Modeling Addition and Subtraction of Fractions with Like Denominators

To make a fraction model in the Gizmo^{tm}, you need to model a numerator and a denominator.

Click Clear, and then drag the Red denominator slider to different values. How does the first (red) fraction model change? Set the Red denominator slider to 4 by entering this value into the box beside the slider, and then pressing ENTER. How many parts is the first fraction model divided into?

Click on different parts within the first fraction model. How many parts get shaded when you click the second part from the left? How many parts get shaded when you click the third part from the left? The number of shaded parts is the numerator of the fraction. What fraction is represented in the Gizmo?

Next, shade in all the parts. What whole number does this model represent? Click Clear to erase all the shaded parts. When none of the parts in a fraction model are shaded, what whole number is the fraction model equal to?

Click Clear, and then click the  button. Set the Red denominator slider to 4, and then shade in one part in the first (red) model to represent the fraction . Set the Blue denominator slider to 4 and shade in one part in the second (blue) model so that it also represents .

Click the + button to model the addition of the fractions and . Notice that the second fraction model moves to the right. Shade in another section of the first model so that it represents . What happens to the position of the second fraction model? Why?

You can use the bottom (green) fraction model to represent the sum of the two fractions. Set the Denominator of sum slider to 4, and then shade in enough parts in this model so that the right edge of its shading lines up with the right edge of the shading in the second fraction model. How many parts did you shade? What is + ?

Click the  button. The fraction models now represent − . Shade the bottom fraction model so that the right edge of its shading lines up with the left edge of the shading in the second model. How many parts did you shade? What is − ?
Modeling Addition and Subtraction of Fractions with Unlike Denominators

Click Clear, and then model the addition of and by setting the first (red) fraction model to , the second (blue) fraction model to , and then clicking the + button.

The sum of these fractions is represented by the right edge of the shading in the second fraction model. Adjust the Denominator of sum slider until one of its divisions lines up exactly with the right edge of the shading in the fraction model. Where is the the slider when this happens? Is this number a common multiple of 3 and 4?

Use the Denominator of sum slider to find another common multiple of 3 and 4, and verify that this value also makes the bottom fraction model line up with the second fraction model. What is the lowest common denominator of and ?

Set the Denominator of sum slider to the lowest common denominator of and . Shade in parts in the bottom fraction model until the right edge of the shading lines up with the right edge of the shading in the second fraction model. What fraction does the bottom fraction model now represent? What is + ?

Be sure the first fraction model represents and the second fraction model represents . Click the  button. The fraction models now represent − . Reduce the shading in the bottom fraction model until its right edge matches the left edge of the second model.

What fraction does the bottom fraction model now represent? What is − ?

In your own words, explain why the same common denominator is used for the sum + and the difference − .