Course Name: 7th Grade Math Unit #4 Unit Title: Comparing and Scaling

BY THE END OF THIS UNIT:

Course Name: 7th Grade Math Unit #4 Unit Title: Comparing and Scaling

Unit Plan / Investigation / Suggested ACE Questions
Standard 7.RP.1
Investigation 1:
Making Comparisons / 1.1 Ads That Sell
1.2 Targeting and Audience
1.3 American Records
Math Reflection 1
Determine the Most Useful Comparison Strategies / ACE 1-3, 11-16, 34
ACE 4-7, 17-33, 34
ACE 8-10, 36-41
Standard 7.G.1, 7.RP.2a
Investigation 2:
Comparing Ratios, Percents, and Fractions / 2.1 Mixing Juice
2.2 Sharing Pizza
2.3 Finding Equivalent Ratios
Math Reflection 2
Compare the Use of Ratios, Decimals and Percents / ACE 1-3, 9-13
ACE 4, 5, 14-18, 22
ACE 6-8, 19-21, 23, 24
Standard 7.RP.2b
Investigation 3:
Comparing and Scaling Rates / 3.1 Technology on Sale
3.2 Time, Rate, Distance
3.3 Comparing CD Prices
3.4 What Does Dividing Tell You?
Math Reflection 3
Compare Rates Using Graphs, Tables and Equations / ACE1-3, 13-18, 33
ACE 4-8, 10, 19-23
ACE 9, 11, 24-26, 34
ACE 12, 27-32
Standard 7.RP.2a, 7.RP.2c
Investigation 4:
Making Sense of Proportions / 4.1 Setting Up and Solving Proportions
4.2 Everyday Use of Proportions
4.3 Developing Strategies for Solving Proportions
Math Reflection 4
Create Multiple Proportions of Equivalence
Looking Back and Looking Ahead
1. Comparing Transportation Survey Results
2. Determining the Better Buy
3.Use Comparison Strategies to Interpret Data / ACE 1, 2, 15-17, 21-23
ACE 3-5, 18-20, 25, 26
ACE 6-14, 24, 27, 28

CORE CONTENT

Cluster Title: Analyze proportional relationships and use them to solve real-world and mathematical problems.
Standard: 7.RP.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.
Concepts and Skills to Master:
·  Ability to describe and identify complex fractions
·  Ability to recognize the difference between unit rate and ratio

SUPPORTS FOR TEACHERS

Critical Background Knowledge
Conceptual
·  Understand ratios as proportional relationships between quantities
·  Understand fractions as a ratio, rate or as a part to whole relationship
·  Understand the relationship between part to whole and part to part
·  Understand the connection between decimals and fractions
Procedural
·  Ability to convert fractions to decimals and percents
·  Ability to determine fractional equivalence
·  Ability to identify common factors or multiples of similar figures
Academic Vocabulary
Ratio, unit rate, compare, describe, explain, relate, quantities, equivalence
Suggested Instructional Strategies:
·  Introduce the concept of ratios by requiring students to write three ratios to represent a jar of marbles that has two colors. This will help the students relate part to part, part to whole and whole to part. Use several real world examples such as boys to girls and girls to number of students in the class.
·  Help students connect arithmetic ratios to algebraic ratios by using variables to represent each quantity in the ratios.
·  Inform the grade level teachers of the comparison language needed for this unit and work together to increase student ability to communicate effective comparative statements. / Resources:
·  Textbook Correlation: Investigation 1
·  MARS Task: E14: Best Buy
·  Graphing Calculator Task: Can You Walk 3 Miles Per Hour?
Sample Assessment Tasks
Skill-based task
In 1990, there were approximately 141,542,000 babies born in the world. About how many births was this per day? Per hour? Per Minute? / Problem Task
Describe a method you could use to estimate how many times your heartbeats in a day, a week, and in a year.

CORE CONTENT

Cluster Title: Analyze proportional relationships and use them to solve real-world and mathematical problems.
Standard: 7.RP.2.a 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.
Concepts and Skills to Master:
·  Ability to graph coordinates interpreted from proportional context
·  Ability to recognize given proportional situations that the two “between ratios” and the two “within ratios” are the same
·  Ability to recognize that two equal ratios represent a proportion and can be represented using ordered pairs and linear representations.

