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4 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.http://www.boneclones.com/SC-018A.htm

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 5

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 6

Go to the TEA TEKS Web page:http://ritter.tea.state.tx.us/rules/tac/chapter 111/index.htmlhttp://ritter.tea.state.tx.us/rules/tac/chapter 111/index.html Identify and list the K–8 mathematics objectives that address the ideas of ratio and proportion. Findings? Patterns? Setting the Stage 7

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 8

Examine prerequisite knowledge throughknowledge packages. Break down the prerequisites. Setting the Stage 9

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. 10

Mathematical Proficiency 1.Conceptual understandingcomprehension of concepts, operations, and relations 2.Procedural fluencycarrying out procedures flexibly, accurately, efficiently, and appropriately 3.Strategic competenceability to formulate, represent, and solve problems 11

Mathematical Proficiency 4.Adaptive reasoningcapacity for logical thought, reflection, explanation, and justification 5.Productive dispositionhabitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and ones own capacity 12

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. 13 Conceptual Understanding

Conceptual understanding goes far beyondhow 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. 14 Conceptual Understanding

The focus is always on a deep understanding of CONCEPTS. Field of DreamsIf 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. 15 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. 16 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. 17 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. 18 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. 19 Classroom Application

These strategies do not occur in isolation. Several can be used simultaneously, making each activity or strategy even more powerful. 20 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. 21 FractionsBeyond 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. 22 FractionsBeyond Slices of a Pizza

Is the above true or false? Justify. 23 FractionsSymbolism and Language

Why 1/2 = 2/4 is false. 24 FractionsSymbolism and Language

Why 1/2 = 2/4 is false. 25 FractionsSymbolism 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 26 FractionsSymbolism and Language

Students Interpretations of Symbols 1.Do students see as 5 x ¼? Do students see 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)? 27 FractionsSymbolism and Language

How can this division problem be interpreted as a proportion? 4 5 28 FractionsSymbolism and Language

Stevie has ½ gallon of gasoline in his container. He uses of the gasoline to mow the grass. How much gasoline did he use? Trace how the whole changes in this scenario: 29 FractionsSymbolism 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) 30 FractionsSymbolism 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? 31 Early Foundations of Proportionality

Show me what 5 more looks like. (Hint: Use vertical bar graphs.) 32 Early Foundations of Proportionality

Show me what 5 more looks like. 33 1 6 15 20 10,000 10,005 Early Foundations of Proportionality

Show me what half as much looks like. (Hint: Use vertical bar graphs.) 34 Early Foundations of Proportionality

Show me what half as much looks like. 35 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? 36 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 38 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 39

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? 41

Requirements for Survival Dietary Capacity Density Environmental Factors Ancient Cat Is ExtinctWhy? 42

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 ExtinctWhy? 43

An optical illusion that makes things seem bigger or smaller than they actually are Who? What? Where? Why? How? Forced Perspective 44

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