6.1 WELCOME TO COMMON CORE HIGH SCHOOL MATHEMATICS LEADERSHIP SUMMER INSTITUTE 2015 SESSION 6 29 JUNE 2015 SEQUENCING BASIC RIGID MOTIONS; THE KOU-KU THEOREM

6.1

WELCOME TO COMMON CORE HIGH SCHOOL MATHEMATICS LEADERSHIPSUMMER INSTITUTE 2015

SESSION 6 • 29 JUNE 2015SEQUENCING BASIC RIGID MOTIONS;THE KOU-KU THEOREM

6.2

TODAY’S AGENDA

Homework Review and discussion

G8 M2 L8: Sequencing Reflections and Translations Discussion

Break

G8 M2 L9: Sequencing Rotations Discussion

Break

Kou-Ku and the Peacock’s Tail (Part I)

Lunch

Kou-Ku and the Peacock’s Tail (Part II)

Daily journal

Homework and closing remarks

6.3

ACTIVITY 1 HOMEWORK REVIEW AND DISCUSSION

6.4

ACTIVITY 1 HOMEWORK REVIEW AND DISCUSSION

Table discussion:

Compare your answers to the “Extending the mathematics” prompt from our last session.

Identify common themes, as well as points of disagreement, in your responses to the “Reflection on teaching” prompt.

6.5

ACTIVITY 2 SEQUENCING REFLECTIONS AND TRANSLATIONS

ENGAGENY/COMMON CORE GRADE 8 MODULE 2, LESSON 8

6.6

ACTIVITY 2 SEQUENCING REFLECTIONS AND TRANSLATIONS

Connecting the lesson goals with pedagogy

Which of the Effective Mathematics Teaching Practices did you see modeled in this lesson?

How did the use of those practices support progress towards the lesson’s stated goals (learning intentions and success criteria)?

1. Establish mathematics goals to focus learning

2. Implement tasks that promote reasoning and problem solving

3. Use and connect mathematical representations

4. Facilitate meaningful mathematical discourse

5. Pose purposeful questions

6. Build procedural fluency from conceptual understanding

7. Support productive struggle in learning mathematics

8. Elicit and use evidence of student thinking

Break

6.8

ACTIVITY 3 SEQUENCING ROTATIONS

ENGAGENY/COMMON CORE GRADE 8 MODULE 2, LESSON 9

6.9

ACTIVITY 3 SEQUENCING ROTATIONS

Connecting the lesson goals with pedagogy

Which of the Effective Mathematics Teaching Practices did you see modeled in this lesson?

How did the use of those practices support progress towards the lesson’s stated goals (learning intentions and success criteria)?

1. Establish mathematics goals to focus learning

2. Implement tasks that promote reasoning and problem solving

3. Use and connect mathematical representations

4. Facilitate meaningful mathematical discourse

5. Pose purposeful questions

6. Build procedural fluency from conceptual understanding

7. Support productive struggle in learning mathematics

8. Elicit and use evidence of student thinking

Break

6.11

ACTIVITY 4KOU-KU AND THE PEACOCK’S TAIL

Session goals

To explore the concept of area

To discover an important theorem about area

To explain the session title

6.12


Turn and talk:

What is “area”?

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Selected CCSSM standards related to area: Grade 3

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.

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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.

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Properties of area

“Moving property”: the area of a shape is not changed under a rigid motion. (Congruent shapes have equal areas.)

“Combining property”: the total area of two (or more) non-overlapping shapes is equal to the sum of their areas.

6.19



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.

6.20


Developing a conjecture about areas

Draw a right-angled triangle near the center of a sheet of grid paper. You should draw the triangle with two of its sides lying along grid lines.

Draw a square on each side of the triangle.

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Draw a right-angled triangle near the center of a sheet of grid paper. You should draw the triangle with two of its sides lying along grid lines.

Draw a square on each side of the triangle.

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Use the area techniques we have developed to find the areas of each of your 3 squares.

Write your answers on the whiteboard.

What pattern do you see?

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Proving your area conjecture

Examine the picture below. How does this picture prove your conjecture? (Or is it not yet a proof?)

Picture credit:http://www.cut-theknot.org/pythagoras/index.shtml

6.24

ACTIVITY 4 KOU-KU AND THE PEACOCK’S TAIL

An argument that counts as proof must meet the following criteria:

The argument must show that the conjecture or claim is (or is not) true for all cases.

The statements and definitions that are used in the argument must be ones that are true and accepted by the community because they have been previously justified.

The conclusion that is reached from the set of statements must follow logically from the argument made.

In addition, a valid proof may vary along the following dimensions:

type of proof (e.g., demonstration, induction, counterexample)

form of the proof (e.g., two-column, paragraph, flow chart)

representation used (e.g., symbols, pictures, words)

explanatory power (e.g., how well the proof explains why the claim is true)

Variance on these dimensions, however, does not matter as long as the preceding three criteria for proof are met.

6.25


Proofs of the Pythagorean Theorem

You can find 111 proofs of the Pythagorean Theorem at the Cut the Knot website, http://www.cut-theknot.org/pythagoras/index.shtml.

The picture proof we have just seen is Proof #9 in its entirety.

Which proof was discovered by a U.S. president?

http://www.cut-theknot.org/pythagoras/index.shtml

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Understand and apply the Pythagorean Theorem.

8.G.6 Explain a proof of the Pythagorean Theorem and its converse.

Lunch

6.28

There are three similar right triangles in the picture. Use your knowledge of similarity to show that a2 + b2 = c2.

Why does this constitute a proof of the Pythagorean theorem? (If it does.)

ACTIVITY 4 KOU-KU AND THE PEACOCK’S TAIL

A similarity proof of the Pythagorean Theorem

Picture credit: EngageNY/Common Core

6.29


Selected CCSSM standards related to the Pythagorean Theorem:

Grade 8

Understand and apply the Pythagorean Theorem.

8.G.6 Explain a proof of the Pythagorean Theorem and its converse.

High School Geometry

Prove theorems involving similarity.

G-SRT.4 Prove theorems about triangles. Theorems include: ... The Pythagorean Theorem proved using triangle similarity.

6.30

ACTIVITY 5 DAILY JOURNAL

6.31

Take a few moments to reflect and write on today’s activities.

ACTIVITY 5 DAILY JOURNAL

6.32

EngageNY G8 M2 L8, Problem Set #1: G8 M2 L9: Problem Set #2.

Extending the mathematics:Each of the proofs of the Pythagorean theorem we looked at today used the Triangle Angle Sum Theorem. Identify where that theorem was used in each proof. Find at least one more proof of the Pythagorean theorem (from your current curriculum, or from the Cut-the-Knot website). Does that proof also use the Triangle Sum Theorem?

Reflecting on teaching:Give a few (2≤x≤4) reasons why learning the Pythagorean Theorem is important for all students. How might we foster stronger connections between the Pythagorean Theorem and related mathematical ideas?

ACTIVITY 6 HOMEWORK AND CLOSING REMARKS

Documents

6.1 WELCOME TO COMMON CORE HIGH SCHOOL MATHEMATICS LEADERSHIP SUMMER INSTITUTE 2015 SESSION 6 29 JUNE 2015 SEQUENCING BASIC RIGID MOTIONS; THE KOU-KU THEOREM