Forgot the Numbers

1015 /www/nrich/html/content/00/09/penta2/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">If you are a teacher click <a href="/1015&part=note">here</a> for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on ... <div style="float: left;"><mdo:image alt="calculator" src="calculator2.gif" style="width: 152px; height: 243px;"></mdo:image></div> On my calculator I divided one whole number by another whole number and got the answer $3.125$. I know that both numbers were less than $50$, but can't remember what they were. Can you work out what they were? Once you've had a chance to think about it, click below to see how four different pupils began working on the task. Here is Gemma and Flo's work: <div class="toggle"><mdo:image src="ForgotNumbers2.jpg"></mdo:image></div> Richard wrote the following: <div class="toggle" style="margin-left: 36pt;"><mdo:image src="ForgotNumbers1.jpg"></mdo:image> He explained: "I multiplied $3.125$ by $1$, then I tried multiplying $3.125$ by $2$, then I multiplied $3.125$ by $3$ ..." </div> Here is the start of Thomas' work: <div class="toggle" style="margin-left: 36pt;">I first looked at the number $0.125$ and worked out what fraction of $1$ it is. It turned out that it was an eighth.</div> Can you take each of these starting ideas and develop it into a solution? <a href="http://nrich.maths.org/8005">Click here for a poster of this problem</a>.</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> We had nearly $100$ solutions sent in for this challenge. We also had one sent in to the <a class="blogbutton" href="http://www.nrich.maths.org/z/infinities">blog</a>. Here are just some of them. From Forest Lake State School in Australia we had contributions from $3$ pupils, Long, Daniel and this is Connor's": At first I did this: $46$ divided by $15=3.0666666$ $44$ divided by $14=3.1428571$ $43$ divided by $14=3.0714285$ $42$ divided by $14=3$ $41$ divided by $14=3.1538461$ $39$ divided by $13=3.25$ $38$ divided by $12=3.1666666$v $36$ divided by $12=3$ $29$ divided by $12=3.2222222$ $28$ divided by $9=3.1111111$ $27$ divided by $9=3$ $26$ divided by $9=3.25$ $25$ divided by $8=3.125$ The last division sum is the correct answer. At first I did $46$ sided by $15=3.06666666$, so I knew the two numbers had to be lower. When I got down to $25$, I knew that the dividing number had to be reasonable. So I tried 8. Hey presto! I got the answer right. Last algorithm - $25$ divided by $8= 3.125$. Isabella from Sharp school, together with Mellisa and Rebecca had a different way of approaching it: Firstly I wrote down the 3.125 times table: $3.125, 6.25, 9.375, 12.5, 15.625, 18.75, 21.875, 25, 28.125, 31.25$ I saw the only whole number was $25$ ($8$ lots of $3.125$) so that means $25$ divided by $8$ is $3.125$. A number of different approaches were shown by pupils from Rykneld School in the UK. The pupils were, Alice, David, Jordan, Kieran, Daniel, Alicia and Alice who suggested a further challenge: After we had worked out the solution to the problem, we made our own! I divided two numbers and got the answer of $13.5$. I can't remember my two numbers but they are both under $75$ and are whole numbers. Can you work out what my numbers were? From Huy at the Australian International School of Vietnam we had the following: We call these two numbers as X and Y. I know that $3.125$ equals to fraction $3 1/8$. $X = (3 1/8)$ times Y and Y is the whole number --> Y must be a multiple of $8$. Because X and Y are under $50$, I figure out $Y = 8, X = 25$ From Varsity Acres in Canada we had the following message, ( they sent in a pictures of their work but unfortunately we were not able to use it.) Student work, done in French , a student discovered the connection between $0.125$ and $125$. $8$ x $125 = 1000$ so - $1.0$!$ A fantastic leap and she brought the class along. Well done all of you. I'm sorry we cannot publish all the solutions! </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Forgot the Numbers</h2> If you are a teacher click <a href="/1015&part=note">here</a> for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on ... <div style="float: left;"><mdo:image alt="calculator" src="calculator2.gif" style="width: 152px; height: 243px;"></mdo:image></div> On my calculator I divided one whole number by another whole number and got the answer $3.125$. I know that both numbers were less than $50$, but can't remember what they were. Can you work out what they were? Once you've had a chance to think about it, click below to see how four different pupils began working on the task. Here is Gemma and Flo's work: <div class="toggle"><mdo:image src="ForgotNumbers2.jpg"></mdo:image></div> Richard wrote the following: <div class="toggle" style="margin-left: 36pt;"><mdo:image src="ForgotNumbers1.jpg"></mdo:image> He explained: "I multiplied $3.125$ by $1$, then I tried multiplying $3.125$ by $2$, then I multiplied $3.125$ by $3$ ..." </div> Here is the start of Thomas' work: <div class="toggle" style="margin-left: 36pt;">I first looked at the number $0.125$ and worked out what fraction of $1$ it is. It turned out that it was an eighth.