## Envelope puzzle and neat little square

I’ve been looking through my saved puzzles again and I found this nice little one in the maths newsletter from Chris Smith (@aap03102):

It’s a nice little question that took me some thinking about.

First I considered the half squares with hypotenuse 2. As these are isoceles RATs, that means their side length is rt2 so each has an area of 1.

Then I thought about the half square with hypotenuse 3. Again it’s an isoceles RAT so pythagoras’s theorem gives us a sidelength of 3/rt2. So an area of 9/4.

These 3 half squares add to 17/4. The area of the rectangle is 6 so the part not covered by the half squares must be 7/4.

When thinking about the overlap, we need to consider what that means. I assume it means the bit that goes over the others, so in this case 9/4 (area of the halfsquare) – 7/4 (area of the gal). Which is 1/2 cm².

I think this is an amazing little question that I cant wait to try out in class. It also got me thinking about the square with diagonal of 2 and area of 2. It’s an interesting square really, and the only one where we see this. Consider a square, side length, x. The diagonal is (2x²)^½, the area is x². If we equate the. And square both sides we have 2x² = x⁴ , or x⁴ – 2x² = 0, so x²(x² – 2) = 0. This generates a solution when x = 0 (which is trivial and discountable as no square can exist without side length. We also get positive and negative square roots if 2. But we can ignore the negative as lengths in this case are scalar, so we have one answer. It’s a neat little square.

## Tilings and areas

My daughter and I had another play around with pattern blocks. Firstly we played around and made some patterns. She made this one that was pretty cool:

We talked about tiling the plane and how shapes tesselate. Looking at which shapes fit together. Then I asked her if she could make a repeating pattern.

She came up with this one. Which wasn’t exactly what I meant but cool non the less.

Then she made this one that was more what I had meant. At this point we discussed which colour had more shapes and which took up more of the area.

We had similar discussions about these two tilings. We discussed how the red and yellow had the same amount if area in the red yellow and green one even though the yellow had twice as many squares. She showed this by making a hexagon put if the trapeziums.

She said the green, blue and purple one looked 3d.

I agree. I mentioned briefly that it was to do with the angles if the lines and that you can get dotted paper to help draw 3d which has dots at these angles. We talked briefly about rotational and reflective symmetry too.

She then made a hexagon:

We talked about how much bigger it was. She said it looked about 4 times bigger. We then discussed what this mean, and looked at the areas. Counting triangles.

I showed her that we could do it without counting triangles. We then looked at the side lengths of the hexagons and discussed how and why this scale factor was different to the area one. I think this photo of the hexagons is an excellent visual to use when looking at similarity in secondary school. Normally I just talk about squares and rectangles but can see an excellent set of visuals using these shapes.

We then started to look at fitting shapes together round points and on a line. And we found that if you put the thin blue rhombuses together on a line you can get some cool patterns:

We didn’t get into angles that much, but I can certainly see this could be a great entry point to those discussions in future. I can also see that as well as similarity there can be further discussions around area and perimeter that build from using these shapes and I hope to explore this more in future sessions.

*This is the tenth post in a series looking at the use of manipulatives in maths teaching. You can see the others **here**.*

## Circles and an octagon

Here’s an interesting puzzle that came via Diego Rattaggi (@diegorattaggi) and involves circles and octagon.

I started as always with a diagram:

I labelled some sides up, then changed my mind and changed labels as I was thinking about taking a coordinate geometry approach and didn’t want to have used x and y. As it happened I didnt use that approach anyway. While looking at the sketch I realised that the triangle AOC was a right angled isoceles. Due to an error a few weeks back I wanted to double check I wasn’t making an assumption here so did some work to justify this was the case:

I was using some similar reasoning to this hexagon puzzle, I could justify that to had to be an isoceles, and that extending the lines gave an isoceles, I could just that the vetex was definitely on diameter I’d drawn and was equidistant from both circles in x and y, but felt that wasn’t enough, and if it wasn’t definitely the centre there could be multiple solutions, then I saw a different version in my screenshot:

Once I had this information it was fairly straightforward using Pythagoras’s Theorem:

At this point I realised the ratio if the radii squared was the same as the area so that’s all I needed:

I got to the end and realised I had my fraction upside down so I flipped it over.

This was an interesting puzzle, and I think I will need to think further on the case where the centre being the exact of the right triangle wasn’t specified. I might need to look on geogebra.

## Circles on a line

I saw this lovely question from Mr Gordon (@MrGordonMaths) the other day:

I looked at it and even though it said it was GCSE maths only it didn’t look at all obvious how to find an answer. It did look interesting though, I wondered how my y11s would get on with it. I thought I’d give it a try:

As always I drew a sketch:

I was looking at straight line shapes I could draw and realised the trapezium was the better option in this case:

From here it was a bit of Pythagoras’s theorem:

Which gave me all I needed for the final answer, which is 1:4. (Obviously I discounted the trival R=0 as it doesn’t make sense in the context of the question).

