How do you see it ?

8296 /www/nrich/html/content/id/8296/ 1 2012-05-01T00:00:00 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>Here are some little problems.<br></br> How would you put them on paper using<br></br> - words?<br></br> - pictures?<br></br> - numbers?<br></br> - objects?<br></br> - other ways?</p> <p style="text-align: left;"><mdo:image src="Zoe%20Sweets%20Pic.jpg" style="vertical-align: middle;"></mdo:image> Zoe had these three sweets.</p> <p style="text-align: left;">Her friend Rajneet gave her two more. How many has Zoe got now?</p> <p><mdo:image src="PIc%20of%208%20Dolls.jpg" style="vertical-align: middle;"></mdo:image> Karala had 8 dolls.</p> <p>She gave 3 of them to her little sister. How many dolls has Karala got now?</p> <p style="text-align: left;"><mdo:image src="all%20cards%20medium.jpg" style="vertical-align: middle;"></mdo:image> It was Toby's birthday and he had 6 birthday cards.</p> <p style="text-align: left;">The postman brought some more cards for him and Toby now has 9 cards.</p> <p style="text-align: left;">How many cards did the postman bring to Toby?</p> <p><mdo:image src="balloons%20in%20space.jpg" style="vertical-align: middle;"></mdo:image>Si had 8 balloons looking bright and bouncy.</p> <p>Some then popped and he only had 5 left that were bright and bouncy.</p> <p>How many had popped?</p> <p> </p> <p style="text-align: left;">Some children were walking to school together.</p> <p style="text-align: left;">Three more joined them.</p> <p style="text-align: left;">Now there were 9 children altogether.</p> <p style="text-align: left;">How many were walking together first of all?</p> <p> </p> <p style="text-align: left;">Marcus had some coins in his pocket.</p> <p style="text-align: left;">Sadly he lost 7 of them, now he only</p> <p style="text-align: left;">has three coins. How many coins</p> <p style="text-align: left;">were there before he lost them?</p> <p> </p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><span class="editorial">We had a number of solutions sent in that provided the "answers" to the calculations. Voltaire  from Ravenor Primary School  in England  sent in the calculations explanation as follows:</span><br></br> <br></br> $1.5$ doing $3+2=5$, $2.5$ doing $8-3=5$ $3.3$ doing $9-6=3$,<br></br> <br></br> $4.3$ doing $8-5=3$, $5.6$ doing $9-3=6$, $6.10$ doing $3+7=10$<br></br> <br></br> <span class="editorial">Our first contribution to the Primary site, not in English came from Jose at the Esapana Istituto;</span><br></br> <br></br> Esto es para dos de los problemas. Mi hermana pequeña, Anya ellos lo hicieron por mí. Ella tiene cinco años.<br></br> Zoe tiene 3 caramelos. Rajneeti le da dos más. 3 +2 = 5 / tres y dos son cinco<br></br> Karala tenía 8 muñecas.<br></br> Ella le dio tres de ellos a su hermana pequeña. ¿Cómo ha Karala muchas muñecas tiene ahora?<br></br> Así que 8-3 = 5 / ocho menos tres es igual a cinco<br></br> <br></br> <span class="editorial">Which I think says:</span><br></br> <br></br> This is for two problems. My little sister, Anya they did for me. She is five years.<br></br> <br></br> Zoe has $3$ candies. Rajneeti gives you two more. $3 +2 = 5$ / three and two are<br></br> five<br></br> <br></br> Kerala had $8$ dolls.<br></br> She gave three of them to his little sister. How many has Karala<br></br> dolls have now?<br></br> <br></br> So $8-3 = 5$ / eight minus three is equal to five<br></br> <br></br> <span class="editorial">Thank you Jose, for that. Thanks also to Voltaire</span><br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>How Do You See it?</h2> <br></br> <p>Here are some little problems.<br></br> How would you put them on paper using<br></br> - words?<br></br> - pictures?<br></br> - numbers?<br></br> - objects?<br></br> - other ways?</p> <p style="text-align: left;"><mdo:image src="Zoe%20Sweets%20Pic.jpg" style="vertical-align: middle;"></mdo:image> Zoe had these three sweets.</p> <p style="text-align: left;">Her friend Rajneet gave her two more. How many has Zoe got now?</p> <p><mdo:image src="PIc%20of%208%20Dolls.jpg" style="vertical-align: middle;"></mdo:image> Karala had 8 dolls.</p> <p>She gave 3 of them to her little sister. How many dolls has Karala got now?</p> <p style="text-align: left;"><mdo:image src="all%20cards%20medium.jpg" style="vertical-align: middle;"></mdo:image> It was Toby's birthday and he had 6 birthday cards.</p> <p style="text-align: left;">The postman brought some more cards for him and Toby now has 9 cards.</p> <p style="text-align: left;">How many cards did the postman bring to Toby?