## A Geometrical Approach to Factors and Primes: Understanding an Algorithm

Reasons for Post:

1. To introduce factors and primes through a geometric approach.
2. To introduce an algorithm to quickly find how many factors a number has.
3. To discuss why the algorithm works based on combinations of factors.

Recently I gave my Year 8 (Grade 7) students a mini-investigation based on factors and primes – without saying it was about factors and primes.

“For this investigation, consider a square to be a special type of rectangle. If the number 10 represents the area of a rectangle, how many different rectangles can you make with a whole number length and width? Investigate this for different numbers.”

Before discussing two examples as a class, I told them that if anyone figured out what this investigation was actually about, then they should keep it to themselves. Of course some realised quite quickly that the width and length of a rectangle or square represented a factor pair of the area. Those students generally went on to look at investigating how many factors each number has and whether there is a way to figure this out. I didn’t expect anyone to get far with this.

I was genuinely surprised when two of the students working together came up with the standard algorithm for finding the number of factors using the indices within the product of prime factors.

For example, to find how many factors the number 30 has.

Step 1: Break 30 into a product of primes:  30 = 2¹ x 3¹ x 5¹

Step 2: Add one to each of the indices and then multiply these numbers together.

(1 + 1) × (1 + 1) × (1 + 1) = 2 × 2 × 2 = 8 factors

Of course, by finding the product of prime factors, you’re essentially finding the fundamental building block of each factor. Still, quite impressed that they’d found an algorithm. I personally don’t know any other way of doing it. Anyone else know a different way?

I told them that as a mathematician – our aim is to understand why an algorithm works but that might be something they’d be able to investigate in a few years once they’d learnt more mathematics. In this case that boils down to why we add one to each of the indicies and then multiply? I spent part of break with a coffee making sure I understood why the algorithm works (on a similar, but less rigorous line to Erdos; “A mathematician is a machine for changing coffee into theorems”). Not hugely difficult to make sense of/justify for numbers with only one base prime factor (e.g 8 = 2³; 2 is the base prime factor) or two base prime factors (e.g 15 = 5 x 3; 5 and 3 are the base prime factors) and then intuitively extend from there.

With one base prime factor, e.g. 8 = 2³. The factors are 2, 2², 2³ and then 1 (you could think of this as 2° if you fancied it). Hence why there are 3 + 1  = 4 factors. So for N=a^x, there will be x + 1 factors in total.

With two base prime factors, take the number 100 = 2² × 5²

By the algorithm, we know that this number has (2 + 1)(2 + 1) = 9 factors.

If you expand out the brackets (2 + 1)(2 + 1) = 2×2+1×2+2×1+1×1 = 4 + 2 + 2 + 1 (*) then we can see that these 9 factors are split up into groups. i.e. 4 factors + 2 factors + 2 factors + 1 factor.

Expanding out any bracket of the form (a + 1)(b + 1)(c + 1)… will always give you a last term which is just a load of one’s multiplied together. So the last 1 in the calculation above represents the fact that 1 is a factor of every single number.

So where do the 4 factors + 2 factors + 2 factors come from in equation (*) above. I think the geometrical representation below sheds the most light on this.

It’s clear to see that the 4 factors are made up of all of the combinations of the two base prime factors. The (2 factors + 2 factors) can be seen in the picture as factors which are only made up from one of the base prime factors.

So for any number with 2 base prime factors, we have:

Of course if each exponent is equal (i.e. x = y), then we have the second binomial expansion (x + 1)². With three base prime factors all being equal (x = y = z), then you have the third binomial expansion (x + 1)³, and so on. There’s the connection to Pascal’s triangle and hence the lead into the different combinations of all of the factors.

So an interesting change in my approach to factors and primes. I’m still not sure whether I’d teach the algorithm above – wouldn’t be too hard to add in some understanding of this but then usually I’d teach factors and primes in Year 7 (Grade 6) which would be before looking at breaking numbers into a product of prime factors. It’s therefore difficult to know where this would all be placed in the curriculum. Possibly worthwhile as a closed investigation in Year 9 as a follow on from expanding brackets.

