7th Grade Unit 2 Stretching and Shrinking Investigations

BY THE END OF THIS UNIT:

Unit Plans / Investigation / Suggested ACE Questions
Standard 7.RP.1; 7.RP.2; 7.G.1
Investigation 1: Enlarging and Reducing Shapes / 1.1Solving a Mystery
1.2 Stretching the Figure
1.3 Scaling up and down
Mathematical Reflections 1 – Enlarging and Reducing Shapes / ACE 1 ,2 , 8-12
ACE 3, 4, 13-18, 21-26
ACE 5-7, 19, 20
Standard 7.RP.1; 7.G.1
Investigation 2: Similar Figures / 2.1Drawing Wumps
2.2 Hats off to Wumps
2.3Mouthing Off and Nosing Around
Mathematical Reflections 2 – Similar Figures / ACE 1, 2, 14-15, 29
ACE 3, 4, 16-18, 30, 31
ACE 5-13, 19-28, 32, 33
Standard 7.RP.1; 7.RP.2; 7.G.1
Investigation 3: Similar Polygons / 3.1 Rep-Tile Quadrilaterals
3.2 Rep-tile Triangles
3.3 Scale Factors and Similar Figures
Mathematical Reflection 3 – Similar Polygons / ACE 1-3, 22-25, 33, 34
ACE 4-6, 26-31, 35-37
ACE 7-21, 32, 38-42
Standard 7.RP.1; 7.RP.2; 7.RP.3; 7.G.1
Investigation 4: Similarity and Ratios / 4.1 Ratios within Similar Parallelograms
4.2 Ratios within Similar Triangles
4.3 Finding Missing Parts
Mathematical Reflection 4 – Similarity and Ratios / ACE 1, 3-13, 15-26, 37
ACE 2, 27-30, 35, 36, 38
ACE 5-14, 31-34, 39
Standard 7.RP.1; 7.RP.2.; 7.RP.3; 7.G.1
Investigation 5: Using Similar Rectangles and Triangles / 5.1 Using Shadows to Find Heights
5.2 Using Mirrors to Find Heights
5.3 On the Ground, but Still Out of Reach
Mathematical Reflection 5 – Using Similar Triangles and Rectangles / ACE 1, 2, 6-21
ACE 3, 4, 22-26, 35, 36
ACE 5, 27-34, 37, 38

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 ratios. / Resources
·  Textbook Correlation: Stretching and Shrinking Investigations 1, 2, 3, 4, and 5
·  Connected Mathematics 2 Additional Practice and Skills Workbook pages 21-23, 24-26, 27-30, 31-34, 35-38
Sample Formative 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 heart beats 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 Recognize and represent proportional relationships between quantities.
Concepts and Skills to Master
·  Ability to describe and identify complex fractions.
·  Ability to recognize the difference between unit rate and ratio.
·  Ability to express unit rates using a variety of representations, given a contextual situation.
·  Ability to recognize that multiplicative relationships are proportional.

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, similar figures
Suggested Instructional Strategies
·  To introduce students to ratios and rates, ask them to give examples of ways to compare two numbers. Organize the examples into comparisons by addition and subtraction and comparisons by multiplication and division. Point out that sometimes when quantities are compare using division, the units are the same (example 3 red apples out of a dozen apples) and sometimes different (example 120 miles in 3 hours). / Resources
·  Textbook Correlation: Stretching and Shrinking Investigations 1, 3, 4, and 5
·  Connected Mathematics 2 Additional Practice and Skills Workbook pages 21-23, 27-30, 31-34, 35-38
Sample Formative Assessment Tasks
Skill-based task
Honey lemon cough drops come in packages of 30 drops per 10-ounce bag. Cherry cough drops come in packages of 24 drops per 6-ounce box. Tell which package has the greater ratio of drops per ounce. / Problem Task
Microsoft Corp. has made an offer to acquire 1.5 million shares of Apple Corp. worth $374 per share. They offered Apple 10 million shares of Microsoft worth $25 per share, but they need to make up the difference with other shares. They have other shares worth $28 per share. How many of the $28 shares (to the nearest share) do they also have to offer to make an even swap?

