Presented byDr Anne M. CollinsLesley University, Cambridge, MA 02339collins2@lesley.edu

Solving Ratio and Proportions on the

Cartesian Coordinate Plane

•Multiple representations for ratios, rates, and proportions.•Examine discrete and continuous representations.•Experience teaching at all grade levels K – 12, undergrads, and graduate

students

Introduction

• Ratio as comparisons• Ratio as part to part• Rates• Answers the question How Many?!

Discrete Representations

• Addition• Subtraction• Fraction as number• Ratio as part to whole• Percents• Answers the question How Much?!

Continuous Representations

Two types of ratio representations.

• Tape Diagrams

• Double Number Lines

• Graphing on the Coordinate Plane

Strategies Include

Delilah and Emmalyse shared a box of beads in the ratio of 5:3.Delilah gave half her share to Emmalyse. Emmalyse then had 30 more beads than Delilah. How many beads did Delilah give to Emmalyse?

Tape Diagram Comparison Problem

BeforeDelilah

Emmalyse

AfterDelilah

Emmalyse

Before – After Tape Diagram Model

BeforeDelilah

Emmalyse

AfterDelilah

Emmalyse

Before – After Tape Diagram Model

•Extremely effective when the quantities have different units.•Effective with percent problems.•Effective with percent of increase or decrease problems.•Extremely effective with rate problems.•Help students make sense of why cross-multiplication works.

Solving Problems Using Double Number Lines

• Anne left home at 6 p.m. to drive the 135 miles to Easthampton. She arrived at Easthampton at 8 p.m. What was Anne’s average rate of speed?

Solve the Problem Using Double Number Lines

Double Number Lines

miles

Time in hours

135

21

?

Double Number Lines- Find Unit Rates/Solve Proportions

miles

Time in hours

135

21

?

Using the values from the number lines we get

? 135 1 2

Step One: Find scale factor or unit rateStep Two: Apply scale factor of ½ to get 67.5 miles per one mile.

From a Double Number Line to a Proportion

Sophie and Jared shared some markers in the ratio of 7:5. Sophie gave Jared 12 markers. If each of them had the same number of markers in the end, how many markers did Sophie have in the beginning? •How many did Sophie have after sharing? •How many did Jared have in the beginning?

Solving Problems on Cartesian Coordinate Plane

Cartesian Coordinate PlaneJa

red

Sophie

Cartesian Coordinate PlaneJa

red

Sophie

12 24 36 48 60 72 84 96 108

12

2

4

36

4

8

60

7

2

84

9

6

108

Lesson 1: Division

Lesson 2: Using Proportions

Lesson 3: Graphing on the Cartesian Coordinate Plane

Comparing Ratios As Percents

Cartesian Coordinate PlaneConvert 3/4 and 4/5 to percents and determine the larger value.

Cartesian Coordinate PlaneConvert 3/4 and 4/5 to percents and determine the larger value.

Mark a square through the point (10, 10).

Notice the square has 100 square units.

The percent is found at the point of intersection between the ray and the x = 10.

Cartesian Coordinate Plane

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Cartesian Coordinate Plane

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Cartesian Coordinate Plane

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Cartesian Coordinate Plane

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•Collins, Anne and Linda Dacey. 2013. It’s All Relative. Stenhouse Publishers. Portsmouth, NH.

•Collins, Anne and Linda Dacey. 2012. Zeroing in on Number and Operations. Stenhouse Publishers. Portsmouth, NH.

References