Area and length models of a fraction
A sheet cake is cut into six equal pieces and four of them get frosting. How do you say, in one number, how much of the cake is frosted?
Where you are 4 in A fraction as part of a whole
Why this exists
In Equal shares and naming the whole a whole was cut into equal shares and one share got a name: one half, one third, one fourth. But one share is rarely all that someone has. Lempi cuts a sheet cake into 6 equal pieces and frosts 4 of them. Pihla eats 3 of the 8 equal lengths of a fruit leather strip. “One sixth” and “one eighth” name a single piece. Nothing so far says “four of the sixths” in one number.
A fraction says it. Behind every fraction is one picture: a whole cut into equal parts, with some of the parts shaded. The whole can be a flat shape, and then the parts are equal areas. Or it can be a length, and then the parts are equal lengths. Both pictures carry the rest of fractions, and the length picture is the one that later becomes the number line.
Definition
A whole is cut into b equal parts, and a of those parts are taken. Together they are the fraction
of the whole. The bottom number counts the equal parts the whole is cut into. The top number counts the parts taken. In this lesson a is never more than b: a whole cut into 6 parts has only 6 to take.
A fraction is read by the name of its parts. The table gives the names used here.
| Equal parts in the whole | One part | The parts are called | For example |
|---|---|---|---|
| 2 | \displaystyle \frac{1}{2}, one half | halves | \displaystyle \frac{1}{2}, one half |
| 3 | \displaystyle \frac{1}{3}, one third | thirds | \displaystyle \frac{2}{3}, two thirds |
| 4 | \displaystyle \frac{1}{4}, one fourth | fourths, or quarters | \displaystyle \frac{3}{4}, three fourths |
| 6 | \displaystyle \frac{1}{6}, one sixth | sixths | \displaystyle \frac{4}{6}, four sixths |
| 8 | \displaystyle \frac{1}{8}, one eighth | eighths | \displaystyle \frac{3}{8}, three eighths |
The top number is called the numerator and the bottom number the denominator. What each one tells you is the next lesson, What the numerator and the denominator each tell you.
A fraction is pictured in two ways.
- The area model. The whole is a flat shape: a cake, a quesadilla, a rectangle drawn on paper. Equal parts are parts with the same area, the same amount of surface (see Area as covering). They need not have the same shape.
- The length model. The whole is a length: a strip, a rope, a line drawn on paper. Equal parts are equal lengths, laid end to end along it. A strip of even width cut straight across is a length model, because its pieces differ only in length.
Every fraction is a fraction of a whole, and the whole has to be named, just as in Equal shares and naming the whole. Three fourths of a big cake is more cake than three fourths of a small one.
Intuition
The lesson runs over four days. Part 1 shades parts of a shape. Part 2 takes parts of a length. Part 3 asks what “equal” means and why the whole matters. Part 4 works back from a part to the whole.
Part 1: shading parts of a shape
Look at the rectangle at the top of Figure 1. First count all the parts: there are 3, and they are the same size, so each is one third. Then count the shaded parts: 2. Two of the thirds are shaded, so two thirds, \displaystyle \frac{2}{3}, of the rectangle is shaded.
Point at the shaded parts and count them by name: “one third, two thirds”. The word thirds says how big each part is; the count says how many of them there are. Counting this way is the start of Counting by unit fractions.
The parts that are not shaded are a fraction too. One of the three is unshaded, so \displaystyle \frac{1}{3} of the rectangle is unshaded. The shaded and unshaded parts together are all 3 thirds, the whole rectangle.
Part 2: a fraction of a length
The line at the bottom of Figure 1 is a length cut into 3 equal lengths, with 2 of them drawn thick. That is \displaystyle \frac{2}{3} of the length. Children can build it: fold a paper strip into three equal lengths (roll it into an S and press flat), open it, and color two of the three.
A length model adds one new idea. Its parts are counted from one end, so a fraction of a length is also a place: the end of the second third is two thirds of the way along. Marking such places on a line that starts at 0 is how fractions land on the number line, in Locating a fraction on the number line.
