Helping Students Become Proportionally Correct: Just Cross ...s3.amazonaws.com/.../pdfs/1342-2498.pdf · •When everyone in the group has completed making their enlargement, as a

Helping Students Become Proportionally Correct: Just Cross-Multiplying is a Thing of the Past!

Dr. Dovie Kimmins & Dr. Jeremy Winters

[email protected] [email protected]

Middle Tennessee State University

NCTM Annual Conference

April 8, 2017

San Antonio

mailto:[email protected]


Consider the Thinking of Bonita

• Problem: A faucet is leaky. Six ounces of water drips every 8 minutes. How much water drips in 4 minutes? • Bonita’s work:

• Is Bonita correct? •What was Bonita’s solution method?

-from page 7, Developing Essential Understanding of Ratios, Proportions & Proportional Reasoning: Grades 6-8

A second problem for Bonita

•A faucet is leaky. Six ounces of water drips every 8 minutes. How much water drips in 16 minutes? But this time solve the problem mentally or use paper and pencil without applying the algorithm.

• Predict Bonita’s answer.


Bonita’s answer

•Problem: A faucet is leaky. Six ounces of water drips every 8 minutes. How much water drips in 16 minutes? But this time solve the problem mentally or use paper and pencil without applying the algorithm.

•Answer: Bonita was at a loss. She said she couldn’t do the problem in her head, and she was unable to do it on paper either. Bonita was apparently unable to perform the simple act of doubling mentally or was unaware that doubling would be a reasonable approach.


A third problem for Bonita

• Problem: Crystal placed a bucket under a faucet and collected 6 ounces of water in 20 minutes. Joanne placed a bucket under a second faucet and collected 3 ounces of water in 10 minutes. Were the faucets dripping equally fast or was one dripping faster than the other?

• Bonita’s solution 1: 6 ounces in 20 minutes because “it took its time.”

• Bonita’s solution 2: 3 ounces in 10 minutes is faster than 6 ounces in 20 minutes because both 6 is greater than 3 and 20 is greater than 10.


Conclusion

•Could Bonita solve the following problem

A faucet is leaky. Six ounces of water drips every 8 minutes. How much water drips in 4 minutes?

using the traditional algorithm?

•Can Bonita reason proportionally?

•“A typical instructional unit or chapter on ratio and proportion shows students different ways to write ratios and then introduces a proportion as two equivalent ratios. Next, students usually encounter the cross-multiplication algorithm as a technique for solving a proportion.”

•“Does this customary development of ratio and proportion promote a deep understanding of these ideas?”

-page 7, Developing Essential Understanding of Ratios, Proportions & Proportional Reasoning: Grades 6-8

Our goal for the workshop

• Learn key essential understandings related to proportional reasoning and gain experience assessing whether students possess these understandings by examining student work. • Learn ways to help students transition to

proportional reasoning by engaging in activities designed to help middle school students develop these essential understandings.

Activity 1: Make a New Puzzle You will be working in assigned groups of six. Each group will have a set of puzzle pieces in an envelope that when put together will look like the puzzle below.

• Each group should assemble the puzzle so that it

looks like the one to the left.

• Each person in the group should then select one piece and should make their piece bigger with the following goal in mind:

The goal is to make a larger puzzle the same shape as the puzzle below, so that each edge of a piece that measures 4 cm in the puzzle below will measure 6 cm in the new puzzle.

• Each person should individually enlarge their own piece without discussing strategy with their group members. No talking at all.

• When everyone in the group has completed making their enlargement, as a group try to put the pieces together to form the larger puzzle. If any pieces of the enlarged puzzle do not fit, discuss the strategies used to enlarge the piece.

From Make a New Puzzle, page 45 of Classroom Activities for Making Sense of Fractions, Ratios, and Proportions, NCTM 2002 Yearbook.

DISCUSSION

•What strategies do you anticipate students using to enlarge the picture?

