164 /www/nrich/html/content/00/01/letme1/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Below is a copy of a calendar for 2008 showing December. <br></br> Put a square box around the 1^st , 2^nd , 8^th and 9^th . <br></br> Choose two of the numbers and add them together. <br></br> Choose another pair and add them. Keep going until you have made all the pairs. <br></br> What do you notice about the answers? <p>Put a box around another set of four numbers and try this again.</p> <p>Look for other patterns.</p> <p>Try multiplying - perhaps you could use a calculator to help.</p> <p>What happens in other months?</p> <p style="text-align: center;"><mdo:image width="370" height="424" alt="" src="cal%20164.jpg"></mdo:image></p> <p style="text-align: center;"></p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <strong style="font-weight: bold;" class="editorial">Anil</strong> <span class="editorial">(Irmak Primary School, Turkey) explored some adding patterns:</span><br></br> <br></br> $1 + 8 = 9$<br></br> $2 + 9 = 11$<br></br> $3 + 10 = 13$<br></br> $4 + 11 = 15$<br></br> $5 + 12 = 17$<br></br> $6 + 13 = 19$<br></br> $7 + 14 = 21$<br></br> $8 + 15 = 23$ <p class="editorial">Alice from Perse School for Girls spotted a pattern:</p> <br></br> <div>There is a pattern, it's that the two numbers diagonal to each other added together make the same number as the other pair of diagonal numbers make.</div> <br></br> <p class="editorial">Chris found another pattern using multiplication:</p> <div>If you cross multiply the set of four numbers, their difference is always $7$.</div> <div>$1 \times 9 = 9$ and $2 \times 8 = 16$, difference $= 7$</div> <div>$3 \times 11 = 33$ and $4 \times 10 = 40$, difference $= 7$</div> <div>$15 \times 23 = 345$ and $16 \times 22 = 352$, difference $= 7$ etc...</div> <br></br> <p class="editorial">If you discovered any other patterns, then do let us know. Please don't worry that your solution is not &quot;complete&quot; - we'd like to hear about anything you have tried.</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><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=164&amp;part=">activity</a> is a great example of how patterns and numbers may be investigated in everyday contexts. If you are looking for opportunities that give your pupils chance to follow things up themselves, then this may be your answer!</div> <br></br> <h3>Possible approach</h3> <div>This investigation would work well with the children in pairs, each pair with their own copy of the December calendar section. It might be helpful to supply them with a paper frame to isolate the set of four numbers. This can easily be slid around the calendar to find new sets of four.</div> <br></br> <div>The problem begins with the lowest set of numbers simply to make the addition tasks easy. Later in the investigation encourage the children to move to the largest numbers they can cope with. Depending on children&#39;s experiences, encourage them to try and explain any patterns that they find.</div> <br></br> <h3>Key questions</h3> <div>Tell me about the numbers you&#39;ve found.</div> <div>What have you done to get these answers?</div> <br></br> <h3>Possible extension</h3> <div>If appropriate, guide the children to try multiplying the numbers and looking for patterns. If the children understand the basic concept of multiplication but can&#39;t readily manage the calculations, using calculators would be appropriate. This investigation could be revisited several times, trying different approaches each time. Encourage the children to discuss discoveries and suggest new things to try. For example, what happens if the square box is enlarged to include nine numbers, or a rectangular frame of six numbers? Test discoveries on other months. What would happen if we lived somewhere where a week consisted of 6, 5, or just 4 days?</div> <br></br> <h3>Possible support</h3> <div>Those who struggle a little may need some help to focus on which numbers they are dealing with at each moment - an adult helper would be good.</div> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> temp-notes<br></br> This investigation would work well with the children in pairs, each pair with their own copy of the December calendar section. It might be helpful to supply them with a paper frame to isolate the set of four numbers. This can easily be slid around the calendar to find new sets of four. <br></br> The problem begins with the lowest set of numbers simply to make the addition tasks easy. Later in the investigation encourage the children to move to the largest numbers they can cope with. The children should soon notice that the diagonally opposite pairs have the same total. Ask the children to make a note of the total of all four numbers as they gradually move down (or across) the calendar. Patterns will emerge.<br></br> If appropriate, guide the children to try multiplying the numbers and looking for patterns. If the children understand the basic concept of multiplication but can't readily manage the calculations, using calculators would be appropriate.<br></br> <br></br> This investigation could be revisited several times, trying different approaches each time. Encourage the children to discuss discoveries and suggest new things to try. For example, what happens if the square box is enlarged to include nine numbers, or a rectangular frame of six numbers? Test discoveries on other months.<br></br> <br></br> Further explorations can be made by changing various parts and carrying out more investigative work.<br></br> <br></br> Here we see what happens when we pretend to have a four-day week and take our block to be 4 by 4. The multiplication is done as follows; <br></br> [multi 4x4]<br></br> <mdo:image width="286" height="219" alt="multi 4 4 " src="multi%204x4.jpg"></mdo:image><br></br> <br></br> This forms the first block of answers in the following table. The table also includes some other pretend weeks. The answers to the 6 multiplications that can be done around the 4 by 4 square are then shown, as well as the digital roots of these answers. <br></br> [blocks of 4]<br></br> <mdo:image width="593" height="581" alt="blocks of 4" src="Blocks%20of%204.jpg"></mdo:image><br></br> Some of the blocks of 4 by 4 would not work because of the short weeks - they are shown with hatching. The numbers 1 to 5 show the starting number for each 4 by 4 block.<br></br> Having given you this starting point, pretend weeks of different lengths could be explored in a similar way. I found it very worthwhile to explore the digital roots that resulted. <br></br> You can then change the size of the blocks and continue with more investigational work. <br></br> When working with your pupils in opening out a challenge it's really good to get some first results from a change in the question and then ask the pupils, &quot;what do you see?&quot;. Then it's a matter of being brave enough to follow some of those leads and not just push for what you have seen. You'll probably be very surprised about all the different things that come up which may lead to further investigations. <br></br></mdoxml> 2 3 0 1 0 0 0 In this section from a calendar, put a square box around the 1st, 2nd, 8th and 9th. Add all the pairs of numbers. What do you notice about the answers? Calculations and Numerical Methods Addition & subtraction Sequences, Functions and Graphs Sequences Using, Applying and Reasoning about Mathematics Investigations