Formula for the Area of a Triangle Objective To guide the development and use of a formula for the area of a triangle.


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1 Formula for the rea of a Triangle Objective To guide the development and use of a formula for the area of a triangle. epresentations etoolkit lgorithms Practice EM Facts Workshop Game Family Letters ssessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Find the areas of rectangles and parallelograms. [Measurement and Reference Frames Goal ] Develop a formula for calculating the area of a triangle. [Measurement and Reference Frames Goal ] Identify perpendicular line segments and right angles. [Geometry Goal 1] Describe properties of and types of triangles. [Geometry Goal ] Evaluate numeric expressions containing parentheses. [Patterns, Functions, and lgebra Goal 3] Key ctivities Students arrange triangles to form parallelograms. They develop and use a formula for finding the area of a triangle. Ongoing ssessment: Recognizing Student chievement Use journal page 4. [Measurement and Reference Frames Goal ] Key Vocabulary equilateral triangle isosceles triangle scalene triangle right triangle base height Materials Math Journal, pp Study Link 8 Math Masters, pp. 5 and 454 transparency of Math Masters, p. 403 (optional) slate centimeter ruler scissors tape index card or other squarecorner object Solving Fraction Problems Math Journal, p. 43 Students identify fractional parts of number lines, collections of objects, and regions. Math Boxes Math Journal, p. 44 Students practice and maintain skills through Math Box problems. Study Link Math Masters, p. Students practice and maintain skills through Study Link activities. Comparing reas Math Masters, p. 7 scissors Students cut apart a regular hexagon and use the pieces to make area comparisons. Finding the rea and Perimeter of a Hexagon Math Masters, p. 8 centimeter ruler Students use a combination of different area formulas to find the area of a nonregular hexagon. EXTR PRCTICE Playing Rugs and Fences Math Masters, pp Student Reference Book, pp. 0 and 1 Students practice finding the perimeter and area of polygons. dvance Preparation For the optional Extra Practice activity in Part 3, consider copying the Rugs and Fences Cards on Math Masters, pages on cardstock. Teacher s Reference Manual, Grades 4 pp , 1, Lesson 93
2 Getting Started Mental Math and Reflexes Dictate large numbers for students to write on their slates. Suggestions: Mathematical Practices SMP, SMP4, SMP5, SMP, SMP7, SMP8 Content Standards 4.NBT., 4.NF.4c, 4.MD.3, 4.G.1, 4.G. Math Message Make a list of everything that you know about triangles. 37, ,0 95,003 1,34,895 0,00,597 35,798,41 3,00,300,004,000,000,500 90,08,351,007 For each number, ask questions such as the following: Which digit is in the millions place? What is the value of the digit x? How many hundred millions are there? Study Link 8 FollowUp Have small groups compare answers and explain their strategies for finding the length of the base when the height and the area are given. 1 Teaching the Lesson NOTE Because isosceles triangles are defined as having at least two sides of equal length, all equilateral triangles are isosceles. So, some students may include the equilateral triangles in the isosceles pile. If this occurs, you may want to have the class decide on a definition of isosceles triangle that would exclude the equilateral triangles. Sample answer: n isosceles triangle has exactly two sides that are the same length. reas of Triangles 1. Cut out Triangles and B from Math Masters, page 5. DO NOT CUT OUT THE ONE BELOW. Tape the two triangles together to form a parallelogram. 1 Triangle B 13 Math Message FollowUp s students share their responses, write them on the board. The list might include: triangle is a threesided polygon. WHOLECLSS The sum of the measures of the angles in a triangle is 180. triangle has three vertices. triangle is a convex polygon. n equilateral triangle is a triangle in which all three sides have the same measure, and all three angles have the same measure. n equilateral triangle is a regular polygon. n isosceles triangle is a triangle that has at least two sides with the same length and at least two angles with the same measure. scalene triangle is a triangle in which there are no sides of equal length and no angles of equal measure. right triangle is a triangle with one 90 angle. base = base = height = 4 height = 4 rea of triangle = 1 rea of parallelogram = 4. Do the same with Triangles C and D on Math Masters, page 5. Triangle C Exploring Triangle Properties (Math Masters, p. 454) WHOLECLSS base = 4 base = 4 height = 4 height = 4 rea of triangle = 8 rea of parallelogram = 1 Math Journal, p. 40 C D Have students cut out the cards on Math Masters, page 454. sk students to sort the triangles into three categories: equilateral, isosceles, and scalene. equilateral: C, F; isosceles: E, G, H, N, P; scalene:, B, D, I, J, K, L, M, O. Discuss the angle properties of each type of triangle: equilateral triangles have three angles of equal measure, isosceles triangles have two angles of equal measure, and scalene triangles have no angles of equal measure. 94 Unit 8 Perimeter and rea
