:
Campus: CJH
Author(s): Black, Dennis, Harvell, Lawrence
Date Created / Revised: 9/28/15
Six Weeks Period: 2nd six weeks
Grade Level & Course: 7th grade math
Timeline: 12 days
Unit Title: Proportional Reasoning with Ratios, Rates and
Percents
Stated
Objectives:
TEK # and SE
Lesson
# 1 of 1
7.1A - Apply mathematics to problems arising in everyday life, society, and the workplace.
7.1B - Use a problem-solving model that incorporates analyzing given information, formulating a plan or
strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and
the reasonableness of the solution.
7.1C - Select tools, including real objects, manipulatives, paper and pencil, and technology as
appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to
solve problems.
7.1D - Communicate mathematical ideas, reasoning, and their implications using multiple representations,
including symbols, diagrams, graphs, and language as appropriate.
7.1E - Create and use representations to organize, record, and communicate mathematical ideas.
7.1F - Analyze mathematical relationships to connect and communicate mathematical ideas.
7.1G - Display, explain, and justify mathematical ideas and arguments using precise mathematical
language in written or oral communication
7.3A - Add, subtract, multiply, and divide rational numbers fluently.
Supporting Standard
7.3B - Apply and extend previous understandings of operations to solve problems using
addition, subtraction, multiplication, and division of rational numbers.
7.4A - Represent constant rates of change in mathematical and real-world problems given pictorial,
tabular, verbal, numeric, graphical, and algebraic representations, including d = rt.
7.4B - Calculate unit rates from rates in mathematical and real-world problems.
7.4C - Determine the constant of proportionality (k = y/x) within mathematical and real-world problems.
7.4D - Solve problems involving ratios, rates, and percents, including multi-step problems involving
percent increase and percent decrease, and financial literacy problems.
7.4E - Convert between measurement systems, including the use of proportions and the use of unit rates.
7.10A Write one-variable, two-step equations and inequalities to represent constraints or conditions within
problems.
7.10B Represent solutions for one-variable, two-step equations and inequalities on number lines.
7.10C Write a corresponding real-world problem given a one-variable, two-step equation or inequality.
7.11A Model and solve one-variable, two-step equations and inequalities.
7.11B Determine if the given value(s) make(s) one-variable, two-step equations and inequalities true.
7.11C Write and solve equations using geometry concepts, including the sum of the angles in a triangle,
and angle relationships.
7.13A - Calculate the sales tax for a given purchase and calculate income tax for earned wages.
Supporting Standard
7.13B - Identify the components of a personal budget, including income; planned savings for college,
retirement, and emergencies; taxes; and fixed and variable expenses, and calculate what percentage
each category comprises of the total budget.
7.13E Calculate and compare simple interest and compound interest earnings.
7.13F Analyze and compare monetary incentives, including sales, rebates, and coupons.
See Instructional Focus Document (IFD) for TEK Specificity
Key
Understandings
Numbers are an efficient way to represent quantities and numeric relationships.
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Why is it important to understand the value of numbers?
What relationships exist between and within numbers, and how are they used?
Why is it important to understand rational numbers?
How are different forms of rational numbers used in everyday situations?
Algebraic reasoning facilitates representing, generalizing, and formalizing patterns and relationships in
everyday life.
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How can situations be identified and described algebraically?
Proportional reasoning can be used to describe and solve problems in everyday life.
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How can proportional reasoning be used to make predictions and comparisons in problem
situations?
Knowledgeable consumers and investors develop an economic way of thinking and problem solving.
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How does financial literacy affect financial responsibility and long-term goals?
A problem-solving model can be applied to critically reason through various problem situations in order to
solve problems and analyze solutions.
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How can the information in a problem be analyzed to determine the question being asked and the
relevant information provided and/or needed?
What types of plans and/or strategies can be used to solve problems?
How can solutions to problems be determined?
How can solutions to problems be justified?
How can the reasonableness of solutions and the problem solving process be evaluated?
Misconceptions
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Key Vocabulary
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Some students may think the constant rate of change and the constant of proportionality are
always the same value rather than understanding the constant of proportionality is represented
by
and may equal the constant rate of change for the linear equation y = mx + b only if b =
0.
Some students may only use additive thinking rather than multiplicative thinking when solving
proportions.
Some students may think ratios and rates may not be represented on a graph rather than
realizing all ratios and rates can be viewed as ordered pairs.
