Ladybird Box

144 /www/nrich/html/content/99/02/letme1/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Some toy ladybirds are kept in this box which has 9 little square compartments. <div style="text-align: center;"><mdo:image width="269" height="103" src="144.gif" alt=""></mdo:image></div> Can you place six ladybirds into the box so that there are just two ladybirds in every column and every row? <div style="text-align: center;"><mdo:image alt="" src="t-crate.gif"></mdo:image></div> <div>You could try this interactivity.</div> <a href="/content/99/02/letme1/MC144.swf">Full screen version</a> <mdo:flash height="400" width="550"><param value="/content/99/02/letme1/MC144.swf" name="movie" ></param><param value="8" name="flashplayerversion" ></param></mdo:flash> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Thank you for all the answers you sent to this problem, however only a few of you described what you did to work out an answer. Sarah from Welton Primary School had a good way of putting the ladybirds in the box: In the first column you put one in the top box and one in the second box. In the middle column you put one at the top so you have two in the first row. Then if you put one in the bottom box you've got one in the bottom row. Then in the last column you've already got two in the top row so you can't put one in there, so you will have to put one in each of the other rows so you've got two in all the rows and columns. Thank you for taking us through your thinking, step by step, Sarah. Here is a picture of the way Sarah put the ladybirds into the box: <mdo:image height="176" width="175" src="ladybirdsol1.gif" alt="first solution"></mdo:image> Stephen found this same arrangement of ladybirds, but went about it in a slightly different way. He said: I worked out that column 1 could have two ladybirds, so row 3 could have two ladybirds, one each at the bottom of column 2 and column 3. Then I worked out where the last two ladybirds could go. Yara, who goes to the British International School of Riyadh wrote: I did it by trial and error. That's a good way to have a go at this probelm, Yara. (Although I like to call it 'trial and improvement'!) This means that you start by putting the ladybirds in any way, and then swap them around so that it works. Several children from the Bishop Harvey Goldwin School sent in well-explained methods too. Here is the first one: I firstly put three bugs diagonally so that there was one bug in each row and column. Then I put one bug under the middle bug, one to the left of the middle bug and the last bug two squares to the right of the top bug. Finally I checked my work to make sure there were two bugs in every row and column. Here is a picture of this solution (which is definitely different from Sarah's, isn't it?): <mdo:image height="161" width="173" alt="second solution" src="ladybirdsol2.gif"></mdo:image> Joel from Carr Green and Holly from Blockley both found four solutions to this problem. (I like the way you looked for more than one answer.) Here are the four that Joel sent which use 'L' to stand for a ladybird. Holly sent the same four solutions, but used a circle for a ladybird which is just as good. It's important to find a way to write things down that helps you. <mdo:image height="114" width="550" src="ladybirdsol3.gif" alt="third solution"></mdo:image> Do you notice anything about these solutions? How do they compare to the one from Bishop Harvey Goldwin School? I would suggest that they are all in fact the same as each other - just turned (rotated) or reflected. However, you may have counted them as different, which is fine. If we do say they are different, might there be other ways to draw Sarah's solution? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <h3>Why do this problem?</h3> <a href="http://nrich.maths.org/public/viewer.php?obj_id=144&part=index">This problem</a> requires both spatial and number understanding. It fits well with work on pairs and counting by twos. To understand the problem, the children will need to be comfortable with the idea that the $3$ by $3$ grid represents a top view of the nine-hole box. They will also need to be familiar with the meaning of rows and columns. The problem provides a good opportunity for developing positional language and utilising ordinal numbers. For example: "I put a ladybird in the middle of the second row" or "Go across three then down one". It is also a good context in which to discuss what makes one solution different from another. <h3>Possible approach</h3> <div>You could place some ladybirds in the box in some way using the interactivity and invite children to describe the positions of the ladybirds. This will give an opportunity for the class to come to an agreement about good ways to describe each 'cell' and to check the meaning of words such as 'row' and 'column'. You could then introduce the problem and encourage the children to work on it individually or in pairs. They could use a prepared $3$ by $3$ grid and six counters or the grid and ladybirds from <a href="/content/99/02/letme1/Ladybird%20box.pdf">this sheet .</a></div> <div>When someone finds a solution quietly ask him/her to explain why it is correct, then have him/her draw the solution onto a grid. Invite him/her to find another way to do it. This could lead to a plenary discussion comparing the different solutions. You can encourage learners to decide what makes a solution different from another one - how about rotating the grid and mirror reversals of solutions?</div> <div>To extend the opportunity for developing positional language, the $3$ by $3$ grid could be marked out with chalk on the floor or playground, and children used as the ladybirds. Children could take turns in giving instructions for the movement of the ladybirds until a solution has been formed.</div> <h3>Key questions</h3> <div>How many counters/ladybirds have you in this row? In this column?</div> Which row and which column can still have a second counter/ladybird? <h3>Possible extension</h3> Children could be encouraged to find all possible solutions and explain why they are sure they have found them all. <h3>Possible support</h3> You could encourage children to start by placing just four counters then seeing which row and which column have less than two counters in them. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> If you are not using the interactivity, it would help to have six counters and a $3$ by $3$ grid drawn on squared paper. Or you could print out this <a href="/content/99/02/letme1/Ladybirdbox.pdf">sheet</a> . </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">David from Histon Junior School said at first he couldn't do this problem. But when he'd found one way to put the milk bottles, he kept finding more and more! Well done! <mdo:image width="200" height="299" src="way-crate.gif" alt=""></mdo:image> </mdoxml> 2 4 1 0 0 0 0 Ladybird Box Place six toy ladybirds into the box so that there are two ladybirds in every column and every row. Using, Applying and Reasoning about Mathematics Working systematically Information and Communications Technology Interactivities Calculations and Numerical Methods Addition & subtraction