Note: Bear in mind that for the above example, we couldn’t use the calculation 4C2 (4 things choose 2 of them) for the 4 factors part because you can’t put 2 and 2² together or 5 and 5² together. You have to see the base prime factors as separate groups (2, 2²) and (5, 5²) and pick an element from one group to pair with an element from another group.

## Shape Review: Pint glasses, Supermarkets, Coding, Population Density, Islamic Art and Bolivia

It’s not easy finding the time nowadays with so much going on (not least with the new puppy – Sir Isaac Newton – to play with) but it’s great to sit down with a cup of tea and write a blog post.  I’m hoping to write another post next week but this post is mainly about some of the shape stuff I’ve been doing this term. Hopefully you’ll find at least one of the ideas interesting and I hope that a few people will be able to add to these ideas or provide new ideas that I can try out next year.

1) A great question when you’re at the pub with friends…

Which is bigger, the length of a pint glass or the circumference of the top of the glass? I bet most people will say the length. Of course, in a maths lesson, I don’t need to use the word “pint” to get my students hooked on a perplexing question. I used my water flask instead…

It turns out in this case that the circumference of the top is slightly longer than the length. Once students have a full understanding of where the circumference formula comes from, they’re amazed to check the results using mathematics (it’s quicker to measure the diameter and calculate over cutting out a piece of string and directly measuring which is where the maths comes in). In future, I might open this up to an investigation into different prisms. E.g.  If the top cross section wasn’t a circle, would this still happen? If so when and why?

2) Surface Area and Structure: Why this shape at the supermarket?

This is the “separating thingy” that separates people’s shopping at the supermarket. We had a great discussion on why they would use this particular shape over any other object. Why this over a cuboid? Nice intro to minimising surface area whilst maintaining the structural importance of the shape.

3) Ipads, Exterior Angles and Hopscotch – Discovery

After doing some work with interior angles, I thought I’d get the students discovering the integral fact about exterior angles of polygons through the use of the iPad app, Hopscotch. Hopscotch is essentially the new Logo. It is a beginner’s coding app in which you can use “ready-made” commands to make a code.

I made this worksheet to go along with the activity.

4) Understanding Area Conversions and Population Density

In a study conducted by the European Commission in 2011, Europeans were asked what they thought the World’s 10 biggest problems are. Obvious stuff like climate change, the economic crisis and terrorism were up there in the top 5. One of the top 10 was the exponentially increasing population of the human species. Hence, I really like doing stuff with population density because it’s relevant and connects to so many areas of mathematics (namely ratio, proportion, standardised measures, area, converting between metric units of area). To introduce this, I like to start with getting the students to internalise how big 1 m² is. How many students can fit inside 1 m²? With the square against a wall, I was quite amazed that 11 students managed to get in there!

5) Understanding 3-D Shapes and Prisms before Volume

If students don’t know the difference between prisms and other shapes then they’ll simply start memorizing formulae for volumes of different shapes with little understanding. Hence, to start this topic I simply got the 3-D shape sets out and asked them in groups of three to organise the shapes in any way they thought was most logical. After 5 minutes, we had a little class discussion about what they’d done and then I encouraged them to try to categorise the shapes into 3 groups. Low and behold, all but 1 group sorted into Prisms, Pyramids and Spheres. This allowed us to move on to properly define prisms using the applet I made a while ago.

5) Islamic Artwork and Transformations Project – Discovery

I decided to get Year 7/Grade 6 discovering transformations on the co-ordinate grid using the hook of Islamic Artwork this year. Since one of my “secondary objectives” is to get students familiar with Geogebra, it was a great way to get them discovering, using and effectively communicating transformations.