</p> <p><mdo:image src="balloons%20in%20space.jpg" style="vertical-align: middle;"></mdo:image>Si had 8 balloons looking bright and bouncy.</p> <p>Some then popped and he only had 5 left that were bright and bouncy.</p> <p>How many had popped?</p> <p> </p> <p style="text-align: left;">Some children were walking to school together.</p> <p style="text-align: left;">Three more joined them.</p> <p style="text-align: left;">Now there were 9 children altogether.</p> <p style="text-align: left;">How many were walking together first of all?</p> <p> </p> <p style="text-align: left;">Marcus had some coins in his pocket.</p> <p style="text-align: left;">Sadly he lost 7 of them, now he only</p> <p style="text-align: left;">has three coins. How many coins</p> <p style="text-align: left;">were there before he lost them?</p> <p> </p> </div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div>This <a href="http://www.nrich.maths.org.uk/8296">activity</a> encourages children to think of different ways of viewing the calculation. Whilst they are doing this the teacher can observe just how their pupils are considering a problem, and this may be somewhat different from the taught approach. Given the opportunity, children often have their own ways of working.</div> <h3> </h3> <h3>Possible approach</h3> <div>Talk to the children about how different people do things in different ways and explain that this actvity is all about that - it's important that the children don't presume that there is one way and one way only to see the calculation.</div> <div> </div> <div>Working within a pair or small group and tackling one problem at a time can help children to focus more deeply on one task rather than racing through them. You could suggest that they think and talk about what they are going to do <em>before</em> they actually begin.</div> <div> </div> <div>They may decide to enact with objects or make a picture and just record the answer. Or use these as a prompt to transfer the problem to a calculation which they would record horizontally. Whichever, your observations will allow you to reflect on the children's confidence, language and understanding and possibly what misconceptions they hold.</div> <div> </div> <div>These particular examples are reasonably straight forward. 'How many more than..?' type questions are much more difficult because the word 'more' usually signifies addition but the calculation here is a subtraction. It's worth introducing word problems in this category separately by taking a lesson to focus on them. </div> <div> </div> <div>There are eight types of change and compare word problems. More examples including the ones that children typically find difficult, can be found here as <a class="doclink" href="/content/id/8296/How%20do%20you%20see%20it%20with%20Su.docx">.doc , </a> <a class="pdflink" href="/content/id/8296/How%20do%20you%20see%20it%20with%20Su.pdf">.pdf</a> You can read more about this in:</div> <div>Harriett C. Bebout, (1990) <em>Children's Symbolic Representation of Addition and Subtraction Word Problems, </em> Journal for Research in Mathematics Education, Vol. 21, No. 2 , pp. 123- 131, NCTM</div> <div> </div> <h3>Key questions</h3> <div>Tell me how you are working this out.</div> <div>What do you think of _____'s way of doing this?</div> <div>How did you know what to do first?</div> <div>Is the way you've done this differently from the way you do other additions/subtractions?</div> <br></br> <h3>Possible extension</h3> <div>Ask the pupils to create stories that involve calculations for their partners to do.</div> <div>Give the pupils a written form of an addition/subtraction eg [] - 5 = 12 and challenge them to create a story around it.</div> <br></br> <h3>Possible support</h3> <div>Some pupils will benefit from working with some small toys/dolls so that they can enact the situation before being able to think about the calculation, for example the final balloon problem might be modelled using Lego figures and counters as the balloons. The physical act of moving objects in different ways (together, away from) can sometimes help less confident children to work out which operation is appropriate.</div> <p><br></br> <br></br> <br></br> </p></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><p>Find some Lego figures and toys or objects such as counters to make sense of the problem.</p></mdoxml> 2 3 1 1 0 0 0 How do you see it ? Here are some short problems for you to try. Talk to your friends about how you work them out. Using, Applying and Reasoning about Mathematics Recording mathematics Admin Lower primary mapping document Admin Upper primary mapping document