Area and perimeter

7280 /www/nrich/html/content/id/7280/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> What can you say about these two shapes?<br></br> <br></br> <div style="text-align: center;"><mdo:image height="82" width="348" alt="2 shapes" src="7280.png"></mdo:image></div> <div> </div> <div>What is the area of each one? What is the perimeter of each one?</div> <br></br> What can you say about the shapes below?<br></br> <br></br> <div style="text-align: center;"><mdo:image height="171" width="342" alt="8 shapes" src="7280B.png"></mdo:image></div> <div> </div> <div>You can print out <a href="/content/id/7280/Pic7280.pdf">a set of the shapes</a> and cut them into separate cards. <a href="/content/id/7280/Pic7280Col.pdf">These cards</a> have the coloured background.</div> <br></br> <br></br> Can you draw a shape in which the area is numerically equal to its perimeter? And another?<br></br> Can you draw a shape in which the perimeter is numerically twice the area?<br></br> Can you draw a shape in which the area is numerically twice the perimeter?<br></br> Can you make the area of your shape go up but the perimeter go down?<br></br> Can you make the perimeter of your shape go up but the area go down? <br></br> <br></br> Can you draw some shapes that have the same area but different perimeters?<br></br> Can you draw some shapes that have the same perimeter but different areas? <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p class="editorial">Matthew from Parkgate Primary School focused on the first two shapes in the problem. He said:</p> The area of the rectangle is $18$cm$^2$ and its perimeter is $18$cm.<br></br> The second shape is $12$cm$^2$ and its perimeter is $22$cm.<br></br> <br></br> <p class="editorial">Thomas from Colet Court examined the eight shapes which were drawn on the cards. He labelled the shapes A, B, C, D, E, F, G and H, going from left to right in the top row, then left to right in the bottom row.</p> <br></br> A: P is 18 , A is 18<br></br> B: P is 20 , A is 17<br></br> C: P is 18 , A is 16<br></br> D: P is 20 , A is 16<br></br> E: P is 22 , A is 12<br></br> F: P is 12 , A is 9<br></br> G: P is 20 , A is 25<br></br> H: P is 16 , a is 9<br></br> <br></br> <p><span class="editorial">Thomas didn't give any units in his solution. I guess we could say the perimeter is measured in 'units' and the area in 'square units', although some of you, like Matthew above, assumed the squares were 1cm long. So, that would mean the perimeter is in cm and the area in cm$^2$. Thomas remarked:</span></p> The perimeter is always bigger except for one (Shape G).<br></br> <br></br> <p><span class="editorial">Noor from Kingsbury Green Primary School answered the question; 'Can you draw a shape in which the area is numerically equal to its perimeter? And another?'. He said</span>:</p> I found if I did $4\times4$ I would get an area of $16$.<br></br> If I counted the sides there would be four on each side: $4+4+4+4 = 16$.<br></br> The area and perimeter are the same.<br></br> <br></br> The same happened if there you have a rectangle that has a length of $6$ and a width of $3$. <br></br> <br></br> <p class="editorial">Table 3 (they didn't give their school) looked at finding a shape which has a perimeter numerically twice the area. They wrote:</p> <div>Yes you can draw a the shape in which the perimeter is numerically twice the area, it is a $2$ by $2$ square, because the area is $4$cm$^2$ and the perimeter is $8$cm.</div> <div> </div> <p class="editorial">Bashayer from Kingsbury Green Primary also found this solution. Miiti from Kingsbury Green created this shape which has a perimeter of $20$ units and an area of $10$ square units:</p> <div> </div> <div> <mdo:image height="110" width="200" src="Miiti.png" alt=""></mdo:image></div> <br></br> <br></br> <p><span class="editorial">Thomas from Colet Court drew a shape in which the area is numerically twice the perimeter:</span></p> <br></br> <br></br> <mdo:image height="206" width="200" alt="" src="Thomas.png"></mdo:image><br></br> <br></br> <p class="editorial">Thomas went on to investigate how to make the area of a shape go up but the perimeter go down. He said:</p> <br></br> ...you go from a dented square to a square shape.<br></br> <br></br> <mdo:image height="219" width="208" alt="" src="Thomas2.png"></mdo:image><br></br> <br></br> <p class="editorial">Thomas also said that you can make the perimeter of a shape go up but the area go down</p> by inserting a dent in your shape the area gets reduced by the dent.<br></br> <mdo:image height="248" width="217" alt="" src="Thomas3.png"></mdo:image><br></br> <br></br> <p><span class="editorial">Sam and Lil from SMS summarised this by saying:</span></p> <br></br> You can say that the more indents in the figure the more perimeter. <br></br> <br></br> <p><span class="editorial">Joe and Charlie from Coniston Primary described the way they worked on this part of the problem:</span></p> First we picked a shape which was a square so we looked at the area and perimeter.<br></br> The area was $25$cm$^2$ and the perimeter was $20$cm.<br></br> We took $1$ chunk out of the top of the square and it did make the perimeter bigger and the area became smaller. The perimeter became bigger because it adds on $2$ more lines so the perimeter became $22$cm. <br></br> <br></br> <p><span class="editorial">Thomas also sent us shapes that have the same area but different perimeters:</span></p> <br></br> <mdo:image height="132" width="275" src="Thomas4.png" alt=""></mdo:image><br></br> <br></br> <p class="editorial">and shapes that have the same perimeter but different areas:</p> <br></br> <br></br> <mdo:image height="127" width="285" src="Thomas5.png" alt=""></mdo:image><br></br> <br></br> <p class="editorial">Very, very well done all of you. You have obviously put a lot of thought into this problem.