The area model
Can a multiplication you do not know yet, like 4 \times 16, be drawn as a floor, cut into pieces you do know, and measured?
Where you are 16 in Multiplication and division
Why this exists
In Area of rectangles and squares, a rectangle’s area was found by multiplying its side lengths. This lesson turns that around. Any multiplication of whole numbers can be drawn as a rectangle: the two factors are the side lengths and the product is the area. That picture is called the area model.
Why turn it around? Because a rectangle can be cut. Kwaku wants to know 4 \times 16, which is not a fact anyone memorizes. Drawn as a rectangle 4 by 16, it can be cut into a 4 by 10 piece and a 4 by 6 piece. Both of those are easy, and the two pieces together are the whole rectangle. The cut is the The distributive property, which was first seen in arrays of dots. Here it is seen as area, in square units, with only the side lengths drawn.
The area model also lets a multiplication be checked (does the picture hold that many squares?) and solved when one factor is missing (how long must the other side be?). And it gives one picture for two kinds of story: things set out in rows, like chairs, and surfaces covered by squares, like a floor.
Definition
The area model of a multiplication a \times b is a rectangle with one side a units long and the other side b units long. Its area is a \times b square units, because it holds a rows of b unit squares (see Area of rectangles and squares). In this lesson a and b are whole numbers.
The squares do not have to be drawn. An area model is usually a plain rectangle with its two side lengths written beside it. The side lengths are the factors; the area inside is the product. The drawing does not need to be to scale, since the labels, not a ruler, say how long each side is.
A rectangle may be split by a straight cut from one side right across to the side opposite. The cut makes two smaller rectangles that fill the first with no gaps and no overlaps, so their two areas add up to the area of the whole.
Intuition
The lesson runs over four days. Part 1 draws a multiplication as a rectangle and moves from tiles to a plain rectangle with only its sides marked. Part 2 cuts a rectangle in two and shows the distributive property as area. Part 3 uses the cut at the ten to multiply a one-digit number by a two-digit number, and uses ten-strips for a one-digit number times a multiple of ten. Part 4 solves word problems about arrays and areas, including one where a side is missing.
Part 1: a multiplication drawn as a rectangle
Adjoa is tiling the kitchen floor of her dollhouse with square tiles 1 inch on a side. The floor takes 7 tiles along one wall and 3 rows of them. On the left below, every tile is drawn. On the right, nothing is drawn inside, and the sides are labeled instead.
The two pictures hold the same information. Along the 7-inch side, 7 inch tiles fit in a row, because each tile is 1 inch wide. Up the 3-inch side, 3 rows fit. So both pictures say 3 \times 7, and both floors are 21 square inches. The plain rectangle is quicker to draw, and it still works when the numbers get too big for anyone to draw every square.
From here on, a multiplication and a rectangle are two ways of writing the same thing:
- the first factor is one side length;
- the second factor is the other side length;
- the product is the area, in square units.
That gives three things to do with the picture. A product can be pictured: 5 \times 9 is a rectangle 5 by 9. A product can be checked: if someone says 3 \times 7 = 24, a rectangle of 3 rows of 7 squares holds 21, not 24. And a missing factor can be solved: in \square \times 8 = 56, one side and the area are known, and the question is how long the other side is (Example 2). That last one is a division in disguise, as Multiplication and division undo each other explains.
Turning a rectangle on its side does not change its area, so 3 \times 7 and 7 \times 3 are the same rectangle drawn two ways: the Properties of multiplication.
Part 2: one rectangle, cut in two
Adjoa wallpapers a dollhouse wall 9 inches long and 7 inches tall. The area is 7 \times 9 square inches. If 7 \times 9 is not yet a known fact, she can cut the rectangle with one upright line after 5 inches.
The cut does not move any wall. The left piece is 7 by 5, and the right piece is 7 by 4, because 9 = 5 + 4. Both pieces are still 7 inches tall: the cut runs straight down, so it leaves the height alone. Their areas are facts she knows, and the two pieces fill the wall exactly:
This is the The distributive property: multiplying by a sum is the same as multiplying by each part and adding. In arrays it was a cut between columns of dots. In the area model the same cut goes through a plain rectangle, and the pieces are measured in square inches. In letters, for any whole numbers a, b and c,
The side that is cut is the one that gets split into a sum. The side the cut runs along stays whole and appears in both pieces. Either side may be cut, and the cut may go anywhere along it. Every choice makes two rectangles that fill the same whole, so every choice gives the same total (Example 4). A good cut is one that leaves two facts that are easy to work out.