Part 3: what “equal” means, and the whole
Equal parts must be the same size. For an area that means the same amount of surface, not the same shape: a square piece and a triangle piece can both be one fourth of a cake. For a length it means the same length.
And the whole still matters. Three fourths of a long strip is a longer piece than three fourths of a short one. The fraction says how the whole was cut and how many parts were taken. How much that is depends on the whole.
Part 4: from a part back to the whole
If you are handed \displaystyle \frac{2}{3} of a strip, you hold two of its three equal parts. Split what you hold into two equal lengths, and one of them is a third. Three thirds make the whole. The fraction tells you how many parts you have and how many the whole needs.
Results
A fraction names equal parts of one whole. \displaystyle \frac{a}{b} of a whole is a parts, when the whole is cut into b equal parts.
The parts taken and the parts left make the whole. If a of the b parts are shaded, then b - a are not. For example, \displaystyle \frac{4}{6} shaded leaves \displaystyle \frac{2}{6} unshaded, and 4 + 2 = 6.
Equal parts have the same size, not always the same shape. Two parts of one whole with the same area are the same fraction of it, whatever their shapes.
The same fraction of a bigger whole is more. \displaystyle \frac{3}{4} of a whole twice as long is a piece twice as long.
One part gives back the whole. If a parts are known, one of them is the known piece shared into a equal parts, and b such parts make the whole.
Why it is true
The parts taken and the parts left make the whole. The whole is exactly its b equal parts, with no gaps and nothing counted twice. Each part is either taken or not, so the taken ones and the others together are all b of them.
Same size, different shape. The name of a part depends only on how many parts of its size make the whole. A triangle with the area of 4 small squares and a square of 4 small squares are the same size, so if four parts of that size make the cake, each is one fourth (Example 5).
The same fraction of a bigger whole is more. If one strip is twice as long as another, each of its fourths is twice as long as a fourth of the other, since four of them make twice the length. Three of its fourths are then twice as long as three of the other’s.
One part gives back the whole. \displaystyle \frac{a}{b} means a equal parts out of b. So the known piece is made of a parts of the same size, and sharing it into a equal parts gives one of them. The whole is b parts that size, put together with no gaps and no overlaps.
Worked examples
Have paper strips, a ruler and some square tiles to hand. Folding and counting answer most of these.
Part 1: a frosted sheet cake
Lempi bakes a sheet cake and cuts it into 6 equal pieces, in 2 rows of 3. She frosts 4 of the pieces, shaded in Figure 2. What fraction of the cake is frosted? What fraction is not?
The whole is the cake. The cuts make 2 rows of 3 pieces, 6 pieces in all, and every piece has the same width and the same height, so they are the same size. The cake is cut into 6 equal parts, so each piece is one sixth, \displaystyle \frac{1}{6}.
Count the frosted pieces: 4. Four of the sixths are frosted, so \displaystyle \frac{4}{6} of the cake is frosted, four sixths.
Count the others: 2 pieces, so \displaystyle \frac{2}{6} of the cake is not frosted.
Check: 4 + 2 = 6, so the frosted and unfrosted pieces together are all six sixths, the whole cake. The frosted pieces do not have to sit next to each other: any 4 of the 6 pieces are \displaystyle \frac{4}{6} of the cake.
Part 1: a quesadilla in fourths
Kaarlo cuts a quesadilla with two cuts through its center, one up and down and one across, and eats the 3 shaded wedges in Figure 3. What fraction of the quesadilla did he eat? What fraction is left? If he then eats the last wedge too, what fraction has he eaten?
The two cuts cross at the center and are square to each other, so the four wedges are the same wedge turned around: each would lie exactly on another. The quesadilla is cut into 4 equal parts, fourths.
He ate 3 of the fourths, so he ate \displaystyle \frac{3}{4}, three fourths. One wedge is left, so \displaystyle \frac{1}{4} is left.
After the last wedge he has eaten all 4 fourths, \displaystyle \frac{4}{4}, which is the whole quesadilla. “Three fourths, four fourths”: the word fourths names the size of each wedge, and the count says how many.
Part 2: a strip of fruit leather
Pihla has a fruit leather strip 24 centimeters long, marked into 8 equal lengths. She eats 3 of them from one end (Figure 4). What fraction of the strip did she eat? How long is the piece she ate, and how much is left?
