Area as a count of unit squares

These teaching notes are for members

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.

Create a free account Sign in

What to assume they know

Perimeter as the distance all the way round: perimeter, so that the new measurement can be told apart from the old one
Multiplication of whole numbers, and division as its inverse
Reading measurements in metres and centimetres from a diagram
Earlier classes: working out how much a rectangle or a square covers, using squared paper — the recall this section opens by asking for

What they should be able to do

State what area measures and distinguish it from perimeter
Explain why area is counted in squares, and read the unit sq m, sq cm or sq unit correctly
Recover the rectangle and square area rules by counting rows of unit squares
Compute the leftover area when one region is laid on another and taken away
Compute how many equal pieces of a stated area fit inside a larger region
Split a rectilinear figure into rectangles and add the parts
Compare the areas of two shapes by superposition alone, without measuring
Express every piece of a set in terms of one chosen piece as the unit
Recover a missing area from the areas of the rectangles around it, with no side length given anywhere — the move the p.148 puzzles are built on

Where it usually goes wrong

"Area and perimeter are two names for the same measurement." They answer different questions and are counted in different units. Put a boundary tape and a tray of unit squares together, once, early.
"Area of a rectangle is defined as length × width." It is a consequence. Counting is the definition, and section 3 should show the multiplication being discovered inside the count — one row of squares, repeated.
"20 sq m is the same as 20 m." The unit changes when the measurement changes. Say the unit aloud in every worked answer.
"You cannot compare areas without measuring them." The whole tangram section is a refutation. Two pieces that cover each other exactly have the same area, whatever their shapes.
"The piece that looks biggest has the most area." D is a square, F a triangle and G a parallelogram, and all three are the same size. Students reliably rank the square smallest.
"Rearranging the pieces changes how much there is." It cannot. This is the quiet assumption behind every splitting problem in the chapter.
"An L-shaped or stepped figure needs its own formula." It needs a cut. Any such outline breaks into rectangles whose areas add.
"Four beds of side 4 m occupy 4 × 4 = 16 sq m altogether." One bed is 16 sq m; four are 64. The printed solution goes through both steps for exactly this reason.

Questions to check understanding

Area of a rectangle or square from its measurements
A missing side from an area and the other measurement
Leftover area after a smaller region is placed inside a larger one
Number of equal plots or tiles that fit a given region
Area of a rectilinear figure by splitting it into rectangles
Cost problems given a rate per unit area
Comparing shapes by area with reasons, not measurements
Expressing several shapes as multiples of one chosen shape
A missing area inside a compound figure, recovered from the areas around it with no side length given — the p.148 puzzle format
The combined area of two rectangles re-expressed as one rectangle, where more than one answer is correct
The answer key appended to the cached PDF answers the p.138 set on its footer page 3, the p.148 puzzles on its footer page 6 and the p.149 questions across its footer pages 6 and 7; it gives no answers for the tangram questions of p.139

