Download 8th Grade Math Lab/Support Curriculum Writing

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Bellmore-Merrick Central High School District
Math 7-8 Support / Lab Curriculum
Aligned to Response to Intervention
Common Core Learning Standards for Math
Number Systems 7-8
Ratios and Proportional Relationships 7
Expressions and Equations 7-8
Statistics and Probability 7-8
Geometry 7-8
Functions 8
Written by
Jody Tsangaris
Supervised by Jane Boyd, Curriculum Developer
March 2012
Dr. Henry Kiernan, Superintendent
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Bellmore-Merrick Central High School District
Math 7-8 Support / Lab Curriculum
This curriculum is designed to meet the Response to Intervention framework for Tier II
which addresses targeting skill deficits with research / evidence based strategies and
tested teacher-designed learning activities. Understanding by Design is incorporated in
the form of Big Ideas and Understandings. The design of the curriculum includes:

Common Core Learning Standards for Grade 7 and 8 Mathematics as Goals.

Pre-Requisite Skills
o
What are possible weak skill areas from previous grades?
o
What skills are foundational and necessary for success?

Common Conceptual Misunderstandings/Struggles
o
What is common in the classroom?
o
What concepts need to be re-visited to construct correct thought processes?

Intervention Strategies/Formative Assessment
o
What support might students need to close the gap?
o
How will skill deficits be addressed?

Grade-Level Concerns
o
What necessary grade-level conceptual developments are necessary for
continued success?

Classroom Connections/Learning Activities/Formative Assessment
o
What learning activities will help students connect and transfer to current
standards and classroom experiences?
o
How will students see the connection between skill and foundation to the
regular education classroom?
Along with this curriculum, students have an individualized program, Academy of Math,
School Specialty, which further identifies and fills in gaps and areas of need.
The curriculum is not meant to be taught in any specific sequence nor is it bound by
specific time factors. It is organized by domain so that the teacher can easily access
curriculum related standards and goals with interventions and activities. The design of
this RtI program is for the teacher to assess and meet the needs of students as they are
uncovered.
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Table of Contents
Common Core Domains and Grade Levels
Page
Number Systems 7 & 8
6
Ratios and Proportional Relationships 7
11
Expressions and Equations 7 & 8
14
Statistics and Probability 7 & 8
20
Geometry 7
24
Functions 8
30
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Common Core Learning Standards
7.NS The Number System
Apply and extend previous understandings of operations with fractions to add,
subtract, multiply, and divide rational numbers.
1. Apply and extend previous understandings of addition and subtraction to add and subtract
rational numbers; represent addition and subtraction on a horizontal or vertical number line
diagram.
a. Describe situations in which opposite quantities combine to make 0.
For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.
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.
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.
d. Apply properties of operations as strategies to add and subtract rational numbers.
2. Apply and extend previous understandings of multiplication and division and of fractions to
multiply and divide rational numbers.
a. Understand that multiplication is extended from fractions to rational numbers by requiring that
operations continue to satisfy the properties of operations, 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 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.
c. Apply properties of operations as strategies to multiply and divide rational numbers.
d. Convert a rational number to a decimal using long division; know that the decimal form of a rational
number terminates in 0s or eventually repeats.
3. Solve real-world and mathematical problems involving the four operations with rational numbers.1
_________________
1 Computations with rational numbers extend the rules for manipulating fractions to complex fractions.
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8.NS The Number System
Know that there are numbers that are not rational, and approximate them by
rational numbers.
1. Know that numbers that are not rational are called irrational. Understand informally that every
number has a decimal expansion; for rational numbers show that the decimal expansion repeats
eventually, and convert a decimal expansion which repeats eventually into a rational number.
2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate
them approximately on a number line diagram, and estimate the value of expressions (e.g., π2).
For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2,
then between 1.4 and 1.5, and explain how to continue on to get better approximations.
Suggested Big Ideas and Understandings
Big Ideas: Systems and Operations
Understandings:
…the order of operations is a universal way of solving a mathematical expression
…there is a relationship between a number and its opposite
…every number has a place on the number line.
…properties provide the basis to rewrite equivalent expressions
…operations make it possible to logically work with numbers
…irrational numbers define a set of numbers that doesn’t function like rational numbers
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Common Conceptual
Misunderstanding/
Struggles
Pre-Requisite Skill
7th:
 Basic understanding
of operations with
fractions and
decimals
 Long division
 Understanding that a
fraction represents a
division problem
 Placement of
positive fractions on
the number line.
 A real world
understanding of
integers.
 Recognize and place
integers on a number
line
 Basic vocabulary
related to fractions
and integers:
reciprocals,
opposites, absolute
value,
 Properties of
operations for
positive rational
numbers









