# What’s the Whole?

Our fourth grade students were having trouble visualizing subtraction of mixed numbers, so I was asked by the fourth grade teachers to do a lesson reviewing this concept.

I decided that we would start with a problem from their math books, but instead of solving it, just talk about and visualize what one whole would look like in the problem.  We started with a problem (3 2/4 – 1 1/4) that did not require regrouping and I asked students,

“What would one whole look like in this problem?”

I asked students to turn and talk to a partner to come to a consensus.  Many of the students immediately started solving the problem and raising their hands to tell me the solution.  I reiterated the fact that I was not looking for a solution, I simply wanted them to think about how we would represent 1 or one whole in fraction form.

Student: “four fourths.”

“How do you know?”  “Can you prove it?”

Student: “Well on a number line that is split up into fourths, it would be four fourths.”

“Can you draw that for us?” Student: “If you split up one whole into four parts, it would be all of the parts.”

“Can you draw that for us?”

“Oh okay, that makes sense.  So will one whole always be represented by four fourths?”

Students: “No.”

“Let’s try another problem to see if we’re right.”

4  3/5 – 2 4/5

“What is one whole in this problem?”

Students: “five fifths.”

“How do you know? How can knowing that help us solve this problem?”

“Do you remember in 1st grade when you learned to make exchanges with base-10 blocks?  What could you exchange one long for?  How many base-10 cubes?”

Student: “10.”

“In this problem what could we exchange one whole for?  Let’s look at it using our virtual manipulatives.”

At this point, we had students go to the fraction tiles manipulatives on ABCya and begin modeling the problem. “So how many whole pieces should I have?”

Student: “Three.”

“And what other tiles do I need?”

Student: “three fifths.”

“Come show me what that looks like.”

“So what could I exchange one of these wholes for?”

Student: “five fifths.”

“Let’s do that.  Let’s lay them right underneath the whole to make sure that works.  Yes, it works.”

“So now can we solve this problem?”

Students: “Yes.”

“How?”

Student: “We can just take away two wholes and four of the fifth tiles.”

“How many does that leave us with?”

Student: “One and four fifths.”

We did this with several examples before having students work independently.  After they modeled it, we asked for volunteers to come show the class what they did.  Here are some examples:

One of the things that we discovered during this lesson was that several students were having difficulty organizing their fraction pieces.  We ended up having them draw a line down the middle of their screen and when they removed pieces, they moved them to the other side.  That way they could easily tell which tiles were left over.  This virtual manipulative tool is a little sensitive, but it is one of our favorites for ease of use. This is a flow chart to help students visualize that there are different types of problems and they should be treated differently.

After the lesson, we had students complete the independent practice sheet on the last page of this packet.  The other pages can be used in small groups with students to show another way to visualize the difference.