Some students may generate an “equivalent” ratio by exchanging the numbers in a ratio without
their appropriate labels rather than interpreting the ratio as a comparison that must maintain the
same relationship. (e.g., 2 girls:3 boys is not equivalent to 3 girls:2 boys)
Some students may think that the order of the terms in a ratio or proportion is not important.
Some students may think that generating an equivalent ratio is different from generating an
equivalent fraction.
Some students may think that all ratios are fractions, rather than understanding that a ratio may
represent a part-to-part or part-to-whole relationship.
Some students may think that rates are not related to ratios.
Some students may think that a unit rate must have a denominator of one rather than
understanding that a unit rate is a ratio between two different units where one of the terms is one.
Some students may not make the connection between the constant rate of change r, in d = rt, to
the constant of proportionality, k, in y = kx.
Some students may not connect the constant rate of change to m in the equation y = mx + b.
Appreciation – the increase in value over time
Budget – a monthly or yearly spending and savings plan for an individual, family, business, or
organization
Commission – pay based on a percentage of the sales or profit made by an employee or agent
Constant rate of change – a ratio when the dependent, y-value, changes at a constant rate for
each independent, x-value
Constant of proportionality – a constant positive ratio between two proportional
quantities
denoted by the symbol k
Expense – payment for goods and services
Fixed expenses – expenses that occur regularly and do not vary month to month
Income – money earned or received
Income tax – a percentage of money paid on the earned wages of an individual or business for
the federal and/or state governments as required by law
Markdown – the difference between the original price of an item and its current price
Markup – the difference between the purchase price of an item and its sales price
Payroll tax – a percentage of money that a company withholds from its employees for the federal
government as required by law
Percent – a part of a whole expressed in hundredths
Percent decrease – a change in percentage where the value decreases
Percent increase – a change in percentage where the value increases
Positive rational numbers – the set of numbers that can be expressed as a fraction
,
where a and b are whole numbers and b≠ 0, which includes the subsets of whole numbers and
counting (natural) numbers (e.g., 0, 2,
etc.)
Principal – the original amount invested or borrowed
Property tax – a percentage of money collected on the value of a property for the local
government as required by law
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Rate – a multiplicative comparison of two different quantities where the measuring unit is different
for each quantity
Ratio – a multiplicative comparison of two quantities
Rational numbers – the set of numbers that can be expressed as a fraction ,
where a and b are integers and b ≠ 0, which includes the subsets of integers, whole numbers, and
counting (natural) numbers (e.g., -3, 0, 2,
,
etc.).The set of rational numbers is
denoted by the symbol Q.
Sales tax – a percentage of money collected by a store (retailer), in addition to a good or service
that was purchased, for the local government as required by law
Savings for college – money saved for continuing education beyond high school
Savings for emergencies – money save for unexpected expenses (e.g., car repairs, emergency
healthcare, etc.)
Savings for retirement – money saved over the period of time an individual is employed to be
spent once the individual retires from their occupation
Simple interest – interest paid on the original principal in an account, disregarding any previously
earned interest
Tax – a financial charge, usually a percentage applied to goods, property, sales, etc.
Taxes – money paid to local, state, and federal governments to pay for things the government
provides to its citizens
Tip – an amount of money rendered for a service, gratuity
Unit rate – a ratio between two different units where one of the terms is
Variable expenses – expenses that occur regularly but vary month to month and can usually be
controlled by an individual
Suggested Day
5E Model
Instructional Procedures
Day 1
Oct. 5, 2015
Topic: Unit Rates
Objective of the Day: I will be able to find and use unit rates.
Engage/Explore
Explain/Engage
Elaborate/
Evaluate
Speed Drill
(Engage, Explore, Explain, Extend/Elaborate, Evaluate)
GO Math: Student
Edition: 2-1 pages
61-63
Student Notes – Unit
Rate
VOCABULARY:
A rate is a comparison of two quantities that have different units, such as miles
and hours. Rates are often expressed as unit rates, that is, with a denominator
of 1 unit.
The constant of proportionality is a constant positive ratio between two
proportional quantities
Materials, Resources,
Notes
denoted by the symbol k.
A complex fraction is a fraction that has a fraction in its numerator,
denominator, or both.
Engage: Students calculate unit rates to find and compare which cheese slices
has the best deal.
 Find the best deal:
Sargento Cheese Slices: $2.48 for 10 slices
Velveeta Cheese Slices: $3.18 for 12 slices
Explore: Commonly used rates like miles per hour make it easy to understand
and compare rates.