A nice little puzzle that I can see could be rather taxing for students despite using only concepts they will have learned in KS3 and 4. It’s the sort of question that can really help with problem solving ability and is one i will definitely try on my year 11s when we are back fully.

I’d love to see how you solved this one, especially if you took a different approach.

## A short area problem

At the weekend I saw this nice question:

It came from @lasalleed

And it looked interesting. Initially I didn’t know what to do, but I realized if I looked at the region with the area 95 I should be able to get an answer:

It was quite a simple form and solve a quadratic problem in the end.

When I got my answer I considered the other solution, the negative 19. If we moved the line from the left hand side do it was 19 away from the other side it would give us a rectangle area 95 on the right of the red rectangle. That’s quite an interesting thing.

19 is also the side length of the square. Which is also interesting.

It makes sense though, you are multiplying a by (a+b) to get the area shown (95) as b is 14 then you are multiplying side length -14 by the sidelength to get 19. A different but similar solution could be obtained.

Quite nice and interesting. How did you solve it?

## A hexagon and some interior lines

Today’s puzzle comes from Eylem Gercek Boss (@_eylem_99) and it’s a nice quick one that I loved, and includes a hexagon, which an awful.lot of puzzles I find at the moment seem to do!:

Initially there wasn’t an obvious solution to me so I sketched it put and labelled a load of things.

Then started writing what I knew:

I had 3 parallel lines equally spaced, so I had 2 similar triangles. I knew the diagonal was double the side length. I had enough to form an eqution:

2x = (1/2)x + 12

3x = 24

x = 8

From here I could easily work out the area:

A nice little problem that got me thinking about problem solving. I didn’t see a solution immediately, although perhaps I should have, so I just started jotting down what I knew until I saw a way forward. This is a key still that students need, just being able to consider what they do know an look at what that means in the context of the question. I think this question would be really good to use with students to model that thought process.

Do you have a different solution? I’d love to hear it.

## Playing with pattern blocks

I’ve been having a lot of fun at home playing with Cusinaire rods with my 7yo daughter. It’s been great, she’s been learning a lot of maths through playing, we’ve been having a lot of fun and I’ve been learning a lot about the rods and how I can use them in lessons. I’ve not only learned how they can be used in the activities we’ve been doing but I’ve seen during the sessions other places they can go which lead to higher level maths that would be more suited to much older students, there will be links to the posts on these sessions at the end of this one if you have missed them.

I’m keen to explore other manipulatives, and when I finished reading the Cusinaire book (Ollerton et al., 2017) I bought another ATM book on Pattern Blocks (Gregg, 2020). I read the first few pages and thought they sounded fun so I purchased a set if block for home (although the colours were wrong again! Turns out typing the name of a manipulative into the search functionof the biggest online retailer doesn’tget youbthe right colours, which knew?).

When they arrived I still hadn’t read too much if the book, but my daughter was interested in the blocks so we got them out. They came with some cards and she wanted to make the shapes on the cards. Here are some of the pictures:

Then we talked about the shapes. She knew what some if them were but not others. She called the rhombuses diamonds. She asked questions about the blocks and I showed her that the side lengths were all the same apart from one of the sides on the trapezium which was double.

She asked if I’d read anything in my book we could do so I told her one of the tasks it suggests was to try and make the shape of the red hexagon out of the other shapes so she tried this:

She made the top 3 very quickly, but then didn’t think she could do anymore. I said she didnt have to use just one colour but she still struggled. I told her to look at the ones she’d made and look for similarities and differences. This was enough of a hint to make her see how to make the rest.

I then asked her if she could make any other hexagons. When she was making one I jumped the gun and said well done when she had made this:

But she said “no I’ve not finished” and added a piece to get this:

I found this quite interesting. She didn’t seem to think that a shape could be finished if it was convex like the first one. We talked about what makes a hexagon and how both versions were.

I then removed 2 purple squares and 2 blue rhombuses (rhombii?) from the second shape and asked if this was a hexagon. She agreed it was and we discussed the similarities and differences in shape between this and the red regular hexagon. Both have 6 equal sides, but they aren’t the same. I did tell her what a regular hexagon was at this point and it wasn’t a term she’d heard before, we didn’t speak long on it though so I don’t know if she will remember.

She also came up with this one:

And this other regular hexagon:

I like this one, and we talked about the similarities and differences between this and a lone red hexagon. I didn’t think she would be ready for a discussion on length and area scale factors yet, but this strikes me as an excellent visual representation of this and it’s certainly one I could see using in a KS3/4 class.

At this point we started talking about how the shapes fit together. I did mention the terms tesselation and tiling but didn’t dwell on them. I asked her if she could find single shape patterns that did and she came up with this:

She said it looked like a honeycomb, and we discussed that bees build them in this shape and talked about why. I also showed her pictures of Giant’s Causeway and the hexagon stones there and discussed how they occur in lots if places in nature.

