Visualizing Ratios and Proportions: Stretching George!

Robert’s lovely 5th grade is about to embark on their annual study of fractions; I know many of you believe that teaching fractions is “forty miles of bad road,” and mark off the days until you finish going over the various “rules” that govern the different operations associated with them.

Ever the contrarian, Robert loves teaching fractions: he sees them as an opportunity to expand his students’ understanding of lots of mathematical topics, including ratios and proportions, probability, divisibility techniques and even geometry. Instead of seeing fractions as a crisis, consider it an opportunity!

No, 16/64 does not equal 1/4 because you crossed out the 6s. This is what's called a "coincidence."

This “technique” for simplifying fractions shows up on my Pinterest boards with alarming regularity.

Robert’s study of fractions also gives him a chance to clear up misconceptions that students might still have about how ratios and proportions work. One of the most prevalent misconceptions is that equivalent fractions can be created by adding or subtracting the same amount from the numerators and denominators. Thus we have students who believe that 3/8 can be turned into 4/9 by adding 1 to both numbers. Of course this is horse hockey, but because Robert is a constructivist (as opposing to being an instructivist), he prefers to create “compelling contexts” that would prove this wrong.

One method to impart the idea that proportional relationships are multiplicative (as opposed to additive) is to teach students by counter-examples. That is, use an example that shows that adding the same amount to the two parts of a ratio will not result in a proportion, and give students to the opportunity to prove the relationship false on their own.

Now some teachers may say, “but this is going to take a long time; wouldn’t it be easier if I just told them?” Yes, it would be “easier,” but you didn’t go into this profession because it was easy, you went into this profession because you want to teach! And by having your students figure it out on their own, you are actually teaching them to think. What’s the danger of that?

You can "non proportionally" change the size of an image on your copy machine.

Do these instructions look confusing? That’s because it’s not for a photocopier, but threading the bobbin on a sewing machine!

Robert likes to use visual models to help students understand this issue and you can do it very easily using a basic drawing program or a photocopy machine. In this case, Robert started with the 5th grade class photo and then put it onto the fancy copy machine with which he does battle on a regular basis. One of the features he has discovered is the panel that allows him to create an enlargement by setting the target size of the copy in inches independently, as opposed to percents proportionally.

Starting with the 5″ x 7″ photo, Robert changed the target sizes by adding 1″ to both the length and width of the photo: his new photos were 6″ x 8″, 7″ x 9″, 8″ x 10″, all the way up to 11″ x 13″ (the photocopier at his school goes up to tabloid size of 11″ x 13″; if yours doesn’t, then go to a copy show, where they can do it for you.) What do you think would happen if you enlarged a photo using this method?

Your copy machine can make a lesson on ratios and proportions come alive!

Poor George Washington gets the mumps treatment when his face is enlarged using the addition feature on the copy machine.

Robert distributed the photos to the students in pairs of 2; they pulled out the rulers and measured the photos to the nearest inch (he makes sure they are done in increments of 1″) and then records them in an organized table of values so that students can see the enlargement pattern. They notice that the width of the photo is always 2″ more than the height, and that enlarging the photo by adding the same amount to the width and height resulted in distortion, particularly as the photo got larger and larger.

His students then “see” for themselves the importance of proportions when enlarging a photo: if you double one side of the photo, you’ve got to do the same to the other side to keep it looking right: if not, the result is a photo that looks “stretched out.”

Robert completes this lesson by reviewing how a ratio is set up and how two ratios can be evaluated to see if they create a proportion by using a “scale factor.” He then gives his students 16 different frames to match up while calculating the scale factor on each one.

Now here’s a question for you: if you’ve ever shopped for photo frames, how come they’re never proportional? A 4″ x 5″ print can’t be enlarged to a 5″ x 7″ print, but it can be made into an 8″ x 10″ print, but that can’t be enlarged to fit an 11″ x 17″ frame, which can’t be enlarged to a 13″ x 17″ frame. What’s the deal with any of this?

This is more than silly mathematics, by the way: our entire technical world is made more annoying by the fact that few of our devices use screens that are proportional to one another: the videos we shoot on our smartphones are not proportional to our tv screens, the widescreen movies we see in theaters are not proportional to our tablets or our televisions. The result is those annoying “letterboxes” that have to be inserted on the top and bottom of movies, or those pervasive and confusing messages that tell us that the original movie had to be modified (read: cropped or stretched) to fit our screens.

