Stretching & Shrinking
Standards & Objectives | |
File Size: | 199 kb |
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Goals for Stretching & Shrinking
Students should...
- Understand what it means for figures to be similar
- Develop strategies for using similar figures to solve problems
Investigation 1
Vocabulary | |
File Size: | 156 kb |
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This Investigation informally introduces ideas about mathematical similarity. Students explore mathematical similarity by relating it to the use of the word similar in everyday life.They estimate the height of a mystery teacher in a picture by taking an actual measurement into account. They use rubber bands to enlarge a figure. They notice that a copy machine can produce similar figures both larger and smaller than the original.
Students explore similar figures in different sizes and compare changes in their side lengths, angle measures, perimeters, and areas. They determine which properties of a figure change and which properties remain the same when a figure is scaled up and down. |
Investigation 2
Vocabulary | |
File Size: | 154 kb |
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Students build a good working definition of similarity in mathematical terms. They begin to see connections between geometry and algebra. Using the coordinate system, they draw several geometric figures representing characters called “Wumps.” They use those figures to compare similar and non-similar shapes. Students will see for the first time that, for non-similar figures, corresponding side lengths do not have the same ratio and corresponding angles are not necessarily the same size. They explore algebraic rules that cause images to change size and to move about the coordinate plane. They also compare angle measures and lengths of corresponding sides informally as they investigate transformations. They make more formal comparisons after they are introduced to the term scale factor. Students find that for two figures to be similar, corresponding angles must be congruent; and corresponding sides must grow or shrink by the same factor, called the scale factor. They begin to recognize the role multiplication plays in similarity relationships.
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Investigation 3
Vocabulary | |
File Size: | 166 kb |
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During Investigation 3, students deepen their understanding of what it means for two figures to be similar. Students experiment with rep‑tiles, shapes where copies are put together to make larger, similar figures. Using rep‑tiles, students explore the relationship between the areas of two similar figures. Students sometimes have difficulty grasping the idea that area does not grow at the same rate as side lengths when a figure is enlarged. Experiences with rep-tiles help them build mental images to support their evolving ideas about the relationship between scale factor and area.
Students also discover that conditions that determine when triangles are similar may not result in similar figures when applied to quadrilaterals and other polygons. For example, equal measures of corresponding angles imply similarity for triangles. This does not hold true for quadrilaterals, even parallelograms or rectangles. Additionally, students learn to use similarity and scale factors to find unknown measures or measures of distances that cannot be easily measured directly. |
Investigation 4
Vocabulary | |
File Size: | 221 kb |
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Students use ratios to determine whether or not two figures are similar. They find ratios of two lengths within each figure, and then compare those ratios. Students learn that they need information about angle measures when testing non-rectangular shapes, such as triangles, for similarity.
Also, students use ratios and scale factors to find missing side lengths of similar figures. In Grade 6, students expressed ratios as comparison statements. In this Investigation, students extend their use of ratios by expressing ratios as fractions. Last, students apply their knowledge of similarity to real-world problems. They use the shadow method to find heights of tall objects. |