Mix up rules for operations: Confuse add
with multiply
Factors – reducing
Multiples – common denominators
Relationships between proper fractions,
improper fractions and mixed numbers.
Setting up and completing long division
problems from a fraction or verbal
statement
Applying the division process to the
calculator
Conceptualizing integers and the number
line – direction
Different ways to symbolize multiplication
Confusion with properties: Distributive
property – multiply the first term only,
incorrect application of integer rules





Understanding of the
set of rational
numbers
Operations with
rational numbers –
fluent in multiplying
decimals
Recognize and place
rational numbers on
a number line
Decimal place value
Correct notation of
repeating decimals
Rounding –
estimating –
approximation skills


Recreate understanding using manipulatives
(bars or circles) and recording to include
exchanging sizes and making number
connections
Recreating part to whole understanding of
fractions and applying to real world
examples like grades
What makes a “whole”?
Arrays for multiplication to show
relationships and develop understanding.
Create rules.
Explicit algorithm – Practice!!!

Verbal equivalent with multiplication

¾.
Division through patterns: 3 hershey



divided in half = 3 
fractions

Misunderstand absolute value as opposite
not the distance


8th

Intervention Strategies/Formative
Assessment

Defining the set of rational numbers to
include that repeating decimals are a part
of the set of rationals not irrationals
 Confusion with properties: Distributive
property – multiply the first term only,
incorrect application of integer rules
 Think decimals must be aligned when
multiplying decimals
 Continuous confusion over rounding
rules!!!!







2 of
3
1
 6 , Change to all
2
3 1 6
  , Pose the question,
1 2 1
How can I do this mathematically? Do
multiple examples until the pattern and rule
for dividing is established. How do I
remember which one to turn over? Would
you turn over the candy bar?
Made strong verbal connections to division.
Long division – Say it, write it, set it up or
put into calculator.
Check division with inverse operation
Commute and Associate connections with
every day understanding – traveling in a car,
combining ingredients from a recipe.
Distributive – Define with every day
understanding. Use real objects or
physically distribute – record
Bag of
candy (boys and girls) c(b+g) = cb + cg
Absolute value – physical counting
differentiating distance from direction.
Box activity – rational and irrational with
numbers on cards. Classify numbers.
Discuss similarities and differences.
You can use arrays for decimal multiplying
also.
Find another way to teach rounding!
Walking to the deli 10 blocks away. When
would you go home if it started to rain?
When would you run to the deli? What if
you were half way? Five blocks behind you
and five blocks in front? How mad would
mom be?
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Grade-Level Concerns
Classroom Connections: Learning Activities
Formative Assessment
Necessary Conceptual
Development
To Meet Grade-Level Standards
7th Grade

Re-develop using alternative approaches to
create understandings of operations
Continue to work with integer rules –
development and application.

Practice integer operations

Additive inverse – rewriting subtraction

Practice operations with positive rationals and
then apply integer rules.

Concept of additive inverse results in “0”


A number plus its inverse = zero.
Write whole numbers as a fraction and represent
its multiplicative inverse

Multiplicative inverse – a number times its
reciprocal = 1. Connect to division of a
number by itself = 1

Index cards with numbers from set of real
numbers and students place in either rational or
irrational box with explanation. Include perfect
and non-perfect square roots.

Signed rational operation concerns

8th Grade
 Complete understanding with examples of
rational numbers

Distinguish between rational and irrational
numbers
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Common Core Learning Standards
7.RP Ratios & 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 a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4
miles per hour, equivalently 2 miles per hour.
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.
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal
descriptions of proportional relationships.
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 be expressed as t = pn.
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.
3. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple
interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease,
percent error.
Suggested Big Ideas and Understandings
Big Idea: Rates, Family & Business Math
Understandings:
…unit rates portray a relationship of two quantities
…unit rate can be observed in many formats
…a constant rate can be used to determine an outcome
…relationships can be used within a proportion to calculate an unknown quantity
…most real world problems involve more than one step
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Common Conceptual
Misunderstanding/
Struggles
Pre-Requisite Skill

Some exposure
to unit rates

Preliminary
background for
proportions

Basic coordinate
plane knowledge

Changing: % to
decimal, % to
fraction

Seeing the relationship between
units in verbal problem and creating
an initial ratio

Students will be given verbal phrases
to identify what is being compared
and create ratios

Establishing a consistent
representation of ratios in a
proportion.

