All About That Base... and Height

Transcription

1 All About That Base... and Height Area of Triangles and Quadrilaterals 2 WARM UP Write 3 different expressions to describe the total area of this rectangle. LEARNING GOALS State and compare the attributes of different shapes. Explain that the area of a parallelogram is the same as that of a rectangle with the same base length and height. Derive the formulas for the areas of triangles, parallelograms, and trapezoids by composing or decomposing the various shapes into rectangles, triangles, and other shapes. Apply the techniques of composing and decomposing shapes to solve real-world and mathematical problems. KEY TERMS parallelogram altitude variable trapezoid You can take a shape apart and put it back together in a different way without changing its area. How can you use composition and decomposition of shapes to reason about the areas of shapes and to derive formulas for the areas of common shapes? LESSON 2: All About That Base... and Height M1-15

2 Getting Started In the 20s Consider each figure. An attribute is a characteristic to describe a figure. 1. Can you name each figure? 2. Describe the attributes of each shape. Are there any attributes that are shared across the different shapes? 3. Each shaded figure shown has an area of exactly 20 square units. Show how you know. M1-16 TOPIC 1: Factors and Area

3 ACTIVITY 2.1 Investigating the Area of a Parallelogram In this activity you will investigate the area of a parallelogram using what you know about the area of a rectangle. A parallelogram is a four-sided figure with two pairs of parallel sides and opposite sides that are equal in length. 1. Cut out a parallelogram from the grid at the end of the lesson. 2. Cut your parallelogram into pieces so that it can be reassembled to form a rectangle. Tape your rectangle in the space provided. A rectangle is a special type of parallelogram. Parallelogram Rectangle c In a parallelogram, any of the four sides can be labeled as the base. The altitude of a parallelogram is another name for the height of a parallelogram. The altitude of a parallelogram is the perpendicular distance from the base of the parallelogram to the opposite side, represented by a line segment. 3. Label the base and height of the parallelogram and rectangle. LESSON 2: All About That Base... and Height M1-17

4 When you write a sentence to explain your reasoning, be sure to express a complete idea. Your sentence should make sense standing alone. 4. How does the height of the parallelogram relate to the height of the rectangle? How does the length of the base of the parallelogram relate to the length of the base of the rectangle? Explain your reasoning. 5. Describe the relationship between the areas of a parallelogram and rectangle that have the same base and height. 6. Use the terms base and height to describe how to calculate the area of a parallelogram. A variable is a letter that is used to represent a number. In mathematics, one of the most powerful concepts is to use a letter to represent a quantity that varies, or changes. The use of letters, called variables, helps you write expressions to understand and represent problem situations. 7. Write the formulas to calculate the areas of a parallelogram and a rectangle. Use b for base and h for height. M1-18 TOPIC 1: Factors and Area

5 ACTIVITY 2.2 Investigating the Area of a Triangle NOTES The base of a triangle, like the base of a parallelogram, can be any of its sides. The height, or altitude, of a triangle is the length of a line segment drawn from a vertex of the triangle to the opposite side so that it forms a right angle with the opposite side. A altitude B altitude altitude C B base C C base A A base B Sailboat racecourses are often shaped like triangles. The course path is defined by buoys called marks. When the course is triangular, the marks are located at the corners, or vertices, of the triangle. Here is a sample course with the marks numbered. MARK 2 MARK 3 START/ FINISH MARK 1 Race officials need to know the area inside the course so that they can plan for the number of spectator boats that can anchor within. LESSON 2: All About That Base... and Height M1-19

6 The triangle represents a sailboat racecourse. Each square on the grid represents 0.1 mile by 0.1 mile Estimate the area of the triangular course in square units. Justify your estimate. Use a straightedge to draw your parallelogram. 2. Use two sides of the triangle to draw a parallelogram on the grid. How does the area of the parallelogram relate to the area of the triangle? 3. Calculate the area enclosed by the triangular course. M1-20 TOPIC 1: Factors and Area

7 4. Label a base and height of the original triangle in the diagram. Describe how to calculate the area of any triangle in terms of the base and the height. 5. Suppose you create a parallelogram using a different side of the triangle. Does this change the area of the triangle? Explain how you know. LESSON 2: All About That Base... and Height M1-21

