Warmsnug Double Glazing

4889 /www/nrich/html/content/id/4889/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Warmsnug calculate the prices of their windows according to the area of glass used and the length of frame needed. Can you work out how Warmsnug arrived at the prices of the windows below? Which window has been given an incorrect price? <mdo:image alt="windows" height="841" src="windows_new.png" width="669"></mdo:image> The information is available on a printable worksheet <a class="pdflink" href="/content/id/4889/Warmsnug%20worksheet.pdf">here</a>. <a class="spreadsheetlink" href="/content/id/4889/Warmsnug%20Pricing.xls">This spreadsheet</a> generates prices for the windows according to different pricing rules. Can you find an efficient strategy for finding how the prices were calculated? <div class="framework">This problem is based on the original Warmsnug Double Glazing problem which appeared in The Language of Functions and Graphs produced by the Shell Centre for Mathematical Education and the Joint Matriculation Board.</div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">We received many solutions in which people used pairs of similar windows to find the prices. One possibility is to find a pair with the same area, but different frame lengths. Harry & Roxana from Thorpe House Langley Preparatory School did this: We looked at K (Area = 12, Frame = 17) and I (Area = 12, Frame = 14) and we used the £60 price difference to find out the cost of the frame (£20 per unit) and the cost of the glass (£10 for each 1 by 1 pane). Millie and Kate's method for finding the costs of the frame and the glass is very neat: K had 3 cm of extra frame and was £60 more so we divided it by 3 to find that each centimetre of frame was £20. Similarly, Jake from Colyton Grammar School started by finding the price of each unit square of glass: J (Area = 4, Frame = 8) and H (Area = 3, Frame = 8) each have the same length of frame, but J has one square of glass more. J costs £10 more than H so that means that a 1 by 1 pane of glass costs £10. Once these prices have been found, the correct price of each shape can be found and compared to its price tag. But what would happen if one of K, I, J or H had the wrong price tag? Would this method work? Luckily, many students checked all the price tags and found that the only incorrect one is E. E has 18 frame squares and 12 glass panes, so it should cost £360 + £120, which equals £480. The price marked is £550, so window E is wrong. Rhea from Loughborough High School used a very systematic approach to make sure she found the window that was priced incorrectly. She used an algebraic method with simultaneuous equations: I used X to represent the price of the frame per square and Y to represent the price of the glass per square. For each window I wrote an equation using X and Y: A. 28X + 32Y = £880 (The frame borders 28 squares and the area of the glass is 32 squares) B. 16X + 15Y = £470 (The frame borders 16 squares and the area of the glass is 15 squares) C. 12X + 8Y = £320 (etc.) D. 20X + 16Y = £560 E. 18X + 12Y = £550 F. 12X + 9Y = £330 G. 26X + 24Y = £760 H. 8X + 3Y = £190 I. 14X + 12Y = £400 J. 8X + 4Y = £200 K. 17X + 12Y = £460 L. 23X + 20Y = £660 M. 24X + 36Y = £840 N. 20X + 24Y = £640 O. 16X + 12Y = £440 I then looked for equations which had equal X or Y figures. I used these equations to explore some simultaneous equations: F. 12X + 9Y = 330 C. 12X + 8Y = 320 F - C: Y = 10 J. 8X + 4Y = 200 H. 8X + 3Y = 190 J - H: Y = 10 N. 20X + 24Y = 640 D. 20X + 16Y = 560 N - D: 8Y = 80 Y = 10 B. 16X + 15Y = 470 O. 16X + 12Y = 440 B - O: 3Y = 30 Y = 10 As all the answers to the simultaneous equations which I investigated are Y = 10, and there is only one incorrect equation, Y must equal £10. It also indicates that equations (and the prices of) F, C, J, H, N, D, B and O must be correct. G. 26X + 24Y = 760 N. 20X + 24Y = 640 G - N: 6X = 120 X = 20 K. 17X + 12Y = 460 I. 14X + 12Y = 400 K - I: 3X = 60 X = 20 E. 18X + 12Y = 550 M. 24X + 36Y = 840 M/3. 8X + 12Y = 280 E - M/3: 10X = 270 X = 27 C. 12X + 8Y = 320 D. 20X + 16Y= 560 D/2. 10X + 8Y = 280 C - D/2: 2X = 40 X = 20 As 3 out of 4 of the answers to the simultaneous equations investigated are X = 20, I assume that X must £20. It also indicates that either equation (and the prices of) E or M is the incorrect one because when they are solved in a simultaneous equation, they produce a different answer to all the others. Let X= £20 and Y= £10 I entered these values into all the equations to see if they fitted in with the figures. All of them except E proved to be correct: E. 18X + 12Y = 360 + 120 = 480; so the price for window E is incorrect. This makes sense because E didn't produce the right answer when put in a simultaneous equation. Well done to everyone who found the solution. Can you see the similarity between the algebraic method and the 'comparing pairs' method? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <h3>Why do this problem?