Category Archives: multiples

US Flags

I’ve been on my travels recently in the United States. I was staying with some friends in Arizona and they had this flag on the wall.  It’s an old flag with 48 stars neatly arranged in 6 x 8 matrix. It was from the days before Hawaii and Alaska joined the Union, sometime before 1959.

US_flag_48_stars.svg

Now, I’m sure that all American school children know most of this stuff already, but it got me thinking about the US flag and specifically the arrangement of the stars and it gave me an idea for a lesson.

So, I know that there are now 50 stars for the 50 states of the USA.  But how are they arranged?  And how could we describe this mathematically?  I might start by showing students this zoomed-in portion to give them a hint.

Screen Shot 2016-07-29 at 07.52.09

It is, of course a 5×4 matrix inside a 6×5 matrix.

Was there a 49? Yes, briefly when Alaska joined in 1959. But what did 49 look like? Clearly it’s a square number, but they decided that to keep the aspect ratio looking more like a rectangle they offset the rows like this.

US_flag_49_stars.svg

There were actually lots of different versions of the stars and stripes detailed on this wikipedia page:

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It might be better to start with a simpler example from the early days when there were only 13 states (2×2 + 3×3). What other numbers can be represented as a sum of two square numbers in this way?

1235px-Hopkinson_Flag.svg

Is there a better way of doing this one? (It’s 36 stars, maybe 6×4 & 4×3)

1235px-US_flag_36_stars.svg

You might want to give this table as a handout (or here as a pdf).  Get students to work in pairs to come up with their own designs and compare those to the ones actually used.

And then, what if we go beyond 50? I don’t want to think too much about the political implications of such of move and who might end up as the 51st state, so let’s stick to this as a mathematical exercise!

 

 

 

 

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Stumped by a Foundation GCSE Maths question

I have just returned from an inspiring morning at #mathsmeetglyn organised by @mathsjem watching Don Steward give a fantastic, brain-stretching whistle-stop tour of some of the great problems on his site.

The theme was “Generalising” – he started off by saying that his current mission is to get some generalising into every single lesson because without it, well, you’re not doing real maths.

I filled 10 pages of notes, some of which I’ll come back to, but wanted to share how he took this Edexcel Exemplar Foundation GCSE question and “generalised” it.

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Now, first up, I don’t mind admitting that after working at it for about 3-4 minutes I was completely stumped.  Don had added “You can’t cut the tiles” to the question which was essential in my view, but didn’t actually help me. I was completely fixated on the fact that all tiles have to be in the same orientation.  They don’t.

I think it’s a pretty poor question, probably mainly because I couldn’t do it. But there is a serious point around what questions like this are really testing. If there is a simple “trick” you need to get, is that fair?

A debate to be had there, I’m sure, but more interesting was what Don did with it next.

What other areas can you fit 40 x 30cm “carpet tiles” into and how many do you need? Start with:

120cm x 60cm
120cm x 70cm
120cm x 80cm
120cm x 90cm

Do you need to go any further?  Can you write a general statement from this? i.e. can you prove that all multiples of 10 will work if the width is 120cm?  What other widths does this work for and why? And then, of course, what happens if you try different sized carpet tiles.

It feels a bit like one of those Maths GCSE coursework questions that were set in the days before I was a teacher.  But I really like the idea of taking what is a pretty bad question and turning into some interesting maths.

Prime Factor visualisations

Datapointed: I like to start this one at the beginning of the lesson, let them watch up to about 20, then teach the lesson leaving it running in the background.  Then come back at various points to see how far it has got.

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Primitives:   This one is more interactive and can be used when teaching Prime Factor Decomposition.

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Chalkface:  And finally a blog post with lots of interesting ideas and this nice graphic.

primesonlycolour

Learning times tables – Number Happy Families

Anyone looking for a simple maths game for end of term Year 7 lessons? Here’s one.

mho maths

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…

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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:

2015-02-19_08h14_52

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!