Simultaneous Equations, refining the procedure

Going over Simultaneous Equations today with Year 11, we all agreed that the thing that is most confusing about solving equations like these are the negative numbers:

4 - 2x = 9 + 6y
6 - 2x = 7 - 2y

We also agreed that we much prefer these types of questions when we heave a sigh of relief realizing that we can add instead of subtract.

4 + 2x = 9 + 6y
6 - 2x = 7 - 2y

Adding is easier. We have one less choice to make and we don’t need to keep track of which is the minuend and which is the subtrahend.

This is subtle.  I would always start teaching simultaneous equations using some real-life examples (the ideas in this post).

I think real-life examples work in this case, because it is apparent what “extra” you are getting for the extra money.  These problems create the need for the algebra – it becomes a way to represent what is going on by writing less.  With these problems, everything is positive so it makes sense to think of the difference between the two situations or the two equations.

From there we can introduce the need to “multiply up” to get the same co-efficient for one of the terms.   In my experience, most students don’t struggle too much with this concept.

What gets tricky is subtracting one equation from the other when negative signs are involved.  We might identify the problem as being one of a lack of mastery of a fundamental concept – in this case negative numbers.  So before teaching these types of simultaneous equations we could do lots of practice and drills on negative numbers.  But I’m not sure that is always a helpful approach.  Some things are just more confusing (present a higher degree of cognitive load if you like).   There is a lot going on, and lots of it needs to be done mentally. So, maybe there is a case here for explicitly teaching an easier technique which is less prone to error.

A simple way to avoid subtraction is by always ending up with equations with opposite coefficients of one of the variables.

So, for example when solving the following:

4x - 3y + 1 = 0  (A)
3x - 7y + 15 = 0  (B)

multiply (A) by 3 and (B) by -4. A valid shortcut / rule to remember here is that multiplying by a negative simply “flips the sign” of any term it is being applied to. So we end up with:

12x - 9y + 1 = 0
-12x + 28y - 60 = 0

It’s easy to check that the sign of each term has been flipped before then proceeding to add the equations.

As any Maths teacher would, I believe strongly in teaching for deep understanding of concepts, not blind memorisation of rules (Nix the Tricks is a frequent reference point). But there is also a place for remembering certain procedures to reduce cognitive load (times tables being the most obvious example). Dani Quinn has written a great post on this in relation to “moving the decimal point” here.   Manipulating Simultaneous Equations like these is another example of when there is a case for a bit of explicit teaching of a method.

 

 

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Negative number grids

It might be good to introduce (or re-introduce) the idea of the multiplication grid first:

before adding some negative numbers into the mix:

Here are some more to practice:

Some more ideas for this topic on Resourceaholic here.

 

Always, Sometimes, Never True for Directed number

These only cover adding and subtracting negative number, i.e. they can be used before going onto multiplying and dividing negative number.

They can be printed (here is the pdf), cut out and then stuck on A3 paper under the three headings with examples and counter-examples to explain why they have been put under that heading.

I should point out that two statements on here are deliberately vague, i.e. “two negatives make a positive” and “a positive and negative make a negative”.  This is often how students remember them and this can lead to problems down the line (e.g. the misconception that -3-2=5).  My idea with these is that the end up in the “Sometimes” column but ultimately we dismiss them as not being very useful.

Simple sequences starters

Here is a simple task which would work well as a starter which practises negative numbers at the same time as (hopefully) leading into a nice discussion about sequences and term-to-term rules.

I created this on a spreadsheet here, so you can change the questions or the order.  If you are feeling ambitious you could do this with decimals or fractions, or also include multiplication, which would generate quadratic sequences.

And here is another one which basically just seeks to reinforce the idea of what “n” means by substituting it into nth term expressions.

 

 

Vertical Number Lines in Books

I made this a while ago for supporting Year 7 students on directed number (i.e. positive and negative numbers).  I think there is something more intuitive about a vertical number line – if you are adding you go up, subtracting you go down.  Having said this, I have always had a horizontal number line across the top of my board!

If the number line is stuck on the inside back cover of the exercise book, it is always there whatever page the child is working on. It can then be folded safely away whether using large or small format books.

Doing some work in primary this week, I realised that the same idea could be useful for supporting younger children learning the essential skills of counting back and counting on when doing addition and subtraction of positive numbers.  So I have made another version just with positive numbers.

The end points of these lines are arbitrary of course.  I have deliberately gone for something a bit random to start a discussion, “Sir, why does it stop at 44?”.  But if that offends your preference for order in life, then feel free to adjust it on the spreadsheet that I used to create the pdfs in the first place.

Vertical number line to stick in books.xls

Vertical number line to stick in books zero to 44.pdf

Vertical number line to stick in books minus22 to plus22.pdf