## Finding the Missing Whole using Bar Models

After students are able to find the percent of a number, they can explore what it means to find the missing whole.  We will look at 3 examples of how to teach this with percent bars.

Example 1

6 is 20% of what ?  First, we need to break our model into 10% sections.  6 is the 20%, so above 20% put the 6.

Now we have to determine what the whole bar is worth.  If 20% is worth 6, then 10% is worth 3.  If we continue to count up by 3, we can see that the whole bar is worth 30.  Therefore, 6 is 20% of 30.

Example 2

12 is 75% of what?  Since this problem is asking for 75%, we can break our model into 25% sections.

12 is the 75% so above 75% we are going to write a 12.  If 75% is worth 12, then every 25% is worth 4.  if we count up by 4 we see that the whole bar is worth 16.

Example 3

12 is 15% of what?  If we use our 10% section model, we can determine that 15% is right between 10% and 20%.

There are 3, 5% sections in 15% so if we divide 12 by 3 we can determine that each 5% section is worth 4.  If we continue counting up, we see that the whole bar is worth 80.  Therefore, 12 is 15% of 80.

Using percent bar models has been monumental for my students.  This visual gives them what they need to understand what it means to find the missing whole.  I hope your students find success with this!

## Fraction divided by a Whole Number

Today we will be focusing on a fraction being divided by a whole number.  I will be using Fraction Tiles which can be purchased here, or you can print and laminate a set of fractions tiles for free here.

Let’s look at the problem 6/8 ÷ 2.  Have students represent 6/8 with their fraction tiles.

We could say, “How many times can 2 fit in to 6/8?”  Have students ponder that question and have a discussion about it.  Students may be puzzled because it can’t fit in at all.  Therefore, it makes sense that our answer is going to be a fraction.  Have students line up 16/8 to represent 2 wholes.  We could say we have 6/8 out of 16/8.  Therefore, 6/16 which simplifies to 3/8.  So, 6/8 divided by 2 = 3/8.

Another way to look at the same problem is with partitive division.  If we have 6/8 ÷ 2, we could say, “What is 6/8 divided into two groups?”  Students can easily split their pieces into to groups to see that there are 3/8 in each group.

Let’s look at another problem, 1/3 ÷ 2.  Have students represent one-third.  We could say, “How many times can 2 fit into 1/3?”  Have students line up 6/3 to represent 2.  We have 1/3 out of 6/3.  Therefore, we have 1/6.  1/3 ÷ 2 = 1/6.

If you want to see the same problem using partitive division, have students represent 1/3.  We can’t break 1/3 into two pieces, but if we look at our fraction tiles, 2/6 = 1/3.  Therefore, 1/3 ÷ 2 = 1/6.

Let’s try one more, 3/5 ÷ 2.  “How many times can we fit 2 into 3/5.”  Have students line up 10/5 to represent 2.  We have 3/5 out of 10/5.  Therefore we have 3/10.  3/5 ÷ 2 = 3/10.

Using partitive division, have students represent 3/5.  We can’t break 3/5 into two pieces, but if we use our tenth pieces, we can see that 3/5 = 6/10 and 6/10 divided by 2 = 3/10.

Students may start to see that you can just multiply the whole number by the denominator.  That is a beautiful thing, because they are starting to see why multiplying by the reciprocal works!!  Don’t let them take short cuts yet!  We are building their fraction sense so when you finally teach the rule, they will have that deep understanding!!

~MN

## Hello!

It is my mission to deliver visual, hands-on lessons that deepen mathematical understanding.  I have been a middle school math teacher for 14 years and have made it my mission to get my students to not just “do” math, but to understand math.  I do this by providing visuals and hands-on lessons to aid in students understanding.  I am writing this blog for all the math teacher out there who are searching for ways to make their math lessons hands-on with a focus on true understanding.  I hope you enjoy!