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Remainders

Age 7 to 14 Challenge Level:

Why do this problem?

The interactive in this task offers a visual representation of the patterns that occur when considering multiples and remainders.
By offering the scaffolding of the visual representation, students can answer more challenging questions than they would otherwise be able to engage with, helping them to become more resilient problem-solvers.
Together with the related problem The Remainders Game, we hope students will begin to develop their own internal representations of numbers and remainders in order to answer such questions without needing the support of the interactive.

Possible approach

You could start with a whole class counting activity:

Start counting together, speaking loudly on the numbers in the two times table, and quietly on the other numbers. Now split the class in two. Ask half the class to continue doing the same and ask the other half to only speak loudly on the numbers in the five times table.

Which numbers were quiet ?
Which numbers were fairly loud and which were very loud?

Now split the class in three. Two groups to continue as before and one group to only speak loudly on the numbers in the three times table.

Can they predict what they will hear?
Which numbers will be quiet?
Which numbers will be fairly loud and which will be very loud?

Try it.

Class could be split in four and the new group could be asked to speak loudly on the multiples of four.
When will everyone speak loudly?

Start again and select two numbers which have a common factor, for example, 4s and 6s.
Ask students to predict which numbers will be spoken loudly.
Try it.

After this introductory activity, the first two questions posed in the problem should be relatively straightforward. 


If you have access to computers/tablets for the students, introduce the interactivity and give them some time to explore. Invite students to suggest how the interactivity could be used to help answer questions like the ones they have just tried.

If you don't have computers, you may find these printable sheets useful:
Number Grids - width 3, 4, 5 and 6 
Number Grids - width 7 and 8 
Number Grids - width 9 and 10

Discuss with the children how they could colour in multiples of different numbers to see where they occur in different widths of grids. 

For the next part of the problem, you may want to use the language of a times table being 'shifted', as introduced in the problems Times Tables Shift and Shifting Times Tables.

Show the students the next question that appears in the problem:
 
I'm thinking of a number that is 1 more than a multiple of 7.
My friend is thinking of a number that is 1 more than a multiple of 4.
Could we be thinking of the same number?

Invite students to use the interactivity or printable sheets to answer the question above, and perhaps pose some of their own. For example, in pairs each student chooses a times table and a shift. Can they find a number that occurs in both sequences?

These sorts of questions are very similar to those generated by Charlie's Delightful Machine, which you may wish to explore with your students.

Bring everyone together and pose the last two questions to discuss as a class:

I'm thinking of a number that is 3 more than a multiple of 5.
My friend is thinking of a number that is 8 more than a multiple of 10.
Could we be thinking of the same number?

I'm thinking of a number that is 3 more than a multiple of 6.
My friend is thinking of a number that is 2 more than a multiple of 4.
Could we be thinking of the same number?

Students might offer the following insights:
"All numbers that are 3 more than a multiple of 5 end in a 3 or an 8."
"If I shift the 10 times table, the last digit is always the same."
"All numbers that are 3 more than a multiple of 6 will be odd because multiples of 6 are always even, and even plus odd is always odd."
"You can't be thinking of the same number as one is always even and one is always odd."

In a follow-up lesson, students could apply what they have learnt by playing The Remainders Game.
 

Key questions

I'm thinking of a number that is 3 more than a multiple of 7.
What size grid would help you identify possibilties?
In which column would my number appear?

Possible extension

The problem offers a challenging extension at the end.

Possible support

Encourage students to use the downloadable printable sheets before starting to use the interactivity.  You could pose questions like:
"Underline all the multiples of 5. Put a ring around all the multiples of 2.  What do you notice about the numbers that are both underlined and ringed?"