</h3> Mathematics lessons can sometimes feel neatly packaged, with information, techniques, patterns all readily accessible. But what do we do when the information arrives all jumbled up (with occasional errors) and we're asked to make sense of it? <a href="http://nrich.maths.org/public/viewer.php?obj_id=4889">This problem</a> offers students a chance to develop strategies for organising and understanding such situations. This is all done within the context of calculating areas and perimeters of rectangles. The problem is also ideal for considering the variety of methods of generating and solving simultaneous equations. <h3>Possible approach</h3> Project <a href="/content/id/4889/windows_new.png">this image</a> of the windows. "Imagine you are the owner of a double glazing business. What variables do you think you would need to consider when deciding on the prices for your windows?" Collect together ideas of the possible relevant variables. "Warmsnug Double Glazing price their windows according to the area of glass used and the length of frame needed. <a class="pdflink" href="/content/id/4889/Warmsnug%20worksheet.pdf">Here</a> is a worksheet showing the prices of different sized windows. Can you work out how Warmsnug arrived at the prices of these windows? Watch out - one of the windows has been priced incorrectly!" Give students time to work together. While they are working, circulate and listen out for useful insights. If students are stuck, here are some helpful prompts: <ul> <li>Are there any windows that use the same amount of glass? How do their frame lengths differ?</li> <li>Are there any windows that use the same amount of frame? How do their glass areas differ?</li> </ul> For anyone who finishes quickly, invite them to consider alternative ways of finding the pricing structure. <a class="pdflink" href="/content/id/4889/Warmsnug%20data.pdf">This worksheet</a> has all the relevant data along with some efficient methods for finding the pricing structure. If your focus is on solving simultaneous equations give the class plenty of time to find as many different ways of finding the pricing structure as they can. Bring the class together to share the new methods they have devised. Next, hand out <a class="pdflink" href="/content/id/4889/Warmsnug%201.pdf">worksheet 1</a>, <a class="pdflink" href="/content/id/4889/Warmsnug%202.pdf">worksheet 2</a>, and <a class="pdflink" href="/content/id/4889/Warmsnug%203.pdf">worksheet 3</a>, which become progressively more demanding. The problems on these sheets offer students the opportunity to apply and refine the methods they shared for the initial problem. Alternatively, if a computer room is available students can work on this <a class="spreadsheetlink" href="/content/id/4889/Warmsnug%20Pricing.xls">spreadsheet</a> version. Finally, use the spreadsheet to generate a Level 1, 2 or 3 problem and challenge the class to use an efficient method to work out the pricing structure and the incorrectly priced window. <div> </div> <h3>Possible extension</h3> <div>For a follow-up problem on area and perimeter, see <a href="/7534">Changing Areas, Changing Perimeters</a>. </div> <h3>Possible support</h3> <div> Along with the prompts above, suggest to students who are struggling with the large quantities of information that they initially ignore the windows with two panes. </div></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Start by looking at the windows with only one pane. Are there any windows that use the same amount of glass? How do their frame lengths differ? Are there any windows that use the same amount of frame? How do their glass areas differ? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> E is the odd one out - it should cost $£48$ <table style="" border="1"> <tbody> <tr> <td style="text-align: center;"> </td> <td style="text-align: center;">Length of frame</td> <td style="text-align: center;">Area</td> <td style="text-align: center;">2 x Length + Area</td> </tr> <tr> <td style="text-align: center;">A</td> <td style="text-align: center;">28</td> <td style="text-align: center;">32</td> <td style="text-align: center;">£88</td> </tr> <tr> <td style="text-align: center;">B</td> <td style="text-align: center;">16</td> <td style="text-align: center;">15</td> <td style="text-align: center;">£47</td> </tr> <tr> <td style="text-align: center;">C</td> <td style="text-align: center;">12</td> <td style="text-align: center;">8</td> <td style="text-align: center;">£32</td> </tr> <tr> <td style="text-align: center;">D</td> <td style="text-align: center;">20</td> <td style="text-align: center;">16</td> <td style="text-align: center;">£56</td> </tr> <tr> <td style="text-align: center;">E</td> <td style="text-align: center;">18</td> <td style="text-align: center;">12</td> <td style="text-align: center;">£48</td> </tr> <tr> <td style="text-align: center;">F</td> <td style="text-align: center;">12</td> <td style="text-align: center;">9</td> <td style="text-align: center;">£33</td> </tr> <tr> <td style="text-align: center;">G</td> <td style="text-align: center;">26</td> <td style="text-align: center;">24</td> <td style="text-align: center;">£76</td> </tr> <tr> <td style="text-align: center;">H</td> <td style="text-align: center;">8</td> <td style="text-align: center;">3</td> <td style="text-align: center;">£19</td> </tr> <tr> <td style="text-align: center;">I</td> <td style="text-align: center;">14</td> <td style="text-align: center;">12</td> <td style="text-align: center;">£40</td> </tr> <tr> <td style="text-align: center;">J</td> <td style="text-align: center;">8</td> <td style="text-align: center;">4</td> <td