Bracelets

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

Cut and Make

Cut a square of paper into three pieces as shown. Now,can you use the 3 pieces to make a large triangle, a parallelogram and the square again?

Is a Square a Rectangle?

How many rectangles can you find in this shape? Which ones are differently sized and which are 'similar'?

Shape Draw

Age 7 to 11 Challenge Level:

Children from St. Peter's Catholic School in Waterlooville submitted explanations and solutions for this problem.  Some worked alone and some in pairs.

Hariny wrote:

On Monday 13th March, we were given the task Shape Draw. I worked with Lydia and started with finding possibilities that will have two parallel lines. I got the possibilities as rectangle, square, parallelogram and rhombus. We then looked if it had two lines of symmetry. We  eliminated parallelogram and rhombus, as they had no lines of symmetry. I also eliminated the square as it had four (4) lines of symmetry.  We debated that the rectangle was an irregular shape, as a regular shape has to have all sides the same and the angles have to be the same. The rectangle had the same angle measurements but not the same sides, which means it is an irregular shape. I then found possibilities to get the area 24cm$^2$ (1, 24, 2,12,  3, 8 and 4, 6). I then added the sides of each one and I got 2cm by 12cm with a bigger perimeter than its area of 28cm. Therefore the answer is a 2cm by 12cm rectangle.

Ann and Lydia wrote:

First, we gathered the information that we had and we came up with few ideas separately. We started to think about the first clue we had been given on our clue list which was to find a shape with an area (24cm$^2$). After trying out our ideas, we took the next clue into account to find a shape with parallel lines and the area (24cm$^2$). Next, we came up with a rectangle and drew it out…. (Along with the next clue to have a numerically bigger perimeter.)  Reading the next statement, we found out that the perimeter on each side had to add up to make a number more than 24. So the ONLY possible way was 2x12. This shape had to be irregular and under the rules we have this shape is technically irregular. This shape had: 2 pairs of parallel lines, 4 right angles, an irregular shape, had 2 lines of symmetry and the shape is a quadrilateral. In total our explanation is correct that it is a rectangle.

Sam and Louis wrote:

We both started by plotting a shape of 24cm$^2$. Then, we looked at the perimeter; it had to be numerically larger than the area. We knew (after a bit of working out) that a rectangle was an irregular quadrilateral – we had to use a quadrilateral to solve the problem, therefore the shape had to have four right angles. The length of each side had to be an even number. The shape also had to have two lines of symmetry. We worked out that the perimeter was 28cm. That is how we found out the shape is a RECTANGLE… which is 12cm x 2cm.

William wrote:

First we had to look at all the clues and see whether all of them were useful.  After working out that the shape's perimeter was larger, it wasn’t useful. We started on the rest: the shape has 2 sets of parallel lines made me think of a Square/rectangle/parallelogram. After that I worked on it having 4 right angles this got rid of a parallelogram. The shape being irregular ruled out a square so it had to be a rectangle.

Ben and Wyatt wrote:

To start with we both focused on the clue: the area of the shape is 24cm$^2$ and the shape's perimeter is numerically larger than its area! The possible answers were: 1cm x 24cm, 2cm x 12cm, 3cm x 8cm and 4cm x 6cm but the only two with its perimeter larger than its own area were: 1cm x 24cm and 2cm x 12cm . Then we looked at the clue: the length of each side is an even number. Then that clue eliminated 1cm x 24cm because 1 isn’t even! After that we checked the other clues and it was: irregular, two lines of symmetry, four right angles, it is a quadrilateral and two pairs of parallel lines; it fits the clues! That was how we found the answer.

Daisy’s explanation for the problem:

If my calculations are correct then the shape should be a 12 x 2 rectangle. The first clue was that it had two pairs of parallel lines so that knocked triangle, circle, semi-circle and pretty much any shape that had more or fewer sides than a square/rectangle, I also found out that it was an irregular quadrilateral with 4 right angles. Another was that the area had to be 24cm$^2$. I narrowed it down so that the only possibilities would be: 1 x 24, 2 x 12, 3 x 8 or 4 x 6 but then I found out that the shape's perimeter was numerically larger than the area and the length of each side was an even number. Those clues knocked off 3 x 8, 1 x 24 and 4 x 6 because 3 and 1 are odd numbers and the perimeter of 6 x 4 is 20 which is smaller than 24. The last one was 2 x 12 and to check that was correct I made sure that the shape was irregular and that it had 2 lines of symmetry. That made it a 2 x 12 rectangle.

Hannah’s Explanation to Shape Draw:

First, I knew that the perimeter needed to be bigger than 24cm$^2$ because that was the area and it says the perimeter is bigger than the area. I tried a kite shape at first, but it didn’t work. Then, I thought a rectangle because it is irregular, has two pairs of parallel lines, 4
right angles, two lines of symmetry and it is a quadrilateral. Immediately, I thought that 12 x 2 which is 24, this has the right area. Lastly, I added up the area (12 + 12 + 2 + 2) and it equalled 28cm which is numerically bigger than 24cm$^2$. This meets all the criteria on the list.

Loren and Ana wrote:

We (Loren and Ana) started thinking about the area (24cm$^2$). We thought about what multiplication we could use that could total to 24. We came up with 2 x 12, 3 x 8 and 6 x 4. The whole perimeter had to be even so it couldn’t be 3 x 8. The numbers then had to add up to higher then 24 so the perimeter had to be 2 x 12. This made a rectangle. The shape we now had, had two pairs of parallel lines, it had four right angles, the shape was irregular (because not all the sides and angles were the same), it had two lines of symmetry and was a quadrilateral.

Our explanation for Shape Draw Frankie and Loren:

We started by looking at the qualities of a rectangle. It’s got two pairs of parallel sides and has got the area of 24cm$^2$. Before we got the answer we tried 4 x 6 which nearly worked but needed one more quality. Next we tried 2 x 12 which was the right answer. It had all the qualities the shape needed!

From Ryan and Ana:

Me and Ryan started by trying out 4 x 6 and got a rectangle but we got stuck at ‘the shape is irregular’. So, later we tried 2 x 12 and it had two pairs of parallel sides, 4 right angles. The length of each side of the rectangle was an even number and the shape was also a quadrilateral. Next we had the check if our shape had two lines of symmetry. So we measured it up and it did. We got stuck when it said the shape was irregular. In the end we worked out what irregular means. It turns that our rectangle was irregular. After that, we had to use our adding knowledge to test if it added up to a number numerically larger than 24cm$^2$. At the beginning we had to get two numbers that when multiplied would add up to 24cm and 12 x 2
does so we knew that we had the right shape.

Talia's explanation:

Firstly we started by writing down all the combinations with parallel lines (24cm$^2$) area and with four right angles. Then I was left with four and I read the bit that said ‘the shape's perimeter is numerically larger than its area'. So I eliminated it down to two shapes. Then it says ‘the length of each side is an even number.’ So I had (1cm by 24cm) and (2cm by 12cm). So the only possible answer is: 2cm by 12cm. In my investigation I didn’t use any operations because it wouldn’t be useful. However I used my knowledge of properties of a shape.

Well done all of you. It was very interesting to read your different accounts of the ways you worked. Thank you very much and keep up the good work. I look forward to reading further accounts of activities from our site, that you worked on.

We also had some good solutions from St. Matthew's C of E School in Redhill, namely Nithish, Jasmine C, Jasmine L, Leon, Daniel and Jonathan. Thank you for those, they were also well explained.