### Mr. Short and Mr. Tall

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Relative and Absolute Thinking
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The Chocolatey Cake Debate
I love chocolate, so I’m
going to get a slice of
the 6-layer cake!
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I want the one that
has more chocolate
flavor, so I’m getting
the 3-layer cake!
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If you wanted to buy a slice of the cake that had the most
chocolate flavor, which slice of cake would you buy?
Chocolatey Cake?
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• 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
that you wrote down?
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Mr. Short and Mr. Tall
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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.
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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?
How many paper clips?
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• 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
that you wrote down?
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Student Work
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Proportional Reasoning
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• Proportional Thinkers understand that
(Van de Walle):
– There is a clear difference between
proportional relationships and nonproportional 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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Proportional Reasoning
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• 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
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California Common
Core
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6.RP3
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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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California Common
Core
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7.RP2 and 7.RP3
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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.
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• Use proportional relationships to solve
multistep ratio and percent problems.
California Common
Core
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8.EE5
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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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Proportional Reasoning
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• Activities for developing proportional
reasoning (Van de Walle):
– Identifying multiplicative situations
– Equivalent-Ratio
– Comparing Ratios
– Scaling with Ratio Tables
– Construction and Measurement
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• What type of activities are “Chocolatey
Cake” and “Mr. Short and Mr. Tall”?
Relative and Absolute Thinking
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Relative and Absolute
Thinking
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• What are you doing or can you do to
distinguish the difference?
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