FractionConcepts1: Lesson: Generate equivalent fractions with an area model: Generate equivalent fractions with an area model

Generate equivalent fractions with an area model

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In this lesson students will begin informally generating equivalent fractions using the area model. At this point, students will not be expected to use the “official” rule for converting $\frac{2}{3}$ into $\frac{8}{12}$ , but students are expected to be looking for patterns (relationships) between two fractions that are equivalent. Within the next few lessons, students will ultimately discover the “official” rule that

$\frac{a}{b}=\frac{a\,\times~m}{b\,\times~m}=\frac{am}{bm}$

Example 1: Find fractions equivalent to $\frac{2}{3}$ using the area model.

Let’s begin by drawing a picture of $\frac{2}{3}$ . Of course, this means two of the thirds are shaded.

If we cut each third into two pieces, we now have a rectangle cut into sixths. In fact, we can see that 4 of the sixths are shaded in.

Since the amount of shaded area did not change, we now know that

$\frac{2}{3}=\frac{4}{6}$

Suppose we wanted to go back to the original $\frac{2}{3}$ rectangle and cut each third into two pieces a different way. Now the picture would look like this…

The important thing to notice is the amount of shaded area still did not change, so we have a second way to show that $\frac{2}{3}=\frac{4}{6}$ .

What would happen if we started with the original $\frac{2}{3}$ rectangle and then cut each third into four pieces?

Our picture would look like this…

Since the amount of shaded area did not change from one rectangle to the next, we now know that

$\frac{2}{3}=\frac{8}{12}$

If necessary, repeat this process again only showing each piece of the $\frac{2}{3}$ rectangle being cut into three pieces. This will demonstrate that $\frac{2}{3}=\frac{6}{9}$ .

So, here are the equivalent fractions we have generated:

$\frac{2}{3}=\frac{4}{6}$

$\frac{2}{3}=\frac{8}{12}$

$\frac{2}{3}=\frac{6}{9}$

Do we see a relationship between the equivalent fractions? If not that is okay...we will continue collecting some more information with the next example.

Example 2: Find fractions equivalent to $\frac{3}{4}$ using the area model.

Start with a picture of $\frac{3}{4}$ and then cut each fourth into two parts. How many eighths are needed to make $\frac{3}{4}$ ?

Start with a picture of $\frac{3}{4}$ and then cut each fourth into three parts. How many twelfths are needed to make $\frac{3}{4}$ ?

Press [Next] to begin the 5 practice problems.