Lesson plan

Lesson 4  The Pythagorean Theorem

By: Shan Huang

Subject: Mathematics Grade: 8		Lesson Number: 4 of 10 Time: 75 minutes
Big Idea: Students will understand that... What does Pythagorean Theorem mean geometrically and algebraically? We can describe, measure, and compare spatial relationships.
Objectives: SWBATs Understand the geometrical and algebraic meanings of Pythagorean Theorem Use the theorem to explain the project
Content: The formula of Pythagorean Theorem: a^2 + b^2 = c^2 . a^2 is the expression of the area of a square with side length a, and so do b^2 and c^2.	Curricular Competencies: Visualize and describe mathematical concepts Explain, clarify, and justify mathematical ideas		Language Objectives: Use mathematical vocabulary and language to contribute to mathematical discussions. Communicate in a variety of ways. Develop mathematical understanding through concrete, pictorial, and symbolic representations
Materials/equipment needed: Pre-designed PPT; projector; scissors, paper, and pen/pencil.
Assessment Plan: Sharing, talking, listening, and summarize group discussion results within class
Adaptations: 1.       Students’ mathematics levels vary 2.       Students may be able to make the large square but fail to explain 3.       Students may not be able to make the large square Modifications: 1.       Ask/invite students to explain with classmates 2.       Appropriately use the pre-designed PPT to demonstrate the process 3.       Allow students to walk around, ask, and observe Extensions: More proofs of the theorem

LESSON COMPONENTS

               

Hook and Introduction (15 minutes)

1.       Quiz on previous topics

2.       Quiz collection and explanation                                                            

Development (50 minutes)

1.       Lecturing (10 minutes)

a.       Vocabulary: leg, hypotenuse, right triangle, Pythagorean Theorem

b.       Formula: a^2 + b^2 = c^2, where a and b are the lengths of the two legs, and c is the length of the hypotenuse.

2.       History of the theorem (10 minutes)

a.       Brief History of the Pythagorean Theorem (video, 5 minutes): https://www.youtube.com/watch?v=PrjTkWGLk2Q

b.       Other story of the theorem (5 minutes)

3.       Cut and make (Blake Peterson’s activity (2009), 30 minutes)

a.       Students in pairs will make two squares. These squares can be with any dimensions, but for convenience, they had better have different and drawable sizes

b.       In each pair, assume the smaller square is with side length a, and the larger is with side length b.

c.        When two squares are put adjacent to each other, the total area of the two squares are a^2 + b^2

d.       Cut the two squares as shown on figure 1. Then each pair will have five shapes.

Figure 1

e.        Move these shapes to make a larger square.

f.        The result will be similar as figure 2.

Figure 2

Closing (10 min)

1.       Feedback of the class

2.       What to do next class: various proofs of the theorem and applications of the theorem

Reference:

Peterson, B. E. (2009). Teaching the Pythagorean Theorem for understanding. The Mathematics Teacher, 103(2), 160-160.

Unit Plan: Grade 8 Chapter 6 Square Roots and the Pythagorean Theorem

EDCP 342A Unit planning: Rationale and overview for planning a 3 to 4 week unit of work in secondary school mathematics

Your name:                             Shan Huang
School, grade & course:          Churchill Secondary, Grade 8

Topic of unit:                          Chap. 6: Square Roots and the Pythagorean Theorem

                                               

Preplanning questions:

(1) Why do we teach this unit to secondary school students? Research and talk about the following: Why is this topic included in the curriculum? Why is it important that students learn it? What learning do you hope they will take with them from this? What is intrinsically interesting, useful, beautiful about this topic? (150 words) In Grade 8 mathematics, after learning integers, fractions, percents/ratios/proportions, as well as decimal numbers, students start to understand different ways of expressing numbers. These numbers are accurately expressed and internally transferable. However, not all the numbers can be expressed accurately (for example, π, e, etc.). When we define “the square of a number” as the product of the number multiply itself, we are also interested in the reversal operation, i.e., if we have a number, what is the number whose square is the given number? Are there any constraints about the given number? Are there any interesting properties about the results? These ideas can be expanded to 3rd power, 4th power, etc. The idea of exponent will be introduced in later course, but in this Chapter, the Pythagorean Theorem will be taught as an application of using square/square roots. Pythagorean Theorem is such an important bridging property between algebra and geometry that it connects these two as a whole and provides useful insight in thinking geometry with an algebraic perspective.