She then wanted to make “honeycombs” of other colours:

She really liked this one as she said there was an extra 4th hexagon hidden in it. Which she in fact noticed before I did.

Her original green shape was:

Which doesnt look like a honeycomb but which she liked because it looked like it was “on fire”.

She didn’t say much about the yellow, but I thought it looked like a set of screws.

That was about it for the session. She played a bit more and I really liked these shapes made:

Then we packed away. It was really fun for both of us to make these shapes, it’s the first time I’ve really played with Pattern Blocks and I can see they will be great for building my daughters maths. I have also started to see where I might be able to use them in my lessons, so a win win all round.

*This is the 7th post in a series about the use of manipulatives in maths teaching. You can read the others **here**:*

Manipulatives – the start of a journey

Meaning making with manipulatives

Patterns, sequences and fractions

Making numbers and quadratic sequences

__Reference:__

Gregg, S. 2020. *Pattern Blocks. *Derby: Association of Teachers of Maths.

Ollerton, M. Gregg, S. And Williams, H. 2017. *Cusineire- from early years to adult.** *Derby: Association of Teachers of Maths.

## Two circles and a trapezium

This puzzle is one that really got me thinking about a number of things, and had me stuck for a little while. I found it on twitter, where it was shared by Elyem Gercek Boss (@_eylem_99) . Here it is:

I started, as always, with a sketch:

I feel that I should mention here that the radius OF wasnt added until much later. In fact adding this line was my breakthrough moment to be honest. I had been struggling for a bit when I thought to add it in. But more of that later. The first thing I did was this:

I saw I come express AC in terms of the 2 radii, I then moved on from that and looked at the similar triangles I had and came up with x = ((r1)^2)/r2

I then worked out I had another similar triangle:

And decided this must be the route to the answer. I did various things:

Including this that was a load of work to get a result I already knew.

After some dead ends and obsessing over the similar triangles I looked back to my original triangle and that’s when I drew the line:

I realised that as I already had an equation in r1, r2 and x and a right angled triangle featuring these on the sides this would be a route to go down.

I realised I had a a quadratic:

At this point I could express the square of one radius over the square of the other, and as area is directly proportional to the square of the radius this was all I needed:

A really nice problem that had me thinking lots. I’m certainly there will be more concise solutions. If you have one, I’d love to see it! Please let me know in the comments or on social media

## 4 squares, find the pink area

I came across this lovely puzzle from Catriona Shearer (@cshearer41) the other day:

I liked the look of it, and the root to a solution wasn’t initially obvious to me so I thought I’d have a go.

First, as usual, I drew a sketch:

I started marking on the side lengths I knew, the started to consider angles. I could see that I had a pair of similar triangles so figured that might be a good route.

I drew out another sketch this time only including the info I needed. I had the 2 short legs on of of the similar right angled triangles, and the hypotenuse of the other. So figured if I used Pythagoras’s theorem I could have values of a matching side:

This allowed me to calculate the scale factor. From here I could use that to find the length of a side of the pink square:

And hence calculate the area (36).

This was a lovely puzzle that doesn’t take a particularly high level of maths to solve, but is not a problem that presents an obvious solution. I think that by getting our students involved with this sort of problem it helps them start to build problem solving strategies, and that part if the reason I love problems like this and like to include them in my teaching.

## A surprising fraction?

The other day I saw this tweet from John Rowe (@MrJohnRowe):

I looked at the picture and decided the answer was probably a half. And thought about it a bit more. The 4 small circles in the top left should be the same as the white quarter circle below, and the 4 quarter circles in the top right should fit over the white circle below them. This was interesting to me, and I thought I’d look at the algebra.

Looking at the top left corner, the circles involved there have the smallest radius, so we will call that radius r. That means each circle has a radius of pi r².

Below it we have a square the same size and a quarter circle radius 4r, so the white area is (16pi r²)/4 or 4pi r² this is the same as the pink area in the square above. We can see from this that half of the left rectangle is pink. Or we can continue with our algebra. The area of each square is 16r² (its sidelength is 4r) so the pink but here is 16r² – 4pi r², so if we add this to the bit above it we have 16r² shaded (and 16r² white).

The top right has 4 shaded quarter circles, each with radius 2r, so the total shaded area is 4pi r². Below it the white circle has the same radius so same area 4pi r², again that makes the shaded area 16r² – 4pi r² so the shaded area in the right rectangle 16r², and in the big square 32r². The total area of the big square if 64r² (radius is 8r) so the shaded fraction is a half. Which is nice.

Now I know I started by saying I thought it looked like a half, which is what I did think at the time I first saw it. But I still think it’s a surprise fraction. I’ve done a lot of geometry puzzles. And many have included shapes like this, so when I look at this that knowledge helps me see. Most students would not have done anywhere near the amount of puzzles I have so wont have that foresight, and I think it would be a very surprising result that could open many up to the wow factor. I also think that it might be a good starting point for some rich class discussions.

I think it’s a great visual and a lovely answer.

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