Rational and Irrational numbers

I only recently properly considered why all fractions are either terminating or recurring decimals.  Fundamentally, this is because there are a finite number of options for the remainder, which is a maximum of 1 less than the divisor.  This is most easily seen when dividing by 7.  All six potential remainders are used and the 7th division goes back to the beginning of the sequence as shown on this task from Don Steward.

Maybe this is an important thing to establish before talking about irrational numbers. With irrational numbers, we are effectively trying to convince students that there is a separate class of numbers on the number line that can’t be expressed as a division of two integers.   

There is a proof for why ∏ is irrational but it’s not pretty. I’m taking Peter Mitchell’s word for it on that who presented on this topic at the recent MEI conference.  He has a proof here, but in his own words “it’s really, really tedious!”  So maybe surds are a better place to look for an example of a proof that irrational numbers exist. Although this is an A level topic, I think with the right class this could be used at KS4.

Proof that √2 is irrational

This is a proof by contradiction, which in itself is a bit strange.  But the logic is sound: if I assume something to be true and then work through it to show that there is something inherent within it that is false, then I have proved that thing cannot be true therefore it must be false.

In this case, we are going assume that √2 is a rational number, prove that that is false, thereby proving that √2 must be irrational.

If √2 is a rational number, then we can write it √2  = a/b where a, b are whole numbers, b not zero.

We additionally assume that this a/b is simplified to lowest terms, since that can obviously be done with any fraction. Notice that in order for a/b to be in simplest terms, both of a and b cannot be even. One or both must be odd. Otherwise, we could simplify a/b further.

Going back to our first statement:

√2  = a/b

we can square both sides to get:

2 = a²/b²

or

a² = 2b².

So the square of a must be an even number since it is two times something.   If a² is even then a itself must also be even. Any odd number time an odd number creates an odd number (some more of these here).

Okay, if a itself is an even number, then a is 2 times some other whole number. In symbols, a = 2k where k is this other number.

If we substitute a = 2k into the original equation 2 = a²/b²:

2 = (2k)²/b²

2 = 4k²/b²

2b² = 4k²

b² = 2k²

 

Again, because b is 2 times something, b must be an even number.

We have shown that a and b are both even numbers, but we started saying that a/b was a fraction in its simplest form.

I might want students to explore what happens with √4 in this same proof, i.e. prove why √4 is not irrational.  From there we could go on to look at √3.  It’s a bit harder, but only really requires that all odd numbers can be written in the form 2n+1.  Here is a spoiler if you are stuck.

 

 

 

 

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Zoomable Number Line

This little gadget on Mathisfun.com is so illuminating for teaching decimals. I always get a positive response from students when I show it.

I think it can be used it in a number of ways:

1. Predict what happens when I zoom in. As you zoom in, first the tick marks appear and then at the next level of zoom, numbers start appearing.  It’s a great way of “bridging” from the familiar to the new.  A number line is likely to be a familiar concept.  However, what is “in between” 0 and 1? An intentionally ambiguous question. Students are likely to say “half” or 0.5.  How else can we show half? What other numbers are in between 0 and 1?
2. Why are there 10 ticks between 0 and 1? We have divided 1 into 10 equal parts, what is each part.  How else can we represent 1/10?

From there we “zoom” into the next level of hundredths

This table might be useful as some practice to relate fractions to decimals.  I would love to hear some comments on this.

 

 

 

 

Sequences in square numbers

List the first 6 square numbers.

What do you notice about the difference between them?

Express the difference between the square numbers as a sequence in terms of n?

Express algebraically the difference between the nth square number and the (n+1)th square number?

Use you algebra skills to show that they are the same thing.

Bus Stop Division

Here’s a big number:

Try different single-digit divisors.  No remainders.

This is an example of purposeful practice – exposing the wonder of mathematics whilst providing a reason to practise lots of  of bus stop division.

You might want to start by asking pupils to come up with their own dividend “in the tens of millions” and try different divisors. (Here for a quick primer on the mathematical language.) Inevitably they will end up with remainders, which they may or may not carry into decimal places. Then let show them this “magic” number.

Questions to ask:

• What divisors does this work for and why? (Purposeful practice)
• What other dividends could I make like this? (Purposeful practice + reasoning)
• What smaller dividends could I make like this? (reasoning)
• What is the smallest dividend I could make that all numbers 1-9 will divide into without remainders? (reasoning)

Whilst I would want everyone in the class to understand the reasoning through a whole-class discussion, you may have some learners who need the practice on bus stop long division and spend most of their time doing this. Those that are confident with this technique can spend their time exploring deeper into the structure of the number.

