Which is quicker?
Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?
Age
7 to 11
Challenge level
Problem
Image
|
Which is quicker, counting up to $30$ in ones or counting up to $300$ in tens? Why? |
Which is quicker, counting up to $40$ in ones or counting up
to $4,000$ in hundreds?
Which is quicker, counting up to $10$ in ones or counting up
to $1,000,000$ in hundred-thousands?
Which is quicker, counting up to $20$ in ones or counting up
to $140$ in sevens?
Which is quicker, counting up to $25$ in French or in
English?
It's much easier to work on this with a partner!
When you have timed yourselves and decided about the reasons
for your results, you could invent some examples for
yourselves.
You could predict which was going to be quicker and then try
them out to test your prediction.
Getting Started
If you are trying to count in something unfamiliar such as hundred thousands it might be useful to write down and read the first two (or more) numbers in the list.
How will you make a prediction? What sorts of things do you need to take into account?
Student Solutions
Natasha from Moorfield Juniors has answered the first part of this problem. She says:
I thought it would take longer to count in $10$s to $300$ because the numbers were longer.
Idress from Al Ameen school in Dubai agrees with Natasha but adds that for the question of counting in sevens:
We have to add $7$ each time in our head then count so it will
take more time than just counting in ones.
Urara and Andra from Canadian Academy in Japan sent in good ideas.
Urara and Andra wrote;
It's much easier to work on this with a partner! When you have
timed yourselves and decided about the reasons for your results,
you could invent some examples for yourselves. You could predict
which was going to be quicker and then try them out to test your
prediction ...
$7$ seconds to get to $30$ in ones. $11$ seconds to get
to $300$ by tens so counting by ones to get to $30$ is quicker
...
$11$ seconds to get to $40$ in ones. $18$ seconds to get to
$4,000$ in hundreds is slower because we normally don't count in
hundreds so it takes longer time. But we count by ones often
...
I am slower at counting by sevens because we don't use it very
often and we only remember the times table until $7\times12$.
It all helps us to realise that when we are solving problems we have to keep our mind on what is happening as well as just looking at the numbers. A lot of folk did the arithmetic and said that there was no difference. They were looking at the difference in the number of numbers used. But of course the numbers were being said in counting and the people above worked on that idea. Well done!
Teachers' Resources
Why do this problem?
This problem could be used as a short one, suitable for the start of a lesson, but it could also be explored more fully and therefore take more time. It will help learners to come to a deeper understanding of how the number system works and can also be extended to cover various multiplication tables.Possible approach
You could start by simply asking the whole class the question
"Which is quicker, counting up to $30$ in ones or counting up to
$300$ in tens?". Give pairs the chance to think together before
discussing conclusions with the whole group. Encourage pupils
to explain how they decided upon their answer, as well as the
reasons.You could ask for volunteers to come up and do the counting
so you can test out the class's predictions.
Learners could then work in pairs on the variations given in
the problem. A stop-watch for each pair could be useful,
although timing can also be done using their own wrist watches or
the classroom wall-clock. Encourage some sort of recording so that
they can participate more fully in final discussions.
When they have done the suggested examples learners could make
up some of their own to work on and then try them out on
others. You might want to place constraints on these, for
example, can they find an example where they predict the two
countings would take the same length of time?
At the end you could ask about their results and the factors
that affected the speed at which they were able to count. It
is likely that as well as mathematical reasons, there will be some
practical considerations too, such as being very familiar with
counting in some ways compared with others.
Key questions
How are you making your predictions?
How are you recording what you're doing?
What sorts of things affect how quickly you can
count?