intervention that supports the educational goals
of the student based on age, needs, and abilities.
Research findings can and do help identify effective
and promising practices, but it’s essential to
consider how well-matched any research actually
is to your local situation and whether or not a
specific practice will be useful or appropriate for
a particular classroom or child. Interventions are
likely to be most effective when applied to similar
content, in similar settings, and with the age groups
intended for them. Teachers and school staff may
make the major suggestions and decisions about
best practices for teaching your child, but you are
your child’s best advocate and it is important for
you to understand what they are suggesting and
deciding for your child and to add your input to
the decision making process.
7. Sample evidence-based strategies and
tools. In a search of evidence-based strategies
and tools from educational research organizations
(see resource citations in number 8 below), we
found several resources that have the following
recommendations for strategies in common:
Providing systematic and explicit instruction:
Systematic instruction focuses on teaching
students how to learn by giving them the tools
and techniques that efficient learners use to
understand and learn new material or skills.
Systematic instruction, sometimes called “strategy
instruction,” refers to the strategies students learn
that help them integrate new information with
what is already known in a way that makes sense
and be able to recall the information or skill later,
even in a different situation or place. Systematic
instruction is particularly helpful in strengthening
essential skills such as organization and attention,
and often includes:
• Memory devices, to help students remember the
strategy (e.g., a first-letter mnemonic created by
forming a word from the beginning letters of
other words);
• Strategy steps stated in everyday language and
beginning with action verbs (e.g., read the
problem carefully);
• Strategy steps stated in the order in which they
are to be used (e.g., students are cued to read the
38
word problem carefully before trying to solving
the problem);
• Strategy steps that prompt students to use
cognitive abilities (e.g., the critical steps needed
in solving a problem). http://nichcy.org/
research/ee/math.
students’ growth and helps them fine-tune their
instruction to meet students’ needs.
Student “think-alouds”: The process of
encouraging students to verbalize their thinking—
by talking, writing, or drawing the steps they used
in solving a problem— was consistently effective.
http://www.nctm.org/news/content.aspx?id=8452.
Teaching visual representation of functions
and relationships, such as manipulatives,
pictures, and graphs: Visual representations
Self-instruction: Self-instruction refers to a
(drawings, graphic representations) are used by
teachers to explain and clarify problems and by
students to understand and simplify problems.
When used systematically, visuals have positive
benefits on students’ mathematic performance.
Manipulatives are objects that can help students
understand abstract concepts in mathematics
(may be actual blocks, coins, rods, or computerbased items), and using pictures and graphs
can help children see and better understand the
relationships between math concepts.
variety of self-regulation strategies that students
can use to manage themselves as learners and
direct their own behavior, including their
attention. Learning is essentially broken down into
elements that contribute to success:
• setting goals
• keeping on task
• checking your work as you go
• remembering to use a specific strategy
• monitoring your own progress
• being alert to confusion or distraction and taking
corrective action
• checking your answer to make sure it makes
sense and that the math calculations were
computed correctly.
Providing peer-assisted instruction: Students
with learning disabilities sometimes receive
some type of peer assistance or one-on-one
tutoring in areas in which they need help. The
more traditional type of peer-assisted instruction
is cross-age, where a student in a higher grade
functions primarily as the tutor for a student in
a lower grade. In the newer within-classroom
approach, two students in the same grade tutor
each other. In many cases, a higher performing
student is strategically placed with a lower
performing student but typically, both students
work in both roles: tutor (provides the tutoring)
and tutee (receives the tutoring).
When students discuss the nature of learning in
this way, they develop both a detailed picture of
themselves as learners (known as metacognitive
awareness) and the self-regulation skills that good
learners use to manage and take charge of the
learning process.
Using ongoing, formative assessment:
Formative assessment is a range of procedures
employed by teachers during the learning process
in order to modify teaching and learning activities
to improve student attainment. Ongoing formative
assessment and evaluation of students’ progress
in mathematics can help teachers measure their
39
8. Other sources of information. To better
understand the evidence base for math and other
educational interventions, use these sources of
information:
• What Works Clearinghouse
http://ies.ed.gov/ncee/wwc/
• Best Evidence Encyclopedia (BEE)
http://www.bestevidence.org
• Center on Instruction
http://www.centeroninstruction.org/index.cfm
Citations and Acknowledgements
Thank you to all of the people who have added to this publication. We appreciate the time put
into the creation of information that can help our students learn!
