Calculating with Scales

Contributor: Marlene Vogel. Lesson ID: 10776

Calculating with scales does not involve fish, unless you are trying to draw a six-foot sturgeon on a piece of notebook paper! Practice hard and play an online game to learn this important math skill!

categories

Middle School

subject
Math
learning style
Visual
personality style
Lion, Otter
Grade Level
Middle School (6-8)
Lesson Type
Dig Deeper

Lesson Plan - Get It!

Audio:

Lee drew a picture of his car using a scale of 1 inch to 4 feet. Print Lee's Scale Drawing of His Car found in the Downloadable Resources in the right-hand sidebar. Using a ruler, measure the length of the drawing Lee made of his car.

We are often required to draw a picture of an object that is either too big or too small to draw at the object's actual size.

For example, consider your home.

If I asked you to draw a picture of your home, would you be able to draw the actual size of your home on one piece of paper? Of course not! What about an ant? Could you draw a picture of an ant, at its actual size, so I could see every part of the ant clearly? Exactly! No you could not!

However, when given a task like drawing a picture of your home or a clear picture of an ant, you can draw your picture using scale. A scale drawing is used to illustrate an object when it is too big or too small to show its actual size.

A scale drawing is drawn in a ratio of the length of the drawing to the length of the actual object.

For example, in the beginning of the lesson, you saw that Lee drew a scale drawing of his car. You read that he drew it in the ratio of 1 in. to 4 ft. The ratio he used for his drawing means that one inch of length in his drawing represents 4 feet of length in the actual car. If the length of the drawing is 2 inches long, then the actual length of Lee's car is 8 feet.

There are two words that will help you explain your scale drawings to someone. Those words are reduction and enlargement.

A reduction is a scale drawing that is smaller than the actual object. An enlargement is a scale drawing that is larger than the actual object. Using the two examples you read about earlier, try to figure out which is the reduction and which is the enlargement. Is the drawing of your home a reduction or enlargement? Is the drawing of an ant a reduction or an enlargement?

Here is an example of how to figure out the size of a scale drawing in comparison to the actual size of the object using the ratio:

  • A bolt is 1.2cm long. Find the length of a drawing of this bolt at a scale of 15:2.
  • When trying to figure out the scale measurement of an actual object or the actual size of an object, use the formula below:
  • You will use this formula for problems where you are asked to find the scale length of an object and for problems where you are asked to find the actual length of an object.
  • In the problem above, about the bolt, fill in the information that you know. The problem tells you that the actual size of the bolt is 1.2cm long:
  • What you don't know is the scale length of the bolt. Replace the words Scale Length with the letter "x." The x represents the length of the bolt in the picture.
  • Did the problem give you any more information? Yes! The problem tells you that the drawing of the bolt was completed by using the ratio of 15:2. This is read as "15 to 2" and can also be written as:

    In order to find the value of x — the scale length of the drawing of the bolt — you will compare the formula to the ratio. Below you will see how to set up the information in order to find the answer to x:
  • Now you are ready to find your answer! In this next step, you are going to cross multiply. Multiply the x in the numerator of the formula and the 2 in the denominator of the ratio. See below:
  • X times 2 equals 2x. Now you cross multiply the denominator from the formula and the numerator of the ratio. See below:
  • 15 times 1.2cm equals 18. Below you will see the two steps that you just completed:

    (original problem) > 2x = 18cm
  • You now know that 2x equals 18, but you still need to find out what x equals. Remember, you cross multiplied 2 and x to get 2x. To get x, you need to do the opposite operation. What is the opposite operation of multiplication? Correct, it's division! You will need to divide 2x by 2 to find out what x equals. This is an important point to remember: whatever you do to one side of the equal sign you need to do to the other side of the equal sign. See below for how this step looks:
  • Now divide! 2x divided by 2 leaves you with x (the 2s can be simplified from 22 to 1). Even though 22 = 1, you do not need to put that 1 next to the x; you can just write "x."
  • When you divide the other side of the equal sign you get 9. See below for the final answer:
    X = 9cm
  • Remember the beginning of this example when you wrote the formula, "x" represented something? X represented the scale length of the drawing of the bolt. Now you know the length of the drawing of the bolt! The scale length of the drawing of the bolt is 9cm.
  • I bet you can answer one more question about the drawing of the bolt. In the problem, you found out the actual size of the length of the bolt is 1.2cm. You just figured out that the scale length of the drawing of the bolt is 9cm. Is the drawing of the bolt an enlargement or a reduction?
  • Below are the steps of this problem. Seeing all of the steps together can make it easier to understand.

    A bolt is 1.2cm long. Find the length of a drawing of this bolt at a scale of 15:2.






    2x = 18cm

    x = 9cm

Whew! Now it is time for you to take this new information and try some problems on your own! Make sure you use the formula you learned about at the beginning of the bolt problem to help you figure out more problems.

The section below has a few activities to help you practice your new math skills. Complete as many of the activities as you would like!

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