6777
/www/nrich/html/content/id/6777/
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2011-02-01T00:00:01
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<div style="text-align: left;">Put in the missing symbols to make
these number sentences correct.</div>
<br></br>
Use $+ , -, \times , \div$ and $ =$ .<br></br>
<br></br>
For example:<br></br>
<div style="text-align: center;"><mdo:image height="25" width="403" src="6777A.png" alt=""></mdo:image></div>
<div style="text-align: center;"> </div>
<div style="text-align: left;">Try these:</div>
<br></br>
<br></br>
<div style="text-align: center;"><mdo:image height="150" width="98" alt="" src="6777B.png"></mdo:image></div>
<br></br>
<br></br>
All these number sentences below, except two of them, have two
solutions.<br></br>
Can you find the symbols to use?<br></br>
<br></br>
<div style="text-align: center;"><mdo:image height="234" width="103" alt="" src="6777C.png"></mdo:image></div>
<br></br>
Which two number sentences have only one answer? <br></br>
Can you see why this is so?<br></br>
<br></br>
You might like to use this interactivity:<br></br>
<br></br>
<a href="/content/id/6777/WhSymbol.swf">Full screen
version</a>:<br></br>
<mdo:flash height="400" width="550"><param value="/content/id/6777/WhSymbol.swf" name="movie" ></param><param value="8" name="flashplayerversion" ></param></mdo:flash><br></br></mdoxml>
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<p><span class="editorial">This problem challenged you to check
your understanding of the building blocks of maths: "plus",
"minus", "divide", and "multiply". You noticed that the</span>
<span class="editorial" style="font-weight: bold;">order</span>
<span class="editorial">of numbers and symbols is important; in
different positions, the "number sentence" can say different
things. This is the same as when you speak or write; the order of
the words matters. However, with some of the number sentences,
there are two different orders, which still mean the same thing.
Again, this is like speaking or writing; even if the order is
different, it can sometimes mean the same thing.</span></p>
<p><span class="editorial">For the first four questions, you were
asked to put the correct symbol into the box. Several students
submitted correct solutions. These include: Jonathan, Jordan, and
Callum from Aycliffe Drive Primary School, Lauren from Princess
Elizabeth, Anna, Isabella, George, Sophie, and Rhiannon from St.
Swithun's, Nathan from Wilson's, Rebecca from Bourne Westfield
primary school, Ayush from Garden Gate Elementary School, Brandon,
Narissa, Jordan, Justin, and Cameron from Village Elementary, and
Charlotte from Manor Preparatory School.</span></p>
<p class="editorial">Narissa and Jordan wrote out the answer:</p>
$16+18=34$<br></br>
$47-28=19$<br></br>
$18\div 2=9$<br></br>
$30=10\times 3$<br></br>
<br></br>
<p><span class="editorial">Rhiannon, from St Swithun's Primary
School approached this problem by trial and improvement: she placed
a different symbol in the box, and looked to see if the calculation
made sense. In this way, she worked out the correct
symbols.</span></p>
<p class="editorial"><span class="editorial">T</span>he next part
of the problem asked you to fill in the symbols, like in the first
section. However, all of the number sentences but two have two
different solutions. Ayush, from Garden Gate Elementary School
submitted the correct solution:</p>
<br></br>
$51- 36 = 15$<br></br>
$51 = 36 + 15$<br></br>
<br></br>
$45\div 5 = 9$<br></br>
$45 = 5\times 9$<br></br>
<br></br>
$27 + 36 = 63$<br></br>
<br></br>
$70-14 = 56$<br></br>
$70 = 14 + 56$<br></br>
<br></br>
$7\times 5 = 35$<br></br>
<br></br>
$50\div 5 = 10$<br></br>
$50 = 5\times 10$<br></br>
<br></br>
<p class="editorial">As Aimee, from Culford School points out:</p>
The two sentences that only have one solution are:<br></br>
<br></br>
$7\times 5=35$ and<br></br>
$30=10\times3$<br></br>
<br></br>
<p><span class="editorial">Rebecca, from Bourne Westfield Primary
School explained</span>:</p>
<br></br>
On the $27+36=63$, the largest number is at the end so if you tried
to make a second number sentence like $27=36-63$ it wouldn't give
the right answer, because if you take away $63-36$ you would go
into negative numbers. So, to get two "working" number sentences,
you have to have the largest number at the front .<br></br>
Here is an example to show a number sentence that works:<br></br>
$50\div 5=10$ and $50=5\times 10$ .<br></br>
This is a an example to show one that you can only do in one way:
$7\times 5=35$.<br></br>
<br></br>
Charlotte, from Manor Preparatory School, also submitted the
correct solution.<br></br>
<br></br>
<p class="editorial">A few people noticed a reason why there can be
two different solutions for the number sentence. As Jonathan,
Jordan and Callum, from Aycliffe Drive Primary School point out,
"plus" and "minus" are inverse operations (processes), as are
"multiply" and "divide". This means that you can do a sum, for
example a multiplication, and then undo it, by doing the reverse,
or "inverse": divide.</p>
<p class="editorial">Thank you to everyone who submitted answers!
Well done to you all!</p>
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<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=6777&part=index">This problem</a> is designed to help young learners to use the symbols plus, minus, multiplied by, divided by and equals to, meaningfully, in number statements. Children frequently meet boxes or similar devices to represent numbers but seldom the actual operational symbols. This problem also helps learners understand
inverse operations.<br></br>
<h3>Possible approach</h3>
<div>You could start with an example like the ones given at the beginning of the problem. It might be useful to use the interactivity with the whole group at this point until everyone has the idea that they are finding symbols, not numbers.</div>
<br></br>
<div>After this the children could work in pairs on the examples either from the screen or <a href="/content/id/6777/6777.pdf">this printed sheet</a> so that they are able to talk through their ideas with a partner.</div>
<br></br>
<div>At the end of the session the group could gather together again and put up their ideas on the board or use the interactivity. You should also discuss why four of the double number sentences have two answers and two only have one answer. Can they see why this is so?</div>
<br></br>
<div>This problem could also be used as a people maths activity during an assembly with children standing in line holding cards, thus forming a human equation. The audience can tell the 'symbols' where to stand.</div>
<h3>Key questions</h3>
<div>What does this symbol mean?</div>
<div>Which symbol tells you to take away?</div>
<div>What do you have to do to "undo" an addition? What about a subtraction?</div>
<div>What do you have to do to "undo" a multiplication? What about a division?</div>
<h3>Possible extension</h3>
Learners could make some more number statements which can be done in more than one way. Can they make one that can be done three or even four ways?<br></br>
<h3>Possible support</h3>
Suggest using the interactivity or, if this is not possible, counters with the five symbols on them which can be moved around on <a href="/content/id/6777/6777.pdf">this sheet</a>.<br></br>
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It might help to have labelled counters with the five symbols $+ ,
-, \times , \div$ and $ =$ on them, which you can move around on <a href="/content/id/6777/6777.pdf">this sheet</a>.<br></br></mdoxml>
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Which symbol?
Choose a symbol to put into the number sentence.
Calculations and Numerical Methods
Multiplication & division
Calculations and Numerical Methods
Addition & subtraction
Information and Communications Technology
Interactivities
Calculations and Numerical Methods
Inverses