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**4.NF.1**

Explain why a fraction a/b is equivalent to a fraction (*n*×

*a*)/(

*n*×

*b*) by using fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

See the Leapfrog problem.

This summer I had the wonderful opportunity to tutor my friend, Dylan. He wanted to better understand fractions, so I challenged him with this MARS Task assessment. After he read the directions, I helped him think of some problem-solving strategies. In order to find the answers, Dylan made a number line graph that helped him visually compare the various fractional parts. These included:

*halves, fourths, thirds, sixths, eighths, and ninths.*

Using 1/2 inch graph paper, Dylan made number lines that were each 18 boxes long, so it was easy for him to make

*halves, thirds, sixths, and ninths.*Then he figured out how to make

*fourths*by cutting the

*halves*in half.

Next, he was able to find the

*eighths*by cutting the*fourths*in half.
In this way, he could easily compare some of the necessary equivalent fractions for this task.

Dylan said that

**Frog 1**was easy to solve. He just had to determine the missing fraction that made the equation add to**one**, and help the frog hop from the lily pad to the island.
So...

here is how

**Frog 1**hops to the island:
Dylan drew a double number line on his paper to compare

*halves*and*fourths*for**Frog 1**.
With each step in the Leapfrog problem, Dylan was able to measure and compare the fractions on his number line graph with the fractions that were on the assessment.

Here is his solution for

This time Dylan's double number line compared

This is Dylan's solution for

His double number line compares

Immediately, he saw the answer! "It's 3/10," he beamed. "Too easy!"

I was so proud of Dylan when he figured out how to make equal parts with

"That's terrific that you knew 6/8 was equal to 3/4," I smiled.

But watch out for

Dylan saw that he had to add

Again, Dylan wasn't sure how to make the last hop to the island, so I showed him the missing portion on his number line and wrote another question mark.

Now for the "big enchilada"! Could Dylan figure out

Here is his solution for

**Frog 2**:

This time Dylan's double number line compared

*thirds*and

*sixths*. You can also see how he helped the frog hop across the number line to solve for the missing fraction.

This is Dylan's solution for

**Frog 3**:

His double number line compares

*fifths*and

*tenths*. Notice that this time these fractions were not on his number line graph sheet, so he had to decide how to compare them. He colored in the 3/5 and 1/10, but then wasn't sure what to do. I marked the missing part and wrote a question mark.

Immediately, he saw the answer! "It's 3/10," he beamed. "Too easy!"

I was so proud of Dylan when he figured out how to make equal parts with

*fifths*and

*tenths*and compare these with the same

**one whole**. Great job, Dylan!!

"That's terrific that you knew 6/8 was equal to 3/4," I smiled.

**"Grandma and I are so impressed!!**But watch out for

**Frog 5**," I continued.Dylan saw that he had to add

*sixths*and*ninths*, so he made another double number line and figured out how to make equal parts with each that represented the same whole.Again, Dylan wasn't sure how to make the last hop to the island, so I showed him the missing portion on his number line and wrote another question mark.

Once more the eyes sparkled and he wrote "2/6 = 1/3."

Fabulous!

Now for the "big enchilada"! Could Dylan figure out

**Frog 6**by himself? I quietly watched as he read the problem out loud, started drawing a double number line as before, when suddenly he stopped and put down his pencil. He thought for a moment and then realized that this time he needed to draw a third number line. First, he added 1/4, then 10/20....but when he highlighted the 1/5 with the orange marker he realized that it was just a little bit short of the island.

Wow!! He did it! I reminded him that he had to use pictures, numbers, and words to prove his answer so he wrote, "no he will not make it because it is short." Way to go, Dylan!! Good job!!

Here are

**Frog 5**and**Frog 6**:
And the studious man hard at work...