28 - upward and onward
Problem
You may like to have a go at the problem So It's 28 before trying this challenge.
How about if we put some small cubes together and count the faces which are visible. Can we arrange the cubes so that there are exactly 28 faces?
Getting Started
How many cubes will you start with?
Try making a shape and then counting the number of faces. Are there more than $28$ or fewer than $28$? Can you see how to change your shape to get exactly $28$ faces?
Student Solutions
We had some lovely ideas sent in. It would have been good to be there when they were created and to know how you got your solutions. Here are just a few. I was interested in both Alexander's and George's good descriptions in words to say what they created.
Alexander wrote:
I used $9$ bricks. I put $8$ of them together in a $3$ by $3$ square with the middle one missing. I then put the $9$th brick on the middle brick of one of the rows. So I had one brick with $5$ faces showing, $7$ bricks with $3$ faces showing and $1$ brick with $2$ faces showing.
and George wrote;
I used $11$ bricks. I had $1$ row of $3$ bricks joined to a row of $4$ bricks and the $4$ brick row was joined to another row of $3$ bricks. On one of the rows of $3$ bricks I put the $11$th brick at the end on top.
Ewan, Fraser, Jenny Lee and Ayan had a good idea of only counting the faces you could see, so they sent in this picture;
Kristy and Amy used the computer and sent in;
Zain sent in two ideas, one of which is;
Finally Lily and Alex sent in:
Well done everyone! I'm sure that many more of you who did not send in solutions came up with some interesting ideas and if your class/group did not have a go try it now!
Teachers' Resources
28 - Upward and Onward
You may like to have a go at the problem So It's 28 before trying this challenge.
How about if we put some small cubes together and count the faces which are visible. Can we arrange the cubes so that there are exactly 28 faces?
Why do this problem?
Possible approach
You could introduce this activity by sharing the examples of 28 in the problem and asking if any of them know something special about the number 28.
Show the group an arrangement of cubes resting on a surface and explain that 28 small faces are visible. Check that everyone is happy that we're not counting the faces that are resting on the surface. Encourage learners to work in pairs to find other arrangements which have 28 faces. Depending on your supplies of cubes, you can suggest children keep each arrangement once it is made, or that they record it in some way. (Isometric paper may be useful.)
You may like to keep this challenge as an ongoing 'simmering' activity which children add to over a period of a number of days/weeks.
Key questions
Tell me about this.
Have you anything to say about this shape that you have made?`
How could you change your shape and still have 28 faces showing?
Possible extension
Children could explore other notable things about 28, for example it's a hexagonal number. 28 is the sum of the first five consecutive primes: 2, 3, 5, 7, 11. You might like to introduce them to perfect numbers.
Possible support
Plenty of cubes will be needed for this activity. If children are struggling to record their arrangements, digital photographs may be the answer.