I started with an introductory lesson (type “Islamic Artwork” into google to find some interesting pics) which got them hooked into thinking about the artwork from a mathematical perspective. We talked about rotational symmetry and then I provided a summarised historical perspective using information from Wikipedia:

Islamic Art began around the 7th century. It was designed and made by people who lived within the Islamic territory that was inhabited by or ruled by Islamic populations. It is thus a very difficult art to define because it covers many lands and various peoples over some 1400 years.

There are often repeating elements in Islamic art which is used to symbolize the infinite nature of God. Mistakes in repetitions are sometimes on purpose to show humility by artists who believe only God can produce perfection.

With it being an introductory lesson, I then allowed for the all important “tinker phase” of introducing new tech/software – you certainly cannot expect much structured learning of mathematics during this important phase(although of course it’s massive for discovery)! They had to perform a rotation, enlargement and reflection without any initial class discussion. Obviously, some of them have almost no idea what’s going on at first but they get there eventually. If anyone found out a good way of using the software, they were encouraged to write it down on sugar paper for the whole class to share solutions. The all important part of this project is how you bring in meaningful mathematical learning without students becoming so memorized by the technology and artwork. That’s where effective classroom pedagogy takes over in planning, class discussions and quality questioning. Furthermore, making the students write ever step down as instructions (using correct mathematical terminology) focuses them more on the mathematics rather than just the artwork (60% grade for instructions, 40% artwork marked with Art teacher). The great thing about the discovery element is that students who understand quickly and finish their work can investigate translations, negative enlargements and dilations. Since I don’t have any of the students’ work available electronically, take a look at a previous post on this.

6) What do tessellations have to do with the Salt Flats in Bolivia? Previous post.

So there’s a load of shape stuff. If anyone has any ideas to improve on these or any extra ideas on anything to do with shape, please share.

Posted in Shape and Measures, Uncategorized | | 3 Comments

## Grade 9 Educational Games on Scratch

A while back, a friend told me that Scratch was too simple a scripting platform for Grade 9 students. As anyone that has some idea of scripting/coding will know, it’s as easy or hard as you want it to be! Here’s some examples of the final scratch games/quizzes which the Grade 9’s have been working on over the last month. Unfortunately, we’ve found that large scripts (over a page in length with many loops) and large numbers of scripts (100 or so scripts) can create bugs in the games – even when there appears to be nothing wrong with the script. Has anyone else found this with Scratch?

(Click the picture to go to the games).

French Conjugation Learning/Quiz:

Character based platform game (always great to code gravity!):

Basic Skills Maths Game (Beat the computer):

## Moving towards more Meaningful Mathematics Assessment in the 21st Century

I’ve always believed that there’s nothing wrong with a sit down test if the test items assess the core philosophy and values of the subject. Hence, I’ve tried to make an assessment for my Year 7 students (grade 6) which I believe is more meaningful in the 21st century. It’s so easy now to do a quick search on Google to find the formula for anything. Indeed, during a planets investigation today with Grade 9, we simplified the problem of planetary orbits to consider them as circles instead of ellipses – pretty standard. However, one of the students put his hand up a minute later to ask why we weren’t just using one of the many formulae he had found through a Google search for the perimeter of an ellipse. This is great but it means that we need to teach students how to derive formulae and analyse formulae instead of simply using them without thinking. I guess this is similar to when students just pick information from a website without critiquing it – so frustrating and meaningless.

Anyhow, here’s the assessment. Any feedback to improve would be appreciated.

## Closed Investigation: Finding the Area of a Triangle using the Geogebra Ipad App

Since I work in a 1:1 Ipad School, I decided to finally try out the Ipad Geogebra App. The main content objectives are obvious from the title. Possibly of more importance, the Mathematical Habit objective was to understand that we could utilize technology to help quickly confirm, but not necessarily prove, our conjectures.

I went through a few of the basic functions of the app on the Apple TV, gave out the worksheet below and let them get going. Whilst I understand that there is always a “tinkering stage” to any new piece of technology, I thought that the use of the app was simple enough so that they wouldn’t have to spend too much time familiarizing themselves with it (Ha, I’m in my seventh year of teaching – I should know better with Year 7/Grade 6!)