3 Draw a right triangle on the board: sk students to tell you the definition of a right triangle. right triangle is a triangle with one 90 angle. sk: Do you see any right triangles in any of the piles? Yes. Triangles, H, I, and P are right triangles. Can an equilateral triangle be a right triangle? Explain. No. ll the angles in an equilateral triangle have the same measure. Can an isosceles triangle be a right triangle? Explain. Yes. Triangles H and P are isosceles and right. Can a scalene triangle be a right triangle? Explain. Yes. Triangles and I are scalene and right. Draw the chart shown in the margin on the board. sk students to help you fill in the chart with and no to show which types of triangles can be right triangles. Tell students that in this lesson they will develop a formula for the area of a triangle using the formula for the area of a parallelogram. Links to the Future The use of a formula to calculate the area of a triangle is a Grade 5 Goal. Developing a Formula for the rea of a Triangle (Math Journal, pp. 40 and 41; Math Masters, p. 5) PROBLEM SOLVING WHOLECLSS Draw a triangle on the board. Choose one of the sides the side on which the triangle sits, for example and call it the base. Label the base in your drawing. Explain that base is also used to mean the length of the base. The shortest distance from the vertex above the base to the base is called the height of the triangle. Draw a dashed line to show the height and label it. Include a rightangle symbol. (See margin.) sk the class to turn to journal page 40 while you distribute copies of Math Masters, page 5. Point out that Triangles and B on the master are the same as Triangle on the journal page. Guide students through the following activity: 1. Cut out Triangles and B from the master. Make sure students realize that the triangles have the same area and are congruent.. Tape the triangles together at the shaded corners to form a parallelogram. 3. Tape the parallelogram in the space next to Triangle in the journal. Equilateral Isosceles Scalene reas of Triangles continued 3. Do the same with Triangles E and F. Triangle E base = 7 base = 7 height = height = rea of triangle = 7 rea of parallelogram = Do the same with Triangles G and H. Triangle G height base The height of a triangle is measured along a line segment perpendicular to the base. s with parallelograms, any side of a triangle can be the base. The choice of the base determines the height. base = 4 base = 4 height = 3 height = 3 rea of triangle = rea of parallelogram = 1 5. Write a formula for the area of a triangle. = (b h) (b h), or = _ Math Journal, p. 41 E Right G no H height length of base F Lesson 95
4 . Draw a line segment to show the height of Triangle SM. Use your ruler to measure the base and height of the triangle. Then find the area. 5 base = height = rea = reas of Triangles continued 7. Draw three different triangles on the grid below. Each triangle must have an area of 3 square centimeters. One triangle should have a right angle. Sample answers: 8. See the shapes below. Which has the larger area the star or the square? Explain your answer. Neither. Both have an area of 1 sq units. rea of square = 4 4 = 1; area of star = area of square in center + (4 area of a triangle) = 4 + (4 3) = 1. S M 13 Discuss the relationship between the area of the triangle and the area of the parallelogram. Triangles and B have the same area. Therefore, the area of either triangle is half the area of the parallelogram. 4. Record the dimensions and areas of the triangle and the parallelogram. Base of triangle and parallelogram = ; height of triangle and parallelogram = 4 ; area of parallelogram = 4 ; area of triangle = the area of parallelogram = 1. Have students repeat these steps with Triangles C and D, E and F, and G and H. Then bring the class together to state a rule and write a formula for the area of a triangle. Since the base and the height of a triangle are the same as the base and the height of the corresponding parallelogram, then: rea of the triangle = the rea of the parallelogram, or Math Journal, p. 4 rea of the triangle = of (base height) Using variables: = of (b h), or = (b h) where b is the length of the base and h is the height. Have students record the formula at the bottom of journal page 41. Solving rea Problems (Math Journal, p. 4; Math Masters, p. 403) PRTNER Fractions of Sets and Wholes 59 lgebraic Thinking Work with the class on Problem. Students can place an index card (or other squarecorner object) on top of the triangle, align the bottom edge of the card with the base (making sure that one edge of the card passes through point S), and then draw a line for the height. Students will need a centimeter ruler to measure the base and the height. 1. Circle of the triangles. Mark Xs on. a. Shade _ 5 of the pentagon. _ 3 of the triangles. index card S b. Shade 3_ 5 of the pentagon. height M 3. There are 5 musicians in the school band: 4 of the musicians play the flute and 8 play the trombone. a. How many musicians play the flute? b. How many musicians play the trombone? Complete. a. 3_ 10 4 of is 90. b. 3 of 7 is Wei had 48 beanbag animals in her collection. She sold 18 of them to another collector. What fraction of her collection did she sell? 3_ 8 _ djusting the ctivity ELL On the board, draw a triangle that has an obtuse angle as one of its base angles. Have students draw the height of the triangle. Demonstrate by extending the base along the side of the obtuse angle and drawing a perpendicular line from the opposite vertex to the extended base. For example: c. 5_ 100 of 10 is. d. 3_ of is 15. e. 3 of 7 is 4. f. 5_ 0 4 of 1 is.. Fill in the missing fractions on the number line Math Journal, p. 43 U D I T O R Y K I N E S T H E T I C T C T I L E V I S U L 9 Unit 8 Perimeter and rea
