Almost two decades across, I helped my school system create and adopt a class in discrete math, (it was becoming trend then) and then took the two sections a year at my school giving up two advanced algebra classes. One of the blocks was about arithmetic in different number bases. My focus was on making as much as possible about applications, so I wrote a unit on "five-money", pennies, nickels, quarters, and Super Dollars (equal to the euro at 1.25 american). Since our school was in England, they were well familiar with the exchange.

I started putting some of their problems on the board to see how "bright" math students in Math Analysis or Stats or Calc would handle them. These are simple versions of some of them:

If a 5 ft 9 in board is cut from a 9 ft 6 in board, what length is the remaining board?

How long is it between 7:40 this morning and 3:25 this afternoon?

A recipe for bread calls for one pound and three ounces of flour for the extra large loaves. The cafeteria needs 15 of these loaves for lunch. How much flour will they need?

What is the area of a rectangular room that is 9 ft 6 in wide and 8 ft 9 in across?

A worker worked 5 hrs 43 min on Monday, 7 hrs 35 min on Tuesday, and 3 hr 20 min on Wednesday. How many hours have they worked so far?

A 16 ft board is to be cut into boards of 2 ft 9 in length. How many boards can be made, and what length piece remains.

The first problem produced lots of 96 -59 = 37 answers, some then converted to 3 ft 1 or 3 ft 7 in. Many students converted everything to inches and subtracted, then converted back to ft and in. None Wrote the problem in the three line subtraction model they used for base ten and simply did it in base 12. Zero out of 34 students.

What surprised me is one of the most common solutions for the time problem was to draw a clock and mark around the perimeter, 20 min + 4 hr + 3 hr + 25 min. Others did the same with what I call number roads, 7:40 >>> (20 min) 8:00 >>>>(4 hrs) noon >>>> (3 hr) 3pm >>> (25 min) 3:25 and get the answer. Nobody on the board problem saw the same solution.

The bread problem had lots of 15 # 45 oz answers, and many adjusted by converting the 45 oz to 2# 13 oz and finishing, while a few got to 15# 45oz and then finished with 19# 5 oz.

The area problem through algebra students into a tizzy because they were sure it would be a problem involving the distributive law, but didn't know what to do with 6x8 and 9x9. As often happens with young students there was little use of units. It was a rare result that included square ft or square inches. Statistics students were more likely than others, perhaps because of my tantrums over unlabeled axes in graphs. I wanted to hug the ones that got something like 72 sq ft, 129 ft inches, 54 sq inches. Many resorted to converting to inches, multiplying and dividing by something (144 and 100 were both common,24 appeared a few times as well. Several just left it as a decimal number of their division, and of those who tried to label it they were split between inches and square inches.

For the last problem I imagined some bright child writing the problem like

2' 9' ) 16 ' 0"

or perhaps 16 / (2 3/4) or even 16 / (2 9/12)

alas, it did not happen. Most correct solutions went back to a sketch of some kind, cutting the length and adding board lengths one at a time. Others converted to inches, but some did the calculator and had no idea of what the .8181... decimal fraction indicated (other than they hadn't thought through how they divided any other problem.

My disappointment made me squeeze time in to talk about counting on to subtract, number roads for all kinds of problems, arithmetic in any base including division of ft/in into square ft/sq inches (and the magical inch-foot that makes them like quadratics. We talked about lot of possible ways to try problems with units, and the beauty and benefit of labels.

And all of that was written to ask a single question. Do any of these kinds of problems appear in the curriculum now??? and should they? If you have teacher friends in middle school or hs, share this with them and ask them to share what they know.

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