Ratio and Proportion. Today’s Goals 1. Get acquainted; 2. Review our learning related to the...

Ratio and Proportion

Today’s GoalsToday’s Goals

1. Get acquainted;2. Review our learning related to the

concept of equality;3. Learn strategies to help students

think proportionally and multiplicatively;

4. Differentiate proportional from non-proportional situations;

5. Understand the PLC tasks related to ratio and proportion strategies.

T or F: 5 x 84 = 10 x 42

By calculating

5 x 84 = 420

10 x 42 = 420

So, 5 x 84 = 10 x 42

By thinking relationally

5 x 84

= 5 x (2 x 42)

= (5 x 2) x 42

= 10 x 42

…or?

TASK Work as a group

Solve your equation in two ways: by calculating by thinking relationally

Show your thinking

Be ready to explain your work to the larger group

Relational Thinking…Relational Thinking…

Focuses on relations rather than only on calculating answers (red)

Uses fundamental properties of arithmetic to relate or transform expressions (green)

Looks at expressions and equations in their entirety rather than as procedures to be carried out step by step (orange)

Flexibly uses operations and relations (yellow)

from Tom Carpenter’s slide show on September 23rd

Walk-about and Discuss Take a ‘wall walk’ with your group

Look for examples of these relational thinking characteristics in these problems.

Use post-its to identify examples when you see them

So….

What is the connection between relational thinking and solving equations?

TASK1. Select a PLC task of your choice:

Baseline Assessment Teaching a lesson Interviewing students Post Assessment

2. Find 2-3 others who share your interest3. Find the related discussion Q’s4. Discuss the Q’s with your small group5. Discuss the Follow-up Q’s 6. Be ready to report out

Summary QuestionsSummary Questions

What do we want our kids to know about equality?

What can we do to improve their understanding of equality as an equivalence relationship?

How will this work impact students understanding of algebra?

Quantitative ComparisonQuantitative Comparison

TASK: Work as a group

Examine the student work and be able to explain the strategy used

Compare the samples of student work What strategies are similar? What strategies are different?

Student Responses Student Responses (from Glenda Lappan)(from Glenda Lappan)

Wes Problem #1Wes Problem #1

Two baseballs cost $5. How much will seven baseballs cost?

Two baseballs cost $5. How much Two baseballs cost $5. How much will seven baseballs cost?will seven baseballs cost?

The unit rate approach is characterized when students find the constant factor across or between the two measure spaces.

baseballs $

2 57 ?

x 2.5

x 2.517.50


The scale factor approach is characterized when students find the multiplicative relationship among elements within the measurement space.

baseballs $

2 57 ?

X 3.5 X 3.517.50


The table strategy is a strategy that is a mix of unit rate and factor of change.

baseballs $

2 54 106 15

8 207 17.50


The graphing strategy has an interesting equation.

baseballs $

2 5

4 10

6 15

8 20 baseballs

Do

llar

s ($

)

0 2 4 6 80

5

10

15

20

y = 2.5x

7 17.50 7

17.50


The fraction strategy is devoid of the problem context.

5

2

?

7=

x 3.5

x 3.5 17.50


Cross-multiplication is also a strategy but is hard for students to explain why it works and they tend not to use it when solving problems on their own.

5

2

x

7=

352 =x baseball-dollar


I can buy 6 baseballs for $20. How much will 18 baseballs cost?

The Marissa ProblemThe Marissa Problem

Marissa bought 0.46 of a pound of wheat flour for which she paid $0.83. How many pounds could she buy for one dollar?

The Track ProblemThe Track Problem

Wendy and Brooke run at the same speed around a track. Wendy was running lap 3 when Brooke was running lap 7. What lap will Brooke be on when Wendy is running lap 9?

Proportional Situation or Proportional Situation or Not?Not?

Taxi cabs in Minneapolis charge $2.50 for the first one-fifth mile and each succeeding fifth will cost $.38. How much will the fare be for a 2 mile trip?


