Middle School Mathematics Grade 7 Unit 2
Subject: MathematicsGrade: 7
Timeline: 23 days
Unit 2 Title: Comparing and Scaling
In this unit, students will build on and formalize previously-learned methods of quantitative comparison, using ratios, fractions, percents, and decimals. They will develop and use the concept of a rate, particularly the unit rate. Additionally, this unit will provide students with practical application of scaling rates and ratios, as well as help to provide an awareness of proportional reasoning strategies and methods. The unit also aims to foster students’ understanding of when such reasoning is appropriate.
At the end of this unit, students should have mastered the following skills and abilities:
- Analyze comparison statements made about quantitative data
- Use ratios, fractions, differences, and percents to form comparison statements in a given situation
- Judge whether comparison statements make sense and are useful
- See how forms of comparison statements are related (i.e. a percent and a fraction comparison)
- Make judgments about which statements are most informative or best reflect a particular point of view
- Decide when the most informative comparison is the difference between two quantities and when it is a ration between pairs of quantities
- Scale a ration, rate, or fraction to make a larger or smaller object or population with the same relative characteristics as the original
- Represent related date in tables
- Look for patterns in tables that will allow predictions to be made beyond the tables
- Write an equation to represent the pattern in a table of related variables
- Apply proportional reasoning to solve for the unknown part when one part of two equal ratios is unknown
- Set up and solve proportions that arise in applications
- Recognize that constant growth in a table is related to proportional situations
- Connect a unit rate to the equation describing a situation
PA.CCSS.Math.Content.2.1.7.D.1 Recognize and represent proportional relationships between quantities. (7.RP.A.2)
PA.CCSS.Math.Content2.1.7.D.1 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. (7.RP.A.2.a)
PA.CCSS.Math.Content.2.1.7.D.1 Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. (7.RP.A.2.b)
PA.CCSS.Math.Content.2.1.7.D.1 Represent proportional relationships by equations. (7.RP.A.2.c)
PA.CCSS.Math.Content.2.1.7.D.1 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. (7.RP.A.2.d)
PA.CCSS.Math.Conetnt.2.1.7.D.1 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. (7.RP.A.3)
PA.CCSS.Math.Content.2.1.7.E.1 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. (7.NS.A.2.d)
PA.CCSS.Math.Content.2.3.7.A.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. (7.G.A.1)
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.
#4 Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
#7 Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Mid-level students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property.
Concepts - Students will know:
- various strategies for presenting quantitative comparison information
- the language of ratios
- comparisons between given data
- equivalent ratios (scaling up and down)
- when it is appropriate to multiply or divide
- part-to-part and part-to-whole relationships
- how unit rates are connected to ratios and linear relationships (slope)
- proportional reasoning
- how insight and flexibility in choosing strategies for solving problems requires proportional reasoning
Competencies -Students will be able to:
- make judgments and choices on given comparative statements about quantities
- analyze and create comparison statements from given data in a table
- solve ratio problems to find the largest ratio
- decide when a situation is part-to-part and part-to-whole
- find unknown parts of ratios using proportions
- compute and interpret unit rates
- fill in rate tables
- write linear equations from data in tables
- convert minutes to hours
- write four versions of proportions
- solve proportions for the missing value
- write proportions from data found in problems or measurements from geometric shapes
- Informal assessments on learning targets
- Check-up Quiz 1
- Partner Quiz
- Common Core Unit Assessment- Comparing and Scaling
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.
Comparing and Scaling expands on the Grade 6 foundation of work with fractions to rational and proportional thinking and the solid foundation of visual imagery laid in Stretching and Shrinking developing the basic concept of scale factor.
Comparing and Scaling confronts students with a series of mathematical tasks that encourage them to make decisions about the quantities relevant to each task, how those quantities can be compared most usefully, and what information is provided by various quantitative comparisons. Another major theme of this unit is scaling.
Next, having experiences with geometric instances of proportional reasoning before concentrating on more numerical situations helps students in two ways: it gives students concrete experiences with visual representation of ratio comparisons, and it begins the work of helping students see the difference between reasoning by taking differences and reasoning by comparing ratios.
Major misconceptions by and struggles for students in this unit include:
- moving from additive reasoning in 6th grade to multiplicative reasoning
- knowing when it’s appropriate to reduce ratios with very large numbers to similar ratios with smaller, easier to compare numbers (Problem 1.1)
- when a relationship is part-to-part and when it is part-to-whole (Problem 1.2 and 2.2)
- how to turn a part-to-part relationship into a part-to-whole relationship (Problem 1.2)
- knowing when one ratio is “more convincing” than the other (Problem 1.2)
- making ratios from data in tables (Problem 1.3)
- reducing ratios to rates (Problem 2.1)
- “greater than or equal to” and “less than or equal to” symbols (ACE page 26 #11)
- converting minutes to hours (Problem 3.2)
- finding average rate (Problem 3.2)
- writing linear equations(Problem 3.3)
- writing four different proportions for a situation (Problem 4.1)
- writing proportions from measurements in geometric figures (Problem 4.1)
- matching up corresponding sides of shapes (Problem 4.1)
- writing fact families with variables (Problem 4.3)
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
- Mathematical Reflections
- “Did You Know?” sections
- phschool.com and web codes
- “Connections” homework questions
- The real-world context embedded in lesson problems
Additional Resources / Games:
CMP2 and/or the Erie School District provide the following additional resources to aid students in achieving mathematical success.
- Additional Practice worksheets per investigation
- Skills Review worksheets to target key components of each investigation
- Parent letter to be sent home prior to beginning the unit to share with parents the skills, goals, and expectations of the coming unit.
- Assessment Resources workbook with extra test items (multiple choice, essay, open ended, question bank, etc)
- Investigation specific pre-generated notebooks that include tables, graphs, problem numbers, and all other items students may need to complete the investigation and all its parts. Students are provided with one per unit.
- Reflection questions at the end of each investigation to assess students’ comprehension of key concepts.
- phschool.com and web codes
- Transparencies of models, graphs, etc used within lesson(s)