Adding Fractions with Unlike Denominators

How do you add fractions with unlike denominators? Find common denominators of course! You know, however, that I'm a big proponent of students understanding why they are using a strategy. If our students don't know why they are using a strategy so many of our learners don't know when to apply that strategy. They become students who have "strong rote skills but struggle with application".

An incredibly effective method of helping our students to understand strategies is to give them concrete and representative experiences to make sense of an abstract strategy. I want to share 2 representative strategies for adding fractions with unlike denominators.

Representative Methods of Adding Fractions 

Begin by representing each fraction. Your students can see that the two fractions can not be combined as is because the pieces of each fraction are not the same size. 

Ask your students what they might be able to do to make it so that the fraction pieces are the same size.

 Your students may suggest creating two like arrays. In the example to the right, You can easily draw eighths over top of the fourths and fourths over top of the eighths to create two arrays each with 32 pieces.

Once the two arrays are equal it is easy to add the two fractions together because you have found a common denominator of 32! 

You may be thinking to yourself that 32nds aren't necessarily easy numbers to work with and that the method is less than efficient. That's ok! Once your students understand that they can combine fractions by making two arrays with the same number of pieces they can explore ways to become more efficient by finding the least common denominator! 

Consider the graphic to the left. Post this picture for your students and ask if they could add these two fractions together using those models. Do your students recognize that both models show eighths even though the shape of the eighths is different in each picture? 



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Teaching Math Strategies that Stick!

Teaching math strategies for addition is commonplace in the first grade classroom. It's so important to be sure that you go beyond the anchor chart and teach strategies such as counting on in ways that allow your students to understand so they can apply the strategy in word problems and other contexts. Teaching math can be fun and productive when you use strategies that stick!
We know that when we teach procedures in math that many of our students lack the ability to generalize those procedures and apply them when it's appropriate. How often have you heard this description of a struggling student:

 "They have great rote procedures but they struggle when it comes to the application". 

It's so common! There are steps we can take as teachers to help teach our students strategies in ways that allow the strategy to transcend "procedure status" and to stick! 

1) Link It Back
2) Context Is King
3) Think CRA

Link It Back

If I am teaching a strategy such as counting on to add, I want immediately link back to what my students already know about addition. If my students know that addition puts parts together, when we talk about counting on we will be talking all about how this is a strategy to put parts together. Down the line when your students see an addition sign, they won't default to the counting on procedure because "the plus sign means I grab the bigger number and count on". They will default to counting on IF it is an appropriate strategy because it is the most efficient strategy to put parts together.

Context Is King

So how do I teach my students to count on without outright stating "We can count on to add parts together. You count on by..."? I give my students a context that lends itself to counting on even though they don't yet know that strategy!
In this example, you can see that I have told my students there are 7 bears in the cave and 3 outside. I want them to figure out how many bears there are altogether without ever getting to see the bears inside the cave. This scenario lends itself perfectly to the counting on strategy. Once my students think they have solved the problem we can get out manipulatives to show 7 bears in the cave and 3 outside and discuss whether or not our counting on strategy was successful in putting the parts together.

Think CRA

When teaching a new strategy that is more abstract in nature such as counting on, it's always helpful to think CRA. Concrete, Representative, Abstract. This means that when I begin teaching the strategy to my students I may start with an activity where I show them a wallet and tell them that there are $6 inside. Outside, there are $5 more. Students can try the counting on strategy but can touch, feel, and manipulate the dollars to solidify their thinking.
At some point, I would want to move to a more representative model such as the pictures of the bears in the cave shown above. In this way, our students are able to "trust" that there are a given number of bears in the cave. They are beginning to be comfortable manipulating the numbers without having to physically manipulate the scenario.

Ultimately, when teaching counting on or when teaching any variety of other math strategies, you will want your students to be able to perform the strategy given an equation with pencil and paper or mentally. This is the abstract level and is best reinforced once your students have solidified the strategy through concrete and representative means.

Following these three steps will help to ensure that the strategies you are teaching are sticking for your students. You are truly teaching strategies rather than a standalone procedure! 
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Research Based Strategies 

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