# 7.2.2B Problem Solving with Proportions

7
Subject:
Math
Strand:
Algebra
Standard 7.2.2

Recognize proportional relationships in real-world and mathematical situations; represent these and other relationships with tables, verbal descriptions, symbols and graphs; solve problems involving proportional relationships and explain results in the original context.

Benchmark: 7.2.2.2 Problems involving Proportional Relationships

Solve multi-step problems involving proportional relationships in numerous contexts.

For example: Distance-time, percent increase or decrease, discounts, tips, unit pricing, lengths in similar geometric figures, and unit conversion when a conversion factor is given, including conversion between different measurement systems.

Another example: How many kilometers are there in 26.2 miles?

Benchmark: 7.2.2.3 Proportions & Reasonableness

Use knowledge of proportions to assess the reasonableness of solutions.

For example: Recognize that it would be unreasonable for a cashier to request $200 if you purchase a$225 item at 25% off.

## Overview

Big Ideas and Essential Understandings

Standard 7.2.2 Essential Understandings

Students have had prior experience with situations involving a change in one quantity effecting a corresponding change in another.  This previous experience has included graphical, tabular, and function rule representations of these relationships. This standard extends the prior understanding to proportional situations. Proportional relationships are a specific linear relationship. When these proportional relationships are graphed, the representation is a line passing through the origin. In other representations of a proportional relationship (tabular, verbal, symbols, or equations), the idea may not be initially evident, but in each representation a constant rate of change can be determined. From translation between these representations, students explore this constant rate of change to determine a unit rate (constant of proportionality or slope). The more connections students can make between these multiple forms of representation, the deeper their understanding of the relationship. With this understanding, students will be able to move to other linear relationships that are not proportional (with a graph that is a line that does not pass through the origin).

All Standard Benchmarks

7.2.2.1
Represent proportional relationships with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another. Determine the unit rate (constant of proportionality or slope) given any of these representations.
7.2.2.2
Solve multi-step problems involving proportional relationships in numerous contexts.
7.2.2.3
Use knowledge of proportions to assess the reasonableness of solutions.
7.2.2.4
Represent real-world or mathematical situations using equations and inequalities involving variables and positive and negative rational numbers.

Benchmark Cluster

Benchmark Group B - Proportional Problem Solving

7.2.2.2
Solve multi-step problems involving proportional relationships in numerous contexts.
For example: Distance-time, percent increase or decrease, discounts, tips, unit pricing, lengths in similar geometric figures, and unit conversion when a conversion factor is given, including conversion between different measurement systems.

Another example: How many kilometers are there in 26.2 miles?

7.2.2.3
Use knowledge of proportions to assess the reasonableness of solutions.
For example: Recognize that it would be unreasonable for a cashier to request $200 if you purchase a$225 item at 25% off.

What students should know and be able to do [at a mastery level] related to these benchmarks

• students should be able to solve multiple-problem types, including missing values, numerical comparison, qualitative comparison, and qualitative prediction;
• assess reasonableness of solutions in the context of the problem.

Work from previous grades that supports this new learning includes:

• Students can fluently translate from percent to decimal;
• Students can assess reasonableness of solutions;
• Students can use proportions (or other strategies) to do measurement conversions;
• Students know the basic metric and standard measurement equivalencies;
• students can solve proportions in a variety of ways (unit rate, factor of change, tabular, graphical, and fraction).
Correlations

NCTM Standards

Relate and compare different forms of representation for a relationship:

• identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations.

Common Core State Standards CCSS

Ratios and Proportional Relationships

7.RP:  Analyze proportional relationships and use them to solve real-world and mathematical problems.

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 ½ /1/4 miles per hour, equivalently 2 miles per hour.

7.RP.2. Recognize and represent proportional relationships between quantities.

• 7.RP.2b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
• 7.RP.2c. 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 as t = np.

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.

## Misconceptions

Student Misconceptions

Student Misconceptions and Common Errors

• the difference between 5 % and 105% of an item;
• increasing by 20% is different than increasing by 20;
• forgetting to check the reasonableness of a solution.

## Vignette

In the Classroom

Math Vignette - The Best-Buy Problem

A sixth grade class is beginning to work with ratio and proportion.  The purpose of this lesson is to use an open-ended investigation to develop the models for thinking about ratios and equivalent fractions.

The teacher gathers the class together and tells the students that he needs their help to solve a problem he encountered in his life.  The teacher tells them that he has recently gotten a kitten from an animal shelter and the kitten needs to eat a special kind of food.  This food is sold at two neighborhood stores and the teacher would like to know which store offers the best buy on cat food.  Bob's store sells 12 cans of kitten food for $15.00 and Maria's store sells 20 cans of the same food for$23.00.

After the teacher poses the problem he invites students to brainstorm their initial ideas about how they might determine which store offers the best deal.  Three students give their ideas:

• "You could think about how many more cans you are getting at Maria's store and if that makes a difference in the price, but that kind of also reflects on how much each can costs."
• "You could try and see if you can make like an equivalent fraction maybe like 3 goes into 12 and it goes into 15 and you can see if like a common fraction or something."
• "You could use the 15 and 5 goes into it so you might say 5 out of the 12 cans might be how much money."

