Scale factor

Here you will learn about scale factor, including enlarging a shape by a scale factor on a grid. You will extend this to learn about fractional scale factors and how to calculate scale factors.

Students will first learn about scale factor as part of geometry in 7 th grade and continue to work with scale factor in high school.

What is a scale factor?

A scale factor is a ratio between two corresponding sides of similar shapes. A scale factor describes how much a shape has been scaled up or down.

To scale a shape up or down, you multiply every side length of a shape by the scale factor to increase or decrease the size. The sizes of the angles do not change. Changing a shape by a scale factor greater than \bf{1} will make the shape a larger figure. Scaled shapes can be referred to as images, while original shapes are referred to as preimages.

For example, shape A that has been enlarged by scale factor 2 to give shape B.

The corresponding angles are identical but each side in shape B is double the size of the original figure.

The two shapes are similar figures. Enlarging a shape by a scale factor between \bf{0} and \bf{1} will make the shape smaller. This is often written as a fraction or a decimal and is known as a fractional scale factor.

Here, shape A has been enlarged by scale factor \cfrac{1}{2} to get shape B.

If you look at the corresponding sides in the scale drawing, each side in shape B is half the size of the original shape. The corresponding angles are identical. The two shapes are similar figures.

What is a scale factor?

Common Core State Standards

How does this relate to 7 th grade and high school math?

• Grade 7 – Geometry (7.G.A.1)
  Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

• High school: Geometry – Similarity, Right Triangles, and Trigonometry (HSG.SRT.A.1)
  Verify experimentally the properties of dilations given by a center and a scale factor:

How to use a scale factor

In order to use a scale factor to increase the size of a shape:

1. Draw a scaled drawing of the first side.
2. Scale another side.
3. Complete the scaled shape.

How to use scale factor examples

Example 1: using a scale factor to enlarge a shape

Enlarge this shape by scale factor 2 :

1. Scale the first side.

Let’s start with the base. The base in the original shape is 1, so the base of the image will be 2.

1\times2=2

You can draw it anywhere on the grid.

2Scale another side.

Let’s choose a side that is attached to the first side. The height of the first shape is 3, so the height of the enlarged shape will be 6.

3\times2=6

3Complete the scaled shape.

Now continue to draw the new shape.

The lengths of the sides of the image are double the lengths of the preimage. You can see the different sizes of the shapes above.

Example 2: using a scale factor to enlarge a shape

Enlarge this shape by scale factor 3 :

Create a scaled drawing of the first side.

The base in the original shape is 2, so the base in the bigger shape will be 6.

2\times3=6

You can draw it anywhere on the grid.

Scale another side.

The left vertical side of the first shape is 3, so the corresponding side of the bigger shape will be 9.

3\times3=9

Complete the scaled shape.

Now continue to draw the new shape. It is easier to do the horizontal and vertical sides first.

The lengths of the sides of the second shape are three-times the lengths of the original shape.

Example 3: using a scale factor to enlarge a shape

Draw this shape with a scale factor of \cfrac{1}{2} :

Create a scaled drawing of the first side.

The base in the original shape is 4, so the base of the new shape will be 2.

4\times \cfrac{1}{2}=2

You can draw it anywhere on the grid.

Scale another side.

The left vertical side of the first shape is 2, so the length of the corresponding side of the new shape will be 1.

2\times \cfrac{1}{2}=1

Complete the enlarged shape.

Now continue to draw the image. It is easier to do the horizontal and vertical sides first.

The length of the corresponding side of the second smaller figure are half the lengths of the original shape.

How to calculate a scale factor

In order to calculate a scale factor:

1. Choose a pair of corresponding sides.
2. Divide the length of the enlarged shape by the length of the original shape.
3. Write down the scale factor.

Calculating scale factor examples

Example 4: calculating a scale factor

Calculate the scale factor for the image of Triangle A to Triangle B :

Choose a pair of corresponding sides.

The two triangles are similar triangles.

Choose a pair of sides, either horizontal or vertical. Here I am choosing the bases.

The base of shape A is 3.

The base of shape B is 9.

Example 5: calculating a scale factor

Calculate the scale factor for the preimage to the image of Shape A to Shape B :

Choose a pair of corresponding sides.

Let’s choose the vertical sides on the left.

The length of shape A is 2.

The length of shape B is 4.

Example 6: calculating a scale factor

Calculate the scale factor for the preimage to image of Shape A to Shape B :

Choose a pair of corresponding sides.

Let’s choose the bases.

The length of shape A is 9.

The base of shape B is 3.

Teaching tips for how to find the scale factor

• Allow students to explore scale factor using physical manipulatives, with a strong focus on fractional scales, before moving on to written examples. This will prepare students for the idea that multiplying side lengths by a fraction will result in a new shape that is smaller than the original figure.

• Bridge the gap between fractions and decimals for students when thinking about scale factor resulting in smaller figures. Posing the idea that 0.5 of the original figure is the same as \cfrac{1}{2} of the original figure will establish a connection that may be rooted in students’ understanding of number sense.

• Use real-world examples to help students understand where and when scale factor may be used in real life. Examples could include creating a scaled model of something, construction blueprint examples, or examples with scales on maps. Providing real-world problems could be accomplished using worksheets or other modalities.

Easy mistakes to make

• Not multiplying all sides of a polygon by the same number
  An easy mistake to make when finding scale factor is not multiplying all sides of a polygon by the same number to enlarge the figure. In order to get scaled copies, all side lengths must be multiplied. The new shape will be a scale model of the original figure.

• Dividing instead of multiplying when the scale factor is less than \bf{1}
  If the scale factor is less than one, students may divide instead of multiplying. This mistake would result in an incorrect answer because dividing by a decimal actually results in a larger number. For this reason, when using scale factor, you must always multiply. When you multiply by a decimal, you get a smaller figure than the original figure.

• Setting up the wrong ratio when finding the scale factor
  When creating the ratio for the corresponding sides of similar geometric figures, the measurement for the original shape is always the denominator and the measurement of the scaled figure is always the numerator. Confusing the two will create an incorrect scale factor.
  For example,
  Below are two similar shapes.
  
  
  The base of shape A is 3 . The base of shape B is 9.
  
  scale \ factor = \cfrac{scaled \ length}{ original \ length}=\cfrac{9}{3}=3 ✔
  
  scale \ factor = \cfrac{scaled \ length}{ original \ length}=\cfrac{3}{9}=\cfrac{1}{3} ✖
  
  ^This is NOT the scale factor, but the inverse of the scale factor of the shapes above

Related transformations lessons

• Rotations
• Translation math
• Reflection in math
• Dilations
• Center of dilation

The next lessons are

• Geometric proofs
• Area
• 3D shapes
• Area of a quadrilateral

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