Helping students make transitions to become proportional reasoners

• Developing Essential Understanding of Ratios, Proportions, and Proportional Reasoning recommends teachers help students make 3 key shifts.

• One of these is the shift from ADDITIVE TO MULTIPLICATIVE COMPARISONS.

• An activity such as Make a New Puzzle can serve as a formative assessment to determine if your students are thinking multiplicatively or additively.

• The beauty of the Make a New Puzzle activity is that students who used additive reasoning to enlarge their puzzle will discover their own misconception when the pieces do not fit together!!

•Consider work of sixth graders for this problem: Before, tree A was 8’ tall and tree B was 10’ tall. Now tree A is 14’ tall and tree B is 16’ tall. Which tree grew more?

From Teaching Fractions and Ratios for Understanding, Lamon, pages 39-40

Who is thinking multiplicatively? Who is thinking additively? Discuss with a colleague.

From Teaching Fractions and Ratios for Understanding, Lamon, pages 39-40

Encouraging multiplicative (relative) thinking

• Discussing problems such as the tree problem in a group setting is often helpful. It is difficult to predict the context in which students begin to think relatively. Even when they do so, it may be in a limited no. of situations. So it is important to present students the multiplicative-additive choice in a variety of contexts.

From Teaching Fractions and Ratios for Understanding, Lamon, pages 43

Encouraging multiplicative (relative) thinking

• Ask both types of questions – those that ask children to think additively and those that ask them them to think about one quantity in relation to the other.

• Questions that require additive thinking: • Who has more cookies, Clint or Marcus? • How many fewer cookies does Clint have? • How many more cookies does Marcus have? • How many cookies do the boys have all together?

From Teaching Fractions and Ratios for Understanding, Lamon, page 44

•Questions that require multiplicative thinking:

• How many times would you have to stack up Clint’s cookies

to get a pile as high as Marcus’s?

• What part of a dozen cookies does Clint have? • Each boy has three chocolate chip cookies. What percent of

each boy’s cookies are chocolate chip? • If Marcus ate one cookie each day, how many weeks would

his cookies last? • Cookies come 6 to a package. What part of a package does

each child have? • Marcus and Clint put all of their cookies together and shared

them at lunchtime with their 3 friends. What part of the cookies will each child eat?

From Teaching Fractions and Ratios for Understanding, Lamon, page 44

•With a colleague brainstorm additional contexts appropriate for the grade level you teach.

Activity 2: Number Talk

•To get a sense of what is entailed in proportional reasoning, try the next problems. Each problem can be answered quickly and mentally if you reason proportionally. Solve the following problems mentally. Use your pen or pencil only to record your answers. Do not perform any computation!!!!!!! You may be tempted to use an equation of the form a/b = c/d, but using those symbols is not reasoning. Think about each problem and explain the solution without using rules and symbols. Solving the problem mentally does not mean do the cross multiply algorithm in your head.

Remember mentally….

1. Sandra wants to buy an MP3 player costing $210. Her mother agreed to pay $5 for every $2 Sandra saved. How much will each contribute?

2. If six chocolates cost $0.93, how much do 22 cost?

3. Bob and Marty like to run laps together because they run at the same pace. Today, Marty started running before Bob came out of the locker room. Marty had run 7 laps by the time that Bob ran 3. How many laps had Marty run by the time that Bob had run 12?

4. Six men can build a house in 3 days. Assuming that all the mean work at the same rate, how many men would it take to build the house in 1 day?

5. Between them, John and Mark have 32 marbles. John has 3 times as many as Mark. How many marbles does each boy have?

6. A company usually sends 9 men to install a security system in an office building, and they do it in about 96 minutes. Today, they have only three man to do the same size job. How much time should be scheduled to complete the job?

7. Bob and Mary like to run laps together because they run at the same pace. Today, Marty started running before Bob came out of the locker room. Marty had run 7 laps by the time that Bob ran 3. How many laps had Marty run by the time that Bob had run 12?

