Mathematics Success Level F

Transcription

1 T598 [OBJECTIVE] The student will find the perimeter and area of rectangles and triangles. [MATERIALS] Student pages S204 S212 Transparencies T612, T614, T616, T618, T620, T622 Ruler Scissors Gridded index cards (1 for each student) Toothpicks Colored pencils Colored paper for foldable (1 sheet per student) [ESSENTIAL QUESTIONS] 1. What is perimeter? 2. What is area? 3. What is the relationship between the area of rectangles and triangles? [WORDS FOR WORD WALL] rectangle, right triangle, perimeter, area, height, base, perpendicular, congruent, square units [GROUPING] Cooperative Pairs (CP), Whole Group (WG), Individual (I) [LEVELS OF TEACHER SUPPORT] Modeling (M), Guided Practice (GP), Independent Practice (IP) [MULTIPLE REPRESENTATIONS] SOLVE, Algebraic Formula, Verbal Description, Pictorial Representation, Concrete Representation, Graphic Organizer [WARM-UP] (5 minutes IP, WG, I) S204 (Answers on T611.) Have students turn to S204 in their books to begin the Warm-Up. Students will work with multiplication arrays. Monitor students to see if any of them need help during the Warm-Up. Give students 3 minutes to complete the problems and then spend 2 minutes reviewing the answers as a class. {Verbal Description, Pictorial Representation} [HOMEWORK] (5 minutes) Take time to go over the homework from the previous night.

2 T599 [LESSON] DAY 1: (50 minutes M, CP, WG, GP, IP) DAY 2: (50 60 minutes M, GP, IP) Day SOLVE Problem (2 minutes GP, WG) T612, S205 (Answers on T613.) Have students turn to S205 in their books, and place T612 on the overhead. 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 find the perimeter and area of rectangles and triangles. They will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE, Algebraic Formula} Perimeter of Rectangles (24 minutes GP, IP, M, WG, CP) T612, S205 (Answers on T613.) 18 minutes M, GP, CP: Organize students into pairs. Use the following activity to help students explore the perimeter of rectangles to determine the perimeter formula. Students will work with toothpicks to build the perimeters of rectangles. {Verbal Description, Concrete Representation, Pictorial Representation, Algebraic Formula} MODELING Perimeter of Rectangles Step 1: Introduce perimeter by showing examples in the room: chair back, desk top, bulletin board, etc. Trace around the book cover to show an example of distance around an object or figure. Ask students to show you the perimeters of their textbooks or pieces of paper, etc. Ask students why it would be important to know the distance around a figure (fencing, bulletin board border, measurement). Step 2: Distribute 14 toothpicks to each pair. On the overhead, build a square using 4 toothpicks, as students build their own squares. Tell students that each toothpick represents one unit of perimeter. Ask students how many units of perimeter there are in the square. (The perimeter is 4 units.) Ask students how they determined the perimeter (counted around).

3 T600 Step 3: Add another square on the overhead using an additional three toothpicks as shown below. Ask students how many units of perimeter this model shows. (It shows 6 units of perimeter.) Ask students how they found the perimeter. (They counted the toothpicks around the rectangle.) Make sure students understand that they are to count the units for the distance around the rectangle, not the toothpick inside the rectangle. Point out the length of the rectangle (2 units). Explain that the length of the rectangle is called the base. Explain that the units on the end are perpendicular to the base and make up the height of the rectangle. There is one unit on each end, creating a height of 1 unit. Step 4: Add another square to the model to create a different rectangle. Ask students how they found the perimeter. (Most likely, they counted the toothpicks around the rectangle.) Ask students what the length of the base of the rectangle is (3 units) and how many units create the height of the rectangle. (The height of each end is 1 unit.) Ask students if there is another way they could determine the perimeter using the dimensions of the base and the height. (They could multiply the base by 2, because it is congruent to the top measurement, multiply the height by 2, because the sides are congruent, and then add the two values together.) Explain to students how to write the formula for finding the perimeter of a rectangle as P = 2b + 2h. Record on T612 (S205). Step 5: Ask students to look at Rectangle 1 on T612 (S205) and record as you model how to solve by asking the following questions: What is the length of the base of the rectangle? (3 cm) Record. What is the height of the rectangle? (2 cm) Record. What is the formula for finding the perimeter of a rectangle? (P P = 2b + 2h) Fill in the unknowns with values and solve: P = 2(3) + 2(2); = 10 cm. Picture Length of Base Height Formula Perimeter 3 cm 2 cm P = 2b + 2h 2(3) + 2(2) = = 10 cm

