Last, but not least, we’re closing out our Working with Fractions series with subtracting fractions with Concrete, Pictorial, Abstract (CPA) means.
When it comes to subtracting fractions, this operation is very similar to adding fractions in how students understand it. We always believe that the CPA approach is the best way to help students gain an in-depth understanding of what we’re conceptually asking them to do so they don’t just learn a procedure for a concept they don’t understand.
Pattern blocks are one of the concrete tools we like to use to help students understand this concept. I found some new pattern blocks that I’m really excited about because they have a brown piece in the set that represents ¼! There’s also a purple piece that represents 1/12! You can get those on Amazon here!
When using the pattern blocks, remember that one whole is the hexagon, the trapezoid is the ½, the rhombus is the ⅓, the triangle is the ⅙, and now we have a brown shape that is ¼ as well.
Subtracting Fractions with Common Denominators
Let’s look at ¾ – ¼, and we’ll start by building ¾ on top of the whole. If three brown pieces are put onto top of the hexagon, and it becomes very easy to see how, if ¼ piece is taken off, then 2/4 will be left.
Of course, that is not the simplest version of that fraction. Building the problems and visualizing them with pattern blocks also makes it easy for students to be able to simplify their fractions. They can match up the pattern blocks and see what piece would cover the total. In this case, it is easy to see that the trapezoid, or the ½, fits over the 2/4 pieces, so ½ is equivalent to 2/4.
The same process works for subtracting fractions larger than one. These “top heavy” fractions where the numerator is larger than the denominator, are often called improper, even though they are still legitimate fractions. They’re just larger than one!
For this example, let’s take 10/6 and build it with pattern blocks. We’ll use two hexagons to show that the fraction is larger than one, and we’ll layer on the other blocks so we can represent sixths. Once I count out all 10 of my ⅙ pieces on top of the hexagons, I can see that one hexagon is full, and I have 6/6 and 4/6. I can see that 10/6 = 1 4/6.
Now that we have that number built with pattern blocks, if I wanted to take 7/6 away from 10/6, I could easily see how that would work. 6/6 fills up one whole, and then we’re left with taking ⅙ from the rest, which gives us 3/6. I can see the three triangle pieces, compare it to two brown pieces to see that 3/6 is equal to 2/4, or put the trapezoid on top and see that 3/6 is equal to ½.
Subtracting Fractions with Uncommon Denominators
You could use pattern blocks for working with this type of fraction, and I always love using patty paper with fraction examples, but the ideal tool here is going to area model paper, and that’s the tool I’ll use in this video.
With area model paper, my whole is going to be red, my ½ will be orange, ¼ will be yellow, ⅛ is pink, and 1/16 is green.
My problem will be ⅞ – 4/16. Let’s pause for a second and make sure that students really understand what ⅞ is and what it’s asking. So let’s build the ⅞ by stacking 7 pink pieces on top of the whole. At a glance, I can notice that one more ⅛ would cover the whole. But I want to take away 4/16.
At this point, we are often quick to teach children a procedure such as “just skip count to find the common denominator.” However, I love taking an inquiry-based approach to problem solving, especially with fractions like this. I want students to think about how they could solve this problem in multiple ways.
I’ll put 16ths on top of the ⅞, and when I do I can easily see that ⅞ is equivalent to 14/16. Then, it’s an easy subtraction problem because I’ve made them all 16ths. I just have to take 4/16 away from 14/16 and I end up with 10/16. Although 10/16 is not in the simplest form, it is definitely correct. I’m not worried about simplifying as much as I am about helping students discover an answer and solve the problem using concrete tools.
Let’s look at the same problem again: ⅞ – 4/16. Some students might notice that I could have used 8ths instead of 16ths. So, I can take the 4/16 and put the ⅛ pieces on top to show that it is equal to 2/8. When the problem becomes ⅞ – 2/8, we can solve it much more quickly and find that we have ⅝. In this case, that answer is in the simplest form, and we eliminated the step of having to reduce or simplify.
Subtracting Fractions with Mixed Numbers
When students start to learn about subtracting mixed numbers, it’s important to make sure they understand it in a conceptual way. We’ll switch back to pattern blocks for this example. While I like using the area model papers, pattern blocks represent whole numbers much more effectively and efficiently, as you don’t need multiple sets to do so.
Let’s take 2 ⅔ – ⅓. Let’s build the two wholes with two hexagons, and then we’ll have 2 rhombuses to show the ⅔. I can simply take off ⅓ and see the answer will be 2 ⅓.
This is pretty simple, but where students sometimes get tripped up is when they have to rename numbers. Sometimes we call it “borrowing” but borrowing really means we’re going to give it back, and that’s not what we’re doing when we’re subtracting!