Part 3: cutting at the ten
The art studio where Kwaku and Yaa paint has a floor 16 meters long and 4 meters wide. Its area is 4 \times 16 square meters. Here the best place to cut is given by place value: 16 is 1 ten and 6 ones, so cut the long side at 10 meters.
The floor is 64 square meters. The tens piece is easy because 4 rows of 10 is 4 tens, which is 40. The ones piece is a small fact.
This works for any one-digit number times a two-digit number. A number like 27 is 2 tens and 7 ones, so a rectangle 3 by 27 is cut into a 3 by 20 piece and a 3 by 7 piece. The first piece is 3 rows of 2 tens, which is 6 tens, or 60; the second is 21; together they are 81.
That first piece used a second idea: a one-digit number times a multiple of ten. A strip 10 units long and 1 unit wide holds 10 unit squares. Call it a ten-strip. Yaa is painting a banner 40 feet long and 3 feet tall. Each row of the banner is 1 foot tall and 40 feet long, which is 4 ten-strips, and there are 3 rows.
The banner holds 3 rows of 4 ten-strips, which is 3 \times 4 = 12 ten-strips. Twelve tens is 120, so the banner is 120 square feet:
The fact 3 \times 4 did the work; the area just happens to be counted in tens instead of ones. The lesson Multiplying by tens and hundreds takes this further.
Part 4: stories about arrays and areas
Two kinds of story lead to the same rectangle.
In an array story, things are set out in equal rows: 5 rows of 14 chairs, 4 rows of 21 paint jars. The product counts things, so the answer is in chairs or jars.
In an area story, a surface is covered by unit squares, or would be if anyone laid them: a floor, a wall, a banner. The product counts square units, so the answer is in square inches, square feet or square meters.
Sometimes a story is both. Square cushions 1 foot on a side, set out in 5 rows of 14 with no gaps, are an array of 70 cushions and a rectangle covering 70 square feet (Example 7). In either kind, the area model is the same: draw a rectangle, label its sides with the two factors, and cut it wherever that makes the arithmetic easy.
A story can also give the area and one side, and ask for the other side. Then the cut is made from the known side: take a piece that is easy to find, see what area is left over, and ask how long the rest of the side must be (Example 8). The full lesson on that question is Finding a missing side from the area.
Results
The area model. For whole numbers a and b, a rectangle a units by b units has area a \times b square units. The factors are the side lengths and the product is the area.
Splitting a rectangle (the distributive property as area). For whole numbers a, b and c,
because a rectangle a by b + c cuts into a rectangle a by b and a rectangle a by c.
Cutting at the ten. A one-digit number times a two-digit number is a tens piece plus a ones piece: for example 4 \times 16 = (4 \times 10) + (4 \times 6).
A multiple of ten. A one-digit number times a number of tens is that many tens: 3 \times 40 = 12 tens = 120.
Why it is true
Take a rectangle with one side a units long and the other side b + c units long, where a, b and c are whole numbers. By Area of rectangles and squares, it holds a rows of b + c unit squares, so its area is a \times (b + c).
Now cut it with one straight line, parallel to the side of length a, at the point b units along the other side. The cut follows a line between columns of unit squares, so no square is cut in half. Every row of b + c squares is split into b squares on the left of the cut and c on the right.
The squares on the left make a rows of b, a rectangle a by b with area a \times b. The squares on the right make a rows of c, a rectangle a by c with area a \times c. Each square of the whole is on exactly one side of the cut, so together the two pieces hold every square of the whole, each exactly once. The count is the same however it is grouped:
Cutting the other side instead gives (b + c) \times a = (b \times a) + (c \times a) by the same argument, with rows in place of columns. \blacksquare
Worked examples
Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, Examples 5 and 6 with Part 3, and Examples 7 and 8 with Part 4.
From tiles to an area model
Adjoa’s dollhouse kitchen floor is tiled with square tiles 1 inch on a side, 7 tiles along one wall and 3 rows of them. Write the multiplication the floor shows, find its area, and describe the area model that shows the same multiplication without drawing any tiles.
The tiles sit in equal rows with no gaps, so they are an array: 3 rows of 7. An array is counted by multiplying the number of rows by the number in each row:
Each tile is 1 square inch, so the floor is 21 square inches.