CORE CONTENT

Cluster Title: Analyze proportional relationships and use them to solve real-world and mathematical problems.
Standard: 7.RP.3 Use proportional relationships to solve multi-step ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
Concepts and Skills to Master
·  Ability to use ratios of corresponding sides within a figure to determine whether two figures are similar
·  Ability to use scale factors to find missing side lengths in similar figures
·  Ability to use ratios to identify similar figures
·  Ability to apply knowledge of similar triangles and similar quadrilaterals
·  Ability to develop a technique for indirect measurement
·  Ability to practice lengths to solve problems

SUPPORTS FOR TEACHERS

Critical Background Knowledge
·  Developing and applying concepts of vertex, angles measures, sides and side lengths.
·  Constructing two dimensional shapes
·  Using symbols to communicate operations
·  Exploring symmetries of a figure
·  Using ratios in fraction form
·  Exploring properties of two-dimensional shapes
·  Finding areas, perimeters, and side lengths of shapes
·  Equivalent ratios and similarity triangles
Academic Vocabulary: Equivalent ratios, Ratio, Scale factor, Similar
Suggested Instructional Strategies
·  Have students find the perimeters of triangles and the ratio of the perimeters. Ask students to make a hypothesis about the relationship between the ratio of the perimeters and the scale factor.
·  Sketch a rough scale drawing of the classroom floor. (You will need to measure the room ahead of time and decide on a scale.) Ask students how much larger they think the actual floor is than the drawing. Have students measure the drawing and the actual floor and determine the scale. / Resources
·  Textbook Correlation: Stretching and Shrinking Investigations 4 and 5.
·  Connected Mathematics 2 Additional Practice and Skills Workbook pages 31-34, 35-38
Sample Formative Assessment Tasks
Skill-based task
The ratio of Holly’s marshmallows to Riley’s marshmallows was 7:4. After Holly ate 25 marshmallows, the ratio became 1:2. How many marshmallows did Holly have at first? / Problem Task
Nancy, Caleb, and Angela paid $30 for a gift for their father. They shared the cost in the ratio 3:2:1. How much money did Nancy contribute?

CORE CONTENT

Cluster Title: Draw, constructs, and describes geometrical figures and describes 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 drawings 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
·  Remind students that translations, rotations, and reflections maintain the shape and size of the original figure. Explain that a dilation is a transformation that preserves the shape but changes the size of the original figure.
·  Revisit fraction equivalence to help students correlate corresponding lengths, sides, widths, and parts. Use cross products to demonstrate proportionally.
·  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: Stretching and Shrinking Investigations 1, 2, 3, 4 and 5.
·  MARS Task: A22: Photographs
·  Connected Mathematics 2 Additional Practice and Skills Workbook pages 21-23, 24-26, 27-30, 31-34, 35-38
Sample Formative Assessment Tasks
Skill-based task
Make an enlargement of a cartoon or picture, using a scale factor of 2. Measure some length on your original and on the enlargement. If you are careful the length in the big picture should be twice the length in the original.
Find the area of some part of your picture and the enlargement. Is the enlargement four times as big in area? / Problem Task
·  Julie shows the scale drawing of her room below. If each 3 cm on the scale drawing equals 5 ft, what are the actual dimensions of Julie’s room? Reproduce the drawing at 3 times its current size.
·  Have students work in groups of three. Ask each student to graph the triangle ABC on a coordinate plane using A (2,4), B (2, -4) and C (-2,2). Give each group three scale factors 0.5, 1.5, and 2.5 on folded scraps of paper. Instruct students to choose one of the scale factors but not to show it to the other group members. Then ask each student to use the scale factor to dilate the triangle. When the drawings are complete, have the other member of each group try to identify the scale factor used to create each drawing.

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Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders. Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.