5632
/www/nrich/html/content/id/5632/
1
2011-02-01T00:00:01
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There are lots of different routes from $A$ to $B$ in this
diagram:<br></br>
<br></br>
<mdo:image width="323" height="273" alt="routes" src="deci.gif"></mdo:image><br></br>
<br></br>
The idea is to work out the product of the numbers on these
different routes from $A$ to $B$. Let's say that in a route you are
not allowed to visit a point more than once. <br></br>
<br></br>
For example, we could have $3\times0.5$ but we couldn't have
$3\times2\times5\times4\times1\times 0.1$ because that route passes
through $A$ twice. <br></br>
<p>Which route or routes give the largest product?</p>
<p>Which route or routes give the smallest product?</p>
<p>Do you have any quick ways of working out the products each
time?</p>
<p><span style="font-style: italic;">[This problem is adapted from
a SMILE Centre card.</span><span style="font-style: italic;">]</span></p>
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<p class="editorial">Neil from Kells Lande Primary School wrote to
say:</p>
The route with the largest product is going up from A along the $5$
line, and down along the $2$ line, and then along the
$0.5$ line to B, which has a product of $5$.<br></br>
The route with the smallest product is to go horizontally along the
$3$ line from A, then down the first $0.5$ line, along the $1$ line
going horizontally, and then up to B along the $0.1$ line, which
has a product of $0.15$.<br></br>
<br></br>
<p class="editorial">You're right, well done, Neil. Class K's Magic
Mathematicians at Charter Primary School told us how they found out
the solution:</p>
We worked in a group of four and found out all the routes from A to
B, then we did all the sums and worked out all the answers.
<p class="editorial">Many of you also said that to find the largest
and smallest products, it helped to look for the largest and
smallest numbers. That's a good strategy, well done.</p>
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<h2>Route Product</h2>
<br></br>
There are lots of different routes from $A$ to $B$ in this diagram:<br></br>
<br></br>
<mdo:image alt="routes" height="273" src="deci.gif" width="323"></mdo:image><br></br>
<br></br>
The idea is to work out the product of the numbers on these different routes from $A$ to $B$. Let's say that in a route you are not allowed to visit a point more than once.<br></br>
<br></br>
For example, we could have $3\times0.5$ but we couldn't have $3\times2\times5\times4\times1\times 0.1$ because that route passes through $A$ twice.<br></br>
<p>Which route or routes give the largest product?</p>
<p>Which route or routes give the smallest product?</p>
<p>Do you have any quick ways of working out the products each time?</p>
<p><span style="font-style: italic;">[This problem is adapted from a SMILE Centre card.</span><span style="font-style: italic;">]</span></p>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>Multiplying decmals is perceived as a tricky task but <a href="http://nrich.maths.org/public/viewer.php?obj_id=5632&part=index">this problem</a> is written to encourage children to make the link between multiplying by a decimal and dividing by a whole number. This challenge also requires learners to work in a systematic way.</div>
<br></br>
<h3>Possible approach</h3>
<div>You might find it useful to leave pupils to have a go at this challenge without saying much at all by way of introduction. It would probably be helpful for them to have a paper copy of the grid.</div>
<br></br>
<div>After a few minutes invite them to comment on how they are going about the task. It would be useful to focus on two aspects. Firstly, the way that they are approaching the problem - how will they know they have looked at all the routes? This would be worth a discussion on having a system, such as starting with all the routes that begin by going up from A along the $5$ path, then those that
start with the $3$ etc. Secondly, you could talk about their methods for calculating the products in each case. Some learners might suggest that multiplying by $2$ 'cancels out' multiplying by $0.5$, for example, and you can encourage them to explain why.</div>
<br></br>
<div>You could ask pupils to write each route (and its product) on a separate strip of paper which could then be stuck on the board in the plenary. Ask the class to organise the strips to help them decide whether any have been missed out. Answering the questions will then be very straightforward.</div>
<br></br>
<h3>Key questions</h3>
<div>How do you know you have looked at all the possible routes?</div>
<div>How are you calculating the product each time?</div>
<br></br>
<h3>Possible extension</h3>
<div>As an extension activity you could invite children to create their own grid using some criteria. For example, could they make the product of all routes $1$? (Perhaps without using the number $1$?) They could start by using the route design given and then make up their own route maps.</div>
<br></br>
<h3>Possible support</h3>
<div>It might help some learners to express the decimals as fractions before finding the products. This may well come up in the whole-class discussion.</div>
<br></br>
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Can you see immediately which routes might have large products and
which ones small products?<br></br>
How will you know that you have checked all the routes?<br></br>
How many routes can you find which use the $5$? What is the product
of each one?<br></br>
How many routes use the $3$? And the $4$? <br></br></mdoxml>
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1. 5 x 0.1 = 0.5<br></br>
2. 5 x 2 x 0.5 = 5<br></br>
3. 5 x 2 x 2 x 0.1 = 2<br></br>
4. 5 x 2 x 0.5 x 1 x 0.1 = 0.5<br></br>
5. 3 x 0.5 = 1.5<br></br>
6. 3 x 2 x 0.1 = 0.6<br></br>
7. 3 x 2 x 0.1 = 0.6<br></br>
8. 3 x 0.5 x 1 x 0.1 = 0.15<br></br>
9. 4 x 1 x 0.1 = 0.4<br></br>
10. 4 x 1 x 2 x 0.5 = 4<br></br>
11. 4 x 1 x 2 x 2 x 0.1 = 1.6<br></br>
12. 4 x 0.5 x 0.5 = 1<br></br>
13. 4 x 0.5 x 2 x 0.1 = 0.4<br></br>
14. 4 x 0.5 x 2 x 0.1 = 0.4<br></br>
<br></br>
Therefore largest is 5 (route 2) and smallest is 0.15 (route
8)<br></br>
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Route Product
Find the product of the numbers on the routes from A to B. Which
route has the smallest product? Which the largest?
Using, Applying and Reasoning about Mathematics
Working systematically
Calculations and Numerical Methods
Multiplication & division
Fractions, Decimals, Percentages, Ratio and Proportion
Calculating with decimals
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Upper primary mapping document