Making Sense of Adding Fractions

If you are introducing fractions in your class, allow students to use unifix cubes to make sense of what fractions really are and look like.

You can start with fractions that have the same denominators so students understand that the whole is made of a number of parts. A fraction is part of a whole. You can have 1 whole divided in 4 parts, and you can have 1 whole divided in 8 parts. The most important thing is for students to understand that the wholes have to be the same size. For instance 1/2 small pizza and 1/2 large pizza do not make 1 whole pizza because the wholes are differen sizes.

After students have had a lot of opportunities to explore adding a lot of fractions with the same denominators, you could start posing problems with “alike” denominators. For instance start with a problem like 1/4 + 1/8. Give students unifix cubes and let them figure out how they could represent these 2 fractions using the same whole as a reference.

After they represent these 2 fractions with the same whole, they can add the 2 fractions. When students work in pairs, they can help each other to make sense of how to represent eights and fourth using the same whole. It might take some time, but they will get there.

Step 1- Ask students: using unifix cubes, how can you make 1 whole that can be divided in “eights” and also in “fourths”?

Ask students to SHOW how their whole can be divided in fourths and eights. Some students might make a whole with 4 unifix cubes only, and when they try and divide that in eights, they will realize it doesn’t work. They will try again. Do not solve for them.

Have students explain their thinking, and share with the class how did they get to that conclusion.

After they have their fractions represented, they can identify 1/8 and 1/4.

Step 2- Ask students to show 1/8 of the first whole, and 1/4 of the second whole. Label each fraction. Students will get a visual representations of what both fractions look like related to one another. It has to make sense to them.

Step 3-Ask students to put their fractions together and find an answer. If students answer 3/4, ask them to go back to what they identified as 1/4, ask: “Does that make sense?” They can discuss and reflect on it, and come to understand that each of those 3 unifix cubes look just like the 1/8 from step 2.

This whole time, students do not need to use any paper or pencil, and defnitely no need to learn a set of rules to find the common denominator. That will come later. Students will make sense of it on their own. Just give it time.

Concrete & Pictorial Representations

Here is one more example to use in your class, to allow students to make sense of adding fractions with alike denominators without using memorized steps. These strategies allow students to understand why if you are adding thirds and sixths, the answer can’t be ninths! Because they have never seen ninths in the representations they are creating.

Step 1 – Pose the problem on the board. 4/6 + 1/3. Ask them to use unifix cubes to represent sixths and thirds using the same size whole. Let them struggle a bit. They can work in pairs, and talk to their partners. They will start with thirds and eventually will make sense of using 6 unifix cubes to also represent thirds. Have students explain to each other how they came up to that conclusion. Explaining their thinking is for their own learning, some students finish making sense of what they learned, when they hear themselves explain, when they organize their thoughts, and make sense of it in words. Learning is fascinating!

Step 2– Ask students to label their fractions.

Step 3– What is the total fraction now? Have students explain why their answers is 1 whole. Have them show how these 2 fractions make 1 whole. This is a perfect moment to ask students if they could come up with other models to represent this same problem. Perhaps they can use fraction tiles, or fraction circles. They can also draw the fractions if you do not have other manipulatives in your class.

Finally, it is time to use pictorial representations. The bar model or tape diagram, is a recomended model to represent fractions. It is easier for students to use a rectangular model than a circular model to be more accurate with the equal number of parts.

Most students will start using 2 tape diagrams, and that is ok, as long as the 2 wholes are the same size.

Eventually they should start partitioning the same whole in both sixths and thirds. They can use 2 different color markers to see their work. This is a good strategy for when they start sbtracting fractions.

Use a variety of good problems to help students see where we use fractions in the real world. Here are some ideas:

If Billy has 35 cookies and shares 1/5 of them, what fraction does he have left?

If I read ¾ of my book, and I have 80 pages left, how many pages is my book?

I am running a 3 km race. I am almost there. I have 1/6 left. How much do I have to run?

Is the sum of 4/5 and 7/8 under or over one? Explain how you know. 

Enjoy playing fractions with your students!