PrepShorts · Teaching notes · Class 6 Mathematics · Chapter 6, Perimeter and AreaPrepShorts

Chapter 6 · Perimeter and Area

Same area, many perimeters: why one does not determine the other

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What the video covers, what to say before it, where a class usually goes wrong, and what to set afterwards. An account is free, and it opens every chapter of every book.

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What to assume they know

What they should be able to do

  • State that two figures may share an area and differ in perimeter, and the other way round
  • List every rectangle with whole-number sides for a given area and rank them by perimeter
  • Identify which rectangle of a given area has the greatest and which the least perimeter, and say why
  • Build figures of a fixed number of unit squares with different perimeters
  • Find the smallest and largest perimeter attainable from a fixed number of unit squares, and justify the extremes
  • Predict the change in perimeter when one unit square is attached, from the number of edges it shares
  • Compare two floor plans of equal area and different perimeter
  • Construct a pair of shapes to order, given constraints on both measurements

Where it usually goes wrong

  • "A bigger area means a bigger perimeter." The single misconception this entire module exists to remove. Two shapes of 18 and 20 square units, the smaller with the longer boundary, is the printed counter-example on p.149.
  • "Equal areas must have equal perimeters." The tangram square and rectangle refute it with no arithmetic at all.
  • "Rearranging pieces changes the amount of stuff." It changes the boundary and nothing else. Keep the two facts in view together.
  • "The rectangle with the biggest perimeter must be the biggest rectangle." Every rectangle in the list has the same area. The long thin one has the longest boundary and encloses no more.
  • "Adding a square always makes the perimeter longer." It can leave it unchanged and it can shorten it. This is the fact students find genuinely surprising, and the reason it is surprising is that they are attending to the area and not to the join.
  • "Two shapes with the same area and the same number of squares must be the same shape." The nine-square puzzle produces several shapes for a perimeter of 20 and several for 18; the answer key appended to the cached PDF notes that only the smallest perimeter pins the shape down.
  • "Two houses on plots of the same area cost the same to wall." Walling is paid for by the perimeter. Charan's and Sharan's plots hold the same area and need different lengths of boundary.
  • "Cutting a square in half halves both measurements." It halves the area of each piece but the two pieces' boundaries together come to more than the square's. Question 8 on p.149 is built on exactly this.

Questions to check understanding

  • Draw two figures with equal areas and unequal perimeters, or the reverse
  • List the rectangles of a given whole-number area and rank them by perimeter
  • Identify the least and greatest perimeter for a given area, with reasons
  • Build a figure of n unit squares with a stated perimeter
  • Predict the perimeter change when a unit square is attached at a stated place
  • Fill in missing room measurements on a plan and total the area
  • Compare two plans by area and by perimeter
  • Multiple-choice items of the p.149 question 8 type, where three of four statements fail on any square
  • The answer key appended to the cached PDF answers the nine-square questions on its footer page 4, the two house plans on footer pages 5 and 6, and the p.149 set on footer pages 6 and 7

Examples worth working on the board

  • The tangram, question 8 (p.139). The same seven pieces are assembled once into a square and once into a rectangle, and the reader is asked whether the two boundaries match, with reasons. Area is obviously unchanged — nothing was added or removed. This is the cleanest possible opening: the surprise is that a question can even be asked.
  • Rectangles of area 24 (p.141). On squared paper, make every rectangle whose sides are whole numbers of units and whose area is 24 square units, then say which has the greatest perimeter and which the least. The factor pairs are 1 and 24, 2 and 12, 3 and 8, 4 and 6.
  • Area 32, and the general question (p.142). The same exercise for 32 sq cm, followed by the real question: given any area, can you say in advance which rectangle will have the greatest perimeter and which the least, with examples and reasons. The appended answer key gives no answer for either part.
  • Nine unit squares, two figures (p.145). The book prints a 3-by-3 block and a second figure made of the same nine squares — read off the printed page, four in a top row, four in a bottom row and one joining them at the left end — and states their perimeters as 12 units and 20 units. It then sets the rules for building more: every square must meet another along a whole side, the figure must be in one piece, and it must have no holes. Four questions follow: smallest perimeter, largest perimeter, make one of 18, and whether more than one shape achieves each. The appended key gives 12 and 20 for the first two and answers the last one yes except for the smallest.
  • The eleven-square figure (pp.145–146). A figure of unit squares is printed with its perimeter stated as 24 units, and one extra square is drawn beside it waiting to be attached. Counted off the printed page, the figure is made of eleven unit squares. The reader is asked to say what the perimeter becomes without recomputing from scratch, then to place the same square in different positions so that the perimeter rises, falls, and stays the same. This is the topic's central mechanism and deserves its own two sections.
  • Charan's plan (p.146). A rectangular plot with 30 ft marked down one side. Printed room measurements: Master Bedroom 15 ft by 15 ft, area 225 sq ft; Toilet 5 ft by 10 ft; Kitchen 15 ft by 12 ft, area 180 sq ft; Small Bedroom 15 ft by a blank, area 180 sq ft. Utility, Hall, Garden and Parking are printed with every measurement blank. Reading the plan off the printed page, the bedrooms occupy a 15 ft column on the left, the toilet a 5 ft strip beside it and the kitchen a 15 ft block on the right, so the plot is 35 ft across.
  • Sharan's plan (p.147). A rectangular plot with 42 ft marked across the top. Printed room measurements: Master Bedroom 12 ft by 15 ft, area 180 sq ft; Small Bedroom 12 ft by 10 ft; Kitchen 18 ft by 10 ft, area 180 sq ft; Utility area 70 sq ft; Hall 23 ft by a blank. Toilet and Entrance are blank throughout. The bedroom column is 12 ft wide and 15 ft plus 10 ft tall, so the plot is 25 ft deep. The book then asks the reader to compare the areas and the perimeters of the two houses — which is the whole topic in one instruction.
  • What the two plots come to. Both work out to 1050 square feet, from 35 by 30 and from 42 by 25; the boundaries come to 130 ft and 134 ft. I derived the two missing plot dimensions from the printed room measurements and then found the same four numbers in the answer key appended to the cached PDF.
  • Question 5 on p.149. Two shapes are wanted: the first covering 18 square units, the second 20, with the first carrying the longer boundary of the two. A construction task rather than a computation, and the strongest single assessment item for this topic.
  • Question 8 on p.149. A square is folded in half and cut along the fold. One of four statements is always true, whatever the square's size — three concern areas and perimeters that do not add the way students expect.
  • The Summary (p.150). The book states both halves of the independence outright: equal areas can carry unequal boundaries, and equal boundaries can enclose unequal areas. Section 12 should show the sentence and check both halves against examples the explanation has already built.

Figures to have open

  • The nine unit squares in their two printed arrangements from p.145, plus a rearrangement track between them. This is the topic's central figure and must match the book's two shapes, since their perimeters are the printed values.
  • The eleven-square figure from p.145 with its detached extra square, and three copies showing the square attached in three different ways. The three placements are not in the book; the book prints one and asks for the others.
  • The two house plans from pp.146–147, redrawn with the printed measurements in place and the blanks left blank so the explanation can fill them. These must follow the book's room layout or the derived plot dimensions will not follow.
  • A strip of every whole-number rectangle of area 24, drawn to scale on one grid. Standard schematic; the book asks the reader to make these rather than printing them.
  • No photograph or textbook data table is needed.

Where this sits in the book

The book

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