Whilst we are on the subject of “Bus Stop”, maybe this technique actually has nothing to do with standing in line waiting for a bus:

“It’s not bus stop, it’s actually the inverse of a model of area for multiplication” @petegriffin7 pic.twitter.com/E1Fc7ogUwT

— Mark Horley Maths (@mhorley) January 30, 2017

Palindromes

No 2 in a series of posts based on Thinking Mathematically (1985) by Mason, Burton, Stacey

This is the sort of exercise I can envisage taking a number of different paths depending on what my students do with it which is exciting. The book walks through a generalisation by looking for the lowest 4-digit palindromic number, 1001 and then noting that subsequent palindromic numbers can be found by adding 110. Since 1001 and 110 are multiples of 11, then all numbers in this series are multiples of 11. However, this series misses out other palindromic numbers, e.g. 7557 so we need to refine it further.

I am intrigued to see if this is indeed a path my students would follow or if we would discover something else in these numbers. Depending on the class, I might start by asking “how many 4 digit palindromic numbers are there?”  Before getting into the general, I would see this as an opportunity for purposeful practice of long division if that was something that my students require.  Some students might need a fair amount of direction to reach a proof, but I would aim to make sure that all students left this lesson with an appreciation of that proof even if I had to lead them through it.

 

Place value game

I often play this simple game with younger pupils to help them build a stronger understanding of place value. It’s simple and requires no resources.

Start off by drawing a place value chart on the board.  Depending on where you are at, you could do just hundreds, tens, ones or you could include digits either side of the decimal point, e.g. tens, ones, tenths and hundredths.  Use this as an opportunity to target questions at students as you are drawing the table “if we are doing place value, which column is this?”

Each group then has a row on the place value chart.  You want to limit it to about 6-7 groups otherwise it takes too long to get round to your go again.  It works really well with small classes working in pairs or threes.

Then we start randomly generating digits.  You could just use a dice (it doesn’t matter if you only have digits up to six), or using something like this from Classtools.net

There are a couple of twists that I have added to this over the years.

1, You don’t just get to place the digits in your own row, you can also place them in someone else’s row. So if you get a low digit, you can scupper someone else’s chances by putting it in their thousands column. This depends very much on the culture within the classroom. If you think there might be existing friendship issues amongst the group then it may be best to avoid this twist.

2, You could add some extra options as shown above, e.g. multiply or divide by ten.  This can make it a bit more interesting.  Other options might be to be able to erase a digit.

It’s good fun, but to get the most out of it, it is good to discuss at various stages who will definitely have the highest number / lowest number, etc.  You don’t always need to fill the grid completely to determine the order of the numbers. Why is this?

If you have used this or have any ideas for other “twists”, please drop me a line in the comments.

Sofia’s Ribbons

I’m not sure who Sofia is or where this originated, but it was presented by Liz Henning at the recent MTN hosted by La Salle Education in London and it struck me as a great way to introduce bar modelling at all levels, and could really help with ratio and fractions.

Start with 2 equal strips of paper, “ribbons”. Ask questions like:

• If each ribbon cost 10p how much do they cost altogether?
• If both ribbons together weigh 6g, how much does one weigh?
• If both ribbons represent one hour, how much time is one ribbon?
• etc.

Representation is a key concept here.  The ribbons can represent something else but that representation can be useful to work things out.  It also uses the idea of part-part-whole.

Next, take one of the ribbons and fold it in half.  Tear along the fold, so you have this:

Now you can ask questions like:

• If the orange ribbon is 10p, how much is the white ribbon?
• If the ribbons weigh 15g overall, how much does the orange one weigh?
• What fraction of this is the total?
• If white represents 12 hours how much does orange represent?
• What is the ratio of orange to white?

Next, take the orange ribbon and fold it in half again, so you have this:

Now ask:

• If orange is now worth 10p, how much is white?
• If both represent 2½ hours, how much does white represent?

and from here you can get into drawing bar model to represent what is going, e.g.

  

I used the bar modelling tool Thinking Blocks to create these images. Once you get used to the interface, it is a quick way of creating bar models for use in the classroom and contains a number of problems that you can use with learners.

 

 

Do you really understand addition and subtraction?

