Number Pyramids

2281 /www/nrich/html/content/id/2281/ 1 2011-05-09T12:33:25 <mdoxml version="1.0"> <div>Choose three single-digit numbers and write them in the bottom row of the pyramid.</div> <div><a href="/content/id/2281/pyramid3.swf">Full Screen Version</a></div> <a href="https://nrich.maths.org/content/id/2281/pyramid3.swf" class="abp-objtab-041946423334873895 visible ontop" title="Click here to block this object with Adblock Plus" style="left: 370px ! important; top: 0px ! important;"></a><mdo:flash height="350" width="370"><param value="/content/id/2281/pyramid3.swf" name="movie" ></param><param value="7" name="flashplayerversion" ></param><param value="350" name="height" ></param><param value="370" name="width" ></param></mdo:flash> Try some different numbers. Can you work out how the numbers in the upper layers are generated? Here are some questions to consider: <ul> <li>Given the numbers on the bottom layer in order, can you find a quick way to work out the number at the top?</li> <li>If you change the order of the numbers on the bottom layer, will the top number change? </li> <li>If you can rearrange the numbers on the bottom layer, can you find a quick way to work out the largest possible number that could go at the top?</li> <li>Given the number at the top, how can you come up with possible numbers to go at the bottom?</li> </ul> Test out your observations and insights. You could use big numbers, small numbers, negative numbers, decimals... <div style="text-align: left; margin-left: 40px;">Can you explain what is happening?</div> <div style="margin-left: 40px;">Can you explain why it is happening?</div> <div style="margin-left: 40px;">Can you explain it algebraically? </div> Can you adapt your insights so that they apply to pyramids with more than three layers? Here is a number pyramid with four layers so that you can test some of your ideas: <a href="/content/id/2281/pyramid4.swf">Full Screen Version</a> <div> </div> <div><a href="https://nrich.maths.org/content/id/2281/pyramid4.swf" class="abp-objtab-041946423334873895 visible ontop" title="Click here to block this object with Adblock Plus" style="left: 445px ! important; top: 0px ! important;"></a><mdo:flash height="420" width="445"><param value="/content/id/2281/pyramid4.swf" name="movie" ></param><param value="7" name="flashplayerversion" ></param></mdo:flash></div> To experiment with 5 and 6 layer pyramids, you may find <a href="/content/id/2281/pyramids%20spreadsheet.xls">this spreadsheet</a> useful. You could adapt it to work on even larger pyramids! For other problems that use this idea go to <a href="http://nrich.maths.org/2282&part=">More Number Pyramids</a>. </mdoxml> <mdoxml version="1.0"> Rearranging the bottom digits gave insights into what was going on. Becky, Nicole, Harriet and Cha from NGHS started by changing the order of the bottom numbers: If you change the order of the numbers on the bottom layers it doesn't matter if you swap the two end numbers, but if the middle number is swapped with one of the end numbers it will change the top number, if the numbers aren't the same. To find the top number, Anna from Dame Alice Harpur School and Kieran from Brighouse High began by noticing the significance of the middle numbers. Kieran explained this as: I double the middle number and add it to the two outer numbers on the bottom to get the top number. Alexander from the Wilson's School explained why we need to double the middle numbers: You double all the numbers that aren't the first and last numbers because they are used twice in the pyramid. You don't double the first and last numbers because they are used only once. Hope, Tikki, Shannon, Hannah and Miriam from NGHS and Jinquan from The Chinese High School continued Alexander's idea using algebra. Jinquan said: Consider the $3$-layer pyramid. If the numbers at the bottom row are $a, b$ and $c$, then the number at the top is easily computed to be $a+2b+c$. This is because the middle number contributes a value of $b$ to each of the $2$ squares on the next tier adjacent to it. Consider the $4$-layer pyramid. If the numbers at the bottom row are $a, b, c$ and $d$, then the number at the top is easily computed to be $a+3b+3c+d$. This is again because the middle numbers contribute a value to each of the $2$ squares on the next tier adjacent to it. Notice that the values of the second tier are $a+b$, $b+c$, $c+d$. From the first example, the sum on the top is $(a+b) + 2(b+c) + (c+d) = a+3b+3c+d$. Kate and Simran, also from Dame Alice Harpur School, gave good explanations of how to find the top number. <a href="/content/id/2281/KateExplanation.pdf">Here</a> is what Kate wrote. To find the largest and smallest top number, consider as an example the bottom digits being 1, 2, 4 and 6. Hwang from Wimbledon High School, found the $4$ different arrangements of these numbers along the bottom that produce $33$ at the top, the largest possible total: $2461, 2641, 1642, 1462$ . This is the explanation