Certain multi-step equations, where \(x\) appears exactly once, as the innermost and leftmost operand, can be solved mentally. These are equations like

$$\begin{align} & [(x \cdot a) + b] / c = d \\ & [(x - a) / b] \cdot c = d \\ & \ldots \end{align}$$Notice that to solve such equations, we start with \(d,\) the output, then repeatedly whittle down until only \(x\) remains. So to solve the equation

$$[(x - a) \cdot b] + c = d$$we'd first subtract \(c\)

$$(x - a) \cdot b = d - c$$Then we'd divide by \(b\)

$$x - a = (d - c) / b$$Then we'd add \(a\)

$$x = (d - c) / b + a$$Thus, equations of the form

$$[(x - a) \cdot b] + c = d$$can be solved mentally exactly when \((d - c) / b + a\) can be evaluated mentally. Now you try. Derive a similar formula for equations of the form \((x \cdot a + b) / c = d.\)

Here's the solution:

To solve the equation

$$(x \cdot a + b) / c = d$$we'd first multiply by \(c\)

$$x \cdot a + b = d \cdot c$$Then we'd subtract \(b\)

$$x \cdot a = d \cdot c - b$$Then we'd divide by \(a\)

$$x = (d \cdot c - b) / a$$This strategy of repeatedly doing the inverse can be used to mentally solve both word problems and mathematical ones. Here's a word problem for you to try:

If I multiply my number by 3, then subtract 5, then multiply that by 4, I get 6 less than 202. What's my number?

Answer: [(202 - 6) / 4 + 5] / 3 = 18

For more problems like this, go here. Refresh that page for even more.

Next, give your students these challenges:

- Area and Perimeter by NRICH
- 2018 Math Kangaroo Levels 3-4 Problem #24 by STEM4all

Find the perimeter:

Here's the solution: Refer to the two figures below.

Notice that in the left figure, the lengths of the red segments sum to the length of the blue segment. Likewise, in the right figure, the lengths of the green segments sum to the length of the magenta segment. After making these two observations, finding the length of each segment is easy. Adding them all up gives us the perimeter, \(24.\)

Conclude by leading this investigation:

Square Dance (patterns)

by MathPickle