Expressions and Equations Ratios and Proportions

Cross-multiplying or Cross-cancelling?

SolvingProportionsStudentMistakesCrossCancelError1 CrossCancelError2


The submitter reports that this happened with several different students who went up to the board to solve proportions problems. This was the “Warm Up” exercise.

How would you react to these mistakes in class?

Thanks to Victoria for the submission!

10 replies on “Cross-multiplying or Cross-cancelling?”

What the heck does ‘cross cancel’ mean anyway?! I used to ban ‘cross multiply’ too, but have gotten lax lately about that. A student recently asked me to explain it, and I was happy I had convinced him to ask for reasons.

If several students have done this, the group is ripe for discussing (1) how we can check our work, and (2) how we know a step makes sense.

This is a very common error. I would first ask the students to substitute their answer into the equation to see if it makes sense. I would then use a simpler problem to demonstrate why you can’t do this…maybe 1/2 = 2/x. I think that when students see proportions, they forget that they’re looking at two equal ratios.

I think it’s also important to look at other methods for solving proportions: simplify one of the fractions, find a common denominator (or numerator), guess and check.

This illustrates why it’s not a good idea to use tricks. It’s a short term gain, and long term loss. They have no meaning and kids aren’t sure when to use them. That’s why they confuse things like adding and multiplying negatives, or the difference between rise/run and plotting coordinates.

My goal next year for students solving proportions is to require them to mark down values too low and too high by first writing down a range of possible answers before doing any calculations.
In the first picture where 14/15 = 21/x, I would first do a “notice/wonder” moment. Then, I would ask students for a value of “x” that would be too low and too high. Let them grapple with this for a bit. See what they come up with. Here are some possible observations students might come up with:
-The fraction on the left is less than one.
-The fraction on the right also needs to be less than one.
-A value for “x” that would be too low is 15, because the numerators are unequal, so the denominators can’t be equal.
-A value for “x” that would be too low is 21, because the fraction (on the right) would be equal to one.
-A value for “x” that would be too high is 42, because this would create a fraction equivalent to one-half.
-The fraction 14/15 is close to one whole and the difference between 14 and 15 is one part. Therefore, a possible value for “x” might be 22.
-A range of values for “x” might be between 22 and 30.

I agree with Nathan that students should substitute their answer into the equation to check their work. However, if we’re dealing with a student who already has no idea what they’re doing, chances of plugging in 2.5 will seem even more daunting to the student than the original question itself. If we require students to make an estimate or determine a range of possible answers based on number sense or dare I even say, gut intuition, we will avoid substituting in that 2.5 and immediately (hopefully) go back and rework the question. If we’ve already ruled out 2.5, there’s no need to plug it in.
I’m seeing proportions as free “number sense” gifts that I haven’t taken full advantage of in years past. I think it’s more important that students develop/utilize number sense skills within the flat context of an existing proportion. My ultimate goal would be for students to contextualize a given proportion. Write me a story. Draw me a picture.

Proportions are nothing more than equivalent fractions.

Sadly, proportion problems in American education have devolved into mostly dry calculations questions and contrived word problems, which skips over understanding and making the essential connection to fractions, and subsequently leads to blind (and incorrect) procedure, such as is illustrated. We like introducing the concept of proportions through similar triangles, but Common Core makes it impossible by putting proportions in Grade 7 and similar triangles in Grade 8.

This proportions question can be taught as early as Grade 5, since equivalent fractions are taught before that:

I agree that proportions have unfortunately taken on a role of “dry calculated questions.” However, it presents us teachers with the challenge to find applicable contexts for proportions and allow our students to model them in class. If they are dry, use them as little number sense gifts: find boundaries or ranges of solutions. Contextualize the question by drawing a picture or writing a story.

Re: “Common Core makes it impossible by putting proportions in Grade 7 and similar triangles in Grade 8”
I think your idea of introducing proportions via similar triangles is wonderful. Common Core has definitely moved some content standards around, but don’t feel like it has put constraints on you. Use the necessary tools to support students in their understanding of proportions and if similar triangles is a necessary tool, use it. Think how much those students will benefit from it, both in grade 7 and in grade 8, when they are a year older and have prior experience.

I find that some kids have enough trouble with the idea of similar triangles that it gets in the way of building an understanding of proportions. I prefer introducing proportions through problems like “15 slices of pizza cost $45. How much would 5 slices cost?” If they can see that as an interpretation of one of the examples here, 5/x = 15/45, then they’ll start finding some alternative ways of solving that proportion. For instance, I’d expect kids in the pizza question to see that 5 is 1/3 of 15, and thus that x is 1/3 of 45. Other kids will see that 45 is 3 times 15, and thus that x is 3 times 5. (Of course I wouldn’t use this as the first example, because having both of those being a factor of 3 could get confusing!)

How about measuring distance by counting steps? Students count how many steps it takes to go 30 ft, then predict the number of steps to go 60 ft, 90 ft, 120 ft, 15 ft, 45 ft… Then measure a long hallway by counting steps.

Oh, thank you for telling me that only multiplying and dividing fractions are open to cross canceling. Proportions are not and this this case are supposed to turn into trinomials or polynomials.

If you really want students not to make this mistake put down a problem like 8/21=6/16 and then put down the same problem again, but replace the = sign with the sign for multiplication or division. Solve all three and show your students the difference between all three different problems.

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