Category Archives: decimals

Recurring Decimals

First up, this post is NOT going to be about this:


This is what came to mind when joining Derek Ball’s session at the ATM conference this week, entitled “Recurring Decimals”.

In fact, we were going the other way: converting fractions to recurring decimals, specifically looking at fractions where the denominator is prime.  It was a fascinating session, great for deepening subject knowledge.  This blog post is my attempt to reflect on what I learned and my thoughts about how I might use this in the classroom, probably Year 10 or 11, but really any group that is confident with bus stop division could investigate this.

I was already aware of some pretty cool things that happen with sevenths, mainly from Don Steward’s blog:

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Why should this happen? Why do we only see these six digits with sevenths? The process of bus stop division helps us see why and this is where I feel I would start with a class. This is good practice of a technique that should be secure but often isn’t in Year 9 / 10.  It helps learn the seven times table and I don’t think it is too tedious to ask students to perform these six calculations manually:

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Some learners might want to find 2/7 by doubling 1/7.  And then maybe find 3/7 by adding 1/7 to 2/7 and so on.  Even if they stick with the bus stop division, they will find that they are essentially doing the same six calculations:

10 ÷ 7 = 1 rem 3
20 ÷ 7 = 2 rem 6
30 ÷ 7 = 4 rem 2
40 ÷ 7 = 5 rem 5
50 ÷ 7 = 7 rem 1
60 ÷ 7 = 8 rem 4

The ones digit is always zero and the tens digit must be less than 7 so there are only these 6 options. Can we extend this rule to other fractions with denominator less than 10?  Of course, all others except 3, 6 and 9 will terminate – why is this?

There are also some interesting things to notice about the 1/7 “wheel” before moving onto higher primes.  I won’t spoil it for you, but suggest you include the fractions around the outside of the wheel to spot some of the patterns.

We then moved onto looking at 13ths.  11th are interesting too for different reasons, so I can see why we went onto 13th because there are some surprising relationships between 7th and 13ths. At this point, we started using calculators and I would do the same with a class. Or even better, open a spreadsheet, which I am always keen to do!

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I’m not sure of the value of kids typing 12 calculations into their calculators, so I might give this image above as a print out for them to write on. Hopefully they will soon spot that there are two sets of recurring digits, i.e. 076923 in 1/13 + 5 others and 153846 in 2/13 + 5 others and which, this time can be written as two wheels:

Ideally at this point you’ll have the class hooked and they would be asking all sorts of questions.  Well, maybe enough of them to get everyone else thinking.  I would try really hard to encourage the students to come up with these questions.  This is what I hope they will ask, but I hope they will also ask questions I hadn’t thought of, something which is a really special moment in any lesson.

  • Do we see the same patterns in the sevenths wheel as we do in the thirteenths wheel?
  • Is there something special about 6 points around the wheel?
  • Why are there 2 wheels? Are there any patterns across the two wheels?
  • What about other fractions, do they have multiple wheels?

On the image above, I have shown that as you move clockwise around the wheel, you are effectively multiplying by 10.  This is obvious going from 1/13 to 10/13 but actually occurs for all other hops if we ignore the whole number part.  i.e.

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So if go 3 hops we have multiplied by 1000.  I think this goes some way to explaining why the fractions that are opposite each other must sum to 1.

The next prime fraction: 17ths recur after its maximum of 16 digits, so effectively we have one, rather large wheel. Unfortunately, Excel gives up after 16 decimal places (please let me know if there is a way around this) but you can still see some patterns in here:

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Beyond that, we found that the following fractions recur after the maximum number of digits so have only one wheel: 3,7,17,19,23,43,59,61 – someone had a much better calculator than me that showed 32 digits!

Other fractions worth exploring are:

  • 31ths – contain 2 wheels of 15
  • 37ths – contain 12 wheels of 3
  • 41th – contain 8 wheels of 5

With the fractions that recur after a relatively few numbers of digits, we can find factors.

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Because 1/41 is a decimal that recurs after only 5 digits, if follows that 41 must be a factor of 99999.

And in the larger wheels there are all sorts of patterns – pentagons and triangles in the wheel of 31ths apparently.

I’m not sure how far I’d go with a class, there could be several lessons worth in here. It would depend on how they responded, of course.  Maybe 13ths would be sufficient unless…

This post captures only some of what we worked on in this session and highlights to me the depth of subject knowledge that can be gained by attending the ATM conference. Other sessions were more pedagogical in nature, but this one was pure fascinating mathematics and I was grateful to be surrounded by so many knowledgable and friendly people!

 

Pattern sniffing with Decimal Subtraction

An idea for a mixed attainment class that came to me about 5 minutes before a lesson today:

  1. 3.4 – 3.04
  2. 5.2 – 5.02
  3. 7.8 – 7.08
  4. 8.2 – 8.02

Find other questions like this.  (The “weakest” student in the class told me the pattern before I’d even finished writing the fourth question on the board.)

What do you notice?  Why is the answer to Q2 the same as the answer to Q4?

Can you create a question with 0.54 as an answer?  How many different answers are there to these types of questions?

Then:

  1. 5.7 – 5.007
  2. 8.3 – 8.003
  3. 6.4 – 6.004
  4. etc.

These are more tricky and test the skills of column subtraction, something that should be secure by Year 7 but may not be. Maybe an opportunity for collaboration amongst students to show how.

And then finally, try these two calculations. Which is easier and why?

7 – 1.392

6.999 – 1.391

Show on a number line why this works and then try some more.  I think these questions are interesting to explore.  But I would hesitate to recommend it as a must-do method to solve e.g 8-2.5687.  Whether or not it is easier to turn it into 7.999-2.5686 or not is an interesting discussion and one which I would want my students to form their own opinion on, not be too swayed by mine.

 

 

 

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

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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.

FDP ordering activity

This is a simple card sort activity where students fill in the blanks, practising converting between fractions, decimals and percentages and then placing them in order from smallest to largest.

What I particularly like about it is that you can hand out the cards in 3 sets of 5.  This could provide differentiation, but more importantly (in my view) it gives the teacher the chance to assess the progress of the students as they go. It’s easy to glance at Set A to see if they have worked it out.img_20161207_132457

As you circulate the class, helping students you can give out Set B which interleave with Set A to produce this:

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And then finally with Set C, you get this:

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Here is the pdf as a set of 15 cards, but if you are printing a class set, then I strongly recommend using the Excel version here.  This is set up so you print 5 at a time.  They come out stacked so that as you cut you have a set of 5 already in a pile without needing to sort them.

Also, if you want to change the actual values and which values you show, you can do that on the spreadsheet too.

 

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.

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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:

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