Mathematics Success – Grade 8 T821

LESSON 31: Pythagorean Theorem Part 2

[OBJECTIVE] The student will apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

[PREREQUISITE SKILLS]Pythagorean Theorem

squares

square roots

[MATERIALS]Student pages S405–S415Copy Master T836CalculatorsScissors

[ESSENTIAL QUESTIONS]1. How does knowing the vertical and horizontal length of a right triangle plotted on the

coordinate plane help you to determine the hypotenuse? Justify your thinking.2. How can you determine the length of the hypotenuse of a right triangle when it

is plotted on a coordinate plane? Explain your thinking.3. Describe the process you would use to find the hypotenuse of a right triangle,

given the coordinates of the three vertices.

[WORDS FOR WORD WALL]Pythagorean Theorem, coordinate grid, square, hypotenuse, vertices, horizontal distance, vertical distance

[GROUPING]Cooperative Pairs (CP), Whole Group (WG), Individual (I)

[LEVELS OF TEACHER SUPPORT]Modeling (M), Guided Practice (GP), Independent Practice (IP)

[MULTIPLE REPRESENTATIONS]SOLVE, Verbal Description, Concrete Representation, Pictorial Representation, Algebraic Formula, Graphic Organizer, Graph

[WARM-UP] (IP, WG, I) S405 (ANSWERS ON T833.)Have students turn to S405 in their books to begin the Warm-Up. Students will find the hypotenuse of triangles using the Pythagorean Theorem. Monitor students to see if any of them need help during the Warm-Up. Have students complete the problems and then review the answers as a whole group. {Algebraic Formula, Pictorial Representation}

[HOMEWORK] Take time to go over the homework from the previous night.

[LESSON] [1 – 2 Days (1 day = 80 minutes) – M, GP, WG, CP, IP]

Mathematics Success – Grade 8T822

LESSON 31: Pythagorean Theorem Part 2

MODELING

Review of Vertical and Horizontal Distances on the Coordinate Plane

Step 1: Have students identify the vertices of the triangle. • Partner A, what are the coordinates of Point A? (2,5) Record. • Partner B, what are the coordinate of Point B? (5,2) Record. • Partner A, what are the coordinates of Point C? (2,2) Record.

1 2 3 4 5

1

-1

2

3

4

5

-2

-3

-4

-5

-1-2-3-4-5

A

C B

SOLVE Problem (GP, WG) S406 (Answers on T834.)

Have students turn to S406 in their books. The first problem is a SOLVE problem. You are only going to complete the S step with students at this point. Tell students that during the lesson they will learn how to apply the Pythagorean Theorem to find the distance between two points in a coordinate system. They will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE, Verbal Description, Graphic Organizer}

Review of Vertical and Horizontal Distances on the Coordinate Plane(M, GP, WG, IP, CP) S406, S407 (Answers on T834, T835.)

M, GP, WG, CP: Direct students’ attention to the graph at the bottom of S406. Students will be using the coordinate graph on S406 and the graphic organizer on S407 to answer questions that will review vertical and horizontal distances on the coordinate plane. Assign the roles of Partner A and Partner B to students. {Pictorial Representation, Graphic Organizer, Graph, Verbal Description}

Mathematics Success – Grade 8 T823

LESSON 31: Pythagorean Theorem Part 2

Step 2: Have students turn to page S407. They will use the coordinate graph on S406 to answer questions that will help them to review finding the horizontal and vertical distances on the coordinate plane.

Step 3: Partner A, what are the coordinates of Point B and Point C? [B: (5, 2) C:

(2, 2)] Record. • Partner B, Which of the coordinates are the same for these two points?

(The y-coordinates are the same so we know that this is a horizontal line.) Record.

• Partner A, explain how we can find the distance between the two points by looking at the graph. (We can count from 2 to 5.) Record.

• Partner B, is there another method we can use? Explain your thinking. [Subtract the values of the x-coordinates. (5 – 2 = 3)] Record.

Step 4: Partner A, what is the horizontal distance between Points B and C? (3 units) Record.

• Partner B, what points can we use to determine the vertical distance of the leg that forms the triangle? (Point A and Point C) Record.

• Partner A, what are the coordinates of Point A and Point C? [A: (2, 5) C: (2, 2)] Record.

Step 5: Partner B, what do you notice about the coordinates of the two points that we are using to determine the measure of the vertical leg? (The x-coordinates are the same.) Record.

• Partner A, how can we find the distance between the two points by looking at the graph? (We can count from 2 to 5.) Record.

• Partner B, is there another method we can use to determine the distance? Explain another method we can use. [Subtract the values of the y-coordinates. (5 – 2 = 3)] Record.

• Partner A, what is the vertical distance between Points A and C? (3 units) Record.

IP, CP, WG: Have students complete Problem 12 on the bottom of S407. Students will identify the coordinates of the triangle and determine the measure of the horizontal and vertical legs of the triangle. Review the answers as a whole group. {Verbal Description, Graphic Organizer, Pictorial Representation}

Mathematics Success – Grade 8T824

LESSON 31: Pythagorean Theorem Part 2

Discovery Activity – Pythagorean Theorem (M, GP, WG) S408, S409, T836 (Answers on 837, T838.)

