Ours is not to reason why, just invert and multiply. WHY!

Keep Change Flip WHY!

My district’s focus, this year, has been conceptual understanding. Therefore my coaching colleagues and I have been searching for a clear explanation to why we invert and multiply. In the process, beside our numerous in depth geeky conversations, we’ve become proficient at solving division of fraction problems multiple ways – Drawing pictures being my favorite method. I can describe which parts of the picture represents the division process. But if someone were to ask me, “Why do we have to flip the second fraction?”, I’d still stumble over the explanation.

**My colleagues: Tim, Mari and Karon**

Recently, a series of experiences connected with earlier conversations have me believing I’m getting close to a clear explanation. Here’s my attempt to shed light on the situation …

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### Groups vs Group Size

Synapses started firing when using Nat Banning‘s website, FractionTalks.com. While leading a FractionTalks activity, the language of groups and group size started swirling around in my head. **My “aha” moment: Groups and group size are reciprocal concepts.**

Groups: 2 Groups: 3 Groups: 4

**Group Size: 1/2 Group size: 1/3 ****Group size: 1/4**

The given FractionTalks.com prompt: How many ways can you divide into … lead into other questions.

- How many ways can you section the rectangle into halves? More than 1
- How many groups did you end up with?
**(2) **
- How many squares make up each group?
**(6)**
- How many squares are found in 1/2 the rectangle?
**(6)**
- How many squares are found in the group size of 1/2?
**(6)**

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**Classroom Example**

- How many ways can you divide the rectangle into thirds?
** (Many) **
- How many groups did you end up with?
**(3)** ** **
- How many squares make up each group?
**(4)**
- How many squares are found in 1/3 of the rectangle?
**(4)**
- How many squares are found in the group size of 1/3?
**(4) **

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**Equal Sharing**

At a recent Anni Stipek training, we were presented with the following problem:

**Patrick had 1/4 of his birthday cake left. If he shared it equally among himself and 2 friends, how much of the birthday cake will each one get?**

**Step 1**: I drew my picture and wrote the expression, 1/4 ÷ 3. When a fraction is divided by a whole number, the concept of equal sharing is applied. The shaded part represents 1/4 of the cake.

**Step 2:** The remaining 1/4 of the cake is **split into 3 sections b/c we have 3 people.** Each section represents a **group size of 1/3**. I get to eat 1/3 of the remaining 1/4 of cake.

I split the 1/4 into 3 groups which means my share represents the group size of 1/3. This acknowledgement addresses why we write the reciprocal of the second number.

**1/4 ÷ 3 (3 groups) now becomes**

**1/4 • 1/3 (group size of 1/3 of the 1/4 remaining cake)**

**Step 3:** The portion of the original cake that I get to eat is 1/12

1/4 • 1/3 = 1/12 of the original cake

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**Understanding what’s truly being asked.**

I’d like to go back to the original problem and refine the question:

**Original Problem:** Patrick had 1/4 of his birthday cake left. If he shared it equally among himself and 2 friends, how much of the birthday cake will each one get?

**Rewritten:** Patrick had 1/4 of his birthday cake left. If he shared it equally among himself and 2 friends, **What portion of the remaining cake will my piece represent when compared to the whole cake?**

- Understanding the question: When I rewrote my question, I’m acknowledging that I need to find how my share compares to the whole cake. I have to compare the same ideas, which in this case is group size of my share to the size of the whole cake.

- Step 1: 1/4 ÷ 3. Divided the remaining cake into 3 groups.
- Step 2: Step 2 inverts the language. Groups and group size are reciprocal concepts. To answer our question, our answer needs to be in the context of size. Therefore we have to invert the concept of the groups to use the language of group size.
- Step 3: 1/4 of the remaining cake • my share (1/3) = the part of the whole cake that my piece represents. (1/12) or 1/4 • 1/3 = 1/12

This process emphasizes the importance of relating the answer back to the whole.

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**Repeated Subtraction**

Let’s continue with the cake theme to discuss the repeated subtraction approach.

**Denise brought her remaining 3/4 cake, to work, to share with friends. **

Given this scenario, let’s tackle the problem: 3/4 ÷ 1/8

If we use the equal sharing method, this problem says Denise is going to split the remaining 3/4 of cake among an 1/8 friend. Huh? That statement doesn’t make any sense. There has to be at least 1 whole person in which to share the cake. Since we don’t have at least 1 whole person, the equal sharing method isn’t appropriate in this scenario.

The equal sharing approach is used when dividing by a whole number. 1/8 is not a whole number. It’s a fraction, therefore we have to apply the repeated subtraction process.

