Preinstructional Planning
Objectives
Students will:
- Define the termproportion
- Calculate a scale factor
- Determine the "real life" side lengths and area for geometric figures presented on a scale when given the scale factor
- Determine the side length and area for geometric figures presented on a scale drawing when given the "real life" side lengths
- Use these concepts to calculate distance on a map
Materials
- An Appetite for Design: Scale Drawings of Geometric Figuresprintable
- Answer Key: Designing With Geometry printable
- Standards Chart: Geometry printable
- Whiteboard or large graph paper and markers
During Instruction
Set Up
1. Make a class set of theAn Appetite for Design: Scale Drawings of Geometric Figures printable.
2. Print a copy of the Answer Key: Designing With Geometry printablefor your use.
Lesson Directions
Introduction to Scale Drawings of Geometric Figures
Step 1:Draw a rectangle on the board with width labeled as 3" and length as 6". Indicate to the class that it represents a chocolate bar. Suppose the candy company wanted to make a larger-size bar with the same shape. If they made the length 18", what would the width have to be? (Answer: 9")
Step 2:Show that the sides of the original bar have aratioof 3 to 6 or 3:6. Make sure the class is comfortable with the definition of ratio (the relative sizes of two or more quantities) and the two ways to express a ratio. Indicate that if the company wants to make a larger bar with the same shape, the larger bar must be inproportionto the original bar. To do so, it must have the same ratio of width to length. 3/6 = 9/18. Indicate that a proportion is a statement showing that two ratios are equal.
Step 3:We can calculate thescalingfactorby comparing the length of corresponding sides of the new and the original chocolate bar. The width of the new bar is 9" and the original is 3", so the scaling factor is 9/3 or 3. Note that if we went from the larger bar to the smaller one, the scaling factor would be 3/9 = 1/3, the reciprocal of the scaling factor obtained when going from the smaller bar to the larger bar. Indicate that, although this might seem obvious, it can be quite helpful when trying to make scale drawings of shapes more complicated than a rectangle.
Step 4:Practice with the class, asking what the sides of a 4" x 6" rectangle would be if the scaling factor were 4 (16" x 24") and if the scaling factor were 1/2 (2" x 3").
Step 5:Indicate that scaling factors when going from smaller to larger shapes aren't always whole numbers. Ask what the dimensions of a 4" x 6" rectangle would be if the scaling factor were 2.5 (10" x 15").
Step 6:Point out that the scaling factor doesn't work the same way when applied to area. For example, the area of a 2" x 4" rectangle is 8 square inches while the area of a proportional 4" x 8" rectangle is 32 square inches. If appropriate for your class, point out that, to create a proportional rectangle, the scaling factor must be squared then multiplied by the area of the original rectangle, i.e., 22 x 8 square inches.
Step 7:Relate the concept to maps. Ask how long the actual distance from city A to city B would be if the scaling factor were 1cm = 100km and the distance on a map was 7.5cm (750km).
Guided Practice
Step 9:Distribute graph paper to students. Write the following two points on a grid on the board:A(2, 10) andB(8, 2). Students should pair up and find the distance between the two points.
Answer: The distance = 10 units. Students should construct a right triangle by adding pointCat (2, 2). [Note: (8, 10) would also work but this example uses (2, 2).] The distance ofAC= 8, which is obtained by subtracting the two y-coordinates (10 - 2). The distance ofBC= 6, which is obtained by subtracting the two x-coordinates (8 - 2). Using the Pythagorean Theorem, 82+ 62= (segmentAB)2. 64 + 36 = 100, which means that segmentAB= √100 which equals 10.
Step 10:Checking for Understanding:Review answers as a class and respond to any questions.
Independent Practice
Step 11:Assign theParty on the Patio: Applying the Pythagorean Theorem printable for classwork or homework.
Step 12: Checking for Understanding:Review the answers to theParty on the Patio: Applying the Pythagorean Theorem printable, which are provided on page 1 of the Answer Key: Designing With Geometry printable. Make sure students explain their mathematical thinking. Address any misconceptions that may arise.
Post Instructional
Standards
- Grade 8:Pythagorean Theorem (CCSS: 8.G.7 & 8)
- Grades 6–8:Making Sense of Problems, Reasoning, Constructing an Argument, Using Appropriate Tools, and Attending to Precision (CCSSMP1–3 and 5–6);NCTMGeometry
For more information, download the comprehensiveStandards Chart: Geometry printable.
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