Practice – Unit Rate
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Students complete the Explore Activity page 61 “Student Edition” Go
Math
Explain: You can use unit rates to simplify rates that appear complicated, such
as those containing fractions in both the numerator and denominator. Complete
Student Notes – Unit Rate as a class.
Extend/Elaborate: Students complete Practice – Unit Rate.
Evaluate/Closing: How do you find and use unit rates?
Day 2
Oct. 6
Topic: Constant Rates of Change
Objective of the Day: I will be able to represent constant rates of change.
Engage/Explore
Explain/Engage
Elaborate/
Evaluate
Speed Drill
SPIRAL BACK:
Vocabulary:
Engage: Ask students: When you walk, can you keep a steady pace?
Explore: Many real-world situations can be described by proportional
relationships. Proportional relationships have special characteristics.
 How can you identify and represent proportional relationships?
 Students complete, Explore Activity, page 67-68 “Student Edition” Go
Math
Explain: Students will learn to identify and represent proportional relationships.
Teacher completes with students: Proportional Relationships: pages 68-69
Student Edition “Go Math”
Extend/Elaborate: Students will practice Proportional Relationships on pages
69-72 Student Edition “Go Math”
Evaluate/Closing: How can you identify and represent proportional
relationships?
2-2 Pages 67-69
Student Edition “Go
Math”
Day 3
Oct. 7
Engage/Explore
Topic: Proportional Relationships and Graphs
Objective of the Day: I can represent constant rates of change in graphical
representations.
Speed Drill
SPIRAL BACK:
2-3 Pages 73-74
Student Edition “Go
Math”
Pick A Card
Unit 3 Activities,
BUNDLEproportional
reasoning math
statistics – PG 64 Speed Drill located in
p:share
Graph paper
Engage: Ask: How much water do you think you use when you take a shower?
Take a guess.
Explore:
 Students learn to graph and recognize proportional relationships.
 Point out to students that when they graph the points from the table, the
points form a line that extends through the origin.
 Teacher and students complete together, Pages 73-74 Student Edition
“Go Math”
Activity: Proportional Relationships: Pick a card (Unit 3 activities). Have
students graph on a graph paper if they do not have enough room.
Day 4
Oct. 8
Topic: Proportions
Objective of the Day: I can use proportions to solve problems.
Student Notes –
Perfect Paint
Engage/Explore
Explain/Extend/
Evaluate
Speed Drill
Practice – Perfect
Paint
SPIRAL BACK:
Engage: Read the story in Student Notes – Perfect Paint. Have students write
the ratio of red drops to blue drops.
Explore/Explain: Students will complete the table for #2. Explain that the scale
factor is the number you multiply the numerator and denominator by to create an
equivalent ratio. Teach the students how to set up a proportion using the WON
tic-tac-toe strategy. “W” stands for the word ratio, “O” stands for the original
ratio in the problem, and “N” stands for the New Ratio which includes a variable.
Complete # 3 – 6 using this strategy.
Extend: Students will complete #7-10 with a partner. Then discuss the
answers.
Evaluate/Closing: Students will complete Practice – Perfect Paint.
Day 5
Oct. 9
Topic: Proportions
Objective of the Day: I can solve problems involving ratios.
Reference: Texas
GO Math pgs. 85-90
Explore
Explain/Extend/
Evaluate
Explore/Explain: Solve problems 1-4 of Student Activity – Proportions as a
class.
Student Activity –
Proportions
Extend: Allow the students to work in partners solving problems 5 – 9.
Practice –
Proportions
Evaluate: Practice – Proportions.
Closing: How is the WON strategy used to solve proportions?
Day 6
Oct. 12
Proportions Quiz
Proportions quiz
Topic: Percents
Objective of the Day: I will use the percent bar and percent equation to find
parts of a whole that equal a percentage of the total in order to answer real
world questions with percents.
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Evaluate
Day 7
Oct. 13
Engage/Explain
Extend/
Elaborate/
Evaluate
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Speed Drill: Teacher Discretion
Engage: Using Percent Estimation, the teacher will help students to recognize
the 10% rule and additional benchmark percentages that can be found from the
10% rule – 5%, 15%, 20%, 30%, etc. Teacher will also review common
percentage rules based on equivalent conversions – 25% = ¼ or dividing the
total by 4, 50% = ½ or dividing the total by 2, etc.
Additional Resource:
https://www.bigideasmath.com/protected/content/ipe/grade%206/04/g6_04_05.p
df
Explain:
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Teacher will work through the Student Notes on Percent Proportions
introducing the Percent Bar and the Percent Equation.