provided: In my opinion you can get the largest answer by putting two biggest numbers, in the list of numbers given, in the middle of the bottom of the pyramid. It is because, the two numbers in the middle can affect two results compared to the other two numbers in the edge which only can affect $1$ result. e.g. $4$ in $2461$ can be added with $2$ and $6$. Hwang also found the $4$ different arrangements of the numbers along the bottom that produce $19$ at the top, the smallest possible total: $4216, 4126, 6124, 6214$ Similar reasoning explained the findings: To get the smallest number, you need to put two smallest numbers in the list of numbers given in the middle of the bottom of the pyramid. The reason being, the bigger numbers in the edge can only affect $1$ result and the smaller numbers could affect $2$ results. This is so that the result of the sum could be smaller. e.g. $4$ in $4216$ can only affect $2$. Emily from Grand Avenue Primary School in Surbiton came to the same conclusion: For the biggest number problem the answer is $33$ using the numbers in the order: $2 6 4 1$, and the smallest is $19$, using the numbers in the order: $1 6 4 2$. Jennifer, Anya and Caitlin from Nottingham Girls High (NGHS) made a generalised statement on how to get the largest top number: To ensure that the top number is the largest possible, the largest bottom numbers must be in the centre of the row. Hannah from Leicester High School for Girls and Simran used this idea for the 3-level pyramid, with Hannah correctly noticing the different 'highest top number' when digits are or are not repeated: If we want to make the biggest possible number, it would be logical to put 9 in the bottom middle because it is the largest single-digit number and in this way it can be used twice. The next largest numbers are 8 and 7, so these can go either side of 9, though it doesn't matter which side. The top number would then have a value of 33. If we can use the same number several times, then it would be obvious to put 9 in all three bottom slots and the top number would be 36. Emily continued this for the 4-level pyramid by using algebra: I wrote it as a letter problem using $A, B, C$ and $D$. The total is always : $A+ 3B+ 3C+ D$, so I put the biggest numbers in the middle boxes to get the highest total and the smallest numbers in the middle boxes to get the lowest total. There are $4$ ways of getting each total. Mayowa, Randeep and Kay, from NGHS, devised a method to find the bottom numbers if you were given the top number: You break the top number down into two numbers that could be added together to make it. You then find a number that is smaller than them both that will be the middle number and then break the other numbers down to find the remaining numbers you need to add on. To give an example of breaking down the top number, Zoha, Sophie and Hannah from NGHS used the number 25: Say the number at the top was 25, you would figure out which two numbers would add together to make 25 for example 10 and 15 then figure out which numbers add to make 10 and 15. Sam from The Chorister School in Durham made some interesting observations: If you haven't already noticed, when all the numbers on the bottom are the same, the top number is the bottom number x4. Another thing is that if the bottom numbers are consecutive, such as 1, 2, 3 or 5, 6, 7 or even 4, 3, 2 ; the top number is the middle number on the bottom x4. One last thing, this is quite interesting. I have found a pattern. If you start with the triangle's bottom numbers all at 0, and move the bottom left corner's value up one to 1, the top number will move up one as well. Then move the bottom middle circle's value up one to 1 and the top number will move up 2, to 3. And finally, move the bottom right corner's value up one and the top number will move up one again, now to 4. So the pattern is 1, 2, 1 for Number Triangles and funnily enough, 1, 2, 1 is in Pascal's Triangle too. Mia and Ellie from NGHS made a start at 5-layer and 6-layer pyramids. Their findings can be found <a href="/content/id/2281/Mia%20and%20Ellie%20Explanation.xls">here</a>. Nathan from Rushmore and Hannah began to find general formulae for n-level pyramids. Well done to Laura and Harriet from The Mount School in York and Mayowa, Randeep and Kay for spotting that Pascal's triangle needs to be used to find the general formula for the top number. Laura and Harriet said: In the 4-level pyramid, the middle two numbers appear three times more than the outer two numbers. The amount of times each letter appears is in the same form as in Pascal's triangle. Below is the start of Pascal's triangle. Can you work out what happens for an n-level pyramid? <mdo:image alt="" src="pascals.jpg" style="width: 564px; height: 388px;"></mdo:image> </mdoxml> <mdoxml version="1.0"> <h3>Why do this problem?