M, GP, WG, CP: Give each student a copy of the coordinate graph on T836. Students will be using the coordinate graph to explore the Pythagorean Theorem using the relationship between horizontal and vertical legs and the hypotenuse of the triangle. Be sure students know their designation as Partner A or Partner B. {Pictorial Representation, Graphic Organizer, Concrete Representation, Graph, Verbal Description}

MODELING

Discovery Activity - Pythagorean Theorem

Step 1: Have students plot the three points given on page S409 onto the coordinate system, label them, and connect them to form a triangle.

-10

-10

-5

-5

5

10

5 10

A C

B

Step 2: Partner A, what is the length of AB ? (4 units) Record. Defend your thinking. (I counted the units or I subtracted the y-coordinates.) Record.

• Partner B, what is the length of AC? (3 units) Record. Defend your thinking. (I counted the units or I subtracted the x-coordinates.) Record.

• Partner A, what is the length of BC ?(I can’t tell looking at the graph.) Record.

Step 3: Explain to students that they can make a ruler by cutting the very bottom row of squares off the grid. They can then place the ruler next to the diagonal line to determine its length.

*Teacher Note: Students may use the coordinate graph on S408 or the copy master on T836.

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LESSON 31: Pythagorean Theorem Part 2

Step 4: Have student pairs measure the length of the hypotenuse. • What is the length of the hypotenuse? (5 units) Record.

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-10

-5

-5

5

10

5 10

A C

B

Plac

e H

ere

Cut here

Step 5: Model how to draw a square to the left of AB, so that AB is one side of the square. Have students draw the square on their coordinate grid.

• Partner B, how long is each side of the square? (4 units) Record. • Partner A, what is the area of the square? (16 square units) Record. • Partner B, justify your answer. (I used the formula for the area of a

square and multiplied the side measures. I can also count the number of unit squares inside the larger square.) Record.

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-10

-5

-5

5

10

5 10

A C

B

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LESSON 31: Pythagorean Theorem Part 2

Step 6: Model for students how to draw a square below AC, so that AC is one side

of the square as they draw the square on their coordinate grid. • Partner A, how long is each side? (3 units) Record. • Partner B, what is the area of the square? (9 square units) Record.

Justify your answer. (I used the formula for the area of a square and multiplied the side measures. I can also count the number of unit squares inside the larger square.) Record.

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-10

-5

-5

5

10

5 10

A C

B

Step 7: Model for students how to draw a square so that BC is one side of the square.

• Partner A, how long is each side? (5 units) Record. • Partner B, explain your thinking. (We measured the line segment BC

to find the length which is 5 units. A square has 4 congruent sides, so each side will be 5 units.) Record.

• Partner A, what is the area of the square? (25 square units) Record. • Partner B, justify your answer. (I used the formula for the area of a

square and multiplied the side measures. Since the square is on a slant, counting the unit squares is not possible. I can also count the number of unit squares inside the larger square.) Record.

Mathematics Success – Grade 8 T827

LESSON 31: Pythagorean Theorem Part 2

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-5

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5

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5 10

A C

B

Step 8: Ask students to identify the area of the three squares. (9, 16, 25 square units) Record.

• Partner A, do you see a relationship between the first two and the third numbers? (Yes) Record.

• Partner B, defend your answer. (The sum of 9 and 16 is 25.) Record.

Step 9: Have partners discuss Question 10 and be prepared to share their response with the whole group.

• Explain what this means in relationship to the sides of the triangles. (The measure of AB squared (42) plus the measure of AC squared (32) is equal to BC squared (52) or 32 + 42 = 52; 9 + 16 = 25.) Record.

Applying the Pythagorean Theorem on the Coordinate Plane (M, GP, WG) S408, S410, T836 (Answers on T837, T839.)

WG, CP, M, GP: Students will apply the Pythagorean Theorem to a triangle

on the coordinate system. Be sure students know their designated role as Partner A and Partner B. Students can use the coordinate grid on S408 or Copy Master T836. {Algebraic Formula, Verbal Description, Graphic Organizer, Pictorial Representation}

Mathematics Success – Grade 8T828

LESSON 31: Pythagorean Theorem Part 2

MODELING

Applying the Pythagorean Theorem on the Coordinate Plane

Step 1: Have students plot the three points given on page S410 onto the coordinate system, label them, and connect them to form a triangle.

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-10

-5

-5

5

10

5 10

A C

B

E

D F

Step 2: Partner A, what is the length of DE? Defend your thinking. (5 units - I counted the units or I subtracted the y-coordinates.) Record.

• Partner B, What is the length of DF? Defend your thinking. (12 units - I counted the units or I subtracted the x-coordinates.) Record.

• Partner A, what is the length of EF? (I can’t tell by looking at the graph.) Record.

Step 3: Have students use the strip they cut from the bottom of the page as a ruler to find the length of EF. What is the length? (13 units) Record.