In this problem, the 1/8 represents the size of the piece in comparison to the whole cake. Therefore, we need to determine how many groups of size 1/8 fit into 3/4 of the cake. The process of determining the number of groups makes this is a repeated subtraction problem.

Let’s start over using the repeated subtraction approach. Denise brought to work her remaining 3/4 cake to share with friends. If each person gets a piece representing 1/8 of the whole cake, how many people can get a piece?

### ** 3/4 ÷ 1/8**

**Step 1:** I drew a rectangle and shaded in 3 out of the 4 sections

**Step 2**: **Now to find sections of 1/8 … **

**Question**: But how do I determine how much is 1/8?
**Answer**:
- We need 8 groups to clearly see a group size of 1/8. Here’s the reciprocal aspect coming into play. By inverting the size of 1/8 to 8/1, we are now talking about groups
- Once the number of groups is acknowledged (8), we have a clear visual of the size, 1/8

**Step 3**: 3/4 of 8 groups is 6 groups

- 3/4 ÷ 1/8 means, How many groups of size 1/8 are found within 3/4?
- 3/4 • 8/1 By establishing 8 groups, we can determine how many of those 8 groups is represented by 3/4. Or … What is 3/4 of 8 groups?

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**Visually – Drawing a Picture**

The explanation starts off the same …

**3/4 ÷ 1/8**

**Step 1:** Draw a rectangle and shade in 3 out of the 4 sections

**Step 2: **Now to find sections of 1/8

**Question**: But how do I determine how much is 1/8?
**Answer: **Since we are looking for groups of size 1/8, common denominators are needed. Therefore 3/4 ÷ 1/8 becomes 6/8 ÷ 1/8

**Step 3:** Now it’s clear how to determine the size of 1/8. When the number of groups needed is applied, the process of finding groups of size 1/8 is easier to see.

**Back to our question:** How many 1/8’s fit into 6/8?
**Answer**: 6 groups of 1/8 fit inside 6/8

**Analysis**:

A: The work shown in A doesn’t show a reciprocal Here’s my thinking… By converting to common denominators, the group amount issue is addressed.

B: The work in B shows the reciprocal action of switching from group size (1/8) to number of groups (8).

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## Summary

**Dividing by a whole number **** 2/3 ÷ 4**

- Equal sharing
- Division means separation into equal groups
- Answer is a size compared to the whole

My answer of 2/3 ÷ 4 represents the portion of the whole cake that I receive. 4 people are eating the remaining 2/3 of the cake.

Each person received 1/6 of the cake.

**1/6** • **4** = **4/6 = 2/3**

**(Size of a piece)** • **(Number of people)** = **Amount of cake shared**

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### Dividing by a fraction **2/3 ÷ 1/****4**

- Repeated subtraction
- Division means finding the number of groups of the given size
- Answer represents the number of groups found.

Mari brought in 2/3 of her remaining birthday cake. If the 2/3 is split into a portion size of 1/4? Is it possible for Karon, Tim, JJ & Jenn to all get cake?

**Method 1**

2/3 ÷ 1/4: How many groups of 1/4 fit inside 2/3? Can Karon, Tim, JJ and Jenn all receive cake?

Since we are dividing by a fraction (group size 1/4) we need to determine the group number we are working within. Converting to common denominators will do that.

We now have 8/12 ÷ 3/12, which means: How many groups of 3/12 fit inside 8/12?

The context of 12 has been established, I can look for groups of 3/12ths. I circled 2 full groups of 3/12. There were 2 sections of the remaining 3 needed, resulting in an answer of 2 2/3 groups

**Answer**: 2 2/3 people (groups) can get a piece of cake.

For this problem, Karon and Tim will each receive cake. J.J. will receive a partial portion. No cake for Jenn.

Karon and Tim’s pieces represent 1/4 of the cake. J.J piece represents 1/6 of the cake.

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**Method 2**

Because the above method, used common denominators to acknowledge the group context. I did the problem again to clearly show the reciprocal action of changing the 1/4 size to 4 groups.

This format clearly shows the division of cake among Karon, Tim, JJ and Jenn. Karon and Tim get full pieces. JJ gets a partial portion. Again, no cake for Jenn.

**Conclusion**

The conversations with my colleagues, the trainings I’ve attended, the search to understand keep, change, flip conceptually and the writing of this post have all helped me unravel some of the fascinatingly complex and intriguing aspects of fractions. I can confidently say, my understanding of fractions has grown but I still have a lot to learn.

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