# Problem Design – Disguised Problems

One of my favorite formative assessment tools is a disguised problem. These problems come from two places. First is the tendency of math curriculum to ask certain questions the same way every time, such that students are confused when they see a very slight variation on the same question. Second is the value in probing beyond Depth of Knowledge 1, such that students cannot repeat the same procedure to obtain an answer and are pushed to think about the underlying concept. Disguised problems try to disrupt mindless procedure and force students to think flexibly about concepts. They are not meant to be complex, challenging questions–just to ask a student to apply a concept in a different way to make sure they understand it.

Some examples;

Too many Pythagorean Theorem questions are asked the same way. I was shocked by how many students struggled to find the diagonal from A to E on this rectangle — after showing fluency with straightforward right triangle questions.

Operations with scientific notation are easy to have students over-practice and disconnect from what the numbers actually mean. While any student learning scientific notation is likely to be able to answer this question without using scientific notation, the method they use is an illuminating piece of data on where their knowledge is.
What is 13 million times 300 thousand?

Another question that can be solved several different ways….one of which will save a ton of work. But this looks very different from “simplify this expression”, and I was surprised at how many students did their substitution first and slogged through the arithmetic.
Evaluate the expression below for x = 3 and y = 5

Trapezoids are a pat peeve of mine because all the questions look the same. Ask a student for the area of this figure, and see if they can a) identify that it’s a trapezoid, and b) identify the correct dimensions to use — or break it into a rectangle and triangle, which is also great! Bonus points if you can pick out why the image is geometrically impossible.

Volume is easy to oversimplify into a set of formulas. Throw a slight variation that asks for the volume in a different way, and see if they can still find it.
How many 1cm by 1cm by 1cm cubes are there in the block below?

Disguised problems are not the be-all-end-all of formative assessment, but asking students one or two at key points as they practice a concept or skill is an incredibly value tool in learning where their understanding is. Most importantly, in my view, it forces a student to pause and think for a moment about what a concept actually means, and allows them to apply their knowledge more flexibly in the future.