The Amazing Splitting Plant

159 /www/nrich/html/content/99/10/letme2/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>The splitting plant grows in a special way.</p> <p>In the first week, the stem splits into two branches.<br></br> In the second week, each of these two branches split into another two branches - making four branches altogether.<br></br> <br></br> This keeps happening every week, until at the end of the sixth week each branch grows a flower.<br></br> <br></br> How many flowers will the plant have?</p> <mdo:image alt="" src="tree.gif"></mdo:image><br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p><strong style="font-weight: 400;" class="editorial">Fred from</strong> <span class="editorial">London says.</span></p> <p></p> <p>"The amazing splitting plant will have 64 flowers."</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>The Amazing Splitting Plant</h2> <br></br> <p>The splitting plant grows in a special way.</p> <p>In the first week, the stem splits into two branches.<br></br> In the second week, each of these two branches split into another two branches - making four branches altogether.<br></br> <br></br> This keeps happening every week, until at the end of the sixth week each branch grows a flower.<br></br> <br></br> How many flowers will the plant have?</p> <mdo:image alt="" src="tree.gif"></mdo:image></div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div>This <a href="http://nrich.maths.org/public/viewer.php?obj_id=159&part=">activity</a> offers a situation that is not too complex to understand and yet opens out many possible explorations. It would be a good chance to focus on the different ways children have represented their solutions.</div> <br></br> <h3>Possible approach</h3> <div>You could present the problem orally to the class, rather like telling a story. Ask them how many branches there would be by the end of the second week. You could then invite them to talk in pairs about the number of branches at the end of the third week - giving learners mini-whiteboards would help at this stage. How did they work out the answer? Having got this far, leave them to work on the problem in their pairs.</div> <br></br> <div>In the plenary, you could draw attention to the different ways of representing the problem you have seen. Some children may have drawn pictures of the plant, others may have just drawn lines. There may be some children who have simply noted down numbers. Invite several pairs to talk about their own representation and then you could have a group discussion about the advantages of each way.</div> <br></br> <div>You may want to make a note of the numbers of branches at the end of each week and to ask the children what they notice. Can they explain why this pattern occurs? Could they predict how many branches there would be after seven weeks (if there weren't any flowers) without drawing a picture?</div> <br></br> <h3>Key questions</h3> <div>How many branches will there be after three weeks? Four weeks ...?</div> <div>How will you keep track of the number of branches?</div> <div>What kinds of things have you noticed?</div> <br></br> <h3>Possible extension</h3> <div>Once children have solved the problem as it stands, the activity may be opened out and extended. For example:</div> <br></br> <div>1. If the plant branches in twos each year and we look at the units figure for a few years' growth we see year by year that the number of flowers [2 4 8 16 32 64 as in the problem] is:</div> <br></br> <div>2 4 8 6 2 4 ...</div> <br></br> <div>You can ask the pupils to see what they notice about the pattern.</div> <br></br> <div>2. Then you can pretend that the plant branches in different ways, maybe in 3s, 4s, 5s etc. For example the fours and fives units would look like:</div> <br></br> <div style="text-align: center;"><mdo:image alt="4's&5's" height="99" src="4%27s%265%27s.jpg" width="308"></mdo:image></div> <div>The pupils can then be asked to explore what the patterns show and look at others for 6s 7s etc.</div> <div>Further extension ideas are described on <a href="/content/99/10/letme2/The%20Amazing%20Splitting%20Plant%20Extension.pdf">this sheet</a> .</div> <br></br> <h3>Possible support</h3> <div>You may want to talk to individual pupils about what is happening to the plant at each stage so they understand the context better.</div> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">How many branches will there be after three weeks? Four weeks ...?<br></br> How will you keep track of the number of branches?<br></br> It might help to draw a picture or jot some things down.<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> REMEMBER TO RE LOAD FILES!<br></br> Let's look at some ways in which the activity may be opened out and extended. <br></br> I'm assuming the pupils have got an answer to the Splitting Plants. <br></br> 1/ If the plant branches in twos each year and we look at the units figure for a few years growth we see year by year that the number of flowers [2 4 8 16 32 64 as in the problem] are: <br></br> <br></br> 2 4 8 6 2 4 . . . . . . . <br></br> <br></br> You can ask the pupils to see what they notice about the pattern. <br></br> <br></br> <br></br> 2/ Then you can pretend that the plants can branch in different ways, may be in 3's, 4's, 5's etc. For example the fours and fives units would look like: <br></br> <br></br> 4's&5's <br></br> <br></br> The pupils can then be asked to explore what the patterns show and look at others for 6's 7's etc. <br></br> <br></br> 3/ Then you can ask the pupils to explore further by introducing the thought "What if you had 10 plants - each branching differently [in 1's, 2's 3's . . . .9's] and you lined them up 1 to 9. <br></br> The pupils could then be asked to look at them after two lots of branching and the number of flowers for each plant. <br></br> <br></br> 1 4 9 16 25 36 49 64 81 <br></br> <br></br> So they could explore these [square] numbers. <br></br> <br></br> Look at units: <br></br> <br></br> 1 4 9 6 5 6 9 4 <br></br> <br></br> 1 Look at digital roots: <br></br> <br></br> 1 4 9 7 7 9 4 1 9 <br></br> <br></br> Look at the table of differences: <br></br> <br></br> 1stTableDiff <br></br> <br></br> 4/ Often in opening out activities we move away from the practical situation and get involved with the numbers that are coming out in patterns. <br></br> <br></br> So, they could be asked to look at the numbers you'd get after another year of growth. <br></br> But of course three lots of branching means we're cubing the numbers instead of squaring them. Calculators come in useful, even the simple ones of course allow the children to do things like 6 x 6 x 6. This may be one way in which you want to introduce the whole idea of powers. <br></br> <br></br> Four lots of multiplication [to the power 4] of the numbers 1 to 9 would give: <br></br> <br></br> 1 16 81 256 625 1296 2401 4096 6561 <br></br> <br></br> The units will be: <br></br> <br></br> 1 6 1 6 5 6 1 6 1 <br></br> <br></br> The digital roots will be: <br></br> <br></br> 1 7 9 4 4 9 7 1 9 <br></br> <br></br> The table of differences would be: <br></br> <br></br> 4's Table of Diff <br></br> <br></br> All of which can be explored. <br></br> <br></br> 5/ Referring back to May 1998's activity "Number Squares " use the first 4 figures of the above pattern to start off the square. <br></br> sq4start<br></br> sq4No.4<br></br> <br></br> Then explore, and then try the powers of 5, 6, 7, etc. <br></br> <br></br> So once you've got some ideas as to how to answer the original question, we look at some answers, and get the pupils to think about "I wonder what would happen if we . . . . . ?" . <br></br> <br></br> In extending this activity producing powers of the number 1 to 9 may be the outcome. Then look at the patterns/sequences in a variety of ways - unit figures, digital roots and lastly special arrangements.<br></br></mdoxml> 2 5 1 0 0 0 0 The Amazing Splitting Plant Can you work out how many flowers there will be on the Amazing Splitting Plant after it has been growing for six weeks? Calculations and Numerical Methods Multiplication & division Calculations and Numerical Methods Addition & subtraction Sequences, Functions and Graphs Geometric sequence Admin Lower primary mapping document