The teacher acknowledges each comment and rephrases them without making any value judgments. Next the teacher sends the students off to work in pairs to solve this problem.  Students may use any strategy or approach they choose and must record their solution and their method on a large sheet of poster paper that will later be shared with the class.

Three pairs of students are seen working on the problem:

Helaina and Lucy:  They thought of $1 per can as the "overall price." At each store this leaves$3 extra to be divided among the total number of cans. They divided the $3 among 12 cans and$3 among 20 cans to get the price per can.

Andres and Zach (sitting across from Helaina and Lucy): They figured the price for 60 cans at each store and saw which store had the lower total for 60 cans.

Dylan and Tristan: They made a list showing the cost for various numbers of cans using the original values given in the problem as the starting point.  Their list says

 Bob's Maria's 12 cans = $15.00 20 cans =$23.00 6 cans = $7.50 10 cans = 11.50 3 cans =$3.75 5 cans = $5.75 1 can =$1.25 1 can = $1.15 After students have solved the problem and created posters showing their solutions and strategies, the teacher brings the group back together and asks some groups to share with the class. ## Resources Instructional Notes Teacher Notes Students may need support in further development of previously studied concepts and skills. • A good video lesson on finding the rate of two jets, given in word format. • Proportional relationships are relationships between two equal ratios. For example, oranges are sold in a bag of 5 for$2. The ratio of oranges to their cost is 5:2. If I bought 20 oranges, I could set up a proportion to determine my cost.

$\frac{5}{2}=\frac{20}{x}$

One common way of solving a proportion is to use cross-products. This would give you the equation $2\times 20 = 5\times x$ or $40 = 5\times x$. To solve this, divide both sides by 5 to see that $x = 8$.

$\frac{5}{2}=\frac{20}{8}$

You can also solve this by figuring out how many bags of 5 you need to have 20 oranges. You would need 4 bags of 5 oranges to equal 20. Therefore, you would multiply $2 times 4 to get$8.

• Students extend their understanding of ratios and develop understanding of proportionality to solve single- and multi-step problems. 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 or decrease. Students solve problems about scale drawings by relating corresponding lengths between the objects or by using the fact that relationships of lengths within an object are preserved in similar objects. Students graph proportional relationships and understand the unit rate informally as a measure of the steepness of the related line, called the slope. They distinguish proportional relationships from other relationships (Common Core State Standards for Mathematics, p. 46).
• Remind students that 5% of a number is like 5 out of 100, or correlate to $0.05 of a dollar. • Students need to be constantly reminded to ask themselves "Does this answer make sense?" after they complete a problem. Instructional Resources • Shopping Mall Math Students participate in an activity in which they develop number sense in and around the shopping mall. They solve problems involving percent and scale drawings. • Understanding Rational Numbers and Proportions In this lesson, students use real-world models to develop an understanding of fractions, decimals, unit rates, proportions, and problem solving. The three activities in this investigation center on situations involving rational numbers and proportions that students encounter at a bakery. These activities involve several important concepts of rational numbers and proportions, including partitioning a unit into equal parts, the quotient interpretation of fractions, the area model of fractions, determining fractional parts of a unit not cut into equal-sized pieces, equivalence, unit prices, and multiplication of fractions. • Capture-Recapture In this lesson, students experience an application of proportion that scientists actually use to solve real-life problems. Students learn how to estimate the size of a total population by taking samples and using proportions. The ratio of "tagged" items to the number of items in a sample is the same as the ratio of tagged items to the total population. Additional Instructional Resources ##### Activity: Have the students bring in grocery ads from newspapers. Have the students figure and compare unit prices on given items at two different stores to find the best buy. (2007 Mississippi Mathematics Framework Revised Strategies, p. 37) ##### Activity: Teach students how to solve proportions using factor of change or unit rate. Factor of change: 20 min. n 4 mi. 12 mi. = Multiply by 3 20 x 3 = 60 min. 4 x 3 = 12 Unit Rate: 20 min. n 4 mi. 12 mi. = It takes 20 minutes to go 4 miles. 20 ÷ 4 = 5, so it takes 5 minutes to go 1 mile. 5 min. 1 mi. 12 miles = 60 minutes (2007 Mississippi Mathematics Framework Revised Strategies, p. 37) Consumer Math with Percent Applications. A website with a variety of links regarding consumer math using proportions, percents, etc. Top Ten Ideas about Proportional Relationships (that students should take into High School Mathematics). New Vocabulary proportional. In mathematics, two variable quantities are proportional if one of them is always the product of the other by a constant quantity, called the constant of proportionality. In other words, x and y are proportional if the ratio is constant. We also say that one of the quantities is proportional to the other. For example, if the speed of an object is constant, it travels a distance proportional to the travel time. Professional Learning Communities #### Reflection - critical questions regarding the teaching and learning of these benchmarks • Do the students know WHY they set up the proportion the way they did? • Are students assessing their answers to make sure they make sense in the context of the problem? • Are students using prior knowledge and mental math to solve problems, or are they always "relying" on the algorithm of the proportion to solve the problem? • Can students solve multi-step problems? #### Materials NCTM's book Developing essential understanding of ratios, proportions, and proportional reasoning for teaching mathematics: Grades 6-8 References Bounds, H. M., Chapman, C., Green, T., Kaase, K., Sewell, B.H., & Thompson, M. (2007). Mississippi mathematics framework revised strategies. Jackson, MS: Mississippi Dept. of Education. Cramer, K., & Post, T. (1993, May). Connecting research to teaching proportional reasoning. Mathematics Teacher, 86(5), 404-407. Helping students gain understanding and self-confidence in algebra. www.purplemath.com Insights into Algebra 1: Linear functions and inequalities. http://www.learner.org/workshops/algebra/workshop2/index2.html National Council of Teachers of Mathematics. (n.d.). Developing essential understanding of ratios, proportions, and proportional reasoning for teaching mathematics: Grades 6-8. Reston, VA: NCTM. Top ten ideas about proportional relationships (that students should take into high school mathematics). http://web.me.com/serpmedia/ToolPresentation/Tool___Articulation_2_files/Top%20ten%20list-1.pdf ## Assessment Answer: Choice C Massachusetts Comprehensive Assessment System Release of Spring 2009 Test Items Answer: Choice D Massachusetts Comprehensive Assessment System Release of Spring 2010 Test Items Answer: Choice C TAKS (Texas Assessment of Knowledge and Skills) Mathematics, Grade 8, 2010 released items. • The price of a pair of shoes rose to$45 from \$24.

