Category Archives: factors

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.

The ones that didn’t get away

This week, most of Year 7 are away on a residential trip leaving 20 pupils who, for whatever reason, are in school.  I taught them on Monday and much to my relief they didn’t seem particularly aggrieved by this state of affairs.

I don’t know what they are doing in other lessons, but in Maths, well, they’re going to do some mathematics!

I actually love the freedom of not having to follow a scheme of work for a bit (although I’m sure I would eventually feel lost without it!)  I wanted to do something that was proper maths but felt a bit different from a normal maths lesson.

As luck would have it, I attended an fantastic session run by John Mason and Anne Watson on Friday.  I’m not going to attempt to write about that session. All I will say is that if you ever get the chance to meet these two, you should.  A great opportunity to reflect deeper on the nature of mathematics and mathematics teaching.

So, like all things you pick up at CPD, I decided I would use something soon before I forgot it.  And here is what occupied us for nearly an hour last thing on Monday afternoon.

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This is a classic low ceiling, high threshold task.  Everyone can see some patterns in this, some will notice that the first term of the nth row is n², some will determine that the last term in every row is…

On Friday, however, I picked up something more valuable than this piece of mathematics.  I got to play around with techniques of how to present it to students.  To start with a blank board and write very deliberately and quite slowly stopping and asking questions like, “What am I going to write next?”, “What should we do now?”, “What are you thinking now?”

I really felt that we began to see the power of getting students to explain their reasoning, especially in a mixed group like this.

So, moving on.  What else can I do in the remaining lessons I have with this group?

My first instinct was to look at my own blog under investigations.  This is, after all the main reason I do this blog – as a reference for me!

Some good stuff, and it triggered memories of lessons I would otherwise have forgotten about.  And then I thought, what about those CPD events I went on in the years before I started this blog?  I looked back through my notes and found this from the 2012 ATM conference.

 

Factor Game

Write out numbers 1-12.

1st person choses a number.  2nd person gets all the remaining factors of that number.

1st person choses another number (but it has to be one that still has factors remaining).  2nd person gets all the remaining factors of that number.  Keep going until there are no numbers left to chose that still have factors – the second person gets all the remaining number.  Sum the totals of each – highest score wins.

Nice way to practice factors of a number as well as introduce Gauss’s trick to sum an arithmetic progression of n terms: n(n+1)/2 as you only need to calculate one of the totals, you can then take it away from the sum to get the other total.  So, in this case the total is always 78 because the sum of the first 12 numbers is 78 and that’s all we are doing.

e.g.

Person 1

Person 2

11

1

4

2

9

3

10

5

12

6

7,8

Total = 46

Total = 32

Can you always win? What is your strategy?  What’s the perfect game, i.e. highest possible score.  Try it for other totals.

I tried:

    • 8.  Nice, easy one to start with.
    • 16 a bit more tricky, but the player choosing the numbers should still win
    • 20 was interesting as it was a draw – 95 each!
    • 19 was easy to win

 

1001 is a lovely number!

I like to use this as a sort of a crescendo to teaching prime factor decomposition which is itself a very satisfying experience. IMG_3352

Although it sometimes feels a bit procedural it’s a nice way of:

  1. Practicing times tables
  2. Getting to know your primes
  3. Appreciating the commutativity of multiplication.

Anyway, here’s the trick (everyone needs a calculator in front of them)

  1. Chose any three digit number. Write it down somewhere.
  2. Type your number into your calculator and divide by 7.
  3. Hands up if you got an integer answer. Opportunity here for a nice discussion that we might expect 1 in 7 hands to be up at this point.
  4. Press clear and divide the same number by 11. Repeat again with 13. Right, now they’ve appreciated that not many numbers are divisible exactly by 7, 11 and 13.  Time to blow their minds…
  5. This time type your 3 digit number into your calculator twice so you have a 6-digit number. e.g.
    Image002
  6. Divide by 7. Hands up if you have a whole number. Wow, everyone. Now don’t press clear, but divide by 11. And then 13. Wow. Gets you exactly back to your original 3-digit number.

IMG_3353

How much you chose to explain this will depend on the ability of the class, but the points are:

  1. Whatever 3-digit number you chose, the 6-digit number is 1001 times the 3-digit.
  2. 1001=7x11x13. Weird but true. And this is why it works.

If your students seem to like it, I always ask them to try it out on their family when they get home. I love the idea that I just might have created a discussion about maths around the dinner table – you never know…

Learning times tables – Number Happy Families

Embedded image permalinkI’ve always felt that secure knowledge of times tables at Year 7 is so important simply because it gives kids the confidence to engage in so many maths topics covered in that year.  As such any opportunity to practice is good even when it is in a simple game like this.

A Simple Factors and Multiples Team Game for 3-4 players

I came up with this idea whilst playing the traditional Happy Families card game with my family when on holiday. Kids seem to love this game – could I create a maths game as engaging?

I’ve tried this several times with Year 7 classes, playing in teams of 3 or 4 and they love it.

It takes very little preparation or explanation – in fact the students make the resources themselves!

The Cards

You need a set of 36 blank cards for each team. Anything will do.  I spent about 10 minutes furious chopping on the guillotine for 7 teams, getting 12 cards out of each A4 sheet, so 3 sheets per team, 21 sheets in all.

The learning starts by getting the teams to create their cards using the following instructions:

1, Arrange your cards into 4 columns by 9 rows

2, You need to write the first 4 multiples of each number 2 to 10 so that every card has a number on it.

I put the 36 blank cards and the above on a slip of paper in an envelope and gave an envelope to each team.  With a bit of discussion within the teams, they worked out what they needed to do, but if you feel the task needs a bit more scaffolding you could use this diagram:

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The Game

Once each team has their cards laid out on the table, they can start playing.

  1. Shuffle the cards and deal them all out.
  2. The objective is to collect “families” of numbers, e.g. 3,6,9,12 is the 3 family. The player with the most families wins.
  3. Play starts with the first player asking one of the other players (they decide who) for a particular card, e.g. “Natasha, do you have a 5?” If Natasha has that card, she must hand it over. The first player can ask again (again, they can chose any player). If the answer is no, play moves on to the next player.
  4. When a player has a family they must lay it face up on the table.
  5. Play continues until all the cards are gone – it’s that simple!