To demonstrate concretely how we rename fractions, we’ll use the example 2 ⅓ – ⅔. I can’t take ⅔ away from ⅓ so I’ll have to decompose (like D.C. does) and rename that 2 ⅓. We’ll do this problem concretely and pictorially in the video.
I’ll change one of the wholes into 3/3, add it to the remaining 1 ⅓, so it becomes 1 4/3. By building it with pattern blocks, students can clearly see what’s happening to the numbers when they are renamed. We can also ask students to prove that 1 4/3 is equal to 2 ⅓ by building the numbers as we just did.
Once the number is renamed, students can see how it then becomes possible to take ⅔ away from the new amount of 1 4/3. I prefer to show this horizontally as opposed to stacking it in a traditional algorithm at first, because I want to promote their number sense and what they’re doing, versus just having them memorize a procedure.
In our last example, we’ll look at how we can take away 1 ¼ away from 3. I know that taking away that one will kind of be easy, but how do I take away the ¼ when I have three wholes? Again, using D.C., I’ll rename the 3 to be 2 4/4. This will make it a lot easier visually to take away 1 ¼ because I’m just going to take away one hexagon and one ¼ piece, leaving a nice easy answer of 1 ¾.
Subtraction with Springling
We can’t leave subtraction without a visit from one of our favorite Math Mights – Springling! The fanciest Math Might, she was born with a coily tail and fancy eyelashes. Fractions on a number line is a really important concept for students to learn, especially because using an open number line to find the distance between two numbers is something that they will use into their middle school years.
For Springling’s example, we’ll use 2 ⅓ – 1 ⅔. Springling wants to hop to find the distance between the two numbers, so we’re going to start by putting the subtrahend (the second number in a subtraction problem) on an open number line. Then, we’ll draw a line and put the minuend at the other end.
As we look at the distance between 1 ⅔ and 2 ⅓, it should remind students of looking at the distance between whole numbers. If students aren’t familiar with Springling, however, there’s nothing wrong with taking upper grade students back to whole numbers to understand the concept of the open number line. Watch the video of Springling subtracting whole numbers on YouTube.
Starting at 1 ⅔, we can see we only have to go ⅓ to get to the whole number 2. From two, we want to get to 2 ⅓, so we can see the distance between the two is ⅔.
Let’s take an example with higher numbers – 12 ¼ – 6 ½.
I recommend using friendly fractions with Springling as kids start to develop their understanding before moving on to more complicated ones! Friendly fractions might be ¼, ½, ¾ – things kids might be able to relate to time or even quarters and dollars.
In our example, Springling will start at the subtrahend, the 6 ½, and look ahead all the way to 12 ¼. She can see that 6 ½ is only ½ hop away from the friendly number of 7. We can draw a curved line, like a small hill, to represent a part of a whole.
Once Springling gets to 7 on the number line, she wants to travel to the next highest whole number, so she hops all the way to the number 12. We’ll draw a large peak here to represent a whole number. The same idea applies if you’re using Springling for elapsed time, money, decimals, or even fractions – use a curved line for part of a whole, and a large peak for whole numbers jumps so kids can easily tell the difference.
The distance between 7 and 12 is 5, and then Springling only needs to hop to 12 ¼, which is ¼ hop. Once I add all Springling’s hops together (½ + 5 + ¼) we can see the answer is 5 ¾.
Using Springling to solve subtraction problems will help to build the number sense that we want children to have as they can visualize exactly what is happening as we work with the different numbers.
The big takeaway message for teachers is that we really need to slow down the process of subtraction to help kids build conceptual understanding for what we’re doing with fractions. We hope these videos will help you do just that.
You can use these videos in two ways. One is as a lesson launch, making sure students have concrete tools in hand as they watch so they can explore the concepts simultaneously. After watching the video, see if students can create other problems like the ones we showed in the video to demonstrate their understanding. They can solve their problems with a partner or share out in a small or whole group.
You also can use the videos in a flipped classroom model where students listen to the video either at home or maybe in another station in your classroom as an introduction to what you’ll be talking more about in your Math with a Teacher station. It’s really helpful for students to have more of a background knowledge of what they’re learning about before you actually try to teach them, especially if we’re teaching with an inquiry-based approach. When we’re not serving as the givers of all information, kids can develop a deeper understanding because they can grapple with the concepts they’re learning and how to apply them.
In our next series, we’re going to feature visual models for word problems that you can apply to any of the math programs that you’re working with. We’re excited about bringing you really great reading comprehension strategies to help your children become more proficient in understanding what story problems are asking!