In the last section, you used algebra tiles to set up and solve one-step equations involving addition and subtraction. In this section, you will use a different model composed of cups and counters to set up and solve one-step equations involving multiplication and division.

For the cups and counters model, a cup represents the variable, usually x, and counters are used to represent numbers.

Consider the equation 3x = 15. This equation can be modeled with cups and counters with 3 cups on the left side of an equal sign and 15 positive counters on the right side of an equal sign.

Click through the interactive below to see how Sandy used cups and counters to solve this equation. Click the Next arrow to step through how Sandy did it.

Use the cups and counters model below to set up and solve each of the equations shown.

Now, consider the equation x over 4 x 4 = 5. This equation uses the operation of division.

Click through the interactive below to see how Sandy used cups and counters to solve this equation. Click the Next arrow to step through how Sandy did it.

Pause and Reflect

When you are dividing counters evenly among the cups, what do you do if there is a counter left over?

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Check Your Answer

Left over counters can be broken into fractional pieces. If you are dividing by 2, break them into 2 pieces and distribute the pieces evenly. If you are dividing by 3, break them into 3 pieces and distribute the pieces evenly.

If you were solving the equation, x over 7 x 7 = 2, into how many pieces would you need to divide the picture of the cup in order to solve the equation?

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Check Your Answer

Since you are dividing x by 7, you would divide the picture of the cup into 7 pieces.

How are the operations of multiplication and division related?

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Check Your Answer

Multiplication and division are inverse operations.

How does this relationship help you to solve equations?

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Check Your Answer

If the equation includes multiplication, you use division to solve the equation because you are grouping the counters. If the equation includes division, you use multiplication to solve the equation because you are scaling sets of counters.

Practice

For Questions 1 – 3, use the cups and counters model to determine the value of x.

1. 3x = 18
   
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   Need a hint?

   How many unit counters do you need to remove from both sides of the model in order to get the 3 cups by themselves?
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   Check Your Answer

   x = 6
2. 2x = 13
   

   Use the griddable response interactive below to record and check your answer.

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   Need a hint?

   How many unit counters do you need to remove from both sides of the model in order to get the 2 cups by themselves?
3. x over 3 x 3 = 4
   
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   Need a hint?

   How many sets of 4 counters do you need to represent the entire cup instead of 1 over 3 1 3 of the cup?
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   Check Your Answer

   x = 12
4. x over 2 x 2 = 4.5
   
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   How many sets of 4 counters do you need to represent the entire cup instead of 1 over 2 1 2 of the cup?
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   Check Your Answer

   x = 9