The whole is the strip. It is cut into 8 equal lengths, so each is one eighth. She ate 3 of the eighths, so she ate \displaystyle \frac{3}{8} of the strip.
How long is one eighth? Eight equal lengths make 24 centimeters, and 8 \times 3 = 24, so each length is 24 \div 8 = 3 centimeters. She ate 3 of them:
so she ate 9 centimeters. Left are 5 of the 8 lengths, \displaystyle \frac{5}{8} of the strip, and 5 \times 3 = 15 centimeters. Check: 9 + 15 = 24.
The strip is flat, but it is the same width all along and is cut straight across, so pieces of equal length are equal pieces. That is why a strip is a length model.
Part 2: tape on a jump rope
Kaarlo’s jump rope is 8 feet long from handle to handle. He marks it into 4 equal parts with three pieces of tape and paints the first part, from the left handle to the first tape (Figure 5). How far from the left handle is the first tape? What fraction of the rope is painted? How far along the rope is the third tape?
Four equal parts make 8 feet, and 4 \times 2 = 8, so each part is 8 \div 4 = 2 feet long and is one fourth of the rope. The first tape is 2 feet from the left handle.
The painted part is 1 of the 4 equal parts, so \displaystyle \frac{1}{4} of the rope is painted.
The tapes sit at the ends of the first, second and third parts: 2, 4 and 6 feet from the left handle. The third tape has 3 parts behind it, so it is \displaystyle \frac{3}{4} of the way along the rope, and 3 \times 2 = 6 feet from the left handle.
In a length model a fraction is a place as well as a piece: \displaystyle \frac{3}{4} of the way along is where the third fourth ends. Doing the same on a line marked 0 and 1 is locating a fraction on the number line.
Part 3: fourths of different shapes
Lempi cuts a square sheet cake into 4 pieces as in Figure 6. The left half is cut across the middle into two squares, and the right half is cut corner to corner into two triangles. The dashed lines mark the cake into 16 small squares. Are the four pieces fourths? What fraction of the cake is shaded?
The cake is 4 rows of 4 small squares, 16 in all. Four fourths of it would have 4 small squares each, since 4 \times 4 = 16.
Each square piece is 2 rows of 2 small squares, 4 small squares.
The right half is 4 rows of 2 small squares, 8 small squares, cut corner to corner into two triangles. Turn one triangle halfway around and it lies exactly on the other, so the two triangles are the same size. Two same-size pieces that make 8 small squares have 4 each, since 4 + 4 = 8.
All four pieces are 4 small squares, so they are fourths, even though two are squares and two are triangles. The shaded pieces are one square and one triangle, 2 of the 4 fourths, so \displaystyle \frac{2}{4} of the cake is shaded. (That is also half the cake; one amount with two names is Seeing equivalent fractions in pictures and on the number line.)
Part 3: three fourths of two strips
Kaarlo’s fruit leather strip is 16 centimeters long and Pihla’s is 8 centimeters long. Each eats \displaystyle \frac{3}{4} of their own strip (Figure 7). Did they eat the same amount?
Kaarlo’s strip in 4 equal lengths: 16 \div 4 = 4, so each fourth is 4 centimeters. He eats 3 of them, 3 \times 4 = 12 centimeters.
Pihla’s strip in 4 equal lengths: 8 \div 4 = 2, so each fourth is 2 centimeters. She eats 3 of them, 3 \times 2 = 6 centimeters.
They ate the same fraction but not the same amount: Kaarlo ate 12 centimeters and Pihla 6, so Kaarlo ate twice as much. His whole is twice as long, so each of his fourths is twice as long. “Three fourths” says how each strip was cut and how many parts were eaten; the amount depends on the strip.
Part 4: the whole from two thirds
Lempi has a piece of fruit leather 6 centimeters long. It is \displaystyle \frac{2}{3} of the strip it was cut from. Draw the whole strip. How long was it?
\displaystyle \frac{2}{3} means the strip was cut into 3 equal lengths and Lempi has 2 of them. So her piece is two equal lengths. Split it in the middle (the dashed line in Figure 8): each half of her piece is one third of the strip, and 6 \div 2 = 3, so one third is 3 centimeters.
