Singapore Mental Math (5) – Subtraction: reverse 3-digit numbers

This one comes from the Grade 6 book (I assume this is the equivalent to Year 7 in England).

Week 4 has this:

Good things:  The pupils will be practising following an algorithm.  There is some mental maths involved.

Bad things:  The idea of having an algorithm (that you are expected to remember) to subtract a three-digit number from its reverse is a bit extraordinary!  There are potential problems with the way it is set out.  The opportunity to explore this, to explain why, to use some algebra, to create a better formula and to expand it to other numbers of digits has been missed.

The potential problem:  “Find the difference of the hundreds digits”.  This would mean that 598 – 895 has the same answer as 895 – 598. 

In the classroom:  We could try a few of these out to convince ourselves that this always works.  The pupils might wonder why the middle number isn’t mentioned in the algorithm.  Hold on: if we reverse a 3-digit number then the middle number is in the middle both times, so when you subtract them this gives zero.

Let’s try some algebra.  It might be natural for pupils to want to show this as ABC – CBA (particularly if they have previously worked on letter-substitution puzzles).  It would be good to explore why this doesn’t work with algebra.  (ABC means AxBxC.)

If we use their example of 895 and want to have a = 8, b=9 and c=5 then we actually need 100a + 10b + c. 

The reverse of this will be 100c + 10b + a.  When you subtract you get 100a + 10b + c – 100c – 10b – a, which simplifies to give 99a – 99c.  Factorising this gives 99(a – c).

This explains why we subtract the hundreds digits, and then why we multiply by 100 and subtract that difference again (leaving 99 of them).

Extensions:

We could try 2-digit reverses:

10a + b – 10b – a = 9(a – b), which means that for 2-digit reverses you subtract the digits and then multiply by 9 (which also means that the answer is always a multiple of 9).

We could then do a similar thing for 4-digit reverses…

Singapore Mental Math (4) – adding a series of odd numbers

Do read the previous few blog entries (here) to see the background to this.

This is the strategy for Week 1 in Grade 7 (which I assume is equivalent to Year 8 in England).

Good: Pupils will be following an algorithm and are using vocabulary, while also doing some mental maths.

Bad: Pupils might assume that in general all ‘series’ must start with 1.  It wasn’t made clear that for this algorithm to work it must start at 1.

Missed opportunity:

Why not mention that the answer is a square number?  (Step 3 could easily be: “Square the quotient obtained in Step 2”)

Why not explore why this is a square number?  There are so many ways to do this.  Here are a few I can think of immediately.]

 If there are an odd number of terms we can use the method from my previous post.  With 1 + 3 + 5 + 7 + 9 the middle number is 5 and there are 5 numbers.  The sum is therefore 5 squared.

The same is true for any odd number.

For an even number of terms we can still use the method from the previous post.  With 1 + 3 + 5 + 7 + 9 + 11 there are 6 terms and the middle is (1+11)/2= 6.  The sum is 6 squared.

Ah: so the algorithm is really adding the first and last terms, dividing by 2 and then squaring it.

We could also draw a picture:

Each colour shows a successive odd number, and after each new colour has been added it is still a square.

We can also explore further why this pattern must increase by an extra 2 each time.  If we just add on the same amount as last time, this is what happens:

To fill those gaps we need to add on an extra 2.

Singapore Mental Math (3) – adding a series of numbers in a pattern

See previous posts (here and here) if you are new to this.  Here is the Week 4 strategy.

Good thing: lots of practise of following an algorithm, and of multiplying by 5, 7 or 9 mentally.

Bad thing: while some questions have 5 numbers (‘addends’!), some 7 and others 9, all of the series go up by 2 each time (the method actually works for any linear sequence).

This is a missed opportunity if only used to answer these questions in the way stated.

Why does this method work?  This is a nice example of where algebra isn’t enough by itself: this is so much easier to see and to understand if you choose you algebra in a particular way.  Make the middle number in the series be n, and this is now straightforward.  (To be clear, it still works if you make the first number n, but this is more difficult to generalise from.)