So it turns out that many did have trouble familiarising themselves with the software (Note to self: showing often doesn’t imply learning) and more annoyingly, it turned out that with the red triangle embedded in the rectangle, the area function couldn’t pick out the triangle area from the rectangle area.

Luckily, I quickly saw that it gives the area of both the rectangle and triangle in the side toolbar (poly1 and poly2) so the investigation wasn’t so badly interrupted.

Next step is to get them using the app to show why we can calculate the area of a parallelogram in the same way we do with a rectangle and then move onto sliders and rotations to find the formula for the area of a trapezium (Next Big Objective: The area of many shapes can be developed from the area of a rectangle).

@Geogebra Team – Great work creating this app! I wonder if there’s any chance of doing some thing about the following stuff:

1. The issue mentioned above – when you want the area of a shape inside a shape,  the only way to do it is to look at the algebra view, anyway of distinguishing between shapes?

2. The angle slider maybe shouldn’t use alpha because there’s no way of putting alpha into the rotation function – maybe set the letter a as default?

3. Is there any way of copying and pasting things on the app?

4. It’s currently quite difficult to edit a script by touching the part of the script you want to change. I wonder if a python style cursor can be developed to do this?

## Allowing Students to Discover the “Mystery” of Square Roots (Surds)

I’m starting this post with the end point. Before you start on the post though, you really HAVE to watch the “Double Rainbow” viral youtube clip (Here’s a post I wrote about rainbows for anyone interested).

Right, now you’ve had a good laugh, below is the Maths equivalent. My Year 10 (Grade 9) students have been raving about the Double Rainbow clip for the last few days so I decided to spice up the topic we’ve been doing with my own “Double Root” clip.

So on the topic of square roots, here’s my attempt at investigating operations with square roots (surds) when you’re not prescribed by a GCSE/IGCSE syllabus. This is only a general outline of the lessons we’ve had so far but it’d be great to hear how other people go about teaching surds.

To start off, we hadn’t spent much time discussing irrational numbers but I thought the question below would push us into a discussion/debate about them and then things would start flowing:

(An irrational number) x (An irrational number) = (A rational number). Discuss

Being a Year 10 class (Grade 9), they were aware that pi and √2 are irrational (though one student still believed pi could be written as a fraction – see previous post). We wrote down some definitions for rational and irrational numbers and after some thought, calculator work and discussion, the students were quite comfortable that it is possible (i.e. √2 ×√2 = 2). Even though the calculator confirmed it, some of them were not comfortable with this result

“How can a number which carries on forever with no pattern multiplied by another number which carries on forever with no pattern equal a rational number???”

So I showed the equation like this: (√2)²=2 and one of the students realised that if you square root 2, then square it, you get back to 2 (At the time, he said it was like taking 2 from 10 and then adding 2 again to make 10 – they’re inverse operations).

At that point, I showed the slide below and allowed them to battle through it for a while:

To start off the next lesson, I put the following slide on the board:

After a few minutes of independent thinking, half of the students had no clue what the picture represented (√2 ×√2 = 2) so I asked the other half that did understand to explain. Once that had been established, I shared one of the Habits of a Mathematician with them:

Mathematics can be represented in many different ways and different forms can often help us to understand how and why things work.

(Note that no specific learning objective was shared, just a habit). They then went about doing a similar thing to show geometrically how √5 ×√5 = 5 which wasn’t so difficult so I asked those who had finished to tell me why I went straight from √2 to √5? Why didn’t I ask them to show √3 ×√3 = 3?

After showing the result, I was happy to now show them a rule and let them thrash it out:

√a × √b = √(a×b)

I was really impressed at this point at how well this was analysed. One student made a = 5 and b = 5 and realised that it was correct for that case and therefore for every case where a = b. Then the students quite quickly used other numbers with a calculator to confirm that this worked for a ≠ b (no one mentioned negative values and I decided at this moment not to bring it up). Admittedly, I was slightly disappointed that no one tried to show this using a geometrical representation but I guess I can’t complain – it is easier to verify with a calculator. (Writing this post, I see now that I could have been more insistent on proof but then I think I would have been in danger of having too much discussion).