5 Have students complete Problems 7 and 8. There are many possibilities for Problem 7. You can use a transparency of a 1 grid (Math Masters, page 403) on the overhead projector to display a number of them. It may surprise some students that the star and the square in Problem 8 have the same area. One way to find the area of the star is to think of it as a square with a triangle attached to each of its sides. (See margin.) Ongoing ssessment: Recognizing Student chievement Journal page 4 Problem 8 Use journal page 4, Problem 8 to assess students ability to describe a strategy for finding and comparing the areas of a square and a polygon. Students are making adequate progress if they are able to count unit squares and partial squares to find the areas of these two shapes. Some students may describe the use of a formula to calculate the areas of the triangles. [Measurement and Reference Frames Goal ] 5. Complete. Math Boxes 1. Write three equivalent fractions for each fraction. 8 a. 4_ 0 40_ 9 18, 45, 90 b. 3_ 15 _ , 40, _ c. _ , 5, 50 _ 14 35_ 70_ 7_ d. 10 0, 50, Complete the What s My Rule? table, and state the rule. Rule: 4. a. 43 is half as much as 8. b. 48 is twice as much as 4. c. 150 is 3 times as much as 50. d. 40 is 5 of 00. Sample answers: in out e. 135 is 5 times as much as 7.. Measure the sides of the figure to the nearest centimeter to find its perimeter. Perimeter = If you throw a die 40 times, about how many times would you expect to come up? Circle the best answer. 70 times B 100 times C 50 times D 10 times. Divide with a paperandpencil algorithm. 7,53 = 1,75 3_, or 1, Math Journal, p. 44 Ongoing Learning & Practice Solving Fraction Problems (Math Journal, p. 43) Students identify fractional parts of number lines, collections of objects, and regions. Math Boxes (Math Journal, p. 44) Study Link Master Name STUDY LINK reas of Triangles 13 Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 85. The skill in Problem previews Unit 9 content. Writing/Reasoning Have students write a response to the following: For Problem 4, write two probability questions for which the correct answer would be D 10 times. Sample answers: If you throw a die 40 times, about how many times would you expect an even number to come up? If you throw a die 1,0 times, how many times would you expect a to come up? Find the area of each triangle. 1.. Number model: Number model: rea = square feet rea = square in. Number model: Number model: rea = square in. rea = square Try This 4' ' (8 4) = 1 10 in. (10 ) = ,75 The area of each triangle is given. Find the length of the base. 5 1 (1 5) = (34 75) = 1,75 Study Link (Math Masters, p. ) 5. rea = 18 in. base = 1 in.? 3 in. 5 m? rea = 15 m base = m Home Connection Students calculate the areas of triangles. They continue to work on Math Masters, page, which should be completed before Lesson 91. Practice ,,, 45,, 3, 8., 1,, 3,,, 5 Math Masters, p. 8 Lesson 97
6 8 7 Teaching Master Name Comparing reas 1. Cut out the hexagon below. Then cut out the large equilateral triangle. You should end up with one large triangle and three smaller triangles. 3 Differentiation Options. Use the large triangle and the three smaller triangles to form a rhombus. a. Sketch the rhombus in the space to the right. b. Is the area of the rhombus the same as the area of the hexagon? c. Is it possible for two different shapes to have the same area? Comparing reas (Math Masters, p. 7) PRTNER 5 15 Min 3. Put all the pieces back together to form a hexagon with an equilateral triangle inside. How can you show that the area of the hexagon is twice the area of the large triangle? Sample answer: The three smaller triangles cover the equilateral triangle. Six of the smaller triangles cover the entire hexagon. There are 4 triangles in the hexagon. The large triangle is called an equilateral triangle. ll 3 sides are the same length. The smaller triangles are called isosceles triangles. Each of these triangles has sides that are the same length. To apply students understanding of area, have them compare the areas of a rhombus and a hexagon. Finding the rea and Perimeter of a Hexagon (Math Masters, p. 8) 5 15 Min Math Masters, p. 7 To apply students understanding of area formulas, have them find the area and perimeter of a nonregular hexagon. Counting squares to find the area is not permitted; students are encouraged to divide the hexagon into figures and then use a formula to calculate the area of each figure. One strategy is to partition the polygon as shown below. nother strategy can be found on the reduction of Math Masters, page 8. Teaching Master Name B 8 7 rea and Perimeter 1. Find the area of the hexagon below without counting squares. Hint: Divide the hexagon into figures for which you can calculate the areas: rectangles, parallelograms, and triangles. Use a formula to find the area of each of the figures. Record your work. Total area of hexagon C. Find the perimeter of the hexagon. Use a centimeter ruler. Perimeter 48 Sample answer: EXTR PRCTICE Playing Rugs and Fences (Math Masters, pp ; Student Reference Book, pp. 0 and 1) PRTNER Min To practice calculating the area and perimeter of a polygon, have students play Rugs and Fences. See Lesson 9 for additional information. 3 Math Masters, p Unit 8 Perimeter and rea
7 Name Exploring Triangle Properties B C D E F G H I J K L Copyright Wright Group/McGrawHill M N O P 454
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