Cross Multiplying When Comparing Fractions? What We Should Do Instead!

Why does cross multiplying work when comparing fractions? Where did it come from? Should we be teaching this technique to our students? Is there a better way?
These are the questions we should ask ourselves about every procedure we teach to our students.
We want to fuel sense making in our students. We want them to know how the strategies we use regularly connects to key understandings we've used in the past.
Helping students understand why and how strategies connect will limit that dreaded question, "Why do we have to learn this?"
In this video we'll:
- show you how to compare fractions using digital manipulatives from Mathigon;
- help you so you can help your students use visual models to compare fractions;
- explain why cross multiplication can work sometimes; and,
- discuss a strategy that builds off of prior knowledge so students don't think they are using a new technique.
Resources:
mathigon.org/polypad
learn.makemathmoments.com/tas...
All Tasks: makemathmoments.com/tasks

[@LadyoftheHouseDiaries]

Excellent!

[@h4ks3lr1]

Excellent👏

[@0jimmeyer0]

Yo. This is a wild take on a simple explanation for why cross multiplying works. It doesn’t have to be this complicated. We cross multiply to simply find a common denominator. Not necessarily the LCD but common. Let’s use ½ and ¾ as an example so we have unique digits. Keep in mind, 8 is a common multiple of our denominators - 2 and 4. When we multiply one denominator - 2 by the opposite numerator - 3, the part we don’t see is that we are actually multiplying both the 3 and the 4 by 2. We are taking ¾ x 2/2. There are two things to notice here - first, we are multiplying ¾ by 1 so we get an equivalent fraction (6/8) and second, we have that common multiple of 8 we talked about earlier as a denominator. Repeat for the other ‘butterfly wing’ and you get ½ x 4/4. Again ½ x 1 is an equivalent fraction (4/8) and again we get a common denominator of 8. We can now compare 4/8 to 6/8 and see that 6 of these eighths is greater than 4 of them. And that is the trick of cross multiplying - it is hiding, or taking for granted the multiplication property of equality… that 2x4 will always equal 4x2. All we see is the numerator action because in the end it is all we need to compare. It is a terrible ‘trick’ because the actual process is not that much more notation and it shows so many beautiful and useful concepts: a/a =1, equality property of mult. Identity property of multiplication (when multiplying a by 1 our product = a), equivalent fractions, and helps see the numerator as a quantity of non-whole units.

[@MakeMathMoments]

Yo back, Jim! Thanks for popping by and sharing your thinking! While all you mention above is true (and beautiful), I would argue that what you're describing as "simple" actually isn't all that simple to younger learners who are trying to make sense of mathematics. Your explanation is simple because you know these "truths" or behaviours of mathematics. For our students, we want to help them reason through these ideas so they can make sense of what they are doing. From there, they will be able to "see" what you're describing as simple. Until they see it for themselves and make sense of it, they will think of your explanation as 'complicated'. To be clear: we are not suggesting we 'hide' the properties you have shared above, but rather we work through the doing of mathematics to emerge these properties and behaviours.