</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Area and Perimeter</h2> <br></br> What can you say about these two shapes?<br></br> <br></br> <div style="text-align: center;"><mdo:image alt="2 shapes" height="82" src="7280.png" width="348"></mdo:image></div> <div> </div> <div>What is the area of each one? What is the perimeter of each one?</div> <br></br> What can you say about the shapes below?<br></br> <br></br> <div style="text-align: center;"><mdo:image alt="8 shapes" height="171" src="7280B.png" width="342"></mdo:image></div> <div> </div> <div>You can print out <a href="/content/id/7280/Pic7280.pdf">a set of the shapes</a> and cut them into separate cards. <a href="/content/id/7280/Pic7280Col.pdf">These cards</a> have the coloured background.</div> <br></br> <br></br> Can you draw a shape in which the area is numerically equal to its perimeter? And another?<br></br> Can you draw a shape in which the perimeter is numerically twice the area?<br></br> Can you draw a shape in which the area is numerically twice the perimeter?<br></br> Can you make the area of your shape go up but the perimeter go down?<br></br> Can you make the perimeter of your shape go up but the area go down? <br></br> <br></br> Can you draw some shapes that have the same area but different perimeters?<br></br> Can you draw some shapes that have the same perimeter but different areas? </div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <a href="http://nrich.maths.org/7280&part=">This problem</a> offers opportunities for children to consolidate their understanding of area and perimeter. The exploratory nature of the task means that learners will be grappling with the two concepts at the same time rather than tackling them independently which might usually be the case. The activity is likely to require persistence and a 'tinkering' or trial and improvement approach. <br></br> <h3>Possible approach</h3> <div>It is essential to have squared paper available (preferably $1$ cm squared) and the shapes printed out and cut into eight separate cards. These cards can be downloaded <a href="/content/id/7280/Pic7280.pdf">here</a> in black and white, and <a href="/content/id/7280/Pic7280Col.pdf">here</a> with a coloured background. It might also be helpful to have post-it notes so that pupils could attach details of area and perimeter onto each card, rather than continually having to re-calculate them.</div> <div> </div> <div>You could start with the whole group looking at the two shapes given at the beginning of the problem and invite learners to talk about anything they notice. (These are also two of the shapes given on the set of cards.) If area and/or perimeter doesn't come up naturally, you could ask direct questions to shift their attention to these concepts. </div> <div> </div> <div>Pairs could then explore the shapes on the remaining cards and you can challenge them with the specific questions given in the problem itself. Copies of these for printing out can be downloaded <a href="/content/id/7280/Qs7280.pdf">here</a>. It is important that you stress we are looking at <span style="font-style: italic;">numerically</span> equal values. The area and perimeter cannot be equal because they are measured in different units.</div> <div> </div> <div>When you gather the whole group together again, invite them to share not just solutions (i.e. shapes that fit the criteria), but their methods for creating the shapes. Did they establish any 'principles' that helped them? What actions could they perform on a shape without changing its perimeter? For example, what happens to the area and perimeter if you take a 'corner square' off a shape? What happens if you take an 'edge square' off a shape? What happens if you take a 'middle square' out of a shape? It may be that some children notice that for a given perimeter, a square gives the maximum possible area. </div> <br></br> <h3>Key questions</h3> <div>How will you find out the perimeter?</div> <div>How will you find out the area?</div> <div>Tell me about that shape.</div> <div>It might help to use squared paper to try out your idea.</div> <div>How could you make the area of that shape bigger/smaller? How will that affect its perimeter?</div> <div>How could you make the perimeter of that shape bigger/smaller? How will that affect its area?</div> <br></br> <h3>Possible extension</h3> Learners might like to have a go at the <a href="http://nrich.maths.org/48&part=">Pebbles</a> investigation which makes a good follow-up to this challenge.<br></br> <h3>Possible support</h3> The problems <a href="http://nrich.maths.org/4962&part=">Sizing Them Up</a> and <a href="http://nrich.maths.org/4964&part=">Wallpaper</a> might be useful starting points for those children who are struggling with the concept of area.<br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Using squared paper to sketch shapes might help. <br></br></mdoxml> 5 3 0 1 0 0 0 Area and perimeter What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters. Using, Applying and Reasoning about Mathematics Generalising Measures and Mensuration Perimeters Measures and Mensuration Area Using, Applying and Reasoning about Mathematics Working systematically Using, Applying and Reasoning about Mathematics Trial and improvement Admin Upper primary mapping document