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-10

-5

-5

5

10

5 10

A C

B

E

D F

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LESSON 31: Pythagorean Theorem Part 2

Step 4: Explain to students that they will not have room to draw all the squares next to each side of the triangle, but that they can still use the same steps as in the first example to figure out the area of each square.

Step 5: Partner B, what would the area of a square be that had DE as one of its sides? Explain your thinking. [The area would be 25 square units. I would multiply s × s to determine the area. (5 × 5 = 25)] Record.

• Partner A, what would the area of a square be that had DF as one of its sides? Explain your thinking. [The area would be 144 square units. I would multiply s × s to determine the area. (12 × 12 = 144)]

• Partner B, what would the area of a square be that had EF as one

of its sides? Explain your thinking. [The area would be 169 square units. I would multiply s × s to determine the area. (13 × 13 = 169)] Record.

• Have partners identify the area of the three squares that would be formed by using the measure of each side of the triangle. (25 square units, 144 square units, 169 square units)

• Partner A, explain the relationship between the three numbers. (The sum of the area of the first two squares equals the area of the third square.) Record.

• Partner B, explain what this means in relationship to the sides of the triangles. (The measure of DE squared (52) plus the measure of DF

squared (122) is equal to EF squared (132) or 52 + 122 = 132; 25 + 144 = 169.) Record.

Step 6: Have students look at the coordinate grid on S408 or T836. • Partner A, how did we get the numbers of 9, 16, 25 in Question 9 on

S409, and 25, 144, 169 in Question 9 on S410. (They were the areas of the squares we made from the sides of the triangles.) Record.

• Partner B, explain how to find the area of a square. (Multiply a side by a side, or s2) Record.

• Partner A, if the sides are identified as a and b, and the hypotenuse, which is always the longest side of the triangle and across from the right angle, was side c, what equation could you write to relate the areas of the three squares? [a2 + b2 = c2 (or the area of the square with side a, plus the area of the square with side b, equals the area of the square with side c.] Record.

Mathematics Success – Grade 8T830

LESSON 31: Pythagorean Theorem Part 2

MODELING

Finding the Hypotenuse on the Coordinate Plane Using Edge Lengths or Coordinates

Step 1: Have students look at Problem 1 on page S411.

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-2 2

2

Step 2: Explain to students that they will be using the triangle and substituting the horizontal and vertical lengths of the triangle from the coordinate grid into the Pythagorean Theorem to determine the hypotenuse.

• Partner A, what is the Pythagorean Theorem? (a2 + b2 = c2 ) • Partner B, What is the length of Side a in the triangle? (6 units) Record. • Partner A, what is the length of Side b in the triangle? (8 units) Record. • Partner B, what is the value of 62? (36) Record. • Partner A, what is the value of 82? (64) Record. • Partner B, what is the value of c2? (100) Record. • Partner A, explain how we can determine the length of Side c if we

know the value of c2? (Find the square root of c2.) • Partner B, what is the value of the square root of 100? (10) Record. • Partner A, what is the length of Side c? (10)

Step 3: Have students turn to page S412 and look at Problem 4. • Partner B, how is Problem 4 different from Problems 1 – 3 on page S411?

(Problems 1–3 have a triangle on a coordinate grid and Problems 4–6

give the coordinates of the triangle, but do not show a picture.) • Partner A, how can we determine the side lengths of the triangle that

is created given the coordinates? (We can find the vertical distance by finding the difference between the y-coordinates of two points and we can find the horizontal distance by finding the difference between the x-coordinates.)

Step 4: Partner B, what is the distance from Point A to Point B? (3 units) Record. • Partner A, what is the measure of AB? (3 units) Record. • Partner B, what is the distance from Point A to Point C? (5 units) Record. • Partner A, what is the measure of AC? (5 units) Record.

Finding the Hypotenuse on the Coordinate Plane Using Edge Lengths or Coordinates (M, GP, IP, WG, CP) S411, S412 (Answers on T840, T841.)

Mathematics Success – Grade 8 T831

LESSON 31: Pythagorean Theorem Part 2

IP, CP, WG: Have students complete Problems 2–3 on page S411 and Problems 5–6 on page S412. Students will determine the measure of the horizontal and vertical legs of the triangle and substitute those values into the Pythagorean Theorem to find the length of the hypotenuse. Review the answers as a whole group. {Verbal Description, Graphic Organizer, Pictorial Representation, Graph, Algebraic Formula}

• Partner B, explain how we will find the distance from Point B to Point

C? (Use the Pythagorean Theorem.)

Step 5: Have students substitute in the values for the vertical and horizontal legs of the triangle into the formula in column 3 and complete the Pythagorean Theorem to determine the length of side c of the triangle.

SOLVE Problem (GP, WG) S413 (Answers on T842.)

Remind students that the SOLVE problem on S413 is the same one from the beginning of the lesson. Complete the SOLVE problem with your students. Ask them for possible connections from the SOLVE problem to the lesson. Students should say that they will apply the Pythagorean Theorem to find the length of the hypotenuse of a right triangle on the coordinate plane. {SOLVE, Algebraic Formula, Verbal Description, Pictorial Representation, Graph}