The tiles are 1 inch wide, so 7 of them in a row make a side 7 inches long, and 3 rows make a side 3 inches long. So the area model is a plain rectangle with one side labeled 7 inches and the other 3 inches, and 3 \times 7 = 21 written inside. Check by skip-counting one row at a time: 7, 14, 21, three numbers for three rows.
Solving for a missing factor
Use an area model to find the missing number in \square \times 8 = 56.
Draw the multiplication as a rectangle. One side is 8 units and the area is 56 square units; the other side is the missing factor. Picture the unit squares in rows along the 8-unit side, so each row holds 8 squares and is 1 unit deep. The missing side is as long as there are rows.
Count up by eights, one number per row: 8, 16, 24, 32, 40, 48, 56. That is 7 rows, so the missing side is 7 units and the missing factor is 7. Asking “how many rows of 8 make 56?” is the division 56 \div 8 = 7.
Check: 7 \times 8 = 56.
Adjoa’s dollhouse wall
Adjoa’s dollhouse wall is 9 inches long and 7 inches tall. Find its area by cutting the 9-inch side into 5 inches and 4 inches.
The cut runs straight up the wall, so both pieces keep the full height of 7 inches. The left piece is 7 by 5 and the right piece is 7 by 4:
The two pieces fill the wall with no gaps and no overlaps, so the wall’s area is their sum:
The wall is 63 square inches. In one line, by the distributive property, 7 \times 9 = (7 \times 5) + (7 \times 4). Check another way: ten sevens are 70, and the wall is one seven short of ten sevens, so 70 - 7 = 63.
Cutting either side
Kwaku has a piece of card in the studio 9 centimeters long and 8 centimeters tall. Find its area twice: once by cutting the 9-centimeter side into 5 + 4, and once by cutting the 8-centimeter side into 5 + 3.
Cut A goes up and down, splitting the 9. Each piece keeps the full height of 8:
Cut B goes across, splitting the 8. Each piece keeps the full length of 9:
Both give 72 square centimeters. They must agree: each cut makes two pieces that fill the same card, and the card does not change size when a line is drawn on it. Which side to cut is a choice about which facts are easier.
The art studio floor
The studio floor is 16 meters long and 4 meters wide. Find its area by cutting at the ten.
Place value says 16 = 10 + 6, so cut the 16-meter side at 10 meters. Both pieces keep the 4-meter width.
The tens piece is 4 rows of 10 square meters, which is 4 tens:
The ones piece is 4 rows of 6:
Together they fill the floor, so its area is
The floor is 64 square meters. Check by doubling: 2 \times 16 = 32, and 4 rows is twice 2 rows, so 32 + 32 = 64.
Yaa’s banner
Yaa’s banner is 40 feet long and 3 feet tall. What is its area?
Each row of the banner is 1 foot tall and 40 feet long. Forty is 4 tens, so each row is 4 ten-strips of 10 square feet. There are 3 rows, so the banner is 3 rows of 4 ten-strips:
ten-strips. Twelve tens is 120, so
The banner is 120 square feet. Check by adding rows: 40 + 40 + 40 = 120.
Cushions on the studio floor
For a drawing class, Kwaku sets out square cushions 1 foot on a side in 5 rows of 14, side by side with no gaps. How many cushions are there, and how much floor do they cover?
SolutionThe cushions are an array of 5 rows of 14, so there are 5 \times 14 of them. Cut 14 at the ten, 14 = 10 + 4:
There are 70 cushions.
The cushions also make a rectangle. Each is 1 foot wide, so the 14 in a row stretch 14 feet, and the 5 rows stretch 5 feet. The rectangle is 14 feet by 5 feet, and each cushion covers 1 square foot, so they cover 70 square feet of floor.
The same multiplication answered both questions. Only the unit changed: 70 cushions in the first answer, 70 square feet in the second.
The pool deck
Kwaku is building a deck along one side of a pool. The deck is a rectangle 3 feet wide, and its boards must cover 57 square feet. How long is the deck?
The area model is a rectangle with one side 3 feet and area 57 square feet; the other side is the unknown, 3 \times \square = 57. That is too big for one known fact, so cut the rectangle.
Take a first piece that is easy: 10 feet long. It is 3 \times 10 = 30 square feet. The area left over is
square feet, and it is still 3 feet wide. A piece 3 feet wide with area 27 square feet is 3 rows of 9, since 3 \times 9 = 27. So the second piece is 9 feet long.
