Ko-Kou and the Peacock’s Tail Core Mathematics Partnership Building Mathematical Knowledge and High-Leverage Instruction for Student Success 2:30 – 4:00 July 22, 2016

Learning Intention We are learning an important area relationship. We will be successful when we can Find areas of shapes on grid paper and recognize relationships between some of those areas; Provide an argument that the area relation we recognized in particular examples is in fact always true; Explain the session title.

Review: What is “Area”? Answer this question (silently!) on your own. Describe how the Measurement process is applied to area. Compare your answer with those of others at your table. Come to consensus on an answer to report out to the whole group. Could ask participants to answer the question and make a drawing/visual. Use the drawing to craft a definition of area or an explanation what you would like your students to be able to explain about area. Or it could be come up with an example & non-example of area. Describe area through the use of the Measurement Process – Identify area as an attribute of a shape. Select a unit to measure the area. Count the number of units needed to cover the shape.

CCSSM Area Measurement Standards Geometric measurement: understand concepts of area and relate area to multiplication and to addition. 3.MD.5 Recognize area as an attribute of plane figures and understand concepts of area measurement a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area. b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. 3.MD.6 Measure areas by counting unit squares. Emphasize the measurement process. How do you see the measurement process in the standards?

CCSSM Area Measurement Standards Geometry: Solve real-world and mathematical problems involving area, surface area, and volume. 6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

Properties of Area “Moving principle”: the area of a shape is not changed under a rigid motion. (Congruent shapes have equal areas.) “Combining principle”: the total area of two (or more) non-overlapping shapes is equal to the sum of their areas. Personal benchmarks are in essence based on comparison.

What is the Area of this Figure? Find the area of the parallelogram in as many ways as you can. (At least two.) Explain your reasoning to your group.

What is the area of this Figure? Standard units still are not

Area of a Right Triangle Explain why the area of a right triangle must be equal to ½ the product of its two legs. We should review area measurement briefly before launching into the PT activity, but we may not need to review as much about area formulas--this slide and the next one could easily be omitted.

Area of an Arbitrary Triangle Explain why the area of a general triangle must be equal to ½ times the product of one side (the “base”) and the perpendicular distance from that side to the opposite vertex (the “height”) Personal benchmarks are in essence based on comparison.

Developing a Conjecture about Area Draw a right-angled triangle near the center of a sheet of grid paper. Shade it in. You should draw the triangle with two of its sides lying along grid lines. Draw a square on each side of the shaded triangle. You will need to justify the squares you made. How can you prove that you have drawn 3 squares? Instructional Questions: Remind participants to guide colleagues with questions vs. telling them what to do.

Developing a Conjecture about Area Use the area techniques we have developed to find the areas of each of your 3 squares. Write your answers in the table on the whiteboard. What pattern do you see? Develop a conjecture from the whole class.

Proving your Area Conjecture Examine the picture below. How does this picture prove your conjecture? Picture credit: http://www.cut-theknot.org/pythagoras/index.shtml

A Capstone Standard Geometry: Understand and apply the Pythagorean Theorem 8.G.6 Explain a proof of the Pythagorean Theorem and its converse. Look at pg. 4 of the 6, 7 , 8th grade Progresssion Study the standards progression. What standards are Middle Schools students working on that is leading to the development of Pythagorean Theorem. Is this what you remember about the Pythagorean Theorem? Write a statement of the Pythagorean Theorem. Hint  Think about the work you did during this session.

The Pythagorean Theorem In any right-angled triangle, the area of the square on the longest side is equal to the sum of the areas of the squares on the other two sides. Explain”Kou-Ko” and “Peacock’s Tail” here.

Standards Trace Look at pg. 4 of the 6, 7 , 8th grade Progression. Study the standards progression. What standards are Middle Schools students working on that is leading to the development of Pythagorean Theorem. What standards in elementary grades will set students up for this middle school idea?

The Pythagorean Theorem In any right-angled triangle, the area of the square on the longest side is equal to the sum of the areas of the squares on the other two sides. What is the converse of this theorem?

The Converse of the Pythagorean Theorem Draw an acute triangle on grid paper. Draw a square on each side of the triangle, and find the area of each of the 3 squares. Does the Pythagorean relationship hold? Repeat with an obtuse triangle. What do you find?

Making a Right Angle It is often important in applications to be able to make an accurate right angle. For example, in constructing a room or a building. Use the Pythagorean relationship to build a right triangle with 3 pieces of string. Hint: what lengths of string would work? Are you using the Pythagorean Theorem in this activity, or are you using the converse?

Proofs of the Pythagorean Theorem You can find 118 proofs of the Pythagorean Theorem at the Cut the Knot website, http://www.cut-the-knot.org/pythagoras/index.shtml. The picture proof we have just seen is Proof #9 in its entirety. Extra credit: Which proof was discovered by a U.S. president? (And which President?) Can we go to the website? There is a nice anecdote at the top of the page of geometry students being asked: “Suppose the three squares were made of beaten gold. If you were offered either the large square, or the two smaller squares, which would you choose?” and being amazed when they were told it made no difference.

Learning Intention We are learning an important area relationship. We will be successful when we can Find areas of shapes on grid paper and recognize relationships between some of those areas; Provide an argument that the area relation we recognized in particular examples is in fact always true: Explain the session title.

Core Mathematics Partnership Project Disclaimer Core Mathematics Partnership Project University of Wisconsin-Milwaukee, 2013-2016   This material was developed for the Core Mathematics Partnership project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Other use of this work without prior written permission is prohibited—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors. This project was supported through a grant from the Wisconsin ESEA Title II, Part B, Mathematics and Science Partnerships.