Right now, algebra students are studying unit rates, proportions and dimensional analysis. Geometry students are working on similar figures and triangle proportionality theorems.
All of these topics are hard for the same reason: They involve ratios, which may seem easy for adults but are actually deeply challenging for the learning brain to grasp.
A ratio is the comparison of two numbers, usually using a fraction bar. If there are two dogs and three cats in a room, I could write that the ratio of dogs to cats is 2/3.
Like so many things (reading, driving), ratios become second nature with enough practice, and people lose touch with how difficult they were to learn. And like reading and driving, ratios are hard for the brain because they involve simultaneity of thought. The brain is required to multi-task; it must think about the 2 dogs while at the same time thinking about the 3 cats.
And some kinds of ratios are way harder for the brain than others. What’s the ratio of dogs to animals? 2/5.
Very young children (ages 4 and under) can’t form these kinds of ratios at all, because they haven’t yet developed the mental capacity to think about the same dogs as comprising all of the numerator while simultaneously comprising part of the denominator (2 dogs + 3 cats = 5 animals).
If you ask a preschool child, What is the ratio of dogs to animals?, she will “hear” What is the ratio of dogs to cats? and answer 2/3, because her brain can’t process the actual question!
This part-whole simultaneity challenge never goes away entirely, which is why you can find those dog-to-animals sorts of ratio questions on standardized tests. (In a math class, there were 4 girls to every 5 boys. If there were 12 girls, how many students were in the class?)
I expect my students to need extra practice on topics such as dimensional analysis, which require using multiple ratios (A car travels 60 miles per hour; how many feet per second is that?) and similar figures problems (especially when the figures in the diagram are overlapping and, for example, the hypotenuse of one triangle is also the leg of a larger triangle).
Students will tend to be confused, to need extra explanation, and to “forget” and need to be told again, often several times. They will also need practice over time to help those neural connections form and grow stronger.
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And, here are some good refresher videos on various ratio topics:
What is a ratio?
Photo by jimmiehomeschoolmom