What percent increase is this?

## Differentiation

Struggling Learners

Struggling Students

• Provide calculators.
• Simplify word problems to only include relevant information.
English Language Learners
• Provide calculators and graphic organizers
• Rewrite word problems using fewer and easier words so that your ELL students can practice both their math and language skills without being stuck on what a particular phrase or word means as they read the problem. Keep the sentences short and to the point. Take out extra detail that is not needed to convey the gist of the problem.
Extending the Learning
• LESSON: A GIGABYTE OF MUSIC, HOW MUCH IS THAT?

Using Mathematical Conversions to Better Understand Numbers in the News, A very rich real world problem on converting megabytes of music to understand in more concrete way.

Go Fish Teacher Notes

Conceptual Understanding
Data Collection
Interpret Data
Proportional Reasoning
Procedural Knowledge
Solve Proportions

Problem Solving
Reasoning
Communication
Connections
Representation

Imagine that you are asked to determine the number of fish in a nearby pond. To count the fish one by one, you could remove the fish from the pond and stack them to one side, or mark each fish so you would not count them over and over again. Counting like this could be hazardous to a fish's health!

To determine the number of animals in a population, scientists often use the capture-recapture method. A number of animals are captured, carefully tagged, and returned to their native habitat. Then a second group of animals is captured and counted, and the number of tagged animals is noted. Scientists then use proportions to estimate the number in the entire population.

Students work in groups of three to four.

Each group needs:
●     1 paper lunch sack - represents the "lake"
●     A supply of goldfish crackers - represents the "fish" in the lake
●     A supply of pretzel fish crackers - represents the "tagged fish"
●     1 styrofoam cup - represents the "net"
●     1 paper plate

1.    Collect the data
2.    Capture:
a.    Each team receives a paper lunch bag with goldfish crackers inside.
b.    With the "net," scoop a sample of goldfish out of your "lake" onto the paper plate.
c.    Replace your sample of goldfish with pretzel fish. These are your "tagged" fish.
d.    Count the number of "tagged" (pretzel) fish and then return them to the bag.
e.    There are _____ tagged fish in the entire lake.
3.    Recapture:
a.    Shake the bag gently.
b.    For the first casting, use your net (cup) to remove a sample of fish. Count the number of "tagged" (pretzel) fish in your sample and record the total in the first column below.
c.    Return all of these fish to the lake (bag) and shake gently to mix them up.
d.    Repeat this process until you have gathered information on 10 samples and filled in the table below.

 Sample Number Number of Tagged Fish in Sample Total Number of Fish from Sample 1 2 3 4 5 6 7 8 9 10 Average

Analyze the Data

e.    To find the AVERAGE number of tagged fish, add up all 10 samples of the tagged fish and divide by 10. Do the same thing to find the AVERAGE number of total fish in your samples. (Using the AVERAGE number with 10 samples is more reliable than using any one sample's data.)

f.     Use the proportion below to estimate the total number of fish in your lake:

 Average # tagged in samples g.    = Total # tagged in lake Average # in samples h. Total # fish in lake

ESTIMATED POPULATION: _____________________
Now count the total number of fish in your lake to determine how close your estimate from the "sampling" is to the actual number of fish in the lake.
ACTUAL POPULATION: _____________________
How close were you to the actual number of fish?

As a result of this activity, students learn how to gather information about a large population based on a representative sample whose makeup is similar.

1.    Where else would scientists use this capture/recapture method?
2.    What are some of the factors that could have caused an estimate to be close or not so close to the actual number of fish?
http://fcit.usf.edu/math/lessons/activities/GoFishT.htm

Classroom Observation