I’ve been waging a campaign for several years now against what I consider an enemy of sense making in middle school math. I believe this enemy is a key player in the conceptual vs. procedural debate, and perhaps even the famous Math Wars. It’s a ubiquitous procedure that has its roots in an amazing and useful mathematical relationship. However, if introduced too soon and given too much power this amazing and useful tool sidetracks, and even halts students’ opportunities and inclination to reason.
I’m such a vocal opponent of this procedure that my colleagues speak of it in hushed tones. They tease me with it, and try to get my goat by suggesting it as a solution strategy whenever possible.
The name of my enemy? “Cross-multiplying” to solve missing value proportion problems.
I can appreciate that it’s an elegant thing that if , then . Even Abraham Lincoln touted cross-multiplying. He bragged that he could “read, write, and cipher to the rule of three.” Ciphering to the Rule of Three is another way of saying “solving missing value proportion problems.” If we know three components of the proportion, we can find the fourth through some good old ciphering.
Related to this is the idea of “the product of the means is equal to the product of the extremes” if we write equivalent ratios in the form a:b::c:d. Here, the means are b and c, and the extremes are a and d. So, the products bc and ad are equal. This is the same result we get if we cross multiply with (or with , which is really cool).
What’s my beef with teaching students to do this? Nothing, if it’s preceded by giving meaning to the proportional relationships involved. But, let’s be honest with ourselves. Cross-multiplying is far and away the default in most classrooms.
One of my favorite comments by a teacher came during a professional development session I was leading about proportional reasoning. We had just talked about the within and between relationships in a proportion. If you’re not familiar with “within” and “between” (sometimes called “across”) relationships, here they are in a nutshell:
The “within” relationship is about the multiplicative relationship in each ratio. In the proportion , we can notice that 4 and 12 have the same multiplicative relationship as 20 and 60. 12 is three times larger than 4 and 60 is three times larger than 20. Another way to think about this “within” multiplicative relationship is that 4 divided by 12 yields the same quotient as 20 divided by 60. Yet another way to think about “within” is that if we simplify and , we get the same fraction.
The “between” or “across” relationship is the same as in equivalent fractions. 4 times 5 is 20 and 12 times 5 is 60. In order to preserve the relationship, if one component is 5 times larger, the other has to be 5 times larger as well.
Let’s say students are asked to find the missing value in this proportion: , Would we want students to launch into cr0ss-multiplying, or to notice that there are nice, friendly whole number multipliers to deal with? My guess is the latter. If students pause and check out the numbers involved here, they can reason that since 12 is three times larger than 4, x has to be three times larger than 20. We can do that mentally, and have used the “within” relationship to do it. The “between” relationship is also pretty nice here, so now we have some flexibility.
I think a student’s thought process should be to first look at the within and/or between relationships to get an estimate. Then, use some technique, perhaps cross-multiplying to find the precise answer, if that’s what is called for. Finally, look back at the within and/or between relationships to gauge the reasonableness of the precise answer.
What do the Common Core State Standards say about proportions? They’re introduced in Grade 6 with both the within and between relationships described here, and basic percent concepts. Grade 7 hits proportions hard, incorporating scale drawings and plotting related quantities on graphs.
I’m disappointed that there isn’t more explicit language in the CCSS about how proportional relationships involve multiplication and division. “Ratio reasoning” and “proportional reasoning” are mentioned, but I don’t think there’s much explanation of what that entails. Even so, the developmental trajectory clearly leads in Grade 8 to the conception of slope as a constant “within” ratio connected to the idea of similar figures. I think CMP’s “Stretching and Shrinking” module in Grade 7 develops this quite clearly. I’ll write about this in more detail in a later post.
Back to the comment from the teacher.
I had posed the question about the meaning underlying the cross-multiplying procedure. We did some algebra to show that if , we can make both denominators the same by multiplying each side by a form of 1. On the left we could multiply by and on the right we can multiply by . This gives us . Because the denominators are the same value, the numerators have to be the same value, too.
Another way to think about this is the technique of “clearing” the denominators using the multiplication property of equality. We could multiply both sides of the equation by bd, resulting in ad on the left and bc on the right.
In both cases, we’re using some rules of arithmetic and algebra to maintain equivalence as we rewrite the expressions. These moves give the same result as cross-multiplying, but there’s some meaning behind them.
An “Aha!” moment occurred, and this teacher (one of my very favorites that I’ve ever worked with) said “I never knew there was anything to understand about that procedure.” I know in my soul that he’s not the only teacher who never knew there was anything to understand about cross multiplying. I came through that wilderness myself. But, it seems unwise, and for me unsatisfactory, to teach something about which you think there is nothing to understand (“Ours is not to reason why….”).
When the cross-multiplying procedure is introduced, students can be walked through the algebra moves or the connection to making equivalent fractions, but in the end, the residue is the one-step cross-multiplying procedure of going from to ad=bc. Along with the procedure comes the crossed fingers that the setup was correct.
Lucky for us and our students, there’s a better way! In next week’s post, I’ll present a double number line inspired way to make sense of percents. That will be followed by a discussion of similar figures, then slope, and then a wrap up of everything.
I’m not saying to never use or teach cross-multiplying. I’m simply proposing not making it the end-all-be-all-go-to procedure for solving missing value proportion problems, and certainly not to introduce proportions with the cross-multiplying procedure.
Letting cross-multiplying usurp all the focus is really not a good idea, and actually undermines the developmental sequence of proportional concepts. Yes, it’s quick and convenient, and students can be trained in the technique, but the trade-off is that proportional reasoning is weakened rather than strengthened.
This Week’s Tweak: If you’re working with proportional relationships of any kind, ask your students if the answer they get makes sense. If they say it does because they put everything in the right place and followed a procedure, ask if they notice or can use the within and between relationships to decide if the answer makes sense. You might also work on estimating based on these multiplicative relationships. The bottom line is to work with the multiplication relationships we can find “within” and “between” the ratios to gain some proportion sense.
I would love to hear about your experiences!