Teaching Order of Ops with the 4-Digit Problem

I teach a low-level Algebra class (Algebra Essentials). Half of the thirty students are Special Ed or English Language Learners. I was teaching Order of Operations, and rather than offer notes on a subject that they had seen several times before (and still failed), I taught the topic with the 4-Digit Problem instead. (created by College Prepetory Mathematic program, UC Davis)

The 4-Digit Problem goes something like this: You must use four of the same digit (I use 8) to form expressions that have designated values. You may use any of the standard operations represented in “PEMDAS;” you may combine eights to make 88; and you may use any number other than 8 only as an exponent (which then makes roots legal). I did offer some hints:
8 ÷ 8 = 1          80 = 1          √(8 + 8) = 4          3√8 = 2.

I had the students practice by first attempting to express the value 19 with four 8’s. I then had them independently attempt to express the values 1-5. Here is what this supposedly low-level group came up. (multiple solutions shown)

1 =  8 ÷ 8 · 8 ÷ 8     or     8 – 8 + 8 ÷ 8

2 = 880 + 880      or     8 ÷ 8 + 8 – 8

3 = 8 ÷ 8 + 80 + 80     or     80 + 8 ÷ 8 + 80

4 = 80 + 80 + 80 + 80     or     8 ÷ 8 · √(8 + 8)

5 = 3√8 + 3√8 + 3√8 – 80

Their class/homework that day was to then generate the values 6-10, and submit for a grade.

Great day for learning Algebra!

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