Dividing Unit Fractions


Unit Fraction: A fraction with a 1 in the numerator such as 1/3, 1/4, 1/8. All fractions are built on unit fractions. For example 2/3 is 1/3 + 1/3 and is made from 2 unit fractions.

This week we will change focus a little bit. Last week we were dividing whole numbers into unit fractions. This week we will take a unit fraction and break it into same size pieces.

We can start by asking What do I have? and What do I do with it?

Today we have 1/4 and we are partitioning, or dividing, our amount into 3 same size pieces.


Start by showing what you have: 1/4

Once you show what you have, 1/4 of your whole. You can break your 1/4 into 3 same size pieces.


Breaking your 1/4 into 3 same size pieces

Your model now needs to show how much of your whole each new piece is. You also need to break up the rest of your fourths into 3 same size pieces.


Each one of your new pieces is 1/12 of the whole

This model can now be represented with the equation 1/3 ÷ 4 = 1/12

Creating Your Own Word Problem

Tonight you will be creating a word problem that matches an expression. We can start with the dividend, or the first number in a division expression. The dividend is the whole, or what you have. The second number is the divisor. This is what you are dividing your whole by.

When you are creating your word problem start with what you have. When creating these sort of problems it is easiest to use things that can cut into smaller pieces. Pieces of yarn, rope and ribbon are common. Making necklaces from yarn, bracelets from rope, and wrapping presents with ribbon are examples. Here is an example of a word problem that the students have seen.

The models that you have been creating all week can be used to represent your situations. Each fraction would be, in the example above, one present being wrapped since 1/3 of a meter of ribbon is used to wrap each present.

Dividing with Fractions

Rules that Expire



As students explore math they begin to make rules, and they expect their work to obey these rules. As students progress they bump into math where their rules does not work anymore. This is an opportunity to explore how to alter the rule to accommodate this new information.

A rule that expires this week is that division always makes your dividend smaller. Students will discover that 5 ÷ 1/3 = 15. This quotient will initially be a bit disorienting as it breaks a rule they have created.

Students can make sense of this by exploring what 5 ÷ 1/3 is asking them to do. This expression is asking them to find out how many 1/3s there are in 5 wholes. This connects to the work you did with them last week converting fractions. An example of a model you may see is below:


In this example 5 was broken into thirds, and then the thirds were counted.

At this point we are not yet teaching an algorithm, but we can ask students to start looking for patterns they see. The algorithm will be further explored in coming posts.

Multiplying Mixed Numbers-Converting Improper Fractions to Mixed Numbers

When students are multiplying mixed numbers they generally get an improper fraction as their product. This improper fraction will need to be converted to a mixed number number. Strategies involve making as many wholes as you can, and then adding the left over unit fractions.

Below is an example of ways that you may see your child thinking about his work. Even if the work your child is doing does not look exactly like this, you will see aspects of this example in your child’s work. The 3, 6, 9 on the bottom are keeping track of how many thirds your have. Notice that the 12 is crossed out. This step shows that 12 thirds is more than we have, so we have 3 wholes and now need to find how many more thirds we have left over.


Adults often tell me that they want to help their children with math, but aren’t sure how to. The math seems to look so much different from when we were kids. Students will begin bringing home homework that will allow you to better understand how students are thinking about math on a daily basis.

The best way to understand how math is being done is to spend some time exploring the math. Every Monday-Thursday students will bring home some math for you to both work on together. You will take turns showing your strategies to each other. It is important to talk about what you are doing. Ask questions about each others’ work, point out what you like about what each other did. Be transparent, when your child does something you are not familiar with tell them that. Ask them to explain what they did. This will help to empower your child to become the teacher and help build their math confidence.

Check in here to see examples of work being done and to get updates on information and strategies for the work. Please leave comments if you have questions or if there is math that you would like to see demonstrated.

Have fun and be creative and brave in your explorations!!

Multiplying Mixed Numbers and Fractions- Changing Mixed Numbers to Improper Fractions


Mixed Number: A fraction with a whole number and a fraction.

Improper Fraction: A fraction with a value greater than 1.

Before students are able to multiply mixed numbers they need a strategy to distribute the values of both numbers to each other. An efficient strategy is to place the value of the whole number in the numerator. This will give us an improper fraction. When doing this work with your mathematician have them first make a visual model and explain it. After they can explain the process then they can use an algorithm. An example of a model students may use and an algorithm are below.

Math Explorations

Adults have been telling me for years that they are having difficulty helping their children with math. Students are thinking about math in much different ways than most of the adults did when we were in school. This website is designed to introduce adults to the mathematical concepts being taught in schools throughout the country. You can use the resources to first explore and do some of the math yourself, then to share in your exploration with your little mathematicians at home.

If I am your child’s teacher then checking in here will also help you keep informed about what your mathematician is currently thinking about during school.

Please feel free to comment here with questions or with suggestions for posts or videos.