The Deca Tree

2006 /www/nrich/html/content/03/11/penta3/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> In the forest there is a Deca Tree. A Deca Tree has $10$ trunks, <mdo:image height="148" width="231" src="Deca1.gif" alt="tree with 10 trunks"></mdo:image> and on each trunk there are $10$ branches, <mdo:image height="144" width="111" src="Deca2.gif" alt="trunk with 10 branches"></mdo:image> and on each branch there are $10$ twigs, <mdo:image height="98" width="108" src="Deca3.gif" alt="branch with 10 twigs"></mdo:image> and on each twig there are $10$ leaves: <mdo:image height="104" width="133" src="Deca4.gif" alt="twig with 10 leaves"></mdo:image> One day a woodcutter came along and cut down one trunk from the tree. Then he cut off one branch from another trunk of the tree. Then he cut off one twig from another branch. Finally he pulled one leaf from another twig. How many leaves were left on the tree then? <a href="http://nrich.maths.org/public/viewer.php?obj_id=6828&part="> Click here for a poster of this problem</a>. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> This problem was very well answered. We had many very well explained solutions but Kirsty's solution was particularly clear: There are 10 leaves per twig There are 10 twigs per branch 10 leaves x 10 twigs = 100 leaves per branch There are 10 branches per trunk 100 leaves x 10 branches = 1000 leaves per trunk There are 10 trunks per tree 1000 leaves x 10 trunks = 10 000 leaves on the tree Cut off one trunk: 10 000 - 1000 = 9000 leaves left Cut off one branch: 9000 - 100 = 8900 leaves left Cut off one twig: 8900 - 10 = 8890 leaves left Pull off one leaf: 8890 - 1 = 8889 leaves left There are 8889 leaves left on the tree. This is excellently explained, Kirsty - you have approached the problem very systematically. Rachel who goes to Aldermaston CE Primary School also sent in a fantastic solution. She did it in the same way as Kirsty, but found the total number of leaves which had been pulled off before the total number of leaves on the tree to start with. Here is everyone else who sent extremely clear solutions which were similar to either Kirsty's or Rachel's. You all deserve a mention - thank you! Alistair from Histon and Impington Junior School Benjamin, Becki, Pauline, Chris, Elliot, Luke, Ryo and Jake from Moorfield Junior School Bianca and Kathryn from Devonshire Primary School Goh Bee, Khoo Kian, Tan Yan, Tan Jun, Tao Mai and Ng Hui at Corporation Primary School in Singapore Hannah from South Parade Junior School Jennie at St. Nicolas CE Junior School, Newbury Jonathan at Dobcroft Junior School, Sheffield Joshua from Hillside Community Primary School, Ipswich Rachael from Ardingly College Stephen from Moorlands School Adam and Toby from Chacombe CE Primary School Chad who goes to the Canadian Academy Rushad, Oliver and Sabrina from Tanglin Trust School, Singapore Douglas from Burgoyne Middle School </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>The Deca Tree</h2> In the forest there is a Deca Tree. A Deca Tree has $10$ trunks, <mdo:image alt="tree with 10 trunks" height="148" src="Deca1.gif" width="231"></mdo:image> and on each trunk there are $10$ branches, <mdo:image alt="trunk with 10 branches" height="144" src="Deca2.gif" width="111"></mdo:image> and on each branch there are $10$ twigs, <mdo:image alt="branch with 10 twigs" height="98" src="Deca3.gif" width="108"></mdo:image> and on each twig there are $10$ leaves: <mdo:image alt="twig with 10 leaves" height="104" src="Deca4.gif" width="133"></mdo:image> One day a woodcutter came along and cut down one trunk from the tree. Then he cut off one branch from another trunk of the tree. Then he cut off one twig from another branch. Finally he pulled one leaf from another twig. How many leaves were left on the tree then? <a href="http://nrich.maths.org/public/viewer.php?obj_id=6828&part=">Click here for a poster of this problem</a>. </div> <h3>Why do this problem?</h3> <div>This <a href="http://nrich.maths.org/public/viewer.php?obj_id=2006&part=index">problem</a> can give some substance to children's ideas on place value especially when you are introducing or using numbers greater than $1000$. It shows in a pictorial way what the extra zeros do to a number when multiplying by $10$ and $100$.</div> <div>The problem can also help children to appreciate how big large numbers really are!</div> <h3>Possible approach</h3> <div>You could introduce the problem by telling the class the 'story' about the tree and its trunks, its branches, its twig and its leaves. During, or after this, ask such questions as "How many branches are there altogether?" and "How many leaves will there be on each branch?".</div> <div>Then, working in pairs, the children could work out the details of the problem. It might help them to draw rough sketches and/or make a table/list of the numbers of the trunks, branches, twigs and leaves, both at the beginning and after the woodcutter has been at work.</div> <div>After the number of leaves taken off the tree has been calculated this needs to be subtracted from the total of all the leaves on the tree. This might be the time for a plenary so that you work on the calculation together as as class.</div> <div>Some children may have do the problem differently, such as deducting the leaves removed from the tree as they go. This could lead to an interesting discussion on different methods to tackling problems such as this.</div> <h3>Key questions</h3> <div>If there are ten twigs on each branch, how many will there be on ten branches?</div> <div>How many leaves are there on ten twigs?</div> <div>How many leaves did the Deca Tree have before the woodcutter came along?</div> <div>How many leaves did the woodcutter chop off each time?</div> <h3>Possible extension</h3> Children could be invited to invent a similar tree with different numbers of branches etc. such a five-branched 'Penta' tree or an eight-branched 'Octa' tree. <h3>Possible support</h3> In addition to, or instead of, drawing sketches, it might help some children to make a model of the tree, or at least part of the tree. Pipecleaners could work well for this purpose. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> How many leaves are there on ten twigs? How many leaves did the Deca Tree have before the woodcutter came along? How many leaves did the woodcutter chop off each time? </mdoxml> 2 3 0 1 0 0 0 The Deca Tree Find out what a Deca Tree is and then work out how many leaves there will be after the woodcutter has cut off a trunk, a branch, a twig and a leaf. Calculations and Numerical Methods Multiplication & division Calculations and Numerical Methods Addition & subtraction Numbers and the Number System Place value Admin Upper primary mapping document