The whole strip is 3 thirds, her two and one more of the same length:
so the whole strip was 9 centimeters long. The missing part is one third, which is half the size of her piece.
Check: a strip 9 centimeters long cut into 3 equal lengths gives 9 \div 3 = 3 centimeters each, and 2 of them are 6 centimeters, Lempi’s piece.
Part 4: the whole cake from what is left
After a party, what is left of Pihla’s sheet cake is 6 small squares in 2 rows of 3. It is \displaystyle \frac{3}{4} of the whole cake. How many small squares was the whole cake? Draw one shape it could have had.
\displaystyle \frac{3}{4} means the whole cake was cut into 4 equal parts and 3 of them are left. So the 6 squares left are 3 equal parts. Three equal parts that make 6 squares have 2 squares each, since 2 + 2 + 2 = 6, so one fourth of the cake is 2 small squares.
The whole cake is 4 fourths:
so it was 8 small squares. One shape is 2 rows of 4 (Figure 9): the piece left, plus one more column of 2 squares.
That is not the only shape. The missing 2 squares could have sat along the top or stuck out at a corner. Every such cake has 8 small squares, so its size is fixed even though its shape is not. Check: 8 squares in 4 equal parts gives 2 each, and 3 parts are 6 squares.
Boundaries
The parts must be equal. Counting pieces is not enough. A shape cut into 3 pieces of different sizes, with 1 shaded, does not show \displaystyle \frac{1}{3} (Exercise 4 is a shape like that). For an area, equal means the same amount of surface; for a length, the same length.
The parts must make the whole exactly. No gaps, no overlaps, nothing left over. Otherwise the pieces are not parts of that whole.
A fraction belongs to one whole. \displaystyle \frac{3}{4} of a big cake and \displaystyle \frac{3}{4} of a small cake are not the same amount (Example 6). Fractions of different wholes cannot be compared by their names alone.
The bottom number is never zero. A whole cannot be cut into 0 parts, so a fraction with 0 on the bottom names nothing.
At most the whole, in this lesson. Here the top number is never more than the bottom one, and when they are equal, as in \displaystyle \frac{4}{4}, the fraction is the whole. Taking more parts than one whole has, such as 5 fourths, needs a second whole; that is Fractions greater than one (improper fractions).
Building the whole from a part. For a length there is one answer: a strip of one length. For an area the size of the whole is fixed but its shape is not (Example 8).
Groups of things. A fraction of a bag of apples counts apples, not area or length; that is A fraction of a set.
Where this goes next
The two numbers of a fraction get their own lesson in What the numerator and the denominator each tell you: the bottom number sets the size of a part and the top number counts parts. Counting “one sixth, two sixths, three sixths” is Counting by unit fractions, built on Unit fractions, and counting past the whole leads to Fractions greater than one (improper fractions).
The length model turns into the number line. Put a length from 0 to 1 on a line, cut it into equal parts, and \displaystyle \frac{3}{4} is a point: Locating a fraction on the number line and Fractions on the number line do that. The square cut into fourths in Example 5, with two fourths covering half, opens Seeing equivalent fractions in pictures and on the number line. Pieces of the same whole compared by size become Comparing fractions with the same denominator or numerator, and working back from a part to its whole grows into Finding the whole when only a fraction of it is known.
Exercises
Part 1: shading parts of a shape
Use Figure 10 for Exercises 1 to 4. Say what fraction of each shape is shaded.
- Shape A.
- Shape B.
- Shape C.
- Shape D. Is the shaded part one third? If not, what fraction is it?
- A quesadilla is cut into 8 equal wedges. Kaarlo eats 3 and Lempi eats 4. What fraction of the quesadilla did they eat together? What fraction is left?
- Draw a sheet cake cut into 3 rows of 2 equal pieces, and shade \displaystyle \frac{5}{6} of it. How many pieces are not shaded, and what fraction of the cake is that?
Part 2: a fraction of a length
- A fruit leather strip 12 centimeters long is marked into 6 equal lengths. How long is each length? Pihla eats 4 of them. What fraction of the strip is that, and how many centimeters?
