Logarithms before calculators

I set what I thought was a meaningful task for my grade 10 math students in which they were to perform a complex arithmetic calculation completely without the aid of a calculator.

The task

Students were required to invent a very complex arithmetic problem involving roots, powers, decimals and products.  They were required to take the log of the calculation and then apply the laws of logs to simplify the calculations required.  Finally they needed to find the antilog of the result to obtain the answer to their problem.   Only after completing the calculations were they able to use a calculator to verify their result and determine the accuracy of the manual calculation.

As the students worked through the calculation, they were required to explain their steps, so that they clearly understood how to apply the laws of logs.

After finding the antilog of their result to determine the final answer, they were asked to reflect on the task.  They had to respond to questions asking what they gained from this process and what thoughts did they have as they worked through the task.

The Students’ work

(a)  The calculation

After marking a few students’ work so far, I feel satisfied with the mechanical use of the laws of logs, and it certainly helped cement the laws of logs and an application of those laws.  All students’ answers have been accurate to at least three significant figures, which was rewarding for student and teacher alike.

(b)  The reflections

Some students have commented that it was easier than they had expected it to be and that it had helped their understanding of the laws of logs.

However,  by far the most common reflective comment was along the lines of  “Thank goodness for calculators!”

What I had wanted

I was hoping for comments which appreciated the ingenuity of this process in dealing with complicated (practically impossible) arithmetic before the advent of calculators, as well as a few inquisitive thoughts about who made the log tables and how were they constructed.  One student did comment that someone must have taken a long time to make the table, but had not given any thought to who had made the table, when and how the person did it.  (Although it would be great to have some consideration of the method of constructing the table, the series’ involved are well beyond the scope of grade 10 extended mathematics.)

Thoughts for next time

I had stated that this assignment would be marked against MYP objective C, Communication, but now realise it would have been very suitable for D, Reflection, as well.  Including objective D would have addressed some of the issues mentioned as well, since the assessment levels in criterion D are looking for comments regarding whether the answer makes sense, justifying the degree of accuracy, and commenting on the significance of the results and the importance of this use of logarithms in the “real world”.  (Yes, MYP likes the term “real world”.  I hate it.  It implies that other things we do are not part of the “real world”.)

So next year I will add the reflection component to the task and hope it goes better.

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About eadurkin

Originally a HS Mathematics teacher from New Zealand, currently working as Associate Principal in the Secondary School at Canadian Academy, an international school in Kobe, Japan. Married with two children.
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