The whole deck is the two pieces end to end:
The deck is 19 feet long. The answer is a length, so it is in feet, not square feet. Check with a cut at the ten: 3 \times 19 = (3 \times 10) + (3 \times 9) = 30 + 27 = 57.
Boundaries
Whole-number sides, for now. The area model here counts whole unit squares, so the side lengths are whole numbers. The same rectangle works for fractions of a unit, where the pieces along the edge are parts of squares; that comes in A fraction times a fraction.
A cut must go all the way across. A straight cut from one side to the opposite side makes two rectangles. A cut that stops partway, or a slanted cut, does not make two rectangles, and their areas are not products of the side lengths.
Each piece keeps the whole of the side the cut runs along. In 7 \times (5 + 4), both pieces are 7 tall, because a straight cut up the wall leaves the height alone. Only the side that is cut is split into a sum.
Units go with the story. The area model is a picture of a multiplication, and it works for any product of whole numbers. An answer is in square units only when the story is about a surface. When it counts chairs or jars, the answer is in chairs or jars.
Length times length is area; area and length are not added. In Example 8, 30 square feet and 27 square feet may be added, and 10 feet and 9 feet may be added, but a length is never added to an area.
Two cuts in each factor are next. A two-digit number times a two-digit number, such as 14 \times 16, needs both sides cut, giving four pieces. That is Partial products.
Where this goes next
The cut at the ten is the start of every written method for multiplying larger numbers. In Partial products both factors are split by place value, and the rectangle falls into four pieces, one for each pair of parts. The standard method in Multi-digit multiplication is the same pieces written in a shorter way.
Cutting a rectangle into rectangles also works the other way: a floor shaped like an L is not a rectangle, but it can be cut into rectangles whose areas are added, which is Splitting a figure into rectangles.
Much later, the same picture multiplies expressions with letters in them. A rectangle with sides x + 3 and x + 2 falls into four pieces, which is Multiplying two binomials.
Exercises
Exercises 1 to 5 go with Part 1, 6 to 10 with Part 2, 11 to 16 with Part 3, and 17 to 21 with Part 4. The reasoning group draws on all four.
Part 1: a multiplication drawn as a rectangle
The figure shows rectangles A and B, for Exercises 1 and 2.
- Rectangle A is ruled into centimeter squares. Write the multiplication it shows and find its area.
- Rectangle B has only its side lengths marked. Write the multiplication it shows and find its area.
- Describe the area model for 5 \times 9: what are its side lengths, and what is its area?
- Use an area model to find the missing number in \square \times 3 = 24.
- Adjoa tiles her dollhouse bedroom floor with inch tiles in 5 rows of 6. How many tiles does she use, and what is the area of the floor?
Part 2: one rectangle, cut in two
- Find 6 \times 7 by cutting the 7 side into 5 + 2. Give the area of each piece.
- Find 9 \times 7 by cutting the 7 side into 5 + 2. Give the area of each piece.
- A dollhouse floor is 8 inches long and 3 inches wide. Cut the 8-inch side into 5 inches and 3 inches and find the area.
- Find the area of a rectangle 6 by 9 in two ways: once by cutting the 9 side into 5 + 4, once by cutting the 6 side into 3 + 3.
- A rectangle is cut into two pieces that share a side 5 units long. The pieces have areas 20 and 15 square units. What are the side lengths of the whole rectangle, and what is its area?
Part 3: cutting at the ten
- Find 3 \times 18 with an area model cut at the ten.
- Find 4 \times 23 by cutting 23 into 20 + 3.
- Find 4 \times 50 using ten-strips.
- Kwaku tapes a roll of paper to the studio floor, 38 feet long and 2 feet wide. What area does it cover?
- Find 9 \times 11 with an area model.
- Find 8 \times 30 using ten-strips.
Part 4: stories about arrays and areas
- Adjoa lays square tiles 1 foot on a side in 4 rows of 21 on the pool deck, with no gaps. How many tiles is that, and how much of the deck do they cover?
- A pool deck is a rectangle 4 feet wide that covers 68 square feet. How long is it?
- Yaa sets out chairs in the studio in 6 rows of 15. How many chairs are there? Is the answer in square units?
- Adjoa has two dollhouse rugs, one 12 inches by 5 inches and one 8 inches by 7 inches. Which covers more, and by how much?
- Kwaku will paint a studio floor 24 meters long and 3 meters wide. One can of paint covers 9 square meters. How many cans does he need?