Create equivalent fractions from a
rate – in a table form

Practice plotting points from a table
to create a line. Identify other points
on the line.

Practice ordering and comparing
percents, decimals and fractions

Connect visual understanding of
percent to a 10 x10 grid

Connect percent understanding to
prior student knowledge of money,
benchmark fractions, and classic
percents

Student created real life word
problem dealing with %’s, fractions
and\or decimals

Practice converting

Practice all types of rounding with
money. The nearest dollar and cents.



Fraction to
decimal

Decimal to
fraction

The percent
equation and/or
the percent
proportion
Solving
proportions
using
%
is ( part )
=
of ( whole) 100
Intervention Strategies/Formative
Assessment

Knowing %’s can be represented as
decimals & fractions
Converting percents, fractions, and
decimals into other equivalent
forms
Reading and writing money: .09
versus .9 (Place value and money)


Understanding
the difference
between
rounding and
estimating
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Grade-Level Concerns
Classroom Connections: Learning Activities
Formative Assessment
Necessary Conceptual
Development
To Meet Grade-Level Standards

Understanding of a complex fraction as it
applies to a ratio – writing it as a fraction over
fraction

Read and re-write complex fractions as division
problems

Match the constant of proportionality to various
formats- tables, graphs, equations, diagrams,
and verbal descriptions

Versa Tiles for all types of conversions

Matching activity practice or memory game
with percents, decimals and fractions: find equal
partner

Connect a complex fraction to a division
problem

Exposure to a variety of formats for a single
constant of proportionality

Critical thinking about what a percent word
problem is asking the student to find

Additional word problem practice calculating
and comparing unit prices

Relate a percent less than 1 to a real life
situation

Does the answer make sense?


Does the answer have to be between 1% and
100%?
Present students with a problem and solution
and have them analyze if the solution makes
sense (can do this with or without any
calculations)

Knowing %’s can be greater than 100 or less
than 1


Applying the correct operation to the vocabulary
in the word problem(discount/subtract, Tax/add,
gratuity/add)

Understanding of family and business math
vocabulary
Students analyze word problems to determine if
they are solving for a part whole or percent (ie
Boxes are labeled part whole or percent.
Students individually or with partner will read
and analyze word problem on an index card and
determine which box to place the card into.
Whole class discussion will follow to justify
each decision.)

Support understanding of family and business
math vocabulary through matching and other
activities.
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Common Core Learning Standards
7.EE Expressions & 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 coefficients.
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, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”
Solve real-life and mathematical problems using numerical and algebraic
expressions and equations.
3. 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.
4. 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.
a. Solve word problems leading to equations of the form px + q = 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?
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.
8.EE Expressions & Equations
Work with radicals and integer exponents.
1. Know and apply the properties of integer exponents to generate equivalent numerical expressions.
For example, 32  35  33  1
3
3
1
27
.
2. Use square root and cube root symbols to represent solutions to equations of the form
x 2  p and x 3  p , where p is a positive rational number. Evaluate square roots of small perfect squares
and cube roots of small perfect cubes. Know that √2 is irrational.
3. Use numbers expressed in the form of a single digit times a whole-number power of 10 to estimate
very large or very small quantities, and to express how many times as much one is than the other.
For example, estimate the population of the United States as 3 times 10 8 and the population of the world as 7 times 10 9 , and
determine that the world population is more than 20 times larger.
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4. Perform operations with numbers expressed in scientific notation, including problems where both
decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for
measurements of very large or very small quantities (e.g., use millimeters per year for seafloor
spreading). Interpret scientific notation that has been generated by technology.
Understand the connections between proportional relationships, lines, and linear
equations.
5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two
different proportional relationships represented in different ways. For example, compare a distance-time
graph to a distance-time equation to determine which of two moving objects has greater speed.
6. Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the
equation y = mx + b for a line intercepting the vertical axis at b.
Analyze and solve linear equations and pairs of simultaneous linear equations.
7. Solve linear equations in one variable.
a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no
solutions. Show which of these possibilities is the case by successively transforming the given equation
into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b
are different numbers).
b. Solve linear equations with rational number coefficients, including equations whose solutions require
expanding expressions using the distributive property and collecting like terms.
8. Analyze and solve pairs of simultaneous linear equations.
a. Understand that solutions to a system of two linear equations in two variables correspond to points of
intersection of their graphs, because points of intersection satisfy both equations simultaneously.
b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by
graphing the equations. Solve simple cases by inspection.
For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
c. Solve real-world and mathematical problems leading to two linear equations in two variables.
For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects
the line through the second pair.
Suggested Big Ideas and Understandings
Big Idea: Truth
Understandings:
…multiple solutions may provide “truth” to a given situation
…rewriting an expression or equation using properties can help when trying to understand it
…an equation or expression symbolizes a real world situation
…very large and very small quantities can be symbolized in multiple ways
…manipulating a two linear equations reveals what they have in common
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Common Conceptual
Misunderstanding/
Struggles
Pre-Requisite Skill
7th grade
with one
step equations

