• # Middle School Mathematics Grade 7 Unit 1

Subject: Mathematics
Timeline: 21 days
Unit 1 Title: Stretching and Shrinking

Unit Overview:

Knowledge of similarity is important to the development of children’s understanding of geometry in their environment.  In their immediate environment and in their studies of natural and social sciences, students frequently encounter phenomena that require familiarity with the ideas of enlargement, scale factors, area growth, indirect measurement, and other similarity-related concepts.  Similarity uses multiplication and is an instance of proportionality.  It is generally understood that understanding proportional reasoning is an important stage in cognitive development (Stretching and Shrinking pg. 3).

Unit Objectives:

At the end of this unit, students should have mastered the following skills and abilities:
• Identify similar figures by comparing corresponding parts
• Use scale factors and ratios to describe relationships among the side lengths of similar figures
• Construct similar polygons
• Draw shapes on coordinate grids and then use coordinate rules to stretch and shrink those shapes
• Predict the ways that stretching or shrinking a figure affect lengths, angle measures, perimeters, and areas
• Use the properties of similarity to calculate distances and heights that can’t be directly measured

Focus Standards:

PA.CCSS.Math.Content.CC 2.1.7.D.1 Analyze proportional relationships and use them to model and solve real-world and mathematical problems. (7.RP.2.c, 7.RP.3)
PA.CCSS.Math.Content.CC.2.3.7.G.1 Visualize and represent geometric figures and describe the relationships between them. (7.G.1)
PA.CCSS.Math.Content.CC.2.3.7.A.3 Solve real-world and mathematical problems involving angle measure, area, surface area, circumference, and volume. (7.G.5, 7.G.6)

Mathematical Practice Standards:

#1 Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends.

#2 Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved.

#3 Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

#6 Attend to precision.

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context.

#8 Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Concepts - Students will know:
• similar figures by comparing corresponding parts
• scale factors and ratios as they describe relationships among the side lengths of similar figures
• similar polygons
• shapes on coordinate grids and then use coordinate rules to stretch and shrink those shapes
• the ways that stretching or shrinking a figure affect lengths, angle measures, perimeters, and areas
• the properties of similarity to calculate distances and heights that can’t be directly measured
Competencies - Students will be able to:
• make similar figures using scale factor
• compare measurements of corresponding parts in similar figures
• use percents as a way to describe size change
• use algebraic rules to produce similar figures on a coordinate grid
• focus on both lengths and angles are criteria for similarity
• contrast similar figures with non-similar figures
• use multiplication to stretch or shrink similar relationships
• add a number to the x- and y-coordinate of an image to make it move  on the grid
• provide examples of the relationships of angles, side lengths, perimeters, and areas of similar figures
• use ratios of corresponding sides within a figure to determine whether two figures are similar and find missing lengths

Assessments:

Formative Assessments:
• Informal assessments on learning targets
• Check-up Quiz 1
• Check-up Quiz 2
• Partner Quiz
Summative Assessment:
• Common Core Unit Assessment- Stretching and Shrinking

Elements of Instruction:

In Grade 6, math instructional time focused on four critical areas: (1) connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking.

Stretching and Shrinking expands on the Grade 6 foundation of connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems.

Similarity is an instance of proportionality. For example, if you increase the size of a diagram by 50%, then distances in the enlarged diagram are proportional to distances in the original diagram. Specifically, every distance in the enlargement is a constant multiple (1.5) of the corresponding distance in the original. It is generally understood that understanding proportional reasoning is an important stage in cognitive development.

Students in the middle grades often experience difficulty with ideas of scale.  They confuse adding situations with multiplying situations. Situations requiring comparison by addition or subtraction come first in students’ experience with mathematics and often dominate their thinking about any comparison situation, even those in which scale is the fundamental issue. For example, when considering the dimensions of a rectangle that began as 3 units by 5 units and was enlarged to a similar rectangle with a short side of 6 units, many students will say the long side is now 8 units rather than 10 units.  They add 3 units to the 5 units rather than multiply the 5 units by 2, the scale factor.  These students may struggle to build a useful conception that will help them distinguish between situations that call for addition and those that are multiplicative (calling for scaling up or down).

Major misconceptions by and struggles for students in this unit include:
1. comparing area and perimeter of similar shapes
2. using algebraic rules to enlarge shapes
3. connecting the operations of multiplication with size of a shape and addition to position of a shape
4. writing algebraic rules from real-world scale factor situations
5. rotating shapes to correctly match corresponding sides
6. creating a correct proportion from two similar shapes
7. identifying congruent angles measures in similar shapes
8. creating a correct proportion to find missing measurements in similar shapes

Differentiation:

Each lesson and/or unit offers a wide variety of ways to differentiate for all levels of learners. These include:
• Special Needs Handbook
• Unit Projects
• Spanish Additional Practice and Skills guide
• Strategies for English Language Learners Guide
• “Extension” homework questions
• District-created notebooks

Interdisciplinary Connections:
• Mathematical Reflections
• “Did You Know?” sections
• phschool.com and web codes
• “Connections” homework questions
• The real-world context embedded in lesson problems