Interrogating Integers

I’m approaching the end of another session with another S1 class (the first year of secondary school in Scotland for pupils typically aged 11 or 12 on arrival after seven years of primary schooling). As the years go by I find I’m enjoying more and more working with these younger pupils – this particular group has featured in a previous blog-post and in a number of tweets. Increasingly, my lesson-planning is becoming more like a dialogue: a transactional process whereby not only am I shaping pupil learning, but their responses during lessons are shaping and directing my next steps and ideas as a teacher. This symbiotic relationship has recently given rise to some activities I feel are worth sharing here.

The first is not an original idea, but it has taken on a new significance for me through the way it has helped many of my pupils grasp the multiplication and division operations with integers. Moreover, one of my pupils had a particular insight that proved very valuable in moving understanding forward for the whole class, as well as extending my own conceptual schema.

Having done some groundwork on adding and multiplying positive and negative numbers using a variety of representational forms, I was ready to tackle division. As I’m sure many other teachers do, I like to stress the inverse relationship between these operations. I started off as I usually do:

If 4 x 5 = 20, what else can we conclude?

It didn’t take long to establish that

5 x 4 = 20

(“by the commutative law for multiplication”, they were keen to tell me), and also that

20 ÷ 5 = 4

and

20  ÷ 4 = 5

(both, of course, non-commutative).

My planned next step was to go on to replace the 4 with -4, then the 5 with -5 in order to demonstrate and promote discussion around division involving negative values. It occurred to me in the moment to use a triangle, which I hastily drew on the board, to emphasise the relationships involved. Here’s an extract from a pupil’s jotter to illustrate.

A multiplication and division triangle.

Towards the end of the lesson, the same pupil made a conjecture that surprised and intrigued me. I asked her to write it down in her jotter so we could investigate it during the next lesson. (Her original statement sounded to me like an assertion, although she has written it in the form of a question.)

M conjectured that the presence of one negative in any multiplication and division triangle would necessitate another.

In readiness for that selfsame investigation I prepared these four triangles to help us generalise and to consider M’s conjecture.

These helped us to see and to talk about the results of multiplication and division with integers: was M’s conjecture correct?

Not only did we agree that the conjecture was correct, but we found using it in conjunction with the triangular representations really helped us determine the sign of the answer to a given multiplication or division calculation.

Another pair of related activities was borne out of my frustration at the perennial misconceptions around addition and subtraction that cause pupils to provide answers like

-5 + (-4) = 9

Surely a moment’s thought will tell you that adding minus four to a number cannot increase the first addend’s value by fourteen!

Some readers may be aware that I like to use the date as a source of numbers for my lessons. Since the 13th of June was approaching, I came up with this activity (which I later shared On Twitter) to try and address this irksome situation.

A context for pupils to attempt to create true statements about adding and subtracting integers.

In the time available, my pupils came up with a number of correct solutions, including one I particularly liked:

13 + 6 + 19 + (-19) = 19

(although 19 appears twice; is this allowed?)

The rest of the responses involved combining just two numbers to make a third number, e.g.

19 + (-6) = 13

-19 – (-6) = -13

These were what I had anticipated: examples that tell me whether or not a learner can successfully add and subtract across the full range of combinations of pairs of integers.

On my bus journey home I began to wonder how many correct addition and subtraction statements could be constructed from such a set of six integers and if there was a systematic way of finding them. The next date that would provide me with a ready-made trio with which to work was June 25th. I devised this addition and subtraction grid and accompanying question to allow us to revisit what we had learned after some forgetting over time had occurred.

An addition and subtraction grid that also provides a restricted solution set.

To illustrate how it works, here is the outcome of the discussion we had in class.

Consider the first number in the left-hand column, 6. Working across the top row and either adding or subtracting each number to or from the 6, we are looking for answers of either 6, -6, 19, -19, 25 or -25. This led us to agreeing that the 6 and -6 on the top line were of no use in constructing true statements. However, with the remaining four numbers we were able to write the following:

6 + 19 = 25

6 – (-19) = 25

6 – 25 = -19

6 + (-25) = -19.

It was also instructive to note that

6 -19, 6 + (-19), 6 + 25 and 6 – (-25) were not permissible as their answers were not members of our solution set.

The reader is invited, ideally with a suitable class, to work through the remaining horizontal lines in the table, adding and subtracting each of the numbers along the top and writing down any true statements that use only members of the same set of three positive numbers and their negative counterparts. There is much scope for discussion, elaboration and, perhaps, further opportunities to wonder and conjecture (for example – and to answer my own question – how many true statements will be found by the end of the process?)

Enjoy!

Thanks for allowing me to share my thoughts with you on these aspects of learning, teaching and doing mathematics. I hope your visit to my blog proves worthwhile.

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