How do you do it?
This group activity will encourage you to share calculation strategies and to think about which strategy might be the most efficient.
Age
7 to 11
Challenge level
This problem is designed to be done in a group. So, ideally, you need to find three or four other people to work with you.
Firstly, take some time to work on your own to find the answers to these calculations, without writing anything down. (Although it would be good to write down the answer itself so you don't forget it.)
$19 \times 24$
$198 + 997$
Half of $57.6$
$3841 - 665.3$
$5.2 \div 4$
$101 \times 16 \times 4$
Now, join together with the other people in your group and focus on the first question.
Do you all agree on the answer?
Give EVERYONE in the group time to explain how they worked it out.
As a group, decide whose method you think is most efficient and why.
Do the same for each of the six questions: give everyone the chance to explain their own method and then choose the most efficient for that calculation.
For the final bit of the challenge, you will need a set of these cards - one set for the group. Each card has another calculation on it.
As a group, decide on the most efficient method for each of the cards.
What did you decide and why?
Can you round the numbers up or down to help?
Can you 'picture' the calculation in your head somehow?
Perhaps you can use some number facts that you know well to help?
Does your answer sound reasonable?
Can you 'picture' the calculation in your head somehow?
Perhaps you can use some number facts that you know well to help?
Does your answer sound reasonable?
This problem encouraged you to work as a group, discussing and sharing different calculation strategies. There may be several different ways of working out the answer to a particular calculation; some are more efficient than others. Can you find different ways of working out the answer to the questions? Look at different types of questions, such as the ones given in the problem. Are the same methods always the most efficient?
William and Conner from St Patrick's R.C. Primary School submitted their solution:
1. answer is $456$ because
\begin{align}20\times24&=480\\480-24&=456\end{align}
2. answer is $1195$ because
\begin{align}200+997&=1197\\1197-2&=1195\end{align}
3. answer is $28.8$ because
\begin{align}57\div2&=28.5\\0.6\div2&=0.3\\0.3+28.5&=28.8\end{align}
4. answer is $3175.7$ because
\begin{align}3841-600&=3241\\3241-65&=3176\\3176-0.3&=3175.7\end{align}
5. answer is $1.3$ because
\begin{align}4\div4&=1\\1.2\div4&=0.3\\1+0.3&=1.3\end{align}
6. answer is $6464$ because
\begin{align}101\times16&=1616\\1600\times4&=6400\\16\times4&=64\\6400+64&=6464\end{align}
William and Conner tackled these problems by splitting the numbers into more familiar-and easier to handle- numbers. They then dealt with these separately and combined the answers.
I wonder whether you used the same
methods as William and Conner for each calculation?
Ben, Aeron, Isla and Emily School from St Mary Redcliffe Primary explained how they worked out their answers to the calculations on the cards:
$21\times44$
The easiest way to understand is to add $21$ together $44$
times, but this isn't very quick.
A quick way is to add: $20\times40$, $20\times4$, $1\times40$
and $1\times4$
This is $800+80+40+4 = 924$
$397+998$
We think that the best way is to add the hundreds and then the
tens and then the units, and then add them all together.
$300+900 = 1200$, $90+ 90 = 180$, $7+ 8 = 15$
Adding these all together gives $1200+180+15 = 1395$
Half of $31.4$
Half of $30$ is $15$
Half of $1$ is $0.5$
Half of $0.4$ is $0.2$
Adding these $15.7$
$7841 - 239.7$
We thought that the easiest thing to do was to ignore the
$7000$, since $841$ is bigger than $249.7$.
So starting with $841$ we took away the two hundred and then
the thirty and then the $9$ and then the $0.7$.
So $841$ take away $200$ = $641$
$641$ take away $30$ = $611$
$611$ take away $9$ = $602$
$602$ take away $0.7$ = $601.3$
$7.2$ divided by $5$
To divide by $5$, we divide by ten and then double it.