8 ACTIVITY 2.3 Investigating the Area of a Trapezoid You have seen that taking apart, or decomposing, a parallelogram forms a rectangle. And putting together, or composing, two triangles also forms a parallelogram. Composing and decomposing can help you think about the shapes differently in order to determine their areas. In this activity you will take apart and put together shapes to determine the formula for calculating the area of a trapezoid. The variable b represents a base, but a trapezoid has two bases. So, subscripts are used to distinguish between the two different bases; b 1 and b 2 are not equal in length. A trapezoid is a quadrilateral with two bases, often labeled b 1 and b 2. The bases are parallel to each other. The other two sides of a trapezoid are called the legs of the trapezoid. An altitude of a trapezoid is the length of a line segment drawn from one base to the other and perpendicular to both. 1. Label the bases of each trapezoid as b 1 and b 2. Cut out two of the trapezoids at the end of the lesson to show how to determine each area. M1-22 TOPIC 1: Factors and Area

9 2. Marcus cut out and composed two trapezoids into a parallelogram to figure out the exact area of one trapezoid. Show what Marcus did to determine the area. Think about how the new shapes formed relate to the original trapezoid. 3. Zoe folded the trapezoid so the bases aligned, cut along the fold, and rearranged the parts to form a parallelogram. Show what Zoe did to determine the area. 4. Angela decomposed the trapezoid into two triangles to determine the exact area. Use this trapezoid to recreate Angela s strategy. 5. Describe how to calculate the area of any trapezoid in terms of the two bases and the height. LESSON 2: All About That Base... and Height M1-23

10 NOTES TALK the TALK All Three Shapes 1. Draw each shape and then label a base and height. Next, write the formula to calculate the area of each. Use A for the area, b for the length of the base, and h for the height. parallelogram triangle trapezoid 2. Show that the two triangles have the same area. R P G M 3. Write a paragraph that will convince your readers that the two triangles have the same area. M1-24 TOPIC 1: Factors and Area

11 Shape Cut Outs Extra shapes are included. LESSON 2: All About That Base... and Height M1-25

12

13 Assignment Write 5. Assign_num_list Define Assign_para each term in your own words. 1. height of a parallelogram 2. height of a triangle Remember 6. Assign_num_list The area of a parallelogram, Assign_para a triangle, or a trapezoid can be determined by composing Assign_para or decomposing it into one or more shapes with an equal Assign_para total area. Area of a parallelogram 5 bh Area of a triangle bh Area of a trapezoid (b 1 1 b 2 )h Practice Answer each question for the given figure. 1. Identify a base and corresponding height for the given parallelogram. Determine the area of the parallelogram. 2. Calculate the area of the parallelogram. 32 yd Stretch 1. Assign_num_list 2. Assign_num_list 14 yd 3. Identify a base and corresponding height for the given triangle. Determine the area of the triangle. 4. Calculate the area of the triangle. Review Assign_para 1. Assign_num_list 2. Assign_num_list 3. Assign_num_list 20 mm 11 mm 1.2 A Using Tables to Represent Equivalent Ratios M-27 LESSON 2: All About That Base... and Height M1-27

14 Assignment 5. Identify the two bases and the height for the given trapezoid. Determine the area of the trapezoid. Write 1. Assign_num_list 2. Assign_num_list 6. Yvonne cut a picture into the shape of a trapezoid to place into her scrapbook. The picture is shown. What is the area of Remember the picture? Assign_para 5 in. Assign_mid 4 in. 7 in. Stretch Practice 1. Answer What each is the question area of a for parallelogram the given figures. that has a base of ft and a height of ft? Calculate Assign_num_list the area of the triangle. 2. Assign_num_list 10 m m 8.66 m 11 m Review Use the Distributive Property to write an equivalent addition expression for each. 3. Identify a base and corresponding height 4. Assign_num_list 1. 6(9 1 1) Assign_para 2. (14 1 3)7 Assign_para (7 1 10) Decompose each rectangle into two or three smaller rectangles to demonstrate the Distributive Property. Then write each in the form a(b 1 c) 5 ab 1 ac M1-28 TOPIC 1: Factors and Area