Conceptual understanding of an
integer and what it represents
 Fluency
 Operations
with
fractions and
decimals (integers
are new)
7th & 8th
•Vocabulary of:
Key words
associated with each
operation
Simplifying
Evaluating
Translating
Solving
• Confusion with concepts behind
operations with integers & absolute
value
• Confusion with application of the
integer rules
• Incorrect way of writing
multiplication of a number and
variable
• Incorrect use of vocabulary with
translating
• Incorrect translation of “at most” and
“at least” with inequalities
 Consistency
•Identifying:
Variables
Coefficients
Constants
Expressions
Equations
Inequalities
• Properties of
Inverse and Equality
with the use of the
inequality symbol when solving.
Don’t change it to an equal sign!
• Confusion with fractions in solving
equations
• Struggle with application of integer
rules and inverses to solve equations
exponents correctly: 23 is
not equal to 6.
 Evaluating
Intervention Strategies/Formative
Assessment
• Use hands on integer number lines to
visualize the concept of an integer
and to redevelop understanding of integer rules
• Discuss real world situations that can
represent an integer
• Brainstorm key words for positive and
negative symbols. Emphasize direction and
“opposites”
•Use number line to develop adding and
subtracting of integers
• Use colored tiles or chips to represent groups
of integers when developing multiplying and
dividing
• Discuss the concept of a negative times
negative as the “opposite” of a negative
•Discuss and develop symbolic meanings of
multiplication and standardization of writing
mathematically
Ex. 3X4, 3Xx,3▪x,3x
•Matching column A to column B
Vocabulary word to either an example or
definition
•Practice: Reading and clarify task where
vocabulary comes up
• Using a highlighter to select one key word in
problem. Discuss importance of the chosen to
the task.
•Frayer model:
th
8 Grade
• Background
understanding of
operations with
integers
8th
• Ability to graph
from a unit table
 Understanding
 Multiplying
of base 10
by 10 as not adding a
zero – decimal moves.
Create a model that includes key word in
middle space surrounded by define,
example/picture, solve/evaluate/simplify,
explain or translate
•Example vs non-example
Textbook definition vs student’s definition
 Spend
• Analyze or find the
unit rate – given a
graph.
time expanding an exponent to develop
understanding.
 Play
with base 10, multiplying and dividing
by powers of 10. Create an equivalency table
17
Grade-Level Concerns
Necessary Conceptual
Development
To Meet Grade-Level Standards
7th
 Students might not see that a number in an
expression can be factored.

Real world problems – read and understand,
convert between equivalent number forms as
necessary, extract relevant information, identify
what the problem is asking for, and check that
the solution makes sense.

Fluency in two step equations with distribution

Solution Set of inequalities, graphing, solving
with signed rational numbers.
Classroom Connections: Learning Activities
Formative Assessment
 25
+ 5x = 5(5+x) Students exposed to different
cases of equivalent expressions using GCF (work the
distributive property backwards)
• Start with a simple statement to represent an
equation or expression. Expand the verbal statement
by elaborating and creating a complex – wordy
problem to help students see where an equation or
expression can be uncovered. (7)
• Write and solve equations to represent verbal
problems. Dry erase boards, jeopardy, matching,
competitions
• Compare and contrast equations and inequalities.
• Practice reading the inequality symbols and verbally
translation of solution.