4 T601 Step 6: Have students look at Problem 3. Point out that there is no model, but that the measurements are given. Model how to draw a picture and use the formula to find the perimeter by asking the following questions: What is the length of the base of the rectangle? (11 in.) What is the height of the rectangle? (4 in.) What is the formula for finding the perimeter of the rectangle? (P P = 2b + 2h) Record. Fill in the length and the width using numbers: P = 2(11) + 2(4). Then solve the problem: P = 30 in. 6 minutes IP, CP: Have students complete Problems 2, 4, and 5 with a partner and come back together to share results. Give students 5 minutes to solve the problems and then go over the answers using T613. {Pictorial Representation, Verbal Description, Algebraic Formula} Area of Rectangles (24 minutes GP, IP, M, WG, CP) T614, S206 (Answers on T615.) 18 minutes M, GP, CP: Have students turn to S206 in their books, and place T614 on the overhead. Use the following activity to help students explore the area of rectangles to determine the formula. {Verbal Description, Concrete Representation, Pictorial Representation, Algebraic Formula} MODELING Area of Rectangles Step 1: Ask students why it would be important to know the space that a flat figure covers. Remind students that they found the distance around a rectangle or square the perimeter in the last activity. They will determine the area that a rectangle or square covers in this activity. Find examples in the room: chair back, desk top, bulletin board, etc. Be specific when examining the different objects. For example, on a textbook cover, show space inside the rectangle to show space inside the figure. Step 2: Have students look at the rectangle at the top of T614 (S206). Ask, What is the length of the base in units? (6 units) What is the height? (3 units) How many square units cover the rectangle? (18) How do you know? (counted) Record.

5 T602 Step 3: Remind students that each of the units inside the rectangle is a square. Explain that the squares are called square units and are shown as units 2. Record the area as 18 units 2. Step 4: Ask students to talk with a partner and discuss a way they could use a formula to find the area of a rectangle, similar to the formula for finding the perimeter of a rectangle. (They should be able to see that the length of the base could be multiplied by the height to find the area.) Step 5: Record the formula for the area of a rectangle: A = bh. Fill in the values with 6 3 = 18, and record as square units. Picture Length of Base Height Formula Area 6 units 3 units A = bh 6 3 = 18 units 2 Step 6: Have students look at Problem 1. Point out that Problem 1 does not have a grid. Complete Problem 1 with students by asking the following questions: What is the length of the base? (5 in.) Record. What is the height? (4 in.) Record. Write the formula for finding the area of a rectangle. (A = bh) Fill in the values, and solve. Record in square inches. Picture Length of Base Height Formula Area 5 in. 4 in. A = bh 5(4) = in. 2

6 T603 Step 7: Have students look at Problem 2. Point out that there is no picture. Model how to solve using the steps below: What is the length of the base of the rectangle? (12 in.) What is the height of the rectangle? (4 in.) Draw a picture to show the rectangle. Write the formula for finding the area of a rectangle. (A = bh) Fill in the values and solve. Make sure to use square inches in the answer. Picture Length of Base Height Formula Area 12 in. 4 in. A = bh 12 4 = in. 2 6 minutes IP, CP: Have students complete Problems 3 5 with a partner and come back together to share results. Give students 5 minutes to solve the problems and then go over the answers using T615. {Pictorial Representation, Verbal Description, Algebraic Formula} Day Perimeter of Triangles (20 minutes GP, IP, M, WG, CP) T616, S207 (Answers on T617.) 13 minutes M, GP, WG: Have students turn to S207 in their books, and place T616 on the overhead. Pass out the following to each student: gridded index cards, scissors, and a ruler. Use the following activity to complete S207 with students. The triangles created in this activity will also be used in the triangle area activity. {Concrete Representation, Pictorial Representation, Verbal Description, Algebraic Formula}