Relate the equation created from the percent bar or the percent
equation is a proportion and emphasize that the same process is used
for solving.
Extend/Elaborate: Students will continue working through the examples on the
Student Notes.
Evaluate: Students will work Percent Proportions HW for independent practice
and homework.
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Conversion Chart
Percent
Estimation
Student Notes on
Percent
Proportions
Percent
Proportions HW
Additional
Resource:
 Snack Bar Math
Handout
 Hungry for Math
Handout
 Go Math pp. 81 –
90
Day 8
Oct. 14
Engage/Explore
/Explain/Elabor
ate/Evaluate
Topic: Percent Applications (Percent of Change)
Objective of the Day: I will learn how to use the percent bar to answer
questions that require a percent of change – increasing and decreasing.
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Speed Drill: Teacher Discretion
Engage:
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Students will work through the Warm Up as an introduction to a Percent of
Change.
Warm Up –
Percent
Applications
Student Activity –
Percent
Applications
Practice –
Percent
Applications
Vocabulary:
Additional Resources
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Discount
Markup
Markdown
Commission
Wholesale
Retail
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Go Math pp. 91 –
108
Commission
Notes and Power
Point Slides
Explore:
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Students will work through the Activity Sheet using the percent bar to show
an increase in the bar to represent over 100% or go within the bar to
represent a discount
Explain:
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Teacher will explain that the “change of percent” is the difference between
the original and the new
Elaborate
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On Number 4, students will learn that commission is the “extra” money that
a person makes based on a percentage of the amount of sales
Evaluate
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Day 9
Oct. 15
Engage/Explore
Explain/
Elaborate/
Evaluate
Students will work Practice-Percent Applications for homework
Topic: Menu Activity – Discount, Tax, and Tip
Objective of the Day: I will solve real world problems using percentages to
determine discount, tax, tip, and total bill.
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Speed Drill: Teacher Discretion
Additional
Resources: Go Math
pp. 405 - 410
Engage:
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The students will work through a real world activity with a menu to calculate
discount, tax, and tip
Explore:
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The students will explore the impact that a tax and a tip has on the total bill
Explain:
Menu Activity
Practice Menu
Activity
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Teacher will explain that a percent bar or an equation can be used to
calculate the discount, tax, and tip.
Elaborate:
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Students will think of other real world examples that require a discount, tax,
tip, commission, markup, markdown, etc.—including real estate, car sales,
military discounts, etc.
Evaluate:
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Day 10
Oct. 16
Engage/Explore
Explain
Students will work through Practice – Menu Activity for extended
practice.
Topic: Percent Practice – Bargain Shopping
Objective of the Day: I will work through a real world situation that allows
student choice and incorporates discount, tax, and total cost.
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Speed Drill: Teacher Discretion
Engage:
 Students will be given a list of items to choose to work through with the
activity – student choice is main objective
Bargain
Shopping
Handout
Bargain
Shopping HW
Additional
Resource:
Go Math pp.
405-410
Explore:
 Students use chosen items to work through the activity finding pricing for
given percentage discount
Explain:
 Students explain the difference in discount along with equivalent amount in
dollars
Elaborate:
 Students apply knowledge in lesson to real world examples
Evaluate:
 Students will work through Bargain Shopping HW for homework
Day 11
Oct. 19
Elaborate/
Evaluate
Topic: Bring It All Together
Objective of the Day: I will work through a review lesson on proportions
and percents to prepare for the unit test.
Speed Drill: Teacher Discretion
Engage: Speed Dating Activity – The desks are arranged in two rows facing
each other. Students are given the packet of questions. Teacher randomly calls
out a problem number for students to work with their partner (the student facing
him or her). The partners who answer the question correctly and can explain
their approach gets a treat. Then, one side of the room moves down one desk
to get a new partner to work with for the next question, the other side stays in
their original desk.
Explore: Students share different approaches to solve problems.
Explain: Students explain their approach for solving to the entire group.
Elaborate: Teacher can take the problem worked and expand on other ways
that a question might be asked for students to recognize the correct approach
for solving.
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Bring It All
Together Student
Record Sheet
Handout
Evaluate: Students will complete the packet for homework and use as a study
guide for the test.
Day 12
Oct. 20
Unit 3 Test Percents and Proportions
Unit 3 Test
Evaluate
Accommodations
for Special
Populations
Accommodations for instruction will be provided as stated on each student’s (IEP)
Individual Education Plan for special education, 504, at risk, and ESL/Bilingual.