8. Mac can mow Mr. Greenway’s lawn in 45 minutes. Mac’s little brother takes twice as long to do the same lawn. How long will it take them if they each have a mower and they work together?

Page 10 and 11 of Teaching Fractions and Ratios for Understanding: Essential Content Knowledge and Instructional Strategies for Teachers by Susan J. Lamon

Mason’s work on #3

Note that Mason solved a proportion, but the proportion was not valid. That is, the number of laps that Mary ran and the number of laps that Bob ran are not directly proportional to each other. Why not, since Mary and Bob were running at the same rate? Note that when two quantities are directly proportional, when the value of one is 0, the value of the other is 0. That is why the graph of the set of ordered pairs (A,B) where A and B are directly proportional to each other is a line through the origin. This is not the case for the number of laps Mary ran and the number of laps Bob ran.

Unitizing (Chunking): What Is It? Why Do It?

We want students in middle grades to be able to reason proportionally. How can we help them to think this way? Research has shown that students who are able to unitize tend to be able to reason proportionally. Unitizing is the process of mentally constructing different-sized chunks in terms of which to think about a given commodity.

Great activities to promote unitizing:

• Renaming It

• Can You See It?

From Classroom Activities for Making Sense of Fractions, Ratios, and Proportions, NCTM 2002 Yearbook

Activity 3: Candy Bars • Make colored tiles available should students need these as support.

• Below are diagrams of regions. Consider each region as a candy bar that is being shared between two people. For each candy bar, cut the bar into two pieces, A and B, so that it represents the given description of Part A and Part B.

• 1. 2.

• 3. 4.

Part A is ½ as large as Part B. Part B is ____ times as large as Part A. Part A is how much of the bar? What is the ratio of Part A to Part B?

Part A is 1/4 as large as Part B. Part B is ____ times as large as Part A. Part A is how much of the bar? What is the ratio of Part A to Part B?

The ratio of Part A to Part B is 3:2 Part A is ____ times as large as Part B. Part B is how much of the bar? Part B is how many times as large as Part A?

Part A is 2/3 as large as Part B. Part B is ____ times as large as Part A. Part A is how much of the bar? Part A is how many times as large as Part B?

From Reconceptualizing Mathematics for Elementary School Teachers by Sowder.

Purposeful selection of numbers in problems

Learning Mathematics for Teaching Released Items, University of Michigan

Activity 4: The Gear Activity

Adapted from Developing Essential Understanding of Ratios, Proportions & Proportional Reasoning

Helping students make transitions to become proportional reasoners

• Developing Essential Understanding of Ratios, Proportions, and Proportional Reasoning recommends teachers help students make 3 key shifts.

• Previously discussed the shirt from ADDITIVE TO MULTIPLICATIVE

COMPARISONS. • Other shifts:

• FROM ONE QUANTITY TO TWO • Essential Understanding 1: Reasoning with ratios involves

attending to and coordinating two quantities. • FROM COMPOSED-UNIT STRATEGIES TO MULTIPLICATIVE

COMPARISONS

The Gear Activity can help students make these two shifts.

Data for the gears having 6 and 10 teeth Round Number of turns of the small

gear until dots line up

Number of turns of the large


1 5 3

2 10 6

3 15 9

4 20 12

5 25 15

6 30 18

7 35 21

Discuss: What are some strategies students might use to predict the number of turns of the large gear when the small gear turns 105 times?

Continuing to add 5 turns for the small gear and 3 for the large gear until the number for the small gear becomes 105. Literally doing this or realizing I will have to add 14 five’s to get 105 so I will have to add 14 three’s which will give me 21 + 14(3) = 63. Tripling the 35 gives 105, so you must triple the 21 also to get 63. Realizing that the number of turns of the large gear is 3/5 of the number of turns of the small gear, so the answer is 3/5 of 105 or 63.

