186 /www/nrich/html/content/01/01/letme1/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <p><mdo:image src="clown.gif" align="left" width="231" height="291" alt="Image of a clown"></mdo:image></p> <p>Lolla bought a balloon at the circus. She gave the clown six coins to pay for it.</p> <p>What could Lolla have paid for the balloon?</p> <p>Which of your answers seems a reasonable amount to pay for a balloon?</p> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <span class="editorial">Mrs Fother's class sent us in their answers to this problem:</span> <br></br> <br></br> First we made a list of all the possible coins Lolla might have used to pay the clown. These were 1p, 2p, 5p, 10p, 20p, 50p, £1 and £2. <br></br> <br></br> If Lolla paid using only 1p pieces, she would have paid 6p, which we thought was too little for a balloon. <br></br> <br></br> If she paid using only £2 coins, she would have paid £12, which we thought was far too much for a balloon. <br></br> <br></br> Then we looked at how many different prices Lola could have paid using exactly two different types of coin. <br></br> <br></br> With the 1p and the 2p she could have paid 7p, 8p, 9p, 10p or 11p. <br></br> <br></br> With the 1p and the 5p she could have paid 10p, 14p, 18p, 22p or 26p. <br></br> <br></br> Here we thought we saw a pattern. We started off with five 1p coins and one of the other type of coin, and then to get the next largest amount we took away one 1p coin and added another of the other type of coin. So, as we did above, to go from 10p (five 1p coins and one 5p coin) to 14p (four 1p coins and two 5p coins) we took away 1p and added 5p. This is the same as adding 4p. <br></br> <br></br> With the 1p and the 10p the smallest amount she could have paid was five 1p coins and one 10p coin, which makes 15p. 10p - 1p = 9p so we need to add 9p to 15p to get the next smallest amount of money - this is 24p, which is four 1p coins and two 10p coins. <br></br> <br></br> Having found this pattern, we then split into groups to look at how many different prices Lola could have paid using exactly three different types of coin. <br></br> <br></br> Here are some of our results: <br></br> <br></br> With the 1p, the 2p and the 5p, she could have paid 11p, 12p, 13p, 14p, 15p, 16p, 17p, 19p, 20p or 23p. <br></br> <br></br> With the 5p, the 10p and the 50p, she could have paid 80p, 85p, 90p, 95p, £1.25, £1.30, £1.35, £1.70, £1.75 or £2.15.<br></br> <br></br> <span class="editorial">Thank you very much, Mrs Fother's class!</span> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Buying a Balloon</h2> <p><mdo:image align="left" alt="Image of a clown" height="291" src="clown.gif" width="231"></mdo:image></p> <p>Lolla bought a balloon at the circus. She gave the clown six coins to pay for it.</p> <p>What could Lolla have paid for the balloon?</p> <p>Which of your answers seems a reasonable amount to pay for a balloon?</p> </div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=186&amp;part=index">This problem</a> offers an opportunity for learners to use numerical operations (addition, subtraction and possibly multiplication) and can be used to highlight ways of working systematically.</div> <br></br> <h3>Possible approach</h3> <div>The problem could be introduced through story and a real balloon can also be used to engage the children. Children can be asked if they have had balloons at home, the types of occasions when balloons are used as decorations, where they can be purchased and how much might they cost.</div> <br></br> <div>Give children time to work on the problem for a few minutes with large sheets of paper available for them to record any solutions. Then invite some children to suggest some different amounts, checking that they can be made with exactly six coins. You could ask what the largest amount Lolla could have paid was, and the smallest amount. It might be appropriate for you to narrow down the problem at this stage so that you are able to emphasise ways of working systematically, so challenge the class to find ALL the different amounts which could be made with six $1$p or $2$p coins only. Invite them to record their ways on strips of paper (each way on a separate strip) as this will make it easier later.</div> <br></br> <div>Having given the group time to work on this, draw them together to find out the different amounts they have made. Ask children to come and stick a strip on the board so you begin to collate some different combinations. Once you have quite a few (there are seven altogether), ask the children how they know whether or not they have all the possible solutions. At this stage, you may be able to highlight some methods that you noticed while the children were working, and you can ask learners for their suggestions. Take up one of these (for example starting with all the $1$ps, then swapping one $1$p for a $2$p, then swapping another $1$p for another $2$p etc.) and order the strips of paper to reflect this on the board. In this way, pupils will notice any gaps and having this modelled will help on future occasions.</div> <br></br> <h3>Key questions</h3> <div>What is the largest amount of money we could make?</div> <div>What is the smallest amount we could make?</div> <div>How will we know when we have all the possibilities?</div> <br></br> <h3>Possible extension</h3> <div>Children could go on to find all the possible combinations of six coins in a similar way if $1$ps, $2$ps and $5$ps are available. Some may be able to use the solution for just $1$ps and $2$ps to help.</div> <br></br> <h3>Possible support</h3> <div>Having plastic (or real) coins available will help the children identify, name and sort to find possible answers.</div> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> What is the largest amount of money we could make? <br></br> What is the smallest amount we could make? <br></br> How will we know when we have all the possibilities?<br></br> You could try all the combinations with just $1$ps and $2$ps first. What would happen if you have $5$ps as well?<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Ask the children to find as many combinations of six coins as they can. One challenge is to organise the answers in a way that allows the children to discover if they have included all of the possibilities or if some have been omitted. As the answers range between 6 pence to 6 pounds, the children need to justify the reasonableness of each possibility and decide which is the most likely cost of a balloon. This activity also provides the opportunity to construct a graph showing the possible cost and the number of responses the children make to each choice and then to interpret the results of the findings.<br></br></mdoxml> 2 3 0 1 0 0 0 Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon? Measures and Mensuration Money Using, Applying and Reasoning about Mathematics Investigations Decision Mathematics and Combinatorics Combinations Calculations and Numerical Methods Addition & subtraction Using, Applying and Reasoning about Mathematics Working systematically Admin Upper primary mapping document