If you’re interested in using this activity in your classroom, check out this activity that I’ve uploaded just the other day:

How did George Washington end up with looking like he had a wisdom tooth removed? The answer is: proportions!


Math is Thematic When You Say “Invariance”

Robert always seems to be obsessing about some aspect of his work, and the last few weeks he has been an absolute veytik in der tokhes because he’s discovered something called “invariance.” He points it out to his students, quizzes his teachers on it, and has taken to philosophizing about it at the dinner table. Even on New Years Eve he couldn’t leave it alone: “You know,” he opined, “I don’t know why we purchased these cruddy plastic champagne ‘glasses’, because the taste of champagne is invariant with respect to the type of container from which it is sipped.” Well, pardon moi! 

But since you’re still reading this, you might as well stick around, because this actually is relevant for your teaching practice, and at worst, it would probably make you seem a lot smarter when talking to your supervisors, colleagues or the people you meet at parties who make snarky remarks about “only being a teacher.” If there’s one thing Robert loves is asking a hot shot attorney, “yeah, well, if you’re so smart, tell me, how would you teach invariance to a 5 year old?” and then watch as he takes a gulp of beer and 23 skidoos towards the corn chips.

Okay, here’s the 411 on “invariance” and why you’re going to see mathematics in a whole new light once you “get it.” Eventually, Robert is going to complete his master opus on this topic and then you can buy it at his store, but for now we’ll just dip our toes in the water and leave you with a few questions. But I’m not going to give you a “textbook definition,” because a) that’s boring, b) that’s not the best way to define a concept. Instead, I’ll give you a few examples, and you can work it out for yourself:

"The number of objects in a group is invariant with respect to the order in which it is counted."

“The number of objects in a group is invariant with respect to the order in which they are counted.”

Now you all know that, because our man Piaget described that as one of the earliest forms of mathematical knowledge children exhibit, and which you’ve no doubt observed time and time again. Invariance shows up over and over again in mathematics education. Here’s another example:

"The sum of two groups of ocelots is invariant with respect to the how the ocelots are distributed between the two groups."

“The sum of two groups of ocelots is invariant with respect to the how the ocelots are distributed between the two groups.”

Are you getting the idea? Basically, the concept of invariance shows up whenever we think to ourselves “it makes no difference if….” Robert loves to use this with his first grades when he hands out materials that comes in different colors. “Okay, these geometric shapes come in red, blue and green, but the underlying function of these materials is invariant with respect to their color,” he’ll say, followed by, “can someone just translate what I just said?” The first time he tried this out, a little girl raised her hand and said, “What that means is that you get what you get and you don’t get upset.”

Less we think that this is just something that shows up in the early childhood years, let’s look at an example from the upper elementary grades:

The area of a parallelogram is invariant with respect to the length of the base and height.

The area of a parallelogram is invariant with respect to the measurement of the base and height.

The above insight came to Robert while observing a student work on one of these cruddy worksheets that a teacher had unfortunately downloaded from the interweb. He asked the student whether it was necessary to re-calculate the area of several triangles which had identical bases and heights, to which she shrugged her shoulders and replied, “I guess so; they’re all different shapes….”

Want to see where this is going in the upper grades? Sure thing: according to Robert, show him a student who is having trouble with solving algebraic equations, and he’ll show you a student who is struggling with the concept of invariance:

The equivalence of an algebraic equation is invariant with respect to whether the two sides have the same amount added or subtracted, or are scaled by the same factor.

The equivalence of an algebraic equation is invariant with respect to whether the two sides have the same amount added or subtracted, or are scaled by the same factor.

If you’ve been following Robert for any amount of time, you’ll know that helping teachers find a “theme” that carries over from year to year is an important aspect of his work (as well as giving young children brain-busting unsolvable problems….) But if you understand even a little bit about these underlying themes, it will make you a better teacher.

Your Assignment: What mathematics topic are you teaching right now to your students? Do you see the concept of invariance showing up in that topic? How would it help your students better understand the mathematics that they’re learning?

This post has been brought to you by SamizdatMath. Yes, we want to sell you stuff, but our stuff doesn’t keep your students’ busy or look cutesy. Our stuff actually makes your kids “think!”

Math Concepts: Why Not Try Teaching Them Sometime? Resources to make teachers smarter and more effective.

Textbooks don’t teach concepts, but if you read this and implement all the techniques, you sure will!