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3Photo: © Google. SEDL used in compliance with Google Maps/Google Earth Content Rules and Guidelines. Retrieved January 12, 2012.

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Photos: left: Thinkstock/Hemera Collection. SEDL used in compliance with its Thinkstock license agreement; right: © Bone Clones. Retrieved from http://www.boneclones.com/SC-018A.htm. SEDL used with permission.

Solve the prehistoric cat problem provided on a separate handout.

Discuss your solution approach or strategies with others at your table.

Share with the whole group.

Setting the Stage

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Examine the handout with the sample solution strategy.

With one or more partners, list the prerequisite knowledge needed to solve a problem such as the one just attempted.

Setting the Stage

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Go to the TEA TEKS Web page:http://ritter.tea.state.tx.us/rules/tac/chapter111/index.html

Identify and list the K–8 mathematics objectives that address the ideas of ratio and proportion.

Findings? Patterns?

Setting the Stage

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Materials: Cutouts with possible prerequisites for developing proportional reasoning

With a partner or group, sort the cutouts into a logical framework based on criteria of your choosing.

Whole group share/report

Setting the Stage

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Examine prerequisite knowledge through “knowledge packages.”

Break down the prerequisites.

Setting the Stage

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Mathematical Proficiency

According to the National Resource Council, students are expected to attain “mathematical proficiency.”

Mathematical proficiency is composed of five interwoven competencies.

Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics.

Washington, DC: National Academy Press.

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Mathematical Proficiency

1. Conceptual understanding—comprehension of concepts, operations, and relations

2. Procedural fluency—carrying out procedures flexibly, accurately, efficiently, and appropriately

3. Strategic competence—ability to formulate, represent, and solve problems

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Mathematical Proficiency

4. Adaptive reasoning—capacity for logical thought, reflection, explanation, and justification

5. Productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own capacity

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It may not be an accident that conceptual understanding is the first component listed.

Conceptual understanding is the foundation on which the other four components are grounded.

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Conceptual Understanding

• Conceptual understanding goes far beyond “how to.”

• With conceptual understanding, students know

– what a concept really is,

– why it works,

– how it connects to other concepts, and

– what it looks like symbolically and graphically.

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Conceptual Understanding

• The focus is always on a deep understandingof CONCEPTS.

• Field of Dreams—If you build it, they will come.

– Math Corollary I (for students)—If you build the concepts, the skills will follow.

– Math Corollary II (for teachers)—If you build your content knowledge, the activities/lessons will come.

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Conceptual Understanding

Conceptual understanding enables us to

1.view an idea or concept from multiple perspectives;

2.make critical connections to other fundamental concepts and ideas; and

3. recognize and understand subtleties in language, symbolism, and representation.

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Conceptual Understanding

• Conceptual understanding enables teachers and students to recognize and understand subtleties in language, symbolism, and representation.

• The foundation for the understanding of mathematics is based on deep, fundamental, and conceptual definitions of critical concepts.

• Deep conceptual understanding in turn is connected to the types of questions teachers use as part of classroom instruction.

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Conceptual Understanding and Language

What can we do to teach MATHEMATICS (not arithmetic and efficiency) to ALL students?

• To the extent possible, use concrete examples or manipulatives.

• Emphasize both symbolism and academic language.

• Organize thinking with graphic organizers.

• Use ambiguity to your advantage, not as a disadvantage.

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Classroom Application

What can we do to teach MATHEMATICS (not arithmetic and efficiency) to ALL students?

• Simplify, yet deepen.

• Use the deep knowledge of one topic to make connections to and leverage the learning of related topics.

• Teach mathematics as relationships.

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Classroom Application

These strategies do not occur in isolation. Several can be used simultaneously, making each activity or strategy even more powerful.

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Classroom Application

Students must see the big picture of fractions.

What does ¾ mean? In small groups, develop a tree diagram that illustrates the possibilities and be prepared to share with the whole group.

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Fractions—Beyond Slices of a Pizza

How can we begin to instill the idea of a fraction as a ratio (rather than the traditional “parts of a whole”) in earlier grades?

Use the provided materials to model one approach.

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Fractions—Beyond Slices of a Pizza

Is the above true or false? Justify.

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Fractions—Symbolism and Language

Why 1/2 = 2/4 is false.

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Fractions—Symbolism and Language

Why 1/2 = 2/4 is false.