This was the question Mark McCourt was getting us to ponder in the first session of Maths Teacher Network.  He started off by running through the classic 1089 “trick” which I have written about before here.

These types of activity are such a powerful way to get students to carry out repeated practice to build fluency. If you handed out a sheet with 20 column subtractions and 20 column additions, you would get groans and do no favours for the general popularity of maths as a subject.  If those repeated calculations have a purpose as they do here, the dynamic and energy in the classroom is completely different.

I’d used this before but had never given it the time that it deserves, nor had I extended it:

• Try it with 4 digits, 5 digits, etc.
• Go back to three digits and try it with a, b, c. In other words prove it algebraically.
• Now try in a different base, e.g. Base 7!

I actually got a bit obsessed with this when I got home.  I do love a good spreadsheet challenge, so I attempted to build something that would provide an algebraic proof of different numbers of digits and bases. Here is the result…

My reflection from this session was that even if I have “seen” something before, have I really investigated it deeply and could I use it more effectively with my classes? To which my answers were no and yes!

Another approach, back to 3 digits. The prime factorisation of 1089 is 9×11×11. Why is this? And what are the prime factorisations of the 4 digit magic numbers. Can these be explained?

How many zeros?

A conversation in the Maths department this week about how to extend a particularly strong Year 7 student whilst teaching multiplying and dividing by 10, 100, 1000 led to this idea:

If you multiply all the numbers from 1 to 100 how many zeros are at the end?

I like this question.  It’s actually really quite hard. Have a go. I had previously used is as, “How many zeros are the end of 100! ?” But phrasing it this way gets around the need to explain factorial notation in the middle of a lesson that is not about factorial notation and therefore is can be used as an extension challenge.  Another less challenging, but similar(ish) question:

How many numbers with repeated digits (e.g. 55, 101, 155) are there between 1 and 201?

It is sometimes hard when teaching “basic” topics like decimal place value to find suitable challenge.

Before thinking about finding hard questions, I think it’s worth putting ourselves in the shoes of some of our higher attaining Year 7s and their experience of our mixed attainment classes in the first few months of secondary school.

Primary school experiences vary a lot. Many are taught in sets, especially in larger 2 and 3 form entry schools which are common in London.  My hunch is that the propensity to set children for Maths increases the closer they get to Year 6 although I don’t have any data to back that up (if you know of such data, please share!)  Most of the primaries that I am working with this year that are implementing mastery approaches and so are moving away from setting to mixed attainment teaching, but starting in the lower years.

In many cases, children who felt they were “good at maths” in Primary, developed that self-perception by completing calculations and getting correct answers. After all, that’s what maths is, right?  In some cases they will have established rules in their own minds schemas, etc. An example might be “to multiply by 10, simply add a zero on the end”. And, of course, for integers this works.  Which is why it is so important to highlight misconceptions e.g. 3.4 x 10 = 3.40 as soon as possible before these “false friends” become established.  This is why Year 7 can be such a challenging year for pupils and their teachers.  There is often a need to undo and deconstruct some false ideas that have become well-established before building back up the correct concepts.

So, back to the problem.  Maybe it’s worth saying that there are 24 zeros at the end of that huge number.  That’s the answer.  But as we all know, the answer is just the beginning.

Decimal Place Value Charts

I use these a lot at the early stages of understanding of place value.  In my opinion, place value is the single most important mathematical concept that children need to master in upper KS2 to prepare them for secondary school maths. Many don’t, so it is incumbent on secondary maths teachers to ensure that any gaps are filled in Year 7.

As a pdf.  Or if you want to change things, as a Google Sheets or MS Excel file.

If you print them out on card, you can put them inside transparent A4 plastic wallets and then use mini-whiteboard pens to write on them.  There are different types of plastic wallet – you want the ones that have relatively thick plastic which is smooth, not textured.  Otherwise you will have a hard time rubbing off the pen!

Initial Activities

1. Show me 3 tens.  Now show me Thirty. Why do we need to put the zero in there?
2. Show me 5 tenths. Show me 5 hundreths.  Why do we need the zeros?
3. Show me 34.  What is 34 made up of?
4. Show me 6.75.  What is 6.75 made up of?

From here, of course, you might want to look at multiplying and dividing by powers of 10 and then eventually 4 operations involving decimals.  But don’t rush into that until you are confident they have secured a depth of understanding.  These questions are good for really testing that:

For me these are a crucial AfL tool to ensure the building blocks are in place before doing more complicated things with decimals.