4.1

WELCOME TO COMMON CORE HIGH SCHOOL MATHEMATICS LEADERSHIPSUMMER INSTITUTE 2015

SESSION 5 • 19 JUNE 2015TRIANGLE CONGRUENCE AND SIMILARITY THEOREMS

4.2

TODAY’S AGENDA

Homework Review and discussion

Triangle congruence and similarity theorems Standards reflection

Break

Sticky gum cases

Four visual proofs

Lunch

Peer-teaching planning

Daily journal

Homework and closing remarks

4.3

LEARNING INTENTIONS AND SUCCESS CRITERIA

We are learning …

how the standard triangle congruence and similarity criteria (SAS, ASA, SSS, AA) follow from properties of similarity transformations.

how Effective Mathematics Teaching Practices can support reasoning and mathematical argumentation.

what counts as a mathematical proof.

4.4

LEARNING INTENTIONS AND SUCCESS CRITERIA

We will be successful when we can:

provide a convincing transformation arguments for each of the triangle congruence or similarity criteria.

connect Effective Mathematics Teaching Practices with student outcomes in narrative cases of classroom practice.

apply our criteria for proof to a different forms of mathematical argument.

4.5

ACTIVITY 1 HOMEWORK REVIEW AND DISCUSSION

4.6

ACTIVITY 1 HOMEWORK REVIEW AND DISCUSSION

Table discussion:

Compare your answers to last night’s “Extending the mathematics” prompt.

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

4.7

ACTIVITY 2 TRIANGLE CONGRUENCE AND SIMILARITY CRITERIA

ENGAGENY/COMMON CORE GRADE 10 MODULE 1, LESSONS 22, 23 & 24;

GRADE 10 MODULE 2, LESSON 15

4.8

ACTIVITY 2 TRIANGLE CONGRUENCE AND SIMILARITY CRITERIA

Side-Angle-Side (SAS) triangle congruence

Theorem 1 (SAS triangle congruence criterion)Let ΔABC and ΔA’B’C’ be two triangles, withAB = A’B’, AC = A’C’, and .Then ΔABC and ΔA’B’C’ are congruent.

Hint: Recall Exercises 1-4 in Grade 8 Module 2, Lesson 10 (pages S.51-S.52)€

m∠A = m∠A'

4.9

Complete Exercise 1 (page S.121).

Complete Problem 1 (page S.123).

ACTIVITY 2 TRIANGLE CONGRUENCE AND SIMILARITY CRITERIA

4.10

ACTIVITY 2 TRIANGLE CONGRUENCE AND SIMILARITY CRITERIA

Base angles of isosceles triangles

Theorem 2 (Isosceles triangle symmetry)Let ΔABC be isosceles, with AB = AC.Then the angle bisector at A is the perpendicular bisector of segment .

BC

4.11

ACTIVITY 2 TRIANGLE CONGRUENCE AND SIMILARITY CRITERIA

Angle-Side-Angle (ASA) triangle congruence

Theorem 3 (ASA triangle congruence criterion)Let ΔABC and ΔA’B’C’ be two triangles, with , AB = A’B’, and .Then ΔABC and ΔA’B’C’ are congruent.

€

m∠A = m∠A'

€

m∠B = m∠B'

4.12

ACTIVITY 2 TRIANGLE CONGRUENCE AND SIMILARITY CRITERIA

Side-Side-Side (SSS) angle congruence

Theorem 4 (SSS triangle congruence criterion)Let ΔABC and ΔA’B’C’ be two triangles, withAB = A’B’, AC = A’C’, and BC = B’C’.Then ΔABC and ΔA’B’C’ are congruent.

4.13

Complete Exercises 1, 2 & 3 (page S.133).

ACTIVITY 2 TRIANGLE CONGRUENCE AND SIMILARITY CRITERIA

4.14

ACTIVITY 2 TRIANGLE CONGRUENCE AND SIMILARITY CRITERIA

Angle-Angle (AA) triangle similarity

Theorem 5 (AA triangle similarity criterion)Let ΔABC and ΔA’B’C’ be two triangles, with , and .Then ΔABC and ΔA’B’C’ are similar.

€

m∠A = m∠A'

€

m∠B = m∠B'

Hint: Use the Triangle Side-splitter Theorem/Dilation Theorem.

4.15

Why transformation geometry?

ACTIVITY 2 TRIANGLE CONGRUENCE AND SIMILARITY CRITERIA

“There is at present an almost total disconnect in the standard curriculum between the geometry of middle school and that of high school. On the one hand, congruence and similarity are defined in middle school as ‘same size and same shape’ and ‘same shape but not necessarily same size’, respectively, and transformations (rotations, reflections, and translations) are taught seemingly only for the purpose of art appreciation, .... On the other hand, the concepts of congruence and similarity are defined anew in a high school geometry course, but only for polygons, and at the end of the year, transformations are sometimes brought up as enhancement of the concept of polygon congruence.”

H. Wu, “Teaching Geometry According to the Common Core” (2012)

4.16

Why transformation geometry?

ACTIVITY 2 TRIANGLE CONGRUENCE AND SIMILARITY CRITERIA

H. Wu, “Teaching Geometry According to the Common Core” (2012)

“One cannot overstate the fact that the CCSS do not pursue ‘transformational geometry’ per se. Transformations are merely a means to an end: they are used in a strictly utilitarian way to streamline the existing school geometry curriculum.”

Break

4.18

ACTIVITY 3 STICKY GUM CASES

Calvin Jenson & Natalie Boyer each implemented A Sticky Gum Problem in their respective classrooms. While both teachers had been focusing on providing opportunities for their students to reason and argue mathematically, and selected this problem because they felt that the context would provide another resource for sense making, they enacted the task in very different ways.

What does each teacher do to support (or inhibit) his/her students learning? Pay particular attention to the ways that the use (or lack of) the P2A Effective Mathematics Teaching Practices supported or inhibited student learning.

4.19

ACTIVITY 3 STICKY GUM CASES

What are The Cases of Calvin Jenson & Natalie Boyer cases of?

4.20

ACTIVITY 4 FOUR VISUAL PROOFS (?)

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.

4.21

ACTIVITY 4 FOUR VISUAL PROOFS (?)

Consider the four visual proofs (or explanations) on the sheets provided.

Do these visual explanations meet our criteria for proof?

What aspects of the mathematics do they illuminate, and what aspects of the mathematics do they obscure?

Choose one visual proof and be prepared to explain it to the whole group.

Lunch

4.23

ACTIVITY 5 PEER TEACHING PLANNING

4.24

Work with your group to plan your lesson.

ACTIVITY 5 PEER TEACHING PLANNING

4.25

ACTIVITY 6 DAILY JOURNAL

4.26

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

ACTIVITY 6 DAILY JOURNAL

4.27

Complete Problems 4 & 5 from the Grade 10 Module 2 Lesson 15 Problem Set in your notebook (page S.100).

Extending the mathematics:We have discussed congruence and similarity for plane (2-dimensional) figures extensively. How would you now define congruence and similarity for 3-dimensional figures? How would the 3-dimensional case differ from the 2-dimensional one? What aspects of the theory would remain essentially the same?

Reflecting on teaching:Discuss Wu’s arguments for why we should take a transformation approach to Geometry.

ACTIVITY 7 HOMEWORK AND CLOSING REMARKS