Examples worth working on the board

The definition (p.137). The book states in bold what area is, then says the rectangle and square rules were reached in earlier classes on square grid paper and prints two boxes for the reader to fill: one for a square, one for a rectangle. Both boxes are printed empty. The Teacher's Note beside them asks the teacher to hand out grid paper and let the students arrive at the rules themselves. Section 3. Note that the worked solution a few lines below the boxes — a Teacher's Note and the line introducing the real-life problems sit between — already uses the rectangle rule, so the blanks withhold a statement of it, not the rule itself.
The carpet (p.137). Floor 5 m long and 4 m wide, so 20 sq m. A square carpet with 3 m sides, so 9 sq m. Uncarpeted: 11 sq m. All three printed.
The four flower beds (pp.137–138). Land 12 m by 10 m, so 120 sq m. Four square beds of side 4 m, 16 sq m each, 64 sq m together. Remaining: 56 sq m. All printed.
The p.138 set. (1) a garden 25 m long with an area of 300 sq m, width wanted; (2) tiling a plot 500 m by 200 m at ₹8 per hundred sq m; (3) a coconut grove 100 m by 50 m where each tree needs 25 sq m, greatest number of trees wanted; (4) two rectilinear outlines to be split into rectangles, all measures in metres. Outline (a) is a staircase descending to the left, drawn with eight of its ten edges labelled — reading round from the top edge of the raised right-hand block: 1 down, 3 left, 2 down, 2 left, 4 down, 3 right, 3 up, 4 right — with the block's own top edge and the far right side left unmarked for the reader to work out. Outline (b) is an arch: a block 5 across the top and 3 down the left with a rectangular bite taken out of the bottom middle, the bite marked 3 along its ledge and 2 down its right-hand wall, and 1 at each foot. The appended answer key gives 28 sq m and 9 sq m; I reconstructed both outlines from the printed page and reached the same two values.
The tangram (p.139). The reader is told to cut out the tangram supplied at the back of the textbook. The printed diagram is the standard seven-piece dissection of a square, lettered: A and B the two large triangles, F the medium triangle, C and E the two small triangles, D the square standing on a corner, G the parallelogram. The book's hint states that A and B match, that C and E match, and that D can be covered exactly by C together with E, so D is twice C. Eight questions follow. The last of them compares the boundary of the assembled square with the boundary of the assembled rectangle, and belongs to Same area, many perimeters: why one does not determine the other.
The tangram in units of C. Reading the printed dissection off the printed page and taking the whole square as four units across: C and E come to 1 each, D and F and G to 2 each, A and B to 4 each, and the assembled square to 16. That answers the book's question 6 — the big square is sixteen times shape C — and settles questions 3 and 4 as ties. The book prints only the hint; these values are worked out here from the figure and should be presented as something the student derives by covering, not as a table handed down.
The area mazes (p.148). Four puzzles under the printed heading "Area Maze Puzzles", lettered a. to d. Each shows a compound figure divided into rectangles; some parts carry an area, some edges carry a length, and one value is left blank — an area in a., b. and c., a length in d. Puzzle a. is the cleanest, and it carries no measurement at all: a rectangle cut by one vertical and one horizontal line into four parts, of which three are marked 13 sq cm at the top left, 26 sq cm at the top right and 15 sq cm at the bottom left, with the fourth wanted. Since the two top parts share a height and one is twice the other, the right column must be twice as wide as the left, so the bottom right part is twice the bottom left: 30 sq cm. I worked that out from the printed numbers and then found the same value, together with 9 sq cm, 16 sq cm and 5 cm for b., c. and d., in the answer key appended to the cached PDF.
The closing exercise set (p.149). Eight questions, of which the first four belong here: (1) give both sides of a rectangle that covers as much as one 5 m by 10 m and one 2 m by 7 m together; (2) a garden 50 m long with an area of 1000 sq m, width wanted; (3) the carpet problem again, with the very measurements the p.137 solution used; (4) four beds 2 m by 1 m in the corners of a garden 15 m by 12 m, the area left for a lawn wanted. Question 6 is a perimeter item and belongs to The rectangle and square formulas are shortcuts for the same addition; questions 5, 7 and 8 belong to Same area, many perimeters: why one does not determine the other. Question 1 has more than one right answer and the appended key gives three, which is worth saying out loud. Question 3 repeating p.137 exactly is worth using too: the student has already done it, and what was being learnt was the method, not the number.

Figures to have open

A rectangle filling with unit squares row by row, for section 3. This is the topic's central figure and carries the argument; standard schematic, but it must be countable.
The tangram square with the book's own lettering A to G and its own colours, so that a student holding the cut-out pieces from the back of the textbook sees the same labels. Redraw from p.139 rather than reproducing the printed art.
The two rectilinear outlines from p.138 with their printed edge labels, and a cutting line shown step by step into each.
The four area-maze figures from p.148, with every printed area and length in place and the blank left blank, so the explanation can fill it. Redraw; in these puzzles the figure is the question.
The floor-and-carpet and the land-and-beds diagrams. Standard schematics; the book prints no figure for either.
No photograph or textbook data table is needed.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 6 "Perimeter and Area", §6.2 "Area", pp.137–139 — the definition and the two empty boxes with their Teacher's Note (p.137), the carpet and flower-bed examples (pp.137–138), the Figure it Out set (p.138) and the tangram set (p.139)
§6.3, p.148, "Area Maze Puzzles", and §6.3, p.149, questions 1 to 4 of the closing Figure it Out — rectangle-area work printed after the triangle section
Chapter Summary, p.150, for the book's closing statements about area, its units, and breaking regions into unit squares
The tangram sheet printed at the back of the textbook, which p.139 tells the reader to cut out
Companion topics Why area is measured in squares and not in circles (why the counting unit is a square) and Estimating the area of a shape with no formula (counting when the boundary does not follow the grid)
Solutions appendix bound into the cached PDF after the book's last printed page, footer pages 3, 6 and 7