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Relative and Absolute Thinking

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I love chocolate, so I’m going to get a slice of

the 6-layer cake!

I want the one that has more chocolate flavor, so I’m getting

the 3-layer cake!

The Chocolatey Cake Debate

If you wanted to buy a slice of the cake that had the most chocolate flavor, which slice of cake would you buy?

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0011 0010 1010 1101 0001 0100 1011Chocolatey Cake?

• Work on your own to decide which cake is more chocolatey

• Put your paper aside when you are done so that we know that you have finished

• Then work with your group members to decide on an answer

• Are you tempted to change your answer that you wrote down?

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Mr. Short and Mr. Tall

When Mr. Short is measured in paper clips, he is 6 paper clips tall. When he is measured in buttons, he is 4 buttons tall.

Mr. Short has a friend named Mr. Tall. When Mr. Tall is measured in buttons, he is 6 buttons tall. How many paper clips tall is Mr. Tall?

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0011 0010 1010 1101 0001 0100 1011How many paper clips?

• Work on your own to decide the height of Mr. Tall in paper clips

• Put your paper aside when you are done so that we know that you have finished

• Then work with your group members to decide on an answer

• Are you tempted to change your answer that you wrote down?

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0011 0010 1010 1101 0001 0100 1011Student Work

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0011 0010 1010 1101 0001 0100 1011Proportional Reasoning

• Proportional Thinkers understand that (Van de Walle):– There is a clear difference between

proportional relationships and non-proportional relationships, especially in the real world

– There are a variety of strategies for solving proportions or comparing ratios (that are not prescribed algorithms)

– There are relationships where 2 quantities vary together (covariation)

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0011 0010 1010 1101 0001 0100 1011Proportional Reasoning

• It is important to develop proportional reasoning both in ourselves and our students

• Develop slowly over middle school years, not just a couple weeks in 6th grade

• Common Core addresses this

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Core

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• Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.  

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• Recognize and represent proportional relationships between quantities.  – Decide whether two quantities are in a

proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

• Use proportional relationships to solve multistep ratio and percent problems.

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California Common Core

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• Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

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0011 0010 1010 1101 0001 0100 1011Proportional Reasoning

• Activities for developing proportional reasoning (Van de Walle):– Identifying multiplicative situations– Equivalent-Ratio– Comparing Ratios– Scaling with Ratio Tables– Construction and Measurement

• What type of activities are “Chocolatey Cake” and “Mr. Short and Mr. Tall”?

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Relative and Absolute Thinking

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Relative and Absolute Thinking

• What are you doing or can you do to distinguish the difference?

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