$7.2$ divided by ten is $0.72$ and doubling that gives
$1.44$
$17 \times 102 \times 3$
$17 \times 102$ is the same as $20 \times 102$ take away $3
\times 102$.
This is $1734$.
$1734 \times 3$ is $5202$
The Maths Group at St Teresa's in Surrey had some slightly different methods:
We did the practice calculations in pairs, then presented our
methods to the class and voted for the best method in each case.
Then we applied them to the new set of calcuations, as
follows:
$21 \times 44$:
$10 \times 44 = 440$
Double $440$ = $880$
Add $44$, by adding $20$ and then $24$ = $924$
$397+998$:
Round both: $400 + 1000 = 1400$.
Subtract $5$ = $1395$
Half of $31.4$:
Half of $30$ = $15$
Half of $1.4$ = $0.7$
Add $15 + 0.7 = 15.7$
$7841 - 239.7$
We tried rounding $239.7$ up to $240$, which made the
subtraction easy, but we ran into problems later about how much to
adjust by!
So, we went:
$800 - 200 = 600$
$40 - 39 = 1$
$1 - 0.7 = 0.3$ and added them up to get $7601.3$
$7.2$ divided by $5$:
we divided by $10$ and got $0.72$ and then doubled it =
$1.44$
$17 \times 102 \times 3$:
$17 \times 3 = (10 \times 3) + (7 \times 3) = 51$
$100 \times 51 = 5100$
$2 \times 51 = 102$
Add them up and you get $5202$.
Very well done. I wonder which method you prefer in each case? Or perhaps you have a method that you think works even better for you?
Why do this problem?
This problem is designed for use in a group and it could easily take a couple of lessons. It offers opportunities for learners to discuss their own mental calculation strategies and to make decisions about the effectiveness and efficiency of different strategies in different contexts. The notes below assume that pupils have worked in groups
before and, in particular, that they are good at listening carefully to each other.
Possible approach
This activity will work best if children are in groups of four or five.
Begin by explaining to the whole class that you are going to ask them some questions and you would like them to work out the answers in their heads on their own. (You could allow them to make jottings, but discourage them from writing out calculations.) Once they have come to an answer, ask them to write it down, for example on a mini-whiteboard. Suggest that it would be helpful if they
tried to remember how they worked out the answer as the methods will be the focus of the group work later. Ask each of the following in turn, giving everyone enough time to work on the answer. It might be helpful to write them up on the board as you say each one.
$19 \times 24$
$198 + 997$
Half of $57.6$
$3841 - 665.3$
$5.2 \div 4$
$101 \times 16 \times 4$
Next introduce learners to the group task. Ask them to consider the first calculation and compare answers. It is very possible that they won't agree but in many ways this is a good thing! Explain to them that you would like them to decide what the right answer is by sharing the ways they worked it out. Emphasise that you would like everyone in the group to have a turn at explaining their own method. Once the group has decided on the answer, the second part of the task is to decide which method they think is most efficient. They can then repeat this process for each of the above calculations.
Once the group has completed this, give them a set of these cards. The challenge now is to work out the answers to this new set of calculations, but working together. The questions on the cards are similar in some ways to the questions initially given, so the idea would be for children to notice the similarities. Encourage them, therefore, to use a method that they have identified as efficient in the first part of the task. It would be good to give each group an A3 sheet of paper (or even bigger) to record the methods that they choose for the calculations on the cards. The posters they create can form the basis of a whole-class discussion in the plenary, comparing their chosen methods, and will make an informative display.
Key questions
How will you agree on the answer to this one?
How did you work out the answer to this one?
Has everyone in the group explained their method?
Which method do you think is most efficient? Why?
Possible extension
Some children will enjoy creating their own calculations which they believe could be solved using a particular method.
Possible support
Some learners will benefit from being 'given permission' to jot down steps in their mental calculations. You may wish some to only focus on the methods they would use, rather than actually performing the calculations themselves.