</h3> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=2281">This problem</a> requires students to draw firm mathematical conclusions following a period of experimentation. It provides an interesting context for students to begin to appreciate the value of using algebraic notation.</div> <div>The problem encourages students to ask more general questions beyond the initial context.</div> <h3>Possible approach</h3> <div>Demonstrate how number pyramids work. This could be done with the teacher remaining silent throughout, and expecting the students to make sense of what is happening.</div> <div> </div> <div>"What do you notice?"</div> <div>"What questions do you think a mathematician might ask next?"</div> <div>Give students time to discuss and suggest. If no suggestions are forthcoming, share the questions suggested in the problem:</div> <ul> <li>Given the numbers on the bottom layer in order, can you find a quick way to work out the number at the top?</li> <li>If you change the order of the numbers on the bottom layer, will the top number change?</li> <li>If you can rearrange the numbers on the bottom layer, can you find a quick way to work out the largest possible number that could go at the top?</li> <li>Given the number at the top, how can you come up with possible numbers to go at the bottom? </li> </ul> Give students time to work on the questions that have been raised. Encourage them to experiment before trying to draw more general conclusions. As the lesson draws on, make it clear that they are expected to be able to explain and justify any generalisations they make. If students notice patterns but can't explain them, it may be helpful to introduce algebraic representation. Similar questions can be asked about larger pyramids. <a href="/content/id/2281/pyramids%20spreadsheet.xls">This spreadsheet</a> may be useful. <div>The class could then move on to <a href="http://nrich.maths.org/public/viewer.php?obj_id=2282&part="> More Number Pyramids</a></div> <h3>Key questions</h3> <div>How are these numbers generated?</div> <div>How does the position of a number on the bottom row affect the total at the top?</div> <h3>Possible extension</h3> <div>With the starting numbers $10$, $1$, $6$, $4$, why is it impossible to make a top total which is a multiple of $3$?</div> <div>If a 100-layer pyramid had $1$s in every cell on the bottom layer, how could you work out the number at the top?</div> <div>Students could have a go at <a href="http://nrich.maths.org/public/viewer.php?obj_id=1865&part="> Top-heavy Pyramids</a></div> <h3>Possible support</h3> <div>"Increase the first number on the bottom layer by $1$. What happens to the total at the top?</div> <div>Now try with the other numbers. What do you notice?"</div> </mdoxml> <mdoxml version="1.0">Increase the first number on the bottom layer by $1$. What happens to the total at the top? Now try with the other numbers. What do you notice? </mdoxml> <mdoxml version="1.0"> Using the starter numbers $2, 1, 4$ and $6$ you can get the largest total at the top of the pyramid if you place the largest numbers in the middle: $1, 4, 6, 2$ $1, 6, 4, 2$ $2, 4, 6, 1$ $2, 6, 4, 1$ all produce a total of $33$. Using the starter numbers $2, 1, 4$ and $6$ you can get the smallest total at the top of the pyramid if you place the smallest numbers in the middle - that produces a total of 19. Justification: with $A, B, C$ and $D$ in the bottom row you end up with $A + 3B + 3C + D$ at the bottom. With $5$ tiers you can get the largest total at the top of the pyramid if you place the largest number in the middle and the smallest numbers on the outside - there are $4$ possibilities altogether: $1, 3, 5, 4, 2$ $1, 4, 5, 3, 2$ $2, 3, 5, 4, 1$ $2, 4, 5, 3, 1$ With $5$ tiers you can get the smallest total at the top of the pyramid if you place the smallest number in the middle and the largest numbers on the outside - there are $4$ possibilities altogether. Justification: with $A, B, C, D$ and $E$ in the bottom row you end up with $A + 4B + 6C + 4D + E$ at the bottom. With $6$ tiers you can get the largest total at the top of the pyramid if you place the largest numbers in the middle and the smallest numbers on the outside - there are $8$ possibilities altogether: $1, 3, 5, 6, 4, 2$ $1,3, 6, 5, 4, 2$ $1, 4, 5, 6, 3, 2$ $1, 4, 6, 5, 3, 2$ $2, 3, 5, 6, 4, 1$ $2, 3, 6, 5, 4, 1$ $2, 4, 5, 6, 3, 1$ $2, 4, 6, 5, 3, 1$ With $6$ tiers you can get the smallest total at the top of the pyramid if you place the smallest numbers in the middle and the largest numbers on the outside - there are $8$ possibilities altogether. Justification: with $A, B, C, D, E$ and $F$ in the bottom row you end up with $A + 5B + 10C + 10D + 5E + F$ at the bottom. <a href="/content/id/2281/pyramid2281.zip">Flash code</a> </mdoxml> 2 3 0 0 1 0 0 Number Pyramids Try entering different sets of numbers in the number pyramids. How does the total at the top change? Calculations and Numerical Methods Addition & subtraction Algebra Creating expressions/formulae Using, Applying and Reasoning about Mathematics Generalising Information and Communications Technology Interactivities