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Remainders

Age 7 to 11 Challenge Level:

Remainders




Here is a chance to explore some properties of numbers and then have a go at some questions.

The interactivity below allows you to choose a divisor and then select numbers in one of the columns.

Example 1:
Divide by 4 and select all the numbers in the right-hand column - they should all turn red.
Now divide by 5 and select all the numbers in the right-hand column - they should all turn yellow, but some will turn orange.
What is special about the numbers that turn orange?
Now divide by 3 and select all the numbers in the right-hand column - most should turn blue, but one will turn black.
What is special about the number that turns black?
What is special about the numbers that turn green and purple?

Example 2: (you will need to clear your previous work)
Find the numbers that have a remainder of 2 when divided by 5 - you'll need to divide by 5 and select the numbers in the second column.
Now select the numbers that have a remainder of 1 when divided by 2 (the odd numbers).
What is special about the numbers that turned orange this time?

Try a few examples of your own and try to predict what will happen in each case.

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Now try out the problem generator below. When you click "Start" the computer will select at random an integer between 1 and 100. Can you identify the chosen number?

You can use the interactivity above to help you, but eventually, try to identify the numbers without the aid of the interactivity.


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One final question:

We know that

When 59 is divided by 5, the remainder is 4
When 59 is divided by 4, the remainder is 3
When 59 is divided by 3, the remainder is 2
When 59 is divided by 2, the remainder is 1

Can you find the smallest number with the property that when it is divided by each of the numbers 2 to 10, the remainder is always one less than the number it is has been divided by? Don't forget to explain your reasoning.

 


Why do this problem?

This problem gives a context for the systematic search for solutions, looking for efficient strategies, and reflecting on which aspects of the problem causes it to be harder/easier. The task could provide an interesting context for practising routine tables facts. The random question generator could be used in lesson starters.

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, pose a question from the question generator. Give students a few minutes to think of numbers that fit one or more of the conditions. Gather some answers and explanations until the whole group feel confident suggesting numbers based on statements about divisors and remainders.

With the same, or a new problem, ask students to work in pairs to find:
a number that fits all the conditions,
then to find all the numbers under 100 that fit them all,
then to write two sentences to explain how they know they have got them all.

With the group together, ask for feedback, and put the answer into the answer box. If you have an interactive whiteboard, it might be appropriate to illustrate the logic with the coloured ball interactivity.

Generate a selection of questions, ask students to pick out particular questions that seem easiest/hardest and work on those. On the board, write "what makes a question like this easy/hard?" and tell students that you'll be collecting suggestions after 15 mins.

If a computer room is available, set students to work at computers in pairs. Students can use the coloured ball interactivity to help them, but emphasise that eventually you would like them to identify the numbers without the aid of the interactivity.

You can print this 10 by 10 number grid so that students can keep a record of their working as they narrow down the possibilities.

Then set the students to play The Remainders Game
Who can reach 100 points in the least number of games?
Ask students to explain any strategies they have generated.

Finally, ask them to have a go at the last question in the problem and emphasise that you will expect themto justify their conclusions.

Key questions

Which clues are most helpful?
When does a clue provide no new information?
What is the minimum number of divisions needed to identify the number?

Possible extension

Students, working in pairs, could think of a number themselves and then give their partners three clues to help them identify their number, or, as in The Remainders Game they could allow their partner to choose the divisors.

Students could also have a go at Ewa's Eggs

Possible support

Clapping Times and Music to My Ears suggest how the introductory activity could be extended.
Use the coloured ball interactivity and ask students to predict what will happen.
Flashing Lights may be an accessible starting problem.