SUPPORTS FOR TEACHERS

Critical Background Knowledge
Conceptual
·  Understand how to use equations, linear graphs, tables and ordered pairs to represent patterns
·  Understand how to construct proportions to determine proportionality between multiple quantities
·  Understand how to test for equivalence using multiple strategies
Procedural
·  Ability to plot patterns in the form of linear representations on a coordinate grid
·  Cross multiply to determine proportionality
·  Ability to continue an arithmetic and geometric sequence
Academic Vocabulary
Linear, origin, equivalence, ordered pairs, proportional relationship
Suggested Instructional Strategies:
·  Graphing Proportions: Allow students to discuss the difference between inversely and directly proportional prior to completing the problem task below. Assign the students real world problems and solutions. Task: Create, solve and describe a directly and inversely proportional situation using a tables, graphs and equations. Explain the difference between inversely and directly proportional.
·  Performance Based Tasks with Rubrics: Complete performance based tasks from the following link. Use the sample response to help student understanding of rubrics. #1 Amy’s Vacation
http://www.oercommons.org/courses/proportional-reasoning/view / Resources:
·  Textbook Correlation: Investigation 2, 4
·  MARS Task: E10: A Golden Crown
·  Graphing Calculator Task: Does the Hand Relate to the Foot?
Sample Assessment Tasks
Skill-based task
One of the pipes in your house broke. You are trying to decide which plumbing company to call, Wizards or Easy Clean. Wizards charges $75 to make a house call and then $25 per hour until the job is finished. Easy Clean charges $60 to make a house call and then $30 per hour until the job is finished. Using the information above describe the meaning of each ordered pair.
1.  (2, 175)
2.  (6, 240) / Problem Task
Photographs come in several standard print sizes. Some of the most common print sizes are 4x6, 5x7, and 8x10. (Note: The dimensions are given in inches.) Does a proportional relationship exist between these print sizes? Justify your answer using a graph, proportion and table.

CORE CONTENT

Cluster Title: Analyze proportional relationships and use them to solve real-world and mathematical problems.
Standard: 7.RP.2b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams and verbal descriptions of proportional relationships.
Concepts and Skills to Master:
·  Ability to express unit rates using a variety of representations, given a contextual situation

SUPPORTS FOR TEACHERS

Critical Background Knowledge
Conceptual
·  Understands how to make connections between decimals, fractions and percents represented in real world situations
·  Understands how to make inferences about quantities and develop strategies and techniques to solve for missing values
·  Understands the purpose of expressing values in the form of unit rates
Procedural
·  Ability to construct tables, graphs, equations and diagrams to demonstrate proportionality
·  Ability to determine unit rate using proportions
·  Ability to use division to determine unit rate and multiply to determine unknown data
Academic Vocabulary
Constant, diagrams, independent, dependent, rate tables
Suggested Instructional Strategies:
·  If students are still having difficulty using fraction equivalents rely upon the practice sheet “Ratios and Fractions” to reinforce simplifying and comparing fractions. This is recommended as a warm-up for students that lack fraction foundations.
·  If students haven’t demonstrated mastery at writing finding unit rates rely upon the skills practice sheet “Finding and Using Rates.” Use this as a homework, review or warm-up activity.
·  Go to this site for additional table and graph situations.
http://dooleymath.com/Algebra/SystemsOverview.html / Resources:
·  Textbook Correlation: Investigation 3
·  MARS Task: A21: Sale
·  Graphing Calculator Task: Equations from Unit Rates
Sample Assessment Tasks
Skill-based task
It takes Juan 80 steps on the elliptical machine to go 0.1 of a mile. When his workout is done, he has gone 4 miles. How many steps has she made on the machine? / Problem Task
Surrounding the EPIC Center are several parking lots. Parking lot A is charging $4.50 for the first hour and $1.50 for every after following. Parking Lot B is charging $3.50 for the first hour and $2.50 for every hour following. Create tables, graphs and equations to determine which parking lot has the better rate for customers. Write a paragraph to justify your final decision.