With a 5-number series that increases by 2 we have:  (n-4) + (n-2) + n + (n+2) + (n+4)

When we sum this we get 5n, which is 5 times the middle number (as per the algorithm).  But we can see why this works: there is some symmetry in that the number to the left of n is 2 below n while the number to the right of n is 2 above n, so these cancel each other out.  If you have additional numbers then they will also cancel each other out, leaving only extra ns.

Can we do this if the series increases by 3?  Yes – that gives us: (n-6) + (n-3) + n + (n+3) + (n+6)

What if the series increases by a?  (n-3a) + (n-2a) + (n-a) + n + (n+a) + (n+2a) + (n+3a)

What if there is an even number of terms?
This still works (although the 'middle' is no longer an integer.

We can also use a visual method for this:

The first diagram shows 6+8+10+12+14.  In the second diagram that has been converted to 5 lots of the middle number.

This diagram also shows neatly that we are working out the mean of the numbers in the sequence (the middle number is the mean) and then multiplying by the number of terms. 

As a final different method, it would be nice to show this on a 100-square to illuminate further what is going on.

Singapore mental math (2) - special numbers

More from this book (see yesterday’s post for background information).

Here is the strategy from Week 33:

Good things: If used like this then there is some mental maths going on and the following of an algorithm.  It could keep mental maths sharp and settle pupils at the start of a lesson.  Vocabulary is used (product, factor, multiple).

Bad things: If used just like this then it seems to me to be very unlikely that anyone will remember what the special number is, or what fraction to use.  It doesn’t make it clear that this is only for 2-digit numbers.

Pupils could be asked to explore why this works (to be fair, perhaps this happens even though it isn’t mentioned in the book).

What is going on?

Multiplying by 429 somehow involves multiplying by 3/7 (and then doing some other jiggery-pokery).  What is the link between 429 and 3/7?

Let’s do 429 divided by 3/7 – we get 1001.  Aha, if we write down a 2-digit number, put a zero in the hundreds place and then write the 2-digit number down for the thousands then that is the same as multiplying by 1001.  This explains the final few steps.

Questions for pupils to consider:

• One of the questions is 7x429.  How does the algorithm need to be tweaked to make this one work?
• What happens if you do this with a 3-digit number?  Or a 4-digit number?
• What would be a better way to explain the algorithm?  (Maybe: “multiply by 3/7 and then multiply by 1001”?)
• What happens if you choose a number that isn’t a multiple of 7? 

The strategy for Week 34 is almost identical, except that the ‘special number’ is 715 and the initial instruction is to multiply by 5/7.

Week 36 has special number 858 and ‘multiply by 6/7’.

Both of these are fractions of 1001 too. 

What other ‘special numbers’ could there be?  Can pupils write an algorithm for those? 

(For example, 1/7 of 1001 is 143…)

Singapore Mental Math

I am exploring the Bar Model at the moment and have bought some books to help me get to grips with it.  Most of these books are very interesting and contain illuminating ideas.  When I have digested them I will post a list of the ones I found most helpful.

While browsing through books on Amazon I also bought a Singapore Math ‘Mental Math’ book. This is very different from the bar model materials.

The book describes itself as containing ‘challenging activities based on the world-renowned Singapore Math curriculum’.  It says it includes ‘must-know strategies for solving problems quickly and accurately’.  It states: ‘To help students build and strengthen their mental calculation skills, this book provides strategies that will benefit students as they learn tips to solve math problems quickly and effectively.  After acquiring such invaluable skills, students can apply them to their future, real-life experiences with math, such as in shopping and banking.’

There are 52 strategies in the book (one for each week of the year) and many of them are quite extraordinary. In the rest of this post I will explain one of them and will then discuss what I see as the pros and cons of this approach.  Future blog posts will include other strategies from the book.

Week 24 has this strategy:

The good things about this:

• Pupils will practise following instructions and using an algorithm.
• It might make a good ‘settler’ task at the start of a lesson.
• Some mental maths muscles will be flexed (with lots of adding and doubling going on).

The bad things about this:

• This strategy has very limited application.
• Pupils are being encouraged to learn lots of strategies of questionable usefulness.
• There could be confusion and it could be applied in the wrong setting.
• There is a much easier way to do this!
• There is nothing to explain why this works or to lead pupils to explore it.

