Scale Factors and Making Scaled Copies
2.1: Number Talk: Multiplying by a Unit Fraction
Find each product mentally.
\(\frac14 \boldcdot 32\)
\((7.2) \boldcdot \frac19\)
\(\frac14 \boldcdot (5.6)\)
2.2: Scaled Triangles
Here is Triangle O, followed by a number of other triangles.
Your teacher will assign you two of the triangles to look at.
- For each of your assigned triangles, is it a scaled copy of Triangle O? Be prepared to explain your reasoning.
- As a group, identify all the scaled copies of Triangle O in the collection. Discuss your thinking. If you disagree, work to reach an agreement.
- List all the triangles that are scaled copies in the table. Record the side lengths that correspond to the side lengths of Triangle O listed in each column.
Triangle O 3 4 5
- Explain or show how each copy has been scaled from the original (Triangle O).
Choose one of the triangles that is not a scaled copy of Triangle O. Describe how you could change at least one side to make a scaled copy, while leaving at least one side unchanged.
2.3: Which Operations? (Part 1)
Diego and Jada want to scale this polygon so the side that corresponds to 15 units in the original is 5 units in the scaled copy.
Diego and Jada each use a different operation to find the new side lengths. Here are their finished drawings.
- What operation do you think Diego used to calculate the lengths for his drawing?
- What operation do you think Jada used to calculate the lengths for her drawing?
- Did each method produce a scaled copy of the polygon? Explain your reasoning.
2.4: Which Operations? (Part 2)
Andre wants to make a scaled copy of Jada's drawing so the side that corresponds to 4 units in Jada’s polygon is 8 units in his scaled copy.
Andre says “I wonder if I should add 4 units to the lengths of all of the segments?” What would you say in response to Andre? Explain or show your reasoning.
Create the scaled copy that Andre wants. If you get stuck, consider using the edge of an index card or paper to measure the lengths needed to draw the copy.
The side lengths of Triangle B are all 5 more than the side lengths of Triangle A. Can Triangle B be a scaled copy of Triangle A? Explain your reasoning.
Here are two polygons. Polygon 2 is a scaled copy of Polygon 1.
The scale factor between Polygon 1 and Polygon 2 is 2, because all of the lengths in Polygon 2 are 2 times the corresponding lengths in Polygon 1. The angle measures in Polygon 2 are the same as the corresponding angle measures in Polygon 1. For example, the measure of angle \(JKL\) is the same as the measure of angle \(DEF\).
Creating a scaled copy involves multiplying the lengths in the original figure by a scale factor.
For example, to make a scaled copy of triangle \(ABC\) where the base is 8 units, we would use a scale factor of 4. This means multiplying all the side lengths by 4, so in triangle \(DEF\), each side is 4 times as long as the corresponding side in triangle \(ABC\).
- scale factor
To create a scaled copy, we multiply all the lengths in the original figure by the same number. This number is called the scale factor.
In this example, the scale factor is 1.5, because \(4 \boldcdot (1.5) = 6\), \(5 \boldcdot (1.5)=7.5\), and \(6 \boldcdot (1.5)=9\).
- scaled copy
A scaled copy is a copy of a figure where every length in the original figure is multiplied by the same number.
For example, triangle \(DEF\) is a scaled copy of triangle \(ABC\). Each side length on triangle \(ABC\) was multiplied by 1.5 to get the corresponding side length on triangle \(DEF\).