CORE CONTENT

Cluster Title: Analyze proportional relationships and use them to solve real-world and mathematical problems.
Standard:
7.RP.2.c Recognize and represent proportional relationships between quantities.
c. Represent proportional relationships by equations.
Concepts and Skills to Master:
Ability to recognize that multiplicative relationships are proportional
For example, if total cost t is proportional to the number of 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.

SUPPORTS FOR TEACHERS

Critical Background Knowledge
Conceptual
·  Understands how to relate corresponding units as fractions and ratios
·  Understands how to interpret real life comparisons and translate the quantities into useful representations
·  Understands which formulas to use when solving for unknown quantities
Procedural
·  Ability to use cross products to determine proportionality
·  Ability to use formulas to solve problems
·  Ability to interpret patterns from tables and use the patterns to determine unknown values
Academic Vocabulary
Constant, relationship, correlation, rate, unit rate, rate table, compare, describe, inverse proportion, direct proportion
Suggested Instructional Strategies:
·  If students haven’t demonstrated mastery at setting up and solving proportions rely upon the skills practice sheet “Solving Proportions.” Use this as a homework or warm-up activity.
·  Review formulas
·  Help students visually align corresponding sides, with tracing paper if the figures are not angled in the same direction. It may also be important to color code each edge to help visual learners. https://www.teachingchannel.org/videos/visualizing-geometry-lesson / Resources:
·  Textbook Correlation: Investigation 4
·  Graphing Calculator Task: The Golden Rectangles
Sample Assessment Tasks
Skill-based task
Determine the missing measurements in the given similar triangles.


x = ______y = ______/ Problem Task
The Iditarod is a 1,159-mile sled dog race from Anchorage, Alaska to Nome, Alaska. Susan Bucher is a four-time winner of the race. Her average time was about 11 days. Find her average rate in miles per day. Use a diagram, formula, substitution, and inverse operations to determine her average rate per day.

CORE CONTENT

Cluster Title: Draw, construct and describe geometrical figures and describe the relationships between them.
Standard: 7.G.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.
Concepts and Skills to Master:
• Ability to describe and identify ratios and proportions
• Ability to reproduce scale drawing at a different scale

SUPPORTS FOR TEACHERS

Critical Background Knowledge
Conceptual
·  Understands the difference between similar and congruent figures
·  Understands the rules of angle relationships such as complementary and supplementary angles
·  Understands the difference between area and perimeter and how the scale factor impacts dimensions of the image
Procedural
·  Ability to determine corresponding parts of similar figures
·  Ability to set up proportions and use cross products to determine missing values
·  Ability to determine proportionality using ratios and cross products
·  Ability to isolate the variable and solve two step equations
·  Ability to construct geometric figures using ordered pairs
Academic Vocabulary
Scale drawing, scale factor, dilation, enlargement, reduction, congruence, similarity, corresponding parts
Suggested Instructional Strategies:
·  Help students revisit the concept of similar figures and proportionality using tools such as a protractor. Use actual measurements to demonstrate the difference between congruent and similar.
·  Revisit fraction equivalence to help students correlate corresponding lengths, sides, widths, and parts. Use cross products to demonstrate proportionality.
·  Students should know supplementary, vertical, complementary, alternate interior and alternate exterior angles. This will improve their ability to identify corresponding parts of geometric figures. / Resources:
·  Textbook Correlation: Comparing and Scaling Labs
·  MARS Task: E09:Triangular Frameworks
·  Graphing Calculator Task: Scaling the Geometry
Sample Assessment Tasks
Skill-based task
If you visit the Museum of Science and Industry in Chicago, you are able to examine a scale model of a human heart that is large enough to walk through. The height of the scale model is 16 feet. The scale used is 1 ft: 9/32 in. What is the height of the actual heart? / Problem Task
If given the dimensions of the builder’s floor plans and the actual dimensions of the actual house, explain how you can determine the scale factor. Design a sample floor plan and actual dimensions to explain how you would determine the scale factor. Use corresponding parts to create equations, tables and proportions when possible as evidence of your understanding how to scale ratios.