Reasoning
- An area model is a plain rectangle with no squares drawn inside. Explain why it still shows exactly how many unit squares the rectangle holds.
- Is 6 \times 13 equal to (6 \times 10) + 3? Use a rectangle 6 by 13, cut at the ten, to explain.
- Kwaku says a rectangle 2 by 30 and a rectangle 6 by 10 have the same area. Use ten-strips to explain whether he is right.
- Explain why cutting a rectangle in two, anywhere along one side, never changes the total area.
Answers
- 2 \times 8 = 16 square centimeters. There are 2 rows of 8 squares.
- 3 \times 9 = 27 square centimeters. The sides are 9 and 3 centimeters, so 3 rows of 9.
- A rectangle with sides 5 units and 9 units. Its area is 5 \times 9 = 45 square units.
- 8. One side is 3 and the area is 24; rows of 3 count 3, 6, 9, 12, 15, 18, 21, 24, which is 8 rows, so 8 \times 3 = 24.
- 30 tiles, and the floor is 30 square inches: 5 \times 6 = 30.
- 42. The pieces are 6 \times 5 = 30 and 6 \times 2 = 12, and 30 + 12 = 42.
- 63. The pieces are 9 \times 5 = 45 and 9 \times 2 = 18, and 45 + 18 = 63.
- 24 square inches. The pieces are 3 \times 5 = 15 and 3 \times 3 = 9, and 15 + 9 = 24.
- 54 both ways: (6 \times 5) + (6 \times 4) = 30 + 24 = 54 and (3 \times 9) + (3 \times 9) = 27 + 27 = 54.
- 5 by 7, area 35 square units. The pieces are 5 \times 4 = 20 and 5 \times 3 = 15, so the other side is 4 + 3 = 7, and 20 + 15 = 35.
- 54. Cut 18 = 10 + 8: (3 \times 10) + (3 \times 8) = 30 + 24 = 54.
- 92. 4 \times 20 is 4 rows of 2 tens, which is 8 tens, or 80; 4 \times 3 = 12; and 80 + 12 = 92.
- 200. Each row is 5 ten-strips, and 4 rows make 4 \times 5 = 20 ten-strips, which is 20 tens, or 200.
- 76 square feet. Cut 38 = 30 + 8: 2 \times 30 = 6 tens = 60, 2 \times 8 = 16, and 60 + 16 = 76.
- 99. Cut 11 = 10 + 1: (9 \times 10) + (9 \times 1) = 90 + 9 = 99.
- 240. Each row is 3 ten-strips, and 8 rows make 8 \times 3 = 24 ten-strips, which is 24 tens, or 240.
- 84 tiles, covering 84 square feet. 4 \times 21 = (4 \times 20) + (4 \times 1) = 80 + 4 = 84.
- 17 feet. A first piece 10 feet long is 4 \times 10 = 40 square feet; 68 - 40 = 28 is left, and 4 \times 7 = 28, so the rest is 7 feet; 10 + 7 = 17. Check: 4 \times 17 = 68.
- 90 chairs: 6 \times 15 = (6 \times 10) + (6 \times 5) = 60 + 30 = 90. No: the product counts chairs, so the answer is in chairs, not square units.
- The 12 by 5 rug, by 4 square inches. It is 5 \times 12 = 50 + 10 = 60 square inches, the other is 8 \times 7 = 56, and 60 - 56 = 4.
- 8 cans. The floor is 3 \times 24 = 60 + 12 = 72 square meters, and 72 \div 9 = 8.
- The side lengths say how the squares would sit. A side b units long holds b unit squares in a row, and a side a units long holds a rows, so the rectangle holds a rows of b, which is a \times b squares, whether or not they are drawn.
- No. Cut at the ten, the rectangle is a 6 by 10 piece and a 6 by 3 piece. The second piece is 6 rows of 3, which is 18, not 3. So 6 \times 13 = 60 + 18 = 78, while (6 \times 10) + 3 = 63.
- He is right: both are 60. The 2 by 30 rectangle is 2 rows of 3 ten-strips, which is 6 ten-strips. The 6 by 10 rectangle is 6 rows of 1 ten-strip, also 6 ten-strips. Six tens is 60.
- A straight cut from one side to the other does not add or remove any surface; it only draws a line. Every unit square of the whole lies in exactly one of the two pieces, so the two areas together count every square once, and their sum is the area of the whole.