This is part 2 of my ongoing rewriting of a Year 7 Scheme of Work.

The first was 18 resources for fractions, here.

For addition and subtraction, I’m trying to emphasis the properties of the operations, and the ways in which you can manipulate the numbers whilst retaining the same values in a sum.

Much of this is *usually* taught in primary school, however many (many!) students found these resources enlightening.

I would stress that they need a lot of discussion and teacher-led questioning to be really effective. Handing out the sheets and writing your reports probably won’t have much impact!!

**1. The Properties of Addition**

Here we looked at the commutative and associative properties of addition. We looked at how reordering a sum can help make it easier.

This was particularly effective if, prior to introducing the techniques, you give students a left-to-right sum of several numbers, and ask a student to go through their process out loud. Most will instinctively go left to right, rather than picking out easy pairs.

e.g.

36 + 21 + 17 + 4 + 13

is far easier as

(36 + 4) + (17 + 13) + 21

We then looked at how by aiming for ‘safe numbers’ such as a round ten or hundred, addition can be broken down to become more manageable.I aimed this specifically at finding angles up to 180 or 360, and finding the nearest hour etc, as these are the areas I’ve found weaker students often struggle with.

The file is here:

and some nice sort cards on the same theme here:

**2. Properties of Subtraction**

Much as before, we looked at the properties of subtraction and how sums can be manipulated. It was great to see students spot that subtraction is *almost* commutative, especially if you keep the first number locked down, or at least attach a ‘plus’ in front of it wherever it goes (yes I know why that works before you comment!).

We then moved onto working with estimating – to help develop some kind of number sense as to what answers might look like. It was interesting to see that some students had *no idea* what the answer would look like, especially when doing the sum mentally without seeing it written.

Finally we tried some puzzle style tasks I created, which promote students getting a sense for how numbers can be manipulated. I assigned a points system so that students weren’t just incrementing one number by one, and subtracting one from the other (although I accepted this was a valid mathematical approach).

It was great to hear how all the students differed slightly in their approaches, but how all students used a method of sorts, rather than plucking random numbers out of the air. Using two variables (the first two puzzles) was no-where near as effective as using three.

**3. Regrouping and ****Understanding the column method**

Here the aim was to help students understand what is happening with the column method, and again reinforcing the ability to manipulate numbers without altering the values in the sum.

I like the table below, it allows for different styles of questions all within one template.

When practising the column method, I thought I’d use some more challenging questioning for the higher ability students (see below)

All the files below belong to this section:

Subtraction and Addition Column Method

**4. Other notable contributions**

I made all of the above resources myself, but I really liked and used some of Don Stewards excellent resources.

Specifically,

Cryptarithms for the more able:

Triangle Differences which went down a treat!

and this 1089 thing

The thing about subtraction is that for the natural numbers it is not a binary operation.

Reason: 6 – 2 = 4, but 2 – 6 doesn’t mean anything at all.

The binary operation for natural numbers is “the difference between”, found by subtracting the smaller from the larger. This IS commutative, as the difference between 4 and 6 is the same as the difference between 6 and 4.

The expressions 3 + 5 and 7 – 2 are arithmetical expressions (when first encountered, far too early in my opinion), and mean “add 5 to 3” and “subtract 2 from 7” respectively.

Expressions such as 3 + 4 + 7 and 10 – 4 – 2 + 3 are algebraic expessions, and if they can be evaluated then some rules are necessary. If in the early years we want to get kids to add up a bunch of numbers then words are much more appropriate than symbols: Add up the numbers 3,5,8,11,23. They have clearly realised before reaching this point that the order doesn’t matter (beans in a bowl stuff).

Once we start on expressions like 10 – 4 – 2 + 3 we find that we need negative numbers for completeness. This is definitely not the best argument for having negative numbers. far more satisfactory is to argue from reality, and the need for numbers to represent positions and levels, and the actions with these numbers to represent not only position but changes in position as well. height above sea level, voltage, temperature etc. prior to this numbers were for counting and for measuring amounts of stuff. So the new numbers are signed numbers, positive and negative. if “up” is the positive direction then changing by +3 means going up, and changing by -4 means going down. This we call adding, for signed numbers. Subtracting means doing the opposite to adding.

The most important thing about all this is that algebra deals almost entirely with signed numbers, and positive integers are NOT the same as natural numbers, although they do match up.

See A. N. Whitehead, “” An Introduction to Mathematics” (1911) the chapter on generalisations of numbers.

So an algebraic expression is a list of terms, each of which has a sign attached, which is not interpretable as how a term is to be combined with the previous term.

Sorry, I do go on about this !