The Same Yet Smaller

MTL Kickoff August 26, 2008

Janis Freckmann Melissa Hedges DeAnn Huinker Hank Kepner Connie Laughlin Kevin McLeod Mary Mooney Lee Ann Pruske Session Goals

1. Review the part-whole definition of fraction.

2. Explore fractions, and operations with fractions, in a geometric setting.

3. Surface algebraic reasoning: recognizing patterns and relationships; conjecturing, extensions, & generalizations.

4. Make connections to Mathematical Knowledge for Teaching (MKT). National Mathematics Panel Recommendation #13

The curriculum should allow for sufficient time to ensure the acquisition of conceptual and procedural knowledge of fractions (including decimals and percents) and of proportional reasoning.

The curriculum should include representational supports that have been shown to be effective, such as number line representations, and should encompass instruction in tasks that tap the full gamut of conceptual and procedural knowledge, including ordering fractions on a number line, judging equivalence and relative magnitude of fractions with unlike numerators and denominators, and solving problems involving ratio and proportion.

The curriculum should also make explicit connections between intuitive understanding and formal problem solving involving fractions.

National Mathematics Panel. (2008). Foundations for Success: Report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education. http://www.ed.gov/about/bdscomm/list/mathpanel/reports.html Paper Folding

1. Cut out the triangle printed on the plain sheet of paper.

2. Fold one vertex of your triangle to the midpoint of the opposite side…. Before you do this, you must first determine how to find that midpoint!

3. Open your triangle flat again. You should see a smaller triangle created by the fold. What fraction of the original triangle is the small triangle? How do you know?

Recursive Triangles (Stage 1)

1. Use the triangle drawn on the isometric dot paper; locate the midpoint of each side.

2. Join the three midpoints with line segments.

3. Shade the downward-pointing triangle in the middle, and pretend that it has been cut out of the picture.

4. The original triangle is defined as having an area of 1 unit.

5. What is the area of the remaining non-shaded “upward-pointing triangles?” Write an expression for the total area of these triangles. Stage 0 Stage 1 Stage 1 How did you arrive at your expression?

At your table, share your expressions and reasoning.

Explain the thinking that might lead to each of the following expressions. 1/4 + 1/4 + 1/4 3 x 1/4 1 – 1/4 3/4

How are these expressions the same? How are they different? Recursive Triangles (Stage 2)

1. In each non-shaded, upward pointing triangle, repeat the construction of connecting midpoints.

2. Shade the downward-pointing triangle in the middle of each of those triangles, and pretend they have been cut out of the picture.

3. Again, define the original triangle as having an area of 1 unit.

4. Write an expression for the total area of the remaining non-shaded “upward-pointing triangles.” Be ready to explain your reasoning. Stage 2 Stage 2 Conjecturing

Suppose that we repeated the construction again, and we got a “Stage 3” picture. What would be the total area of the (non- shaded) smallest upward-pointing triangles?

What would the corresponding relationship be that describes the result at Stage 4? Stage 5? Stage n?

What ways of organizing your data might help you to see patterns and make a conjecture?

Does your relationship describe Stage 0? Mathematical Knowledge for Teaching (MKT) MKT involves not only knowing the mathematics, but knowing it in ways that makes it useful for teaching.

• Explanations (accurate & useful) • Definitions (accurate & appropriate) • Representations • Pose strategic problems & questions • Aware of students’ views & misconceptions • Interpret accuracy & rigor of student work MKT Issues: Fractions Student views, misconceptions, and teaching challenges:

• Students often think fractions cannot have the same number for the numerator as for the denominator (e.g., 7/7). Why?

• Which term reflects the mathematics: “reduce” or “rename”?

• Consider the term “improper” versus “a fraction greater than 1” from a student viewpoint.

• Is “Part to Whole” the only way to view fractions? What other ways are there? Big Ideas

• Mathematical strands are not separate. In particular, familiarity with fractions can be built up while teaching other strands. • Operations with fractions can be given meaning in many different contexts. • Multiplicative thinking is an important part of reasoning with fractions and of algebraic reasoning. Biological Recursion

"Great fleas have little fleas upon their backs to bite 'em, and little fleas have lesser fleas, and so ad infinitum.”

-- Augustus De Morgan (after Jonathan Swift)