(2) What is the history of the mathematics you will be teaching, and how will you introduce this history as part of your unit? Research the history of your topic through resources like Berlinghof & Gouvea’s (2002) Math through the ages: A gentle history for teachers and others  and Joseph’s (2010) Crest of the peacock: Non-european roots of mathematics, or equivalent websites. (100 words) Square is a shortcut of expressing the product of a number multiplies itself, while square root is the reversal operation of square. The concepts of them are straight forward but the sign of square root comes from the “r” in the word “root”. The history of Pythagorean Theorem is interesting. Ancient people in Egypt, Babylon, and China at different times somehow figured out a triangle with side lengths of 3 units, 4 units, and 5 units has a “perfect angle”----now we know that is a 90-degree angle, or a right angle. In general, any right triangle will satisfy the following relationship: a^2 + b^2 = c^2. A wonderful YouTube video will be used to demonstrate the history of finding Pythagorean Theorem (https://www.youtube.com/watch?v=PrjTkWGLk2Q ).

(3) The pedagogy of the unit: How to offer this unit of work in ways that encourage students’ active participation? How to offer multiple entry points to the topic? How to engage students with different kinds of backgrounds and learning preferences? How to engage students’ sense of logic and imagination? How to make connections with other school subjects and other areas of life? (150 words) To encourage students’ active participation, on-hand activity is necessary to give students opportunities to think about the meaning of the theorem. By that, students may realize the theorem is not merely a formula, but a bridge that connects geometry and algebra. A shape may have algebraic meanings, while an algebraic expression may some geometrical explanations. Students should also prepare for the standardized exams. Emphasizing on calculation, explanation, and problem solving skills is never too much for students in this age. Various problems within different scenarios will be posed and students will be encouraged to solve them. Scenarios should be diversified to make sense to all ethnic background students. In order to help students understand more about the topic, various subjects like physics, geography, and tech education topics that are relevant to Pythagorean Theorem will be used as demonstrations. All in all, mathematics should be applicable in people’s daily lives.

(4) A mathematics project connected to this unit: Plan and describe a student mathematics project that will form part of this unit. Describe the topic, aims, process and timing, and what the students will be asked to produce. (100 words) One interesting project that encourages students to discover the meaning of Pythagorean Theorem is to do a shape-matching game. This is a game developed by Blake Peterson (2009). (a)    Students in pairs will make two squares. These squares can be with any dimensions, but for convenience, they had better have different and drawable sizes (b)   In each pair, assume the smaller square is with side length a, and the larger is with side length b. (c)    When two squares are put adjacent to each other, the total area of the two squares are  (d)   Cut the two squares as shown on figure 1 (see lesson plan). Then each pair will have five shapes. (e)    Move these shapes to make a larger square. (f)    The result will be similar as figure 2 (see lesson plan). Then it is illustrated that the larger square with side length c has the same area as the two previous squares, or a^2 + b^2 = c^2.  Also, students can also see that this relationship is independent of the values of a and b.

(5) Assessment and evaluation: How will you build a fair and well-rounded assessment and evaluation plan for this unit? Include formative and summative, informal/ observational and more formal assessment modes. (100 words) Formative assessment: (a)    Communicate with students during class time; (b)   Peers’ solution sharing; (c)    Quizzes every week (one quiz every two classes); (d)   Workbook from page 211 to 240; (e)    Mathematical project explanation; (f)    Explanations on different ways of proving Pythagorean Theorem. Summative assessment: (a)    Basically all the formative assessments can be treated as summative assessments, and vice versa; (b)   Various in-class problems based on different scenarios; (c)    The mathematical project itself can be considered as a summative assessment as even though students can make the large square, s/he may not be able to explain how they see it. (d)   Unit exam focus on the applications of the theorem, as well as some different proofs of the theorem.

Elements of your unit plan:

a)  Give a numbered list of the topics of the 10-12 lessons in this unit in the order you would teach them.

Lesson	Topic
1	Square roots
2	Modelling square roots
3	Non-perfect square roots
4	The Pythagorean Theorem
5	Various proofs of Pythagorean Theorem
6	Various applications of Pythagorean Theorem 1
7	Various applications of Pythagorean Theorem 2
8	Extension: irrational numbers and nun-perfect square roots
9	Chapter review
10	Summative unit test