Cypher IV Mathematics Leadership Project

Teaching Student-Centered Math Book Study 5-12 Session 6

Developing Concepts of Ratio & Proportion

(Re)Introductions 5-12 Group

• Kalvin Beuerlein (Gr. 3-9, Telegraph Creek)• Emily Macht (Gr. 6, Whitehorse)• Sheila Merkley (Gr. 8-11, Whitehorse)• Amy Hes (Grades 6-9, Telegraph Creek)• Sarah Baigent-Nixon (Gr. 6-9 Teacher, Atlin)• Trygve Sort (Grades 5-6, Dease Lake)• Asa Berg (Local Union President, Atlin)• Bev Fairful (Principal, Atlin)

Group Norms

• Be Responsible For How & What You Learn– Everyone brings prior

experience & knowledge. Take ownership of your learning by being on time and staying, doing the reading & reflection to prepare for discussion, and be willing to try out new ideas in your classroom.

• Encourage Risk-Taking and Accept All Ideas– When learning and

discussing, everyone needs to feel safe& that ideas will be respected, even if there is disagreement. Discussion of new ideas allows everyone to ? their own beliefs & discover new ways of thinking – an essential focus of this book study.

Group Norms - cont’d

• Be Your Own Watchdog– Monitor and manage

your participation to prevent contributing too much or too little.

• Be An Attentive Listener– Listen to each other

during the discussion. Turn off your e-mail and refrain from surfing the net during the sessions.

Homework Review

• Based on the homework assigned in the previous session, discuss the following questions:– What have you tried in

your classroom as a result of the last session?

– What role did you play in the teaching and learning of math?

– What role did the students play in their learning?

– What discoveries did you and your students make?

– What misconceptions, if any, surfaced about the topic? How did you redirect the students?

– What suggestions do you have for others when they try this?

Objectives

• Focus on the Big Ideas of ratio and proportion

• Identify multiplicative relationships

• Clarify “within” and “between” ratios

• Develop informal strategies for solving proportions

Materials

• Line Master 3: Multiplicative Relationships

• Cut-out rectangles from Line Masters 4 and 5: Rectangles

• Rulers• Articles from NCTM

Journals– Are We Golden?

– Using Literature to Engage Students in Proportional Reasoning

– Exploring Proportional Reasoning Through Movies & Literature

• Other - Human Body - Math and Literature - Measurement

• Evaluation Form

Before - Contextualizing Proportional Reasoning

• In a small group, brainstorm the areas of daily life where we use proportional reasoning and record them one your whiteboard.

• Be prepared to present the ideas from your whiteboard.

During - Big Ideas

• Refer to the big ideas to help determine:– Which class team has

more girls?– In our curriculum, the

outcomes and achievement indicators explicitly refer to ratio or proportion beginning in grade six.

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During - Multiplicative Relationships

• Perform the tasks on Line Master 3: Multiplicative Relationships (next two slides)

• Many adults struggle with proportional reasoning so do not be surprised if you do too. “It is estimated that half of the adult population cannot be viewed as proportional thinkers.” (Lamon, 1999).

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Line Master 3: Multiplicative Relationships (cont’d)

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If You Hopped Like A Frog -Multiplicative Relationships

• In the TPL 513.2 SCH with a math lesson plan or from amazon.ca @ http://www.amazon.ca/If-You-Hopped-Like-Frog/dp/0590098578/ref=sr_1_1?ie=UTF8&qid=1306343033&sr=8-1– A 7.5 cm frog can jump 150

cm! If you could leap like that, how far would you be able to hop? Explain.