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Fractions—Symbolism and Language

• Solve mentally:

• For the above, you got an answer. What was the question that you answered? Explain.*

*Do not use any form of the terms divide or goes into.

10 ½

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Fractions—Symbolism and Language

Students’ Interpretations of Symbols

1.Do students see ⁵⁄₄ as 5 x ¼?Do students see ⁵⁄₄ as 5 one-fourths? Why or why not?

2.Do students see 1½ as 1 + ½? Do students see that 9 = 1½ x 6 means that 9 is 1

and ½ sixes (1 six and half of another six)?

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Fractions—Symbolism and Language

How can this division problem be interpreted as a proportion?

4 5

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Fractions—Symbolism and Language

Stevie has ½ gallon of gasoline in his container. He uses ⅔ of the gasoline to mow the grass. How much gasoline did he use?

Trace how the “whole” changes in this scenario:

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Fractions—Symbolism and Language

A map indicates the following scale:

1 inch = 20 miles

Would that confuse you?

Consider: 1 inch (on map)

20 miles (real life)

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Fractions—Symbolism and Language

1. Define multiplication.  

2. How are multiplication and division the same? 

3. How can a conceptual understanding of multiplication be used as a powerful tool to establish an early foundation for students’ understanding of inverse variation?

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Early Foundations of Proportionality

Show me what “5 more” looks like.

(Hint: Use vertical bar graphs.)

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Early Foundations of Proportionality

Show me what “5 more” looks like.

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1

6 1520

10,00010,005

Early Foundations of Proportionality

Show me what “half as much” looks like.

(Hint: Use vertical bar graphs.)

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Early Foundations of Proportionality

Show me what “half as much” looks like.

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2

1 20

40

2,500

5,000

Early Foundations of Proportionality

Solve:

In the same 6-week period, one pig grew from 5 pounds to 10 pounds. Another pig grew from 100 to 108 pounds. Which pig grew more?

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Early Foundations of Proportionality

Suggested strategy:

Ask students for definitions and give them three options.**

1.Use a formal definition.2.Use your own words.3.Use a drawing, picture, or example.

**Students answer using only one of the three methods. 37

Early Foundations of Proportionality

Define ∏.

1. Formal definition

2. Own words

3. Drawing, picture, or example

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Early Foundations of Proportionality

• Cat-like animal femur bone• Length of femur bone – 10 inches• Length of modern domestic cat femur –

about 3 inches• Weight of a modern domestic cat –

approximately 10 lbs.• Approximate age of bone – 10,000 to 12,000

years

Prehistoric Cat: What We Know

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Photo: © Bone Clones. Retrieved from http://www.boneclones.com/SC-018A.htm. SEDL used with permission.

• Assume similar musculature.• Domestic cat weighs about 10 lbs.• Fossil femur is 10 inches long.• Modern cat femur is 3 inches long.

What might the prehistoric cat have weighed?

How Big Is It?

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Requirements for Survival• Dietary• Capacity

• Density

Environmental Factors

Ancient Cat Is Extinct—Why?

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•http://nationalzoo.si.edu/Animals/GreatCats/catfacts.cfm

•http://www.talktothevet.com/ARTICLES/CATS/feedadultcat.HTM

•http://www.bbc.co.uk/nature/wildfacts/factfiles/3007.shtml

•http://en.wikipedia.org/wiki/Smilodon

•http://www.enchantedlearning.com/subjects/mammals/smilodon/

Ancient Cat Is Extinct—Why?

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An optical illusion that makes things seem bigger or smaller than they actually are•Who?•What?•Where?•Why?•How?

Forced Perspective

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Photo: © SEDL45

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Photo: Vin7474, Picture of the Leaning Tower. Retrieved from http://en.wikipedia.org/wiki/File:Europe_2007_Disk_1_340.jpg. SEDL used in compliance with Creative Commons public domain designation.

47Photo: © SEDL

48Photo: © SEDL

Photo: Robert Slawinski, StinkyJournalism, Monster pig hoax. SEDL used with permission from iMediaEthics.

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• Use the landscape option.• Choose a sunny day; more light is good.• Use the widest angle to exaggerate perspective.• Position the subject in the approximate pose.• Move the camera until you get the shot you want.• Use low angles to exaggerate height.• Take lots of shots.

Forced Perspective Tips

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