</div> Can you take each of these starting ideas and develop it into a solution? <a href="http://nrich.maths.org/8005">Click here for a poster of this problem</a>. </div> <h3>Why do this problem?</h3> <div><a href="/1015">This task</a> has the potential to address many aspects of number and calculation, including the inverse relationship between multiplication and division, and the relationship between division and decimal fractions. The richness of the activity comes in the many different approaches which could be used to solve it and discussion of these different methods is emphasised in these notes. You may well need to spend a couple of lessons on this activity.</div> <h3>Possible approach</h3> <div>Pose the problem orally, or project the text onto a screen, without mentioning the examples of how some children started. Give the class a few minutes to consider, individually, how they might go about tackling the problem, then pair them up and suggest that they talk to their partner about their ideas so far. Try to stand back and observe, and resist the temptation to make helpful suggestions! Allow pairs to work on the task so that you feel they have made some progress, but do not worry if they have not completed it or if they report being stuck. The aim at this stage is for everyone to 'get into' the problem and work hard on trying to solve it, but not necessarily to achieve a final solution. At a suitable time, hand out this (<a href="/content/00/09/penta2/BBforgot%20the%20number%20examplesWord.doc">doc</a> <a href="/content/00/09/penta2/BBforgot%20the%20number%20example%20NEW.pdf">pdf</a> ) to pairs. Suggest to the class that when they've finished or can't make any further progress, they should look at the sheet showing three approaches used by children working on this task. Pose the question, "What might each do next? Can you take each of their starting ideas and develop them into a solution?". You may like pairs to record their work on large sheets of paper, which might be more easily shared with the rest of the class in the plenary. Allow at least fifteen minutes for a final discussion. Invite some pairs to explain how the three different methods might be continued. You may find that some members of the class used completely different approaches when they worked on the task to begin with, so ask them to share their methods too. You can then facilitate a discussion about the advantages and disadvantages of each. Which way would they choose to use if they were presented with a similar task in the future? Why?</div> <div> </div> <h3>Key questions</h3> <div>Tell me about this approach. What do you think he/she/they were doing? How do you think this will help to solve the problem? What do you think he/she/they would have done next?</div> <h3> </h3> <h3>Possible extension</h3> <div>Can they apply what they have learnt to a similar problem, where the answer is $3.375$, $4.5$, $4.1$? Can they make up a similar question and solve it?</div> <h3> </h3> <h3>Possible support</h3> <div>Have calculators available for pupils to use, should they wish.</div></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> What do you know about division that might help? Having a calculator to hand will be useful. Have a look at each approach. What do you think he/she/they were doing? How do you think that will help to solve the problem? What do you think he/she/they would have done next? </mdoxml> 5 4 0 1 0 0 0 Forgot the Numbers On my calculator I divided one whole number by another whole number and got the answer 3.125 If the numbers are both under 50, what are they? Calculations and Numerical Methods Multiplication & division Information and Communications Technology Calculators Fractions, Decimals, Percentages, Ratio and Proportion Calculating with decimals Fractions, Decimals, Percentages, Ratio and Proportion Calculating with fractions Using, Applying and Reasoning about Mathematics Working systematically Admin Upper primary mapping document