Connected Mathematics Stretching And Shrinking Answers

Stretching and Shrinking

In Stretching and Shrinking, students learn the mathematical meaning of similarity and explore the properties of similar figures.

The following materials (pdf's) and information may be helpful:

Stretching and Shrinking mathematical and problem solving goals :

Stretching and Shrinking was created to help students:

• Enlarge figures using rubber-band stretchers and coordinate plotting

• Informally visualize similar and distorted transformations

• Identify similar figures visually and by comparing sides and angles

• Recognize that lengths between similar figures change by a constant scale factor

• Build larger, similar shapes from copies of a basic shape

• Divide a shape into smaller, similar shapes

• Recognize the relationship between similarity and equivalent fractions

• Learn the effect of scale factor on length ratios and area ratios

• Discover that areas of similar figures are related by the square of the scale factor

• Observe and visualize ratios of lengths and areas

• Recognize that triangles with equal corresponding sides are similar

• Recognize that rectangles with equivalent ratios of corresponding sides are similar

• Determine and use scale factors to find unknown lengths

• Collect examples of figures and search for patterns in the examples

• Use the concept of similarity to solve real-world problems

• Draw or construct counterexamples to explore similarity transformations

• Make connections between algebra and geometry

• Use geometry software to explore similarity transformations

The overall goal of the Connected Mathematics curriculum is to help students develop sound mathematical habits. Through their work in this and other geometry units, students learn important questions to ask themselves about any situation that can be represented and modeled mathematically, such as:How does the mathematical idea of "similar" differ from the everyday use of the word? In similar figures, what is exactly the same? What is different? How is it different? How can we find a way to describe the sizes of two similar figures? When figures are similar? what relationship can we find in their areas? In their perimeters? Where can we apply these similarity concepts in the everyday world~ Can the coordinate system help us understand similarity? To understand ratios? How do ideas of stretching and shrinking tie algebra and geometry together?

The Mathematics in Stretching and Shrinking

The activities in the beginning of the unit elicit students' first notion about similarity as two figures with the same shape. Students may have difficulty with the concept of similarity because of the way the word is used in everyday language—family members are "similar," houses are "similar." Through the activities in Stretching and Shrinking students will grow to understand that the everyday use of a word and its mathematical use may be different. For us to determine definitively whether two figures are similar,similarity must have a precise mathematical definition.

Tests for Similarity

Two figures are similar if (1) the measures of their corresponding angles are equal and (2) the lengths of their corresponding sides increase by the same factor, called thescale factor.

For rectangles, since all angles are right angles, we need only check the ratios of the lengths of corresponding sides. For example, rectangles A and B are similar, but neither is similar to rectangle C.

The scale factor from rectangle A to rectangle B is 2 because the length of each side of rectangle A multiplied by 2 gives the length of the corresponding side of rectangle B. The scale factor from rectangle B to rectangle A is ½ because the length of each side of rectangle B multiplied by ½ gives the length of the corresponding side of rectangle A. Rectangle C is not similar to rectangle A, because the lengths of corresponding sides do not increase by the same factor.

The perimeter from rectangle A to rectangle B also increases by a scale factor of 2, but the area increases by the square of the scale factor, or 4. This can be seen by dividing rectangle B into four rectangles congruent to rectangle A.

For triangles to be similar, the measures of corresponding angles must be equal and the lengths of corresponding sides must increase by the same factor. Through their experiments with reptiles in Investigation 3, students discover a special property of triangles: angles are what determine a triangle's shape, and we only have to check the angles to determine whether two triangles are similar. The ratio of corresponding sides will be equal if corresponding angles are equal.

For polygons other than rectangles and triangles, we must make sure that corresponding angles are congruent and that lengths of corresponding sides increase by the same scale factor.

Equivalent Ratios

In similar figures, there are several equivalent ratios. Some are formed by comparing lengths within a figure. Others are formed by comparing lengths between two figures. For example, for the rectangles above, the ratio of length to width is 4/8 or ½ for rectangle A and 8/16 or ½ for rectangle B. We could also look at the ratios of corresponding sides: width to width and length to length are 8/16 and 4/8 mrespectively, which are equivalent ratios.

Equivalent ratios can be used to solve interesting problems. For example, shadows made by the sun can be thought of as sides of similar triangles, because the sunlight hits the objects at the same angle. Shown below is a building of unknown height and a meterstick, both of which are casting a shadow To find the height of the building, we can use the scale factor between the lengths of the shadows. Since going from 0.25 to 10 involves a scale factor of 40, we multiply the height of the meterstick by 40 to obtain the height of the building, or 40 meters. We could also think of this as and use equmaknt fractions to find the value of x that would make the ratios equivalent.

Similarity Transformations

The rubber-band stretcher introduced in Investigation 1 is a tool for physically producing a similarity transformation. It does not give precise results, but it is an effective way to introduce students to similarity transformations. More precision is gained in transformations governed by algebraic rules that specify how coordinates are to be changed.

In this unit, students will create figures on a coordinate system and use algebraic rules to transform them into similar figures. For example, if the coordinates of a figure are multiplied by 2, the algebraic transformation is from(x, y) to(2x, 2y). In general, if the coordinates of a figure are(x, y), algebraic rules of the form(nx + a, ny + b) will transform it into a similar figure with a scale factor ofn. These algebraic rules are called similarity transformations.