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If You Hopped Like A Frog

• Lesson idea from Carole Fullerton: http://mindfull.wordpress.com/2008/02/24/if-you-hopped-like-a-frog-proportional-reasoning/– http://

mindfull.files.wordpress.com/2008/02/if-you-hopped-like-a-frog.pdf

• http://www.k-state.edu/smartbooks/Lesson031.html

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Other Books to Consider from the TPL

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Jim and the Beanstalk Lesson

• http://books.google.ca/books?id=yuwdL9eukPYC&lpg=PA60&ots=72fc3KZhhG&dq=jim%20and%20the%20beanstalk%20proportional%20reasoning&pg=PA15#v=onepage&q=jim%20and%20the%20beanstalk%20proportional%20reasoning&f=false

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Sir Cumference and the First Round Table

• http://www.amazon.ca/SIR-CUMFERENCE-FIRST-ROUND-TABLE/dp/1570911525/ref=sr_1_1?ie=UTF8&qid=1306356831&sr=8-1

During - Clarifying “Within” and “Between” Ratios

• Take out your cut-out rectangles from Line Masters 4 and 5

• Visually sort the rectangles into groups that “look alike”– Talk about your sorting

rationale– Stack the rectangles, as in

Figure 6.3 (p. 160), to demonstrate the proportional relationship between them.

• Measure (to the nearest quarter cm) and compare the rectangles– Form small groups. Each

group will have one of the sets of “look-alike” rectangles to analyze more closely.

– Measure your rectangles in as many ways as possible (length, width, area, diagonals, angles, etc.) and record your measurements.

During - Clarifying “Within” and “Between” Ratios (cont’d)• Create “within” and

“between” ratios to compare (p. 169)– Give an example of a

“within” comparison.– Give an example of a

“between” comparison.

– Compare ratios of proportional rectangles and confirm that they are the same.

• The between ratio for Diagonal A to Diagonal G is the same as that of Perimeter A to Perimeter G; the within ratio for Length A to Width A is the same as that of Length G to Width G.

During - Clarifying “Within” and “Between” Ratios (cont’d)• Graph the ratios

– Construct graphs where the measures of similar rectangles (e.g., length, width) become coordinates on a grid (see Figure 6.8, p. 165).

– Describe the pattern formed by the plotted coordinates of all the similar rectangles

– How can this pattern be expressed algebraically?

During - Clarifying “Within” and “Between” Ratios (cont’d)

• From If You Hopped Like a Frog - Deep inside the eye, on a surface called the retina, are many tiny cells called cones. The more cones an eye has, the sharper its vision. You have about 200 000 cones on each mm2 of your retina. Eagles have about 1 000 000 cones in the same area. So an eagle has about 5 times as many cones. It see about 5 times as well as you do.

• From the air, you can spot a moving rabbit about 300 m away. With eagle eyes, you could spot it about 1500 m away. That would put you in the clouds.– Consider creating “within”

and “between” ratios to compare (p. 169)

• Give an example of a “between” comparison.

• Give an example of a “within” comparison.

Informal Strategies to Solve Proportions

• Solve the “pencil” & “candy” problems (p. 171) & use them to describe the strategies of unit price & factor of change.– When would one strategy be more useful

than the other?– Make realistic problems that could be solved

intuitively by finding the unit price or the factor of change. Record on your whiteboard to share.

Informal Strategies to Solve Proportions (cont’d)

• Read the “Brian” problem (p. 171) and notice how the numbers are no longer “neat.”– Can the strategies of unit rate or factor of

change be used to solve this problem?– Is one strategy easier than the other?

After - Round-table Sharing

• Reflect Stems– Do you plan to teach the cross-product algorithms? If not, why? If so, when

would it be appropriate? How would you make the algorithm “sensible”?– I feel good about…– I used to… but now I…– My goal is… I will know I am on my way when…– One thing that worked today was…– One question I have…– Two things I remember are…– If I could do something again differently, I would…– Our discussions tonight have made me realize that …– One thing that I will try in my classroom after this session is …

After - Evaluation & Self-Assessment Form

• Please take a few minutes to complete the evaluation & self-assessment form at this time and e-mail it to me at Paula.Thompson@yesnet.yk.ca or fax to 867-393-6339.

After - Homework

• Choose one activity from the chapter to do in your classroom and be ready to share your experiences at the next session.

• Look through the remainder of the text and identify any areas of learning that you would like to explore. Reflect on the Book Study experience and be prepared to share your thoughts at the next session.