I then gave them a few questions to do on this with the endpoint being that if we can combine square roots by multiplying, could we “break them apart”?. I gave these 4 questions for them to break apart in all of the ways they could:

√20 = √10√2 = √20√1 = √4√5 = 2√5

√45 = √9√5 = 3√5 = √15√3

√23 = (Important to stick one in that can’t be broken apart)

√30 = √3√10 = √15√2 = √6√5 (and important to stick one in that can be broken apart but neither of the roots will contain a square number)

Question: Why do they always ask students to simplify roots in this manner – are the answers actually simpler? How is 2√5 simpler than √20? Neither I nor my students feel this is the case.

After they’d made up some of their own questions which could be broken apart into at least one square number – we moved onto adding square roots:

Does √2 +√2 = √4?

They had an intuitive guess first, at which over half felt it wasn’t true but some felt it might be. Then they were easily able to verify on calculators that this isn’t correct. So I issued an 8 minute challenge:

1. Can you find out what √2 + √2 is equal to as a square root?

2. Can you use Pythagoras’ Theorem to show with a picture representation what √2 + √2 is equal to as a square root?

Everyone answered the first one – no one answered the second one (in fact many were trying to represent it in a similar way to √2 × √2). I was stuck at this point, give those interested a chance to have a go for homework or show a representation and allow them to thrash out the meaning. I went with the latter (without the equations at the bottom or the √2’s and √8 on the diagonal lengths):

Once that was done, I asked them to have a go at √5 + √5 and √7 + √7.

Most verified on the calculator (again, not using a visual representation) that √5 + √5 = √20 and √7 + √7 = √28

A number of students simultaneously conjectured that √a + √a = √(4a) and so I asked one of them to explain what they meant using more examples on the whiteboard.

To help them explain this, I put this on the board: √3 + √3 = 2√3 which helped one student take it further into this: √3 + √3 = 2√3 = √4√3 = √(4×3)

Hence √a + √a = 2√a = √4√a = √(4a)

I then gave a sheet out with more square root addition, but in the questions, the numbers inside the square roots weren’t the same E.g. √8 + √2, √90 + √40  etc.

They were pretty livid at this point: I could tell that they were internally shouting: “Stop making us think so hard about everything – just give us something easy to get on with for a while.” Next lesson we’ll be doing a review/reflection on what we’ve covered so far so it should be less strenuous.

I was really happy though that it wasn’t long before a few students started to figure this out by changing one of the roots so the “base roots” are the same:

√8 + √2 = √2 + √2 + √2 = 3√2 = √9√2 = √18

There’s so much more to investigate and discover! Unfortunately there will come a point when it’ll be time to move on but I’ve been massively enthused and excited about this unit – there’s just so many deep aspects to it and so many questions you can pose!

Having said all that, can’t wait to have a bit of fun on Monday with the Double Root Video.

## Discovering and Creating Mathematics Part 2: Does investigating problems make it easier to pose interesting problems?

After my first post on discovering and creating mathematics, I decided it was about time I sat down again with a random topic to see where I could take it in 30 minutes.

I decided to try and define a triangle root and see what happens. Below are some scribbles with the direction it took me (Apologies that’s it’s messy but it’s hopefully still readable).

I wonder if anyone else fancies exploring triangle roots in a different way and seeing what they can come up with in 30 minutes?

One of the most interesting aspects I’m finding of open investigation is that you naturally start asking more and more questions. Is it easier to pose problems in Mathematics during an investigation than if you simply try to pose a problem just by thinking about a topic?

Since being a Mathematician is so heavily dependent on asking interesting, fruitful questions to investigate, this to me is an important question. I regularly encourage students to pose mathematical questions and I’m sure I have a way to go in gaining a deeper pedagogical understanding of student problem posing in mathematics.

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