7 T604 MODELING Perimeter of Triangles Step 1: As you model, have students create a rectangle with a base length of 8 cm and a height of 6 cm on the gridded paper. Have students shade the rectangle using a blue colored pencil. Have students cut out the rectangle. Step 2: Using a green colored pencil, model for students how to draw a line using a ruler from the top right corner to the bottom left corner of the rectangle they have drawn. Point out the diagonal line and ask students what two polygons are created by the diagonal (two right triangles). Model how to cut along the green diagonal line that divides the rectangle into two congruent triangles. Step 3: Remind students that the perimeter of a rectangle is the distance around the rectangle. Ask, What is the perimeter of a triangle? (the distance around the triangle) Step 4: Ask students if they can determine the lengths of the sides of the triangle by looking at the units on the grid. (They can determine the lengths of two sides, but not the diagonal.) Step 5: Model how to find the length of the third side of the triangle using a ruler. Be sure to measure from 0, not 1, on the ruler, as students use their rulers to measure. The length of the side is 10 centimeters. Step 6: Model how to find the perimeter of the triangle by adding the three side lengths ( = 24 cm).

8 T605 Step 7: Ask students to help you create a formula for finding the perimeter of a triangle. Label the base of the triangle a, another side b, and the longer side c. Explain to students that they are adding side a, side b, and side c, so P = a + b + c. Step 8: On T616 (S207), model how to find the perimeter of the triangle at the top of the page, as students record. The lengths of each side are given: a = 8 cm, b = 6 cm, and c = 10 cm. Record in the appropriate column. Record the formula: P = a + b + c. Record the perimeter: = 24 cm. Picture Length Side a Length Side b Length Side c Formula Perimeter 8 cm 6 cm 10 cm P = a + b + c P = = 24 cm Step 9: Look at Problem 1 on T616 (S207). Point out that it is not on a grid. Ask students how it is different from the first triangle. (All three sides are equal and there is not a right angle.) Point out to students that if all sides are equal, then each side equals 4 inches. Record the dimensions. Fill in the formula to use and find the perimeter. Step 10: Have students look at Problem 2. The measurements are given. Model how to draw a picture of the triangle and find its perimeter by asking the following questions: What is the length of side a, in units? (7 in.) What is the length of side b, in units? (5 in.) What is the length of side c, in units? (5 in.) What is the formula for finding the perimeter of the triangle? (P = a + b + c) Fill in the lengths in the formula using number values: P = Solve the problem: P = 17 in.

9 T606 7 minutes IP, WG, CP: Have students complete Problems 3 5 with a partner and come back together to share results. Give students 5 minutes to solve the problems and then go over the answers using T617. {Pictorial Representation, Verbal Description, Algebraic Formula} Area of Triangles (7 minutes GP, IP, M, WG, CP) T618, S208 (Answers on T619.) 7 minutes M, GP, WG: Pass out the following to students: gridded paper, scissors, ruler. Have students turn to S208 in their books, and place T618 on the overhead. Use the following activity to help students explore the areas of triangles. {Concrete Representation, Pictorial Representation, Verbal Description, Algebraic Formula} MODELING Area of Triangles Step 1: Explain to students that they will be using the area of a rectangle to discover the area of a triangle. Use the triangles created for the perimeter activity. Ask students to put the triangles back together to form the rectangle. Ask students what the area of the rectangle is (48 cm 2 ). Step 2: Ask students to place the triangles one on top of the other to show that they are exactly the same size, or congruent. Step 3: Point out to students that a triangle is exactly one half of the rectangle. Ask, If this is so, what would the area of one triangle be? (24 cm 2 ) Step 4: Place the triangle in front of you, with the right angle of the triangle on the right. Remind students that the area of the triangle is 24 cm 2. Record on the triangle.