The action figure of a famous wrestler has a scale of 1/12. If the doll is 7 inches tall, how tall is the actual wrestler?


Dara swims four laps across the pool in the same time as Michael swims three laps. They begin swimming at the same time. How far has Michael swum when Dara completes lap 10?


Jake typically exchanges 200 text messages a day. The plan he signed up for charges a $30 flat fee per month and does not charge for the first 1000 text messages exchanged. Each message exchanged after the first 1000 costs $0.05. What is the amount for Jake’s bill for the month of October?


Lance is riding his bicycle in France. He rides 18 km in half an hour. If he rides at the same rate, how long will it take him to ride 162 km?


Two U.S. dollars can be exchanged for three Euros. How many U.S. dollars can be exchanged for 21 Euros?

What is proportional thinking? What is proportional thinking? (partial list adapted from Lamon)(partial list adapted from Lamon)

the ability to use proportionality as a mathematical model to organize information in appropriate contexts;

the ability to distinguish situations in which proportionality is not an appropriate mathematical model from situations in which it is useful;

knowing that the graph of a direct proportional situation is a straight line through the origin;

the ability to explain the difference between functions of the form y = mx and functions of the form y = mx + b. In the latter function, y is not proportional to x, but rather the change in y is proportional to the change in x.

Minnesota State Algebra Minnesota State Algebra Standards (7Standards (7thth grade) grade)

Understand the concept of proportionality in real-world and mathematical situations, and distinguish between proportional and other relationships.

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.

Apply understanding of order of operations and algebraic properties to generate equivalent numerical and algebraic expressions containing positive and negative rational numbers and grouping symbols; evaluate such expressions.

Represent real-world and mathematical situations using equations with variables. Solve equations symbolically, using the properties of equality. Also solve equations graphically and numerically. Interpret solutions in the original context.

Upcoming PLC Tasks

Administer a Baseline Assessment on Ratio and Proportion to at least one class

Teach a Lesson Interview at least 3 students Administer a Post Assessment Meet in your PLCs to discuss your

observations/insights

Be a Student…

Work the Baseline Assessment

Predict how your students will respond to the questions

Ask Q’s about anything that is unclear

Baseline Assessment #1Baseline Assessment #1

2 baseballs cost $5. How much will 7 baseballs cost? Answer: Use drawings, words, or numbers to show how you got your answer.

Baseline Assessment #2Baseline Assessment #2A sixth-grade class needs 5 leaves each day to feed its 2 caterpillars. How many leaves would they need each day for 12 caterpillars? (NAEP, 1996)

Answer: Use drawings, words, or numbers to show how you got your answer.

Baseline Assessment #3Baseline Assessment #3The length of Mr. Short is 4 large buttons. The length of Mr. Tall is 6 large buttons.

When paper clips are used to measure Mr. Short and Mr. Tall: The length of Mr. Short is 6 paper clips.What is the length of Mr. Tall in paper clips?


Wendy and Brooke run at the same speed around a track. Wendy was running lap 3 when Brooke was running lap 7. What lap will Brooke be on when Wendy is running lap 9?


Doughman Instant Pancake mix requires 3 cups of water for every 4 cups of mix. Xiong needs to make a large number of pancakes. She has 23 cups of mix and wants to use it all. How much water does Xiong need to add to make her pancakes?

TASK: Prepare to teach a lesson

Turn to page 7 in the handout.

Work with a partner to create sample problems for each of the strategies.

Summary for the Day Keep reviewing equality -- students develop their

understanding over time. Use multiple strategies to model problems involving

proportional thinking. Unit rate Scale factor Tables Graphs Fractions

Think deeply before you teach cross multiplication. Teach what proportions ARE and what they ARE NOT

Ratio and Proportion. Today’s Goals 1. Get acquainted; 2. Review our learning related to the...

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