• Practice rewriting inequalities to put in the x>y form.
8th Grade
• Holt On-line Textbook resource:
Find pages using key vocabulary words and recall the
meaning or definition
 Decide
• Versa Tiles for: translating, writing,
evaluating, and/or solving
Reading and manipulating the inequality into a
x>y form where x is on the left.
& identify key words that translate into:
Equations – as a precursor to verbal problems
•Key operation language
•Strong understanding of distributive property
with signed rational numbers, monomials, and
exponent rules
•Understanding of like terms
•Check answer by using substitution
•How to manipulate terms within a multi-step
equation
•Understanding the difference between the solution
of an equation and an inequality
• Students will understand the truth of the solution
to an inequality may depend on reversing the
inequality symbol (7th)
• Partner activity: Guess the word or picture from
other student’s description
• Physical Activity:
Student will get up and find the verbal phrase
or algebraic expression that matches their card
to another student
Also apply to like terms
•Dry erase boards:
Writing examples of:
 Expressions
 Equations
 Inequalities
 Using boards to also:
 Evaluate Expressions
 Order of operations
 Solve equations
• Partner solve and check for truth of equation and
inequality solutions
•Calculator lessons for order of operations:
Comparing answers for pencil & paper vs
four function calculator vs Scientific calculator
18
Grade-Level Concerns -- continued
Necessary Conceptual
Development
To Meet Grade-Level Standards
8th Grade
• Understanding of unit rate
• Misunderstanding of slope and the algorithm.
• Vocabulary – y intercept. How does it relate to a
line or graph?
Naming the coordinates for the y intercept
Classroom Connections: Learning Activities
Formative Assessment - continued
• Use Smart Board software – Gallery: Students
are given different pictures of graphs of lines they
can manipulate and make observations.
• Practice graphing and writing equations.
• Use of dry erase boards with coordinate planes
• Correlating the components of the equation to the • Give students models of the process and
solution to a system of equations. Have them
graph of the line.
analyze and make observations about the process.
• Know how to transform equations into y = format From there create a set of prompting questions to
ask to use the process.
Are the equations in y = format?
• Understand substitution – expands from a single
How do I transform my equation?
value to a expression
How do I set up my system?
If I find y, how do I get x? or x then y?
• Define solution or solution set
How do I represent my solution?
• How do I check my solution with a graph?
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Common Core Learning Standards
7.SP Statistics & 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.
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.
For example, 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.
Draw informal comparative inferences about two populations.
3. Informally assess the degree of visual overlap of two numerical data distributions with similar
variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of
variability.
For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the
soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the
two distributions of heights is noticeable.
4. Use measures of center and measures of variability for numerical data from random samples to draw
informal comparative inferences about two populations.
For example, decide whether the words in a chapter of a seventh grade science book are generally longer than the words in a
chapter of a fourth-grade science book.
Investigate chance processes and develop, use, and evaluate probability models.
5. Understand that the probability of a chance event is a number between 0 and 1 that expresses the
likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0
indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely,
and a probability near 1 indicates a likely event.
6. 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, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably
not exactly 200 times.
7. 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.
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.
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?
21
8. Find probabilities of compound 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 sample spaces for compound events using methods such as organized lists, tables and tree
diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the
outcomes in the sample space which compose the event.
c. Design and use a simulation to generate frequencies for compound events.
For example, use random digits as a simulation tool to approximate the answer to the question: 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?
8.SP Statistics & Probability
Investigate patterns of association in bivariate data.
1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of
association between two quantities. Describe patterns such as clustering, outliers, positive or negative
association, linear association, and nonlinear association.
2. Know that straight lines are widely used to model relationships between two quantitative variables.
For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the
model fit by judging the closeness of the data points to the line.
3. Use the equation of a linear model to solve problems in the context of bivariate measurement data,
interpreting the slope and intercept.
For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of
sunlight each day is associated with an additional 1.5 cm in mature plant height.
4. Understand that patterns of association can also be seen in bivariate categorical data by displaying
frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table
summarizing data on two categorical variables collected from the same subjects. Use relative
frequencies calculated for rows or columns to describe possible association between the two variables.
For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or
not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?
Suggested Big Ideas and Understandings
Big Idea: Data, Chance
Understandings:
...random samplings gives a picture of the population as a whole
…random sampling can be used to make predictions
…generalizations can be made about data and populations based on how it is distributed
…a number can be created that represents the chance that something will occur
…a model can be used to develop the probability of multiple events
…visually representing two sets of data reveals if there is a relationship between them
…a line can be used to analyze and predict the relationship between two sets of data
…converting data into percents can be used to compare with other sets of data
22
Common Conceptual
Misunderstanding/
Struggles
Pre-Requisite Skill
7th Grade

