Fitted
3 out of 3
Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?
Problem
Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15 and 18 cm can be fitted together with no gaps and no overlaps, to form a rectangle.
What are the dimensions of the rectangle?
Once you've had a chance to think about it, click below to see how three different pupils began working on the task.
This is how Anna started:
Here is what Brendan tried:
Here is Chandra's initial approach to the problem:
Can you take each of these starting ideas and develop them into a solution?
Getting Started
What do you know about the rectangle?
Have a look at each approach. What do you think he/she was doing?
How do you think this will help to solve the problem?
What do you think he/she would have done next?
Student Solutions
We have had some good solutions for this challenge.
Here is what Ella from Neil Cummins school in the USA wrote:
I figured out the problem by first finding the area of all of the squares. Once I figured out the area of the squares, I added all of the areas together. At first I got 956 for the area, but when Mr. Owens showed us what other students had done, the right answer for the area was shown on the screen. It turned out that I had forgotten to add in the area of the 10 by 10 square (100).
I then figured out that you have to find all of the factors for the total area. Mr. Owens showed us all of the factors for 1056, which is the total area. He also mentioned that the sides of the rectangle could be bigger than 18 (which is the dimensions of the biggest square). I tried dividing that by 18, but the answer turned out to be 58 r 12. After I had listed them down, I thought that 32 x 33 would be the most likely of all. I then drew a rectangle with those dimensions, and tried experimenting. At first, I wasn't sure where to put the 18 square, but my friends Dakota and Hannah suggested that I put it in a corner. So I started by putting the 18 square in the top left corner. Since 18 + 18 is 36, I knew that 18 + 14 is 32. Therefore, underneath the 18 square, I put the 14 square. I put the 4 square right underneath the 18 square on the right side. Then, since 4 + 10 is 14, I added the 10 square right underneath it. I realised that the 9 square could fit beside the 10 square, so I put it there. I made a mistake at first and made the dimensions of the 9 square 9 by 10. Since it looked equal, I thought I was on to something. After that, I tried to put the 15 square on top of the 10 square, but I knew it wouldn't work. So then I tried the 7 square. It fitted well, and there were 15 squares left on top of it. There were 8 squares left over. It was perfect for the 8. But, I realised that the 8 was a square taller than the 7. That was when I figured out that I had made a mistake. So I corrected the 9 and 8 squares, and found the perfect spots for the 15 and 1 squares. Then, I realised that it all fited!
Here's the result:
Ella, that is written and explained very well. Well done!
Phoebe and Isobel from Greenacre School for Girls wrote the following:
We worked out the area that the rectangle would be and then put some squares together to find possible lengths of the side of the rectangle. We knew we had to be able to divide the total area by the length of a side exactly, and checked that. When we put the 15x15 square next to the 18x18 square we got a length of 33cm and found this was a factor of the total area and that the other side had to be 32 cm, so we rearranged the squares until we had made that rectangle.
Here's the result:
Molly and Elizabeth from The Academy of Cuxton Schools sent in their thoughts:
We are proud to have solved this problem. We are exhilarated that we completed this task; it made us feel that any goal can be accomplished.
We completed this investigation by cutting out the squares and physically moving them into place. It took a lot of trial and improvement but we did it in the end.
We found one solution but if we kept trying, we believe we may find more. How many have you found?
Here's the result:
Drew from Y6 Keston Primary sent in the following:-
I've worked out a solution for Fitted. I cut out the squares and managed to complete a seamless square - actually a rectangle. Adding the dimensions gives 32cm x 33cm.
After I'd done this, I calculated the areas of all 9 squares and added them which totalled 1056cm$^2$. I multiplied my shape's side lengths together i.e. 32x33 and it totalled 1056cm$^2$!
Well done to everyone who sent in their results. It was good to read the different methods for getting a solution.
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
This problem is a challenging context in which to explore area. It can be solved in many different ways and reflection on different methods is encouraged through inclusion of the three sample approaches. You may well need more than one lesson for this activity.
Possible approach
Pose the problem orally, or project the text onto a screen, without mentioning the examples of how some children started. Give the class a few minutes to consider, individually, how they might go about tackling the problem, then pair them up and suggest that they talk to their partner about their ideas so far. Try to stand back and observe, and resist the temptation to make helpful suggestions!
Allow pairs to work on the task so that you feel they have made some progress, but do not worry if they have not completed it or if they report being stuck. The aim at this stage is for everyone to 'get into' the problem and work hard on trying to solve it, but not necessarily to achieve a final solution. Make sure that the children have easy access to any resources that they are likely to need but don't put anything out on tables already otherwise this may lead them down particular routes.
At a suitable time, hand out this sheet to pairs. Suggest to the class that when they've finished or can't make any further progress, they should look at the sheet showing three approaches used by children working on this task. Pose the question, "What might each do next? Can you take each of their starting ideas and develop them into a solution?" You may like pairs to record their work on large sheets of paper, which might be more easily shared with the rest of the class in the plenary.
Allow at least fifteen minutes for a final discussion. Invite some pairs to explain how the three different methods might be continued. You may find that some members of the class used completely different approaches when they worked on the task to begin with, so ask them to share their methods too. You can then facilitate a discussion about the advantages and disadvantages of each. Which way would they choose to use if they were presented with a similar task in the future? Why?
Key questions
Tell me about this approach. What do you think he/she was doing?
How do you think this will help to solve the problem?
What do you think he/she would have done next?
Possible extension
Tiles on a Patio is an investigation based on similar ideas which would make a good follow-up activity.
Possible support
It might help some pupils to have sets of squares cut out ready for use. However, only give them out if children explicitly decide to approach the task in this way at the start, or if they are continuing Brendan's solution, and you think that they would spend too long accurately drawing and cutting the squares themselves.