Unit 4

Percents and Proportional Relationships

 

Grade 7 Math 

Unit Length and Description:

 

20 days

 

Unit 4 extends the concepts of ratio and proportion covered in Unit 1 to include percents.  Students use their understanding of ratios and proportionality to solve a wide variety of percent problems, including those involving discounts, interest, taxes, tips, and percent increase and decrease. Additionally, students solve problems involving scale drawings and percents.

 

Standards:

 

7.RP.A.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 ½ / ¼ miles per hour, equivalently 2 miles per hour.

 

  • Standards Clarification:  Students are revisiting this standard from Unit 1 as they study percents and proportional relationships.

 

7.RP.A.2  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.

b.    Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

c.    Represent proportional relationships by equations.  For example, if total cost, t, is proportional to the number, n, of items purchased at a constant price, p, the relationship between the total cost and the number of items can be expressed at t = pn.

d.   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.

 

  • Standards Clarification: Students are revisiting this standard from Unit 1 as they study percents and proportional relationships.

 

7.RP.A.3  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.

 

  • Standards Clarification:  Students are revisiting the “percent” part of this standard as they focus on percents and proportional relationships.  In Unit 6, students will use proportional reasoning and percents with probability and statistics.

 

7.EE.A.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”

 

o   Standards Clarification:  In this unit, this standard is applied to expressions with rational numbers in them.

 

7.EE.B.3  Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically.  Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.  For example:  If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50.  If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 ½ inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

 

o   Standards Clarification:  The equations and inequalities in this unit should provide the students an opportunity to work with all types of rational numbers, not just integers, and especially including those involving percents and conversions to decimals.

 

7.G.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.

 

  • Standards Clarification: The concept of similarity is not introduced until the 8th grade and is not intended to be discussed when working with scale drawings in 7th grade. Scale drawings are introduced in this unit to support the coherence of mathematical topics within this grade, as scale drawings provide direct application of proportional relationships.  Students are revisiting this standard as they delve deeper into percents, and again in Unit 5 as they work with topics in geometry and measurement.

 

Focus Standards of Mathematical Practice:

 

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

MP.2 Reason abstractly and quantitatively.

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

MP.4 Model with mathematics.

MP.5 Use appropriate tools strategically.

MP.6 Attend to precision.

MP.7 Look for and make use of structure.

MP.8 Look for and express regularity in repeated reasoning.

 

Instructional Outcomes:

Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:

 

7.RP.A.1

·         I can compute unit rates associated with ratios of fractions in like or different units.

·         I can compute fractional by fractional quotients.

·         I can apply fractional ratios to describe rates.

 

7.RP.A.2a

·         I can determine that a proportion is a statement of equality between two ratios.

·         I can analyze two ratios to determine if they are proportional to one another with a variety of strategies (e.g. using tables, graphs, pictures, etc.)

 

7.RP.A.2b

·         I can define constant of proportionality as a unit rate.

·         I can analyze tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships to identify the constant of proportionality.

 

7.RP.A.2c

·         I can represent proportional relationships by writing equations.

 

7.RP.A.2d

·         I can recognize what (0,0) represents on the graph of a proportional relationship.

·         I can recognize what (1,r) on a graph represents, where r is the unit rate.

·         I can explain what the points on a graph of a proportional relationship means in terms of a specific situation.

 

 7.RP.A.3

·         I can recognize situations in which percentage proportional relationships apply.

·         I can apply proportional reasoning to solve multistep ratio and percent problems, e.g., simple interest, tax, markups, markdowns, gratuities, commissions, fees, percent increase and decrease, percent error, etc.

 

7.EE.A.2

·         I can write equivalent expressions with fractions, decimals, percents, and integers.

·         I can rewrite an expression in an equivalent form in order to provide insight about how quantities are related in a problem context.

 

7.EE.B.3

·         I can convert between numerical forms as appropriate.

·         I can solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically.

·         I can apply properties of operations to calculate with numbers in any form.

·         I can assess the reasonableness of answers using mental computation and estimation strategies.

 

7.G.A.1

·         I can use ratios and proportions to create scale drawing.

·         I can identify corresponding sides of scaled geometric figures.

·         I can compute lengths and areas from scale drawings using strategies such as proportions.

·         I can solve problems involving scale drawings of geometric figures using scale factors.

·         I can reproduce a scale drawing that is proportional to a given geometric figure using a different scale.

 

Enduring Understandings:

 

·         Rates, ratios, percentages and proportional relationships can be applied to solve multi-step ratio and percent problems.

·         Proportional reasoning using rates, ratios, percentages and proportional relationships can be applied to problem solving situations such as interest, tax, discount, etc.

·         Several ways of reasoning, all grounded in sense making, can be generalized into algorithms for solving proportion problems.

·         Scale drawings can be applied to problem solving situations involving geometric figures.

·         Geometrical figures can be used to reproduce a drawing at a different scale.

 

Essential Questions:

 

·         How can percent help you understand situations involving money?

·         How can I use proportional relationships to solve ratio and percent problems?

·         What are the types/varieties of situations in life that depend on or require the application of ratios and proportional reasoning?

      In what situations is an understanding of proportional reasoning crucial?

      What characteristics define the graphs of all proportional relationships?

      How can I use geometric figures to reproduce a drawing at a different scale?

·         How can geometry be used to solve problems about real-world situations, spatial relationships, and logical reasoning?

·         How does Geometry help us describe real-world objects?