Interpreting information (in a table)
Know forms of
statistical plots

Not putting numbers in numerical
order before finding median
Be able to
construct a basic bar
or line graph

Mixing up definitions

Thinking mean is the best because
its asked the most often

That range is a measure of central
tendency

●

Operations with
rational numbers

Definitions of
mean, median,
mode

Rounding
Intervention Strategies/Formative
Assessment
Students will analyze data in a
variety of formats
 Compare and contrast measures of
central tendency with range
 Create visuals relating to words:
MODE = MOST
MEDIAN = MIDDLE
MEAN  “you’re average!”
 Practice defining and computing
mean, median, mode and range
▪ Develop explicit procedures for
determining central tendencies and
range
 Hands on manipulation of putting
things in order
 Practice reading and analyzing
problems which asks for the most
accurate central tendency
 Holt Textbook interactive activities
 Interactive Websites including:
Brainpop, www.purplemath.com,
www.mathbits .com,
www.kidsolr.com

8th Grade
• Set up the first
quadrant of the
coordinate and label
the axis
appropriately.
• Plot ordered pairs
• Interpreting data
• Graphed a line
• Knowledge of
slope

Labeling the axis to include scales
that pertain to the data and plotting
accurately

Not be able to read or interpret data
from a graph
• Use first quadrant transparencies with
dry erase markers to practice
representing bivariate data
• Write a story about why these two
sets of data would be compared. What
was the investigator looking for?
23
Grade-Level Concerns
Classroom Connections: Learning Activities
Formative Assessment
Necessary Conceptual
Development
To Meet Grade-Level Standards

Critical thinking about decision making and problem
solving

Making predictions

Mean absolute deviation

Students’ ability to analyze and compare two sets of
data from two different populations.

Make inferences about the populations.

Vocabulary – random sampling, etc.
Probability
 Develop an understanding of what is an “event”
 Have students look for keywords that indicate
independent/dependent events (such as w/
replacement, w/o replacement)
 Importance of thinking methodically about how to
represent choices.
 Definitions and procedures for finding theoretical
probability, outcomes, sample space
 Use of visual representations to communicate ideas
(i.e.-Make organized lists to solve problems)
 When constructing tree diagrams for compound
events, students list each event then list all the
possible outcomes for each event.
 Students are not clear about whether the events are
independent or dependent.
 Students are confused that the number of outcomes
decrease by 1 for dependent events.
 Multiplying fractions
8th Grade

Drawing the line

Interpret what the line means

Interpreting the slope and intercept

Converting data into percents

Whiteboard questioning in pairs
Versatiles (including workbooks)
 Let students put themselves in voice order (alto –
soprano) – define range.
 Now practice calculating with real data. (poll students
on: number of siblings, amount of hours spent
studying each night, number of books they carry …)
 Rounding songs: “5 or above, give it a shove, 4 or
below, let it go!”
 Real life context: collect and work with student data
to create a depth of understanding with sampling
 Create a survey and design an experiment for two
different populations. Create measures of center and
variability and compare.
 Reinforce concept and practice calculating the mean
absolute deviation