Reviewers: David Williams, Ryan Mathis, Jami Garner, Margy Ragsdale
Graphic Arts: Mary Revenig
Sources used in the creation of this document:
n A Parent’s Guide to Response to Intervention http://www.ncld.org/learning-disability-resources/
ebooks-guides-toolkits/parent-guide-response-intervention
n A Parent’s Guide to Response to Intervention (RTI) by Susan Bruce, Regional Education Coordinator
http://www.wrightslaw.com/info/rti.parent.guide.htm
n Arizona Department of Education: http://www.azed.gov/azcommoncore/mathstandards/
6-8math/
n Delaware Common Core Assessment Examples, 2012: http://www.doe.k12.de.us/aab/
Mathematics/assessment_tools.shtml
n Glencoe Online Pre-Algebra Examples: http://www.glencoe.com/sec/math/prealg/prealg05/
extra_example
n Graham, S., Harris, K. R , & Reid, R. (1992): Developing self-regulated learners. Focus on Exceptional
Children, 24(6), 1-16.
n Home and School Math: Homeschool.math.net
n http://nichcy.org/research/ee/assessment-accommodations
n http://www.atto.buffalo.edu/registered/ATBasics/Foundation/Assessment/sett.php
n http://www.ccsesa.org/index/documents/CCSParentHandbook_020411_000.doc
n IDEA Parent Guide http://www.ncld.org/learning-disability-resources/ebooks-guides-toolkits/
idea-parent-guide
n Implementing the Common Core State Standards Initiative: http:www.corestandands.org
n K-8 California’s Common Core Standards Parent Handbook: http://www.ccsesa.org/index/
documents/ccsparenthandbook_020411_000.doc
n Kansas Department of Education Math Flip Books: http://www.ksde.org/Default.aspx?tabid=5646
n Learn Zillion.com: http://learnzillion.com/common_core/math/k-8
n Lenz, B. K., Ellis, I.S. & Scanlon, D. (1996). Teaching learning strategies to adolescents and adults
with learning disabilities. Austin, TX: Pro-Ed.
n Maccini, P., & Gagnon, J. (n.d.). Mathematics strategy instruction (SI) for middle school students with
learning disabilities. Retrieved November 20, 2007, for the Access Center Web site: http://www.
k8accesscenter.org/training_resourses/massini.asp
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n Nets and Surface Areas; http://www.cimt.plymouth.ac.uk/projects/mepres/book8/bk8_6.pdf
n North Carolina Department of Education: http://www.ncpublicschools.org/acre/standards/
common-core-tools/#unmath
n Online Accommodations Bibliography: http://www.cehd.umn.edu/NCEO/OnlinePubs/
AccommBibliography/AccomStudies.htm
n Properties of Addition and Subtraction | eHow: http://www.ehow.com/info_8434648_propertiesaddition-subtraction.html#ixzz2TTMnPkw5
n Sixth Grade Math. Utah District Consortium (Jordan, Davis, Granite, Salt Lake), 2012.
http://www.graniteschools.org/depart/teachinglearning/curriculuminstruction/math/
elementarymathematics/Pages/Math6.aspx
n Spotlight on the Common Core State Standards: A series published by Education Northwest
to keep regional stakeholders informed about the Common Core initiative, March 2011. http://
educationnorthwest.org/resource/1547
n The ABC’s of RTI: http://www.nrcld.org/free/downloads/ABC_of_RTI.pdf
n TNCore: Tennesse Department of Education Common Core Standards. http:// www.tncore.org/
n Utah Department of Education Eight Grade Statistics and Probability: http://elemmath.
jordandistrict.org/files/2012/06/CF9.pdf
n Utah Department of Education Geometry: http://www.graniteschools.org/depart/
teachinglearning/curriculuminstruction/math/elementarymathematics/K6%20Support%20
Documents/6th%20Grade%20Support/Concept%20Foundation/Conceptual%20
Foundations%20-%20The%20Number%20System%20Part%202.pdf
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Notes/Questions to Ask Teachers
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Tennessee State Personnel Development Grant
Tennessee Department of Education,
Division of Special Populations
710 James Robertson Parkway
Nashville, TN 37243
This project is supported by the U.S. Department of Education, Office of Special Education
Programs (OSEP). Opinions expressed herein are those of the authors and do not necessarily
represent the position of the U.S. Department of Education.