Some terminology Round Number of turns of the small


Number of turns of the large


1 5 3

2 10 6

3 15 9

4 20 12

5 25 15

6 30 18

7 35 21

5 3

Is a COMPOSED UNIT. Figuring out unknown values in the table by continuing to add the composed unit by is called ITERATING. Finding unknown values in the table by multiplying the number of turns of the small gear by 3/5 to get the number of turns of the large gear or multiplying the number of turns of the large gear by 5/3 to get the number of turns of the small gear is called using a MULTIPLICATIVE COMPARISON.

Essential Understanding 2: A ratio is a multiplicative comparison of two quantities, or it is a joining of two quantities in a composed unit. Key Shift 3 - From Composed-Unit Strategies to Multiplicative Comparisons

Students will iterate before they develop the multiplicative comparison strategy. When students iterate, they ARE reasoning proportionally. Just not in the most efficient manner. Students need to progress at their own rates. For a student who only uses the iterative strategy, consider posing problems whose solutions would make the iterating strategy inconvenient rather than suggesting a more efficient strategy.

Activity 5: Brainstorm all of the patterns in the data below.

Round Number of turns of

the small gear until

dots line up

Number of turns of

the large gear until

dots line up

1 5 3

2 10 6

3 15 9

4 20 12

5 25 15

6 30 18

7 35 21

Doubling, tripling, halving etc. Adding 5 in the first column and adding 3 in the second. Adding two rows gives another row. Five times the number of the round is the number of turns of the small gear. Three times the number of the round is the number of turn of the large gear. 3/5 of the number of turns of the small gear is the number of turns of the large gear. 5/3 of the number of turns of the large gear is the number of turns of the large gear. Graph is a line through the origin (constant slope of 3/5) Ratio of any two numbers in the 1st column = the ratio of the corresponding numbers in the 2nd column = the ratio of the corresponding numbers in the 3rd column. Ratio of the number of turns of the large gear to the number of turns of the small gear is 3/5.

Given these situations, in which is B directly proportional to A? (That is, for which set of data do the patterns previously found hold?)

A

Number of Laps

Marty Ran

B

Number of Laps

John Ran

7 3

8 4

9 5

10 6

11 7

12 8

13 9

A

Mother’s

Contribution

B

Sandra’s

Contribution

5 2

10 4

15 6

20 8

25 10

30 12

35 14

Given these situations, in which is B directly proportional to A?

A. Number of Laps

Marty Ran

B. Number of Laps

John Ran

7 3

8 4

9 5

10 6

11 7

12 8

13 9

A. Mother’s

Contribution

B. Sandra’s

Contribution

5 2

10 4

15 6

20 8

25 10

30 12

35 14

Bob and Marty like to run laps together because they run at the same pace. Today, Marty started running before Bob came out of the locker room. Marty had run 7 laps by the time that Bob ran 3. How many laps had Marty run by the time that Bob had run 12? Sandra wants to buy an MP3 player costing $210. Her mother agreed to pay $5 for every $2 Sandra saved. How much will each contribute?

Activity 6: Numberless Problems

Cocoa Activity from Classroom Activities for Making Sense of Fractions, Ratios, and Proportions, page 39.





Activity 6: Perfect purple Paint

Grade 6: Perfect Purple Paint is made by mixing 2 cups of blue paint and 3 cups of red paint. That makes one batch of purple paint. How can you create 20 cups of perfect purple paint? How could you model with linking cubes? Explain your model. Create as many models as you can in your group. As a group decide on models that make sense to you. Then share with me. (No writing) Grade 8: Perfect Purple Paint is made by mixing 1/3 cup blue paint to ½ cup red paint. How much of each color is needed to make 20 total cups of Perfect Purple Paint. Solve in more than one way. Solve in as many ways as possible. https://www.teachingchannel.org/videos/ratios-and-proportions-lesson-sbac https://www.illustrativemathematics.org/content-standards/tasks/2049

[email protected]

[email protected]




Documents

Helping Students Become Proportionally Correct: Just Cross ...s3.amazonaws.com/.../pdfs/1342-2498.pdf · •When everyone in the group has completed making their enlargement, as a