Probability






Practice making a tree diagram to count the number
of possible outcomes; Identifying an event as
independent/dependent events; Finding the
probability of independent/dependent events
Students are given manipulatives to find all possible
outcomes of 2 or more events to help distinguish
between independent/dependent events.
Explicit Instruction given on criteria for
dependent/independent events.
Graphic Organizer with criteria embedded for
decision making.
Students will use manipulatives (i.e – spinners,
playing cards, counters of various colors, marbles,
dice) to find the probabilities of
independent/dependent events
Construct tree diagrams to confirm results from the
use of the manipulatives
8th Grade
 Use stories to extend to drawing a line and
interpreting the line, slope and intercept. Use data
that relates to students. Sleep and outcome on a test.
 Activities to re-address creating percents from raw
data. Use of proportions or conversions of fractions
to decimals to percents.
24
Common Core Learning Standards
7.G 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.
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.
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.
Solve real-life and mathematical problems involving angle measure, area, surface
area, and volume.
4. 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.
5. Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem
to write and solve simple equations for an unknown angle in a figure.
6. 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.
8.G Geometry
Understand congruence and similarity using physical models, transparencies, or
geometry software.
1. Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
2. Understand that a two-dimensional figure is congruent to another if the second can be obtained from
the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a
sequence that exhibits the congruence between them.
3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures
using coordinates.
4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the
first by a sequence of rotations, reflections, translations, and dilations; given two similar twodimensional figures, describe a sequence that exhibits the similarity between them.
25
5. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about
the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity
of triangles.
For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line,
and give an argument in terms of transversals why this is so.
Understand and apply the Pythagorean Theorem.
6. Explain a proof of the Pythagorean Theorem and its converse.
7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world
and mathematical problems in two and three dimensions.
8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Solve real-world and mathematical problems involving volume of cylinders, cones,
and spheres.
9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world
and mathematical problems.
Suggested Big Ideas and Understandings
Big Ideas: Rates, Relationships, Formulas
Understandings:
…Similarity is related to scale drawings
…the characteristics of a shape define it and its name
…understanding of the formulas for the area and circumference of a circle has connections to ratios and
the area of parallelograms
…angle relationships can be used to calculate unknowns
…movement can be tracked by a set of defined steps
…discovering and applying triangle relationships makes it possible to solve problems
26
Common Conceptual
Misunderstanding/
Struggles
Pre-Requisite Skill
7th Grade
 Ratio


Basic
understanding of
shapes and
properties
Vocabulary
knowledge of
area, perimeter,
and types of
angles, etc.
8th Grade
 Coordinate
geometry
knowledge
 Basic plane
geometry
vocabulary (line,
line segment,
parallel,
congruent,
similar)
 Solving 2-step
algebraic
equations
 Understanding
inverse
operations,
especially
squaring/square
root.
 Rounding
 Understanding of
basic geometrical
shapes. 2D and
3D Names and
Properties
 Use of formulas
 Knowledge of
pairs of angle
vocabulary

Confusion with basic facts of
shapes and angles – acute, obtuse,
straight

Units to describe measurements

Calculator use

Rounding

Confusing legs with hypotenuse

Dividing by 2 instead of taking the
square root when using Pythagorean
theorem

Forgetting to put the largest value
as the “c” variable.

Mixing up volume with area and
perimeter
Intervention Strategies/Formative
Assessment

Hands on, visual representations
of basic 2D and 3D shapes: Real
objects – define characteristics,
classify

Activity to reinforce Vocabulary
for measurement – area, perimeter,
volume, surface area,
circumference, pi, appropriate
units
27
Grade-Level Concerns
Classroom Connections: Learning Activities
Formative Assessment
Necessary Conceptual
Development
To Meet Grade-Level Standards
7th Grade
 Set up and solve proportions

Use of tools to draw – rulers and protractors





Pi


Knowledge and use of formulas – substitution
and proper notation in the formula

Rounding

Ability to take information from a verbal
problem and make decisions about what it is
describing and asking.





7th Grade
Practice use of tools – rulers, protractors, free
draw.
Reading measurements
Use of SmartBoard tools. Geometer’s
Sketchpad and/or other on-line tools.
Practice with circle formulas
Symbolic representation for naming angles and
relationships:
Fully describe a geometric picture with angles
using the correct vocabulary. Solve for any
unknowns and find the measures of the angles.
Interpret verbal problems – identifying area,
versus, volume and surface area.
Practice breaking down complex figures to
include 3D shapes.
Vocabulary for angle relationships
th
8 Grade
 Vocabulary for transformations

Sequencing a series of transformations to
describe the process.

Compare and contrast similarity and congruency

Experience with angles created by parallel lines
cut by a transversal.

Develop the understanding that a formula is used
to solve for an unknown.