10 T607 Step 5: Model how to determine a formula for finding the area of a triangle by asking students the following questions: What is the length of the base of the triangle? (8 cm) Record on the triangle. What is the height of the triangle? (6 cm) Record on the triangle. What fractional part of the whole rectangle does the triangle represent? ( 1 2 ) Can we find a numerical way to determine the area of this triangle using the measurements we have? Record the formula on the triangle. (A = 1 2 bh) Step 6: Model how to find the area of the triangle at the top of T618 (S208) by asking the following questions: What is the length of the base of the triangle? (10 ft) Record in the graphic organizer. What is the height of the triangle? (8 ft) Record. What is the formula for finding the area of a triangle? (A = 1 2 bh) Record. Put in the values for b and h: A = 1 (10) 8. Record the answer in square 2 feet: 40 ft 2. Picture Length Base Height Formula Area 10 ft 8 ft A = 1 2 bh A = 1 (10) 8 A = A = 40 ft 2 Step 7: Point out to students that Problem 1 on S208 is not a right triangle. However, the formula for finding the area of a triangle will work for any triangle. Use the following directions to model how to solve: What is the length of the base of the triangle? (6 meters) What is the height of the triangle? (4 meters) What is the formula for finding the area of any triangle? (A = 1 2 bh) Put in the values for b and h: A = 1 (6) 4. Record the answer in square 2 meters (12 m 2 ).

11 T608 Step 8: Point out that only measurements are provided for Problem 2. Use the following directions to model how to solve: What is the length of the base of the triangle? (12 inches) What is the height of the triangle? (8 inches) Draw a picture of the triangle. What is the formula for finding the area of a triangle? (A = 1 2 bh) Record. Put in the values for b and h. (A = 1 (12) 8) Record the answer in 2 square inches (48 in. 2 ). 10 minutes IP, CP, WG: Have students complete Problems 3 8 on S208 S209 with a partner and come back together to share results. The problems on S209 provide mixed practice for perimeter and area of triangles and rectangles. Give students 8 minutes to solve the problems and then go over the answers using T619 and T621. {Pictorial Representation, Verbal Description, Algebraic Formula} Foldable (6 minutes GP, WG, M) Follow the steps below to have each student make a foldable. Students will create a foldable for perimeter and area of triangles and rectangles. {Verbal Description, Pictorial Representation, Graphic Organizer, Algebraic Formula }

12 T609 MODELING Foldable Step 1: Fold one corner of the piece of paper down to the edge of the other side of the paper. Cut off the strip at the bottom. A square should be left. Step 2: Open the square. Fold each corner into the center. Write Triangle Perimeter and Triangle Area on one outside flap. The other side will have Rectangle Perimeter r and Rectangle Area. Two outside flaps will be blank to be written on later. Step 3: Pull up the flap that says Triangle Perimeter and Triangle Area. On the triangle that sticks up, draw two triangles with the base and height labeled. Underneath the triangles, write the formulas. On the square portion, draw another triangle, with values for the height and base. Find the area and perimeter using the formula. See your foldable for the information. Step 4: Pull up the flap that says Rectangle Area and Rectangle Perimeter. On the triangle that sticks up, draw two rectangles with the base and height labeled. Underneath the rectangles, write the formulas. On the square portion, draw another rectangle, with values for the height and base. Find the area and perimeter using the formula. See your foldable for the information.

13 T610 SOLVE Problem (5 minutes GP, WG) T622, S210 (Answers on T623.) Remind students that the SOLVE problem 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 will work with area of triangles.) {SOLVE, Algebraic Formula, Verbal Description} If time permits (10 minutes IP, I) S211 (Answers on T624.) Have students complete Problems 1 10 independently. {Algebraic Formula, Pictorial Representation, Verbal Description} [CLOSURE] (2 minutes) To wrap up the lesson, go back to the essential questions and discuss them with students. What is perimeter? (The distance around a figure.) What is area? (The space that a figure covers.) What is the relationship between the area of rectangles and triangles? (If the area of a rectangle is divided into two congruent triangles, the area of each of the triangles is one half the area of the rectangle.) [HOMEWORK] Assign S212 for homework. (Answers on T625.) [QUIZ ANSWERS] T626 The quiz can be used at any time as extra homework or to see how students did on the perimeter and area of rectangles and triangles.