Perhaps I am being unfair, and maybe pupils will want to know how and why it works.  Maybe teachers using this resource _do_ explore why this works.  If so then I think this could be a wonderfully positive resource.  For example:

Let’s use some algebra.  Call the two numbers n+1 and n.

We have (n+1)² – n².  We could expand the brackets, to give n² + 2n + 1 – n²

Alternatively we could use the difference of two squares:

This gives us (n+1)² – n² = (n+1 + n)(n+1 - n), which simplifies to (2n+1).

Going back to the original:

The original has (n+ ½ ) multiplied by 2, which is 2n+1.  

The easier way:

But 2n+1 is also (n+1) + n, so this is actually the two numbers added together!  11+10 = 21

An alternative method could involve drawing some diagrams:

This shows that the non-yellow part contains (n+1) + n squares, and properly demonstrates that we are adding the two numbers together.

I prefer the difference of two squares method because it seems to me to be the most easily generalizable:

If the numbers are two apart then we have (n+2)² – n², which is (2n+2)x2

If the numbers are three apart then we have (n+3)² – n², which is (2n+3)x3.

Every time (and using (n+a)² – n²) makes this obvious), we need to add the two numbers and then multiply by their difference. 

I can see some really fertile material here for KS3 and KS4 pupils to explore, so as a starting point this turns out to be an exciting resource.  It also leads back to the important idea that there is no such thing as a ‘good resource’ or a ‘bad resource’, but that it depends on how you use them.

Constructions: it's all about the rhombus

Constructions seems to be an ideal topic in which links can be made with other parts of geometry.  It also seems a good opportunity to teach for understanding (in a Skemp-type way) as well as leaving pupils with a method they can use.

I learned this when I was at school (in the 20th century).  I remembered where to put the point of the compasses and which arcs to draw so I could carry out the constructions accurately, but I think there is more to it than that.

First off: what counts as a 'construction' and what doesn't?

If you are asked to draw an "accurate" version of a triangle when given the three sides, it is usual to use a pair of compasses.  (Draw one side - then make an arc from each end - where the arcs cross is the third vertex.)  This will give a pretty good version of the triangle (assuming the compasses don't slip and you have a sharp pencil), but this won't ever be entirely accurate because measurement is involved.  This isn't usually considered to be a proper construction for this reason.

Euclidean constructions involve using a 'straight edge' (a ruler - but not using it for measuring) and a pair of compasses (for drawing arcs).  [These also won't be exactly accurate because the compasses might slip, there is the thickness of the pencil to consider, etc, but the idea here is that this method will, in an ideal situation, give an idealised answer, so it considered to be a purer construction.]

The main constructions involve bisecting an angle and creating the perpendicular bisector to a line segment.  Arguably, all of the others build on these.

The big idea here is that they both involve constructing a rhombus.

There are three properties of a rhombus that we need:  

1. A diagonal of a rhombus bisects the angle
2. The two diagonals of a rhombus bisect each other
3. The diagonals of a rhombus are perpendicular

Consider how to construct an angle bisector.  Here is an angle:

When we draw in the arcs using a pair of compasses then we are actually constructing a rhombus:

We know this is a rhombus because we have set the compasses to a particular length, and then each of the sides of the rhombus is the same as that length.

Because of the properties of a rhombus, if we join the opposite vertices we bisect the angle.

Now consider how to draw the perpendicular bisector of a line segment:

Again, we need to draw arcs and construct a relevant rhombus:

The original line segment becomes one diagonal of the rhombus, with the line we will draw in becoming the other diagonal.  The properties of a rhombus tell us that the diagonals bisect each other at right angles, so we have got the perpendicular bisector.

There are lots of other constructions, such as constructing the perpendicular to a line that passes through a particular point, but this can be done by putting the compass point on the point and drawing an arc that intersects the line twice.  Now use the line segment between the places where the arcs cross the line and use the second construction above.

One way to teach this is to start by getting pupils to construct lots of rhombi.  Then they can sketch the rhombus they need onto the diagram, and finally they can construct the relevant rhombus.

Remember: it's all about the rhombus!