style="text-align: center;">£20</td> </tr> <tr> <td style="text-align: center;">K</td> <td style="text-align: center;">17</td> <td style="text-align: center;">12</td> <td style="text-align: center;">£46</td> </tr> <tr> <td style="text-align: center;">L</td> <td style="text-align: center;">23</td> <td style="text-align: center;">20</td> <td style="text-align: center;">£66</td> </tr> <tr> <td style="text-align: center;">M</td> <td style="text-align: center;">24</td> <td style="text-align: center;">36</td> <td style="text-align: center;">£84</td> </tr> <tr> <td style="text-align: center;">N</td> <td style="text-align: center;">20</td> <td style="text-align: center;">24</td> <td style="text-align: center;">£64</td> </tr> <tr> <td style="text-align: center;">O</td> <td style="text-align: center;">16</td> <td style="text-align: center;">12</td> <td style="text-align: center;">£44</td> </tr> </tbody> </table> We received lots of solutions to this problem. Rebecca and Katherine from Stanley Park Junior School, Emily from Durham Johnston Comprehensive School, Jamie and James from Gillingham School, Fraser from Wallingford School, Charlotte, Luke, Jordan, Stephen and Katie from Bosworth College, Anna and James from Desford Community College, Henry from Hitchin Boys' School, Gary from St Margaret's High School, Peter from Fulford and Kayleigh, Rachel from Edwinstree Middle School and Terence from Brumby Engineering College all got the formula right and worked out which window had been incorrectly priced. Well done all of you! Here's what Fraser wrote: The way to do it is first to find all the factors needed to work out the formula. So there's Area and Perimeter. In fact some of the shapes like shape A have an extra frame down the middle, so we should call it 'length of the window frame'. If you work out the area and add the perimeter, then add the perimeter again, you will work out how much this window company charge for each window. So $A = \text{area}, P = \text{window frame length}, C = \text{cost}$ $10A + 20P = C$ So if you take example A for instance. It costs $£880$, its window frame length is $28$ and the area is $32$. $10\times32 + 20\times28= 880$. Gary and Peter told us that they drew tables. For example, Peter says: Make a table showing the perimeter (including the inside lines) and area of each shape, and the cost of the windows. From this, you can then see that the cost of the windows follows the general formula of: $$\text{Cost} = 10\times\text{Area} + (20 \times \text{Perimeter})$$ Some of you phrased your answer slightly differently. For example, Jordan says: Each line of frame costs $£20$, and each square of window costs $£10$. Can you see why this is the same as the formula? Here's what Anna and James told us about the incorrectly priced window: By applying our formula to all the shapes we found that the shape that has been incorrectly priced was shape E. We can say this because the perimeter of the shape is $18$ and the area of the shape is $12$. When we put this into our formula we get $\text{Price} = (10\times12) + (20 \times18)$ which should equal $550$ if the price is correct. However the price is incorrect as the formula shows that the price should be $£480$, therefore window E has been incorrectly priced. Katie noticed something else about window E. Also because the area of the pane of glassis a multiple of $20$and so is twice the length of the frame (because you've multiplied it by $20$you can divide it by $20$) then the cost for the window must be also be a multiple of $20$. $550$ is not a multiple of $20$ so this also means that it is wrong. Well spotted! Terence used simultaneous equations to find the formula, and checked it (and found the window with the wrong price) using a spreadsheet. $f$ is how many units of frame there are, and $g$ is how many units of glass there are. I found the formula by using simultaneous equations: Window C has $12$ units of frame, $8$ units of glass, and costs $£320$. So $12f+8g=320$. (Equation 1) Window F also has $12$ units of frame, but has $9$ units of glass and costs $£330$. So $12f+9g=330$. (Equation 2) So by doing (Equation 2)$-$(Equation 1), I get: $12f-12f+9g-8g=330-320$, which simplifies into: $$g=10$$ So, I put 10 instead of $g$ in Equation 1 to get: $12f+80=320$, so $12f=240$ and $f=20$, which gives the formula for the cost of a window as $c=20f+10g$. Window E is incorrectly priced, because it has $18$ units of frame and $12$ units of glass. So, the price should be $(18\times20)+(12\times10) = £480$. Instead, it is shown as $£550$, $£70$ more! You can see this in my spreadsheet. <div style="text-align: center;"> </div> <div style="text-align: center;"><mdo:image alt="Spreadsheet" height="202" src="doubleglazingspreadsheet.gif" width="657"></mdo:image></div> </mdoxml> 2 3 0 0 1 0 0 Warmsnug Double Glazing How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price? Using, Applying and Reasoning about Mathematics Working systematically Measures and Mensuration Perimeters Measures and Mensuration Area Handling, Processing and Representing Data Comparing data Secondary Mapping Document MD Equations and formulae US Secondary Mapping Document MD Perimeter, Area and Volume LS Secondary Mapping Document DisplayCabinet Secondary processes PM - Thinking Strategically