Photo by powerbooktrance.
Paraphrased from a homeschool math discussion forum:
“Help me teach fractions! My son can do long subtraction problems that involve borrowing, and he can handle basic fraction math, but problems like give him a brain freeze. To me, this is an easy problem, but he can’t grasp the concept of borrowing from the whole number. It is even worse when the math book moves on to .”
Several homeschooling parents replied to this question, offering advice about various fraction manipulatives that might be used to demonstrate the concept. I am not sure that manipulatives are needed or helpful in this case. The boy seems to have the basic concept of subtraction down, but he gets flustered and is unsure of what to do in the more complicated mixed-number problems.
The mother says, “To me, this is an easy problem” — and that itself is one source of trouble. Too often, we adults (homeschoolers and classroom teachers alike) don’t appreciate how very complicated an operation we are asking our students to perform. A mixed-number calculation like this is an intricate dance that can seem overwhelming to a beginner.
I will go through the calculation one bite at a time, so you can see just how much a student must remember. As you read through the steps, pay attention to your own emotional reaction. Are you starting to feel a bit of brain freeze, too?
Afterward, we’ll discuss how to make the problem simpler…
7 Steps to Subtract Mixed Numbers
Step 1. Ignore the whole numbers for a moment, and focus on the fraction parts.
Step 2. Convert both fractions to a common denominator. That in itself involves several steps:
- Look at the two denominators. Can you see a super-easy common denominator? For example, is one of the numbers a multiple of the other, or are they both divisors of some common number like 12?
- If not, then realize that the Least Common Denominator is an irrelevant sidetrack and don’t worry about it.
- Instead use the Easiest Common Denominator: Multiply the denominators of your fractions. It always works.
- Multiply the numerators of your fractions by the same amount to create equivalent fractions.
- Cross out the original fractions and write your new fractions next to them.
Step 3. Compare the fractions. Notice that the one being subtracted (which is called the subtrahend) is larger than the one you are taking it away from (the minuend). Panic. Take a deep breath. Calm down and go on to…
Step 4. Borrow/rename from the whole number part of the minuend to make a bigger fraction. This involves several more steps:
- Subtract 1 from the whole number, crossing it out and writing your new number above.
- Mentally convert the “borrowed” 1 to an improper fraction with the common denominator.
- Mentally add that converted fraction to the fraction part of the minuend.
- Cross out the fraction part of your minuend (yes, again!) and write your newer value beside it.
Step 5. Subtract the fraction part of the subtrahend from the fraction part of the minuend. Write this answer down below.
Step 6. Check to make sure the fraction part of your answer is in lowest terms. Whew! This time it is.
Step 7. Now look at the whole number part of the calculation and follow the standard rules for whole number subtraction…
I can see how a student might get confused, can’t you? And transferring the problem to manipulatives will only add steps, making it even more complicated!
Photo by gotplaid?
Make It Simpler, Part 1
I would like to try a different approach. Sometimes, taking the long way around a problem actually leads to a shorter solution. As an added bonus, the round-about method may help a student make sense out of the calculation, rather than just trying to memorize and follow a complex recipe of steps.
Here is the basic principle:
Whenever you get stuck on a problem, think about anything you can do to make it simpler.
With that in mind, let’s look again at our mixed-number calculation:
How can we make it simpler? Ignore the fractions for now. If it was 10 – 2, that would be easy:
Now put the fractions back where they were:
- Can you see that this calculation will have exactly the same answer as the problem before?
This is a key principle of working with arithmetic — do whatever you can to make your calculation simpler, without changing the basic problem. Most of the first year of algebra is spent learning how to simplify equations in this way.
Make It Simpler, Part 2
Okay, now what else can we do? If we had a really short problem like , that would be easy, right? I am sure you could do it in your head. So let’s “steal” 1 from the 8 and ignore everything else:
Now we have to put that left-over back with the other part that we were ignoring. This is what our calculation really looked like:
So far, we have made the problem simpler (only one mixed number instead of two), and we have taken away everything we were supposed to subtract. Our last step is a relatively straight-forward addition problem, putting the two fraction pieces together.
At this point, we are forced to find a common denominator — but at least we got the hard subtraction work out of the way first!
Show Your Work
How would this work in the “real life” of math homework? Start by crossing out the 10 – 2 and writing in the 8. It is easy to figure 8 – 1 = 7 mentally. I think most students can do in their heads, but teachers will probably want them to show that step — especially at first.
The main thing students would need to write down would be that last line of addition:
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