### I'm Eight

Find a great variety of ways of asking questions which make 8.

### Repeaters

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

### Oh! Hidden Inside?

Find the number which has 8 divisors, such that the product of the divisors is 331776.

### Why do this problem?

Most of us can carry out long multiplications using a standard method. But do we understand what we are doing? Here is a chance to find out...

### Possible approach

Rather than showing the videos from the problem, you may choose to study the methods and recreate them on the board at the start of the lesson (in silence, as in the videos, so that students are expected to make sense of it without any explanation offered). The latter approach has the advantage of preserving a record of the four methods, if your board is big enough. The rest of the lesson follows in the same way however you choose to introduce it.

"Here are two multiplication calculations that I'd like you to do, using whatever method you like:"
$$23 \times 21$$ $$246 \times 34$$

Give students a short time to carry out the multiplications, perhaps on individual whiteboards.

"Now I'm going to show you four methods that could be used to work out those multiplications. Some of you used some of these methods, but there might be methods that you haven't seen before. Watch carefully, and see if you can work out what's going on."

Show the videos, or recreate the calculations from the videos on the board.

Then hand out this worksheet showing each finished method for $246 \times 34$.
"With your partner, try to recreate the methods and make sense of the different steps. Once you think you have made sense of them, make up a few calculations of your own and work them out using all four methods until you feel confident that you can use each method effectively."

Towards the end of the lesson(s), bring the class together and invite students to explain why each method works, and to discuss the benefits and disadvantages of each method.

Finally, this 2 minute video clip from the film Ma and Pa Kettle could be shown to provoke discussion about the misconceptions in Ma and Pa's methods.

### Key questions

Will each method always work?
Where is the $20 \times 20$? Where is the $20 \times 3$? ...

### Possible extension

Challenge students to adapt each method for multiplying decimals.

Show students this video clip of another multiplication method that requires no intermediate writing down, and invite them to make sense of it.

### Possible support

Offer students this worksheet with the methods for $23 \times 21$ to make sense of first, as there are no 'carry' digits so it is clearer to see what is going on.

Stu Cork created this GeoGebra file to use with this problem, which he has kindly given us permission to share.