Visualize shapes and dimensions

Development of Pythagorean theorem

An understanding of when to use Pythagorean
theorem and correctly draw and identify the
components

Be able to apply the theorem in a two and three
dimensional situations (3D example of a prism
problem: See Holt On-Core Grade 8 Workbook
page 130)
8th Grade

Physically transform to create understanding of
the different transformations and whether the
objects properties are preserved

Use cutouts of shapes to manipulate on a
coordinate plane. Record the transformations to
go from concrete to abstract.

Opportunities to explain step by step process of
transforming a shape from one place to another
demonstrating its congruence or similarity.
(symbolically and in words)

Use physical representations of angles (cut out
or colors, pattern blocks, Smart Board) for
students to manipulate and visually see the
angle relationships

Have students cut out area of a leg (a²) and
another leg (b²) and fit them into the area of the
hypotenuse (c²)

Group students into pairs and have them solve
problems as a team.

Smartboard Interactions including jeopardy &
Hollywood squares style assessment games

Versatiles (including workbooks)
 Shoot an arrow thru the box (at the right angle)
explaining that it will always “hit” the
hypotenuse.
 Display right triangles in many different
28
Grade-Level Concerns -- continued
Classroom Connections: Learning Activities
Formative Assessment - continued
Necessary Conceptual
Development
To Meet Grade-Level Standards

rotations & identify legs.
Apply to a coordinate geometry problem to find
the distance between two points


Matching definitions game
Understanding why units are squared or cubed

Using 3-D geometrical shapes as manipulatives
to develop understanding.

Practice with verbal problems and create
diagrams, properly label, correct substitution,
calculator usage, solve equation, find the square
root, interpret the solution.
29
Common Core Learning Standards
8.F Functions
Define, evaluate, and compare functions.
1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a
function is the set of ordered pairs consisting of an input and the corresponding output.1
2. Compare properties of two functions each represented in a different way (algebraically, graphically,
numerically in tables, or by verbal descriptions).
For example, given a linear function represented by a table of values and a linear function represented by an
algebraic expression, determine which function has the greater rate of change.
3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give
examples of functions that are not linear.
For example, the function A = s2 giving the area of a square as a function of its side length is not linear because
its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
Use functions to model relationships between quantities.
4. Construct a function to model a linear relationship between two quantities. Determine the rate of
change and initial value of the function from a description of a relationship or from two (x, y) values,
including reading these from a table or from a graph. Interpret the rate of change and initial value of a
linear function in terms of the situation it models, and in terms of its graph or a table of values.
5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g.,
where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the
qualitative features of a function that has been described verbally.
_________________
1Function
notation is not required in Grade 8.
Suggested Big Ideas and Understandings
Big Ideas: Visual, Symbolic
Understandings:
…graphs represent real world situations
…linear expression has a constant change between points.
…change can be calculated and used to predict future progression of the linear situation
…a line can only pass through the y-axis at one distinct point.
…a unique point can be found either graphically or algebraically
30
Common Conceptual
Misunderstanding/
Struggles
Pre-Requisite Skill





Ability to graph
from a unit table
Analyze or find
the unit rate –
given a graph.
Understanding of
a linear equation
Coordinate plane
experience
Seeing unit rate as a constant so it
can be related to slope
Intervention Strategies/Formative
Assessment

Ask students to graph from a table
representing real world data and
make observatioms
31
Grade-Level Concerns
Classroom Connections: Learning Activities
Formative Assessment
Necessary Conceptual
Development
To Meet Grade-Level Standards

Defining a function

Use of coordinate geometry dry erase boards

Creating a table of values from a function

Large post-it coordinate planes

Interpret the equation of a line. Connect unit
rate with constant of proportionality with slope
– rate of change. Initial value – intercept

Use of on-line sites that include visual and
auditory components


Creating a function given two points, a table or
a graph
Using larger representations allowing students
to count and compare


Examine and describe linear versus non-linear
Use the graphing calculator to check results of
graphing a line

Use graphing calculator and Smart Board to
examine non-linear equations and describe
observations and the differences.

Analyzing positive and negative slope and
comparing two lines where equations are the
same except opposite slopes

Given a linear equation, identify slope and yintercept. Graph it from the equation. Check it
by substituting and creating a table. Create a
step by step process.

Given the graph of a line, identify the slope, yintercept and create the equation.

Graph two lines and describe the relationship
and “solution”