- A jump rope 9 feet long is marked into 3 equal parts with two pieces of tape. How far from the left handle is each piece of tape? What fraction of the way along the rope is the second?
- A fruit leather strip 18 centimeters long is marked into 6 equal lengths, and 1 length is eaten. What fraction of the strip is left, and how long is it?
- Lempi’s jump rope is 6 feet long. She puts tape 2 feet from the left handle. What fraction of the way along the rope is the tape?
Part 3: equal parts and the whole
- A square sheet cake is marked into 16 small squares, 4 by 4. How many small squares are in \displaystyle \frac{1}{4} of the cake? How many are in \displaystyle \frac{3}{4} of it?
- Pihla eats \displaystyle \frac{2}{3} of a strip 9 centimeters long. Kaarlo eats \displaystyle \frac{2}{3} of a strip 12 centimeters long. How many centimeters does each eat?
- Which is longer: \displaystyle \frac{3}{4} of a strip 4 centimeters long, or \displaystyle \frac{1}{2} of a strip 8 centimeters long?
- A sheet cake of 24 small squares, 4 rows of 6, is cut into 6 equal pieces. How many small squares are in each piece? How many are in \displaystyle \frac{5}{6} of the cake?
- Use Figure 11. Are the four pieces fourths of the cake? What fraction of the cake is shaded?
Part 4: from a part back to the whole
- A piece of fruit leather 12 centimeters long is \displaystyle \frac{4}{6} of the strip. How long is the whole strip?
- A piece of fruit leather 9 centimeters long is \displaystyle \frac{3}{4} of the strip. How long is the whole strip?
- What is left of a sheet cake is 4 equal pieces, and that is \displaystyle \frac{2}{3} of the cake. How many pieces of that size made the whole cake?
- A piece of tape is \displaystyle \frac{5}{6} of the way along a jump rope, 10 feet from the left handle. How long is the rope?
- A piece of 2 small squares is \displaystyle \frac{1}{4} of a sheet cake. How many small squares was the whole cake? Draw two different shapes it could have had.
Reasoning
- A rectangle is cut into 4 pieces and 1 is shaded. Is the shaded piece always \displaystyle \frac{1}{4} of the rectangle? Explain, with a drawing.
- Kaarlo and Lempi each eat \displaystyle \frac{2}{3} of a quesadilla. Did they eat the same amount? Explain.
- Two squares are the same size. One is cut into 4 equal strips, the other corner to corner both ways into 4 triangles. Explain why one strip is the same amount as one triangle.
- Can \displaystyle \frac{1}{4} of one sheet cake be more than \displaystyle \frac{3}{4} of another? Explain with small squares.
- A strip that is \displaystyle \frac{2}{3} of a whole strip gives exactly one whole strip, but a flat piece that is \displaystyle \frac{2}{3} of a cake could have come from cakes of different shapes. Explain why, and say what all those cakes have in common.
Answers
- \displaystyle \frac{5}{8}. Two rows of 4 same-size pieces make 8 eighths, and 5 are shaded.
- \displaystyle \frac{5}{6}. The cuts all pass through the center and are evenly spaced, so the 6 wedges are the same size; 5 are shaded.
- \displaystyle \frac{1}{6}. Two rows of 3 same-size pieces make 6 sixths, and 1 is shaded.
- No. The three pieces are not the same size, so they are not thirds. The dashed lines cut the shape into 6 equal strips, and the shaded piece is 1 of them, so it is \displaystyle \frac{1}{6}. (The middle piece is 2 strips and the wide piece 3: 1 + 2 + 3 = 6.)
- \displaystyle \frac{7}{8} eaten and \displaystyle \frac{1}{8} left. 3 + 4 = 7 of the 8 wedges were eaten, and 8 - 7 = 1 is left.
- 1 piece, \displaystyle \frac{1}{6} of the cake. The cake has 3 \times 2 = 6 equal pieces; any 5 may be shaded, and 6 - 5 = 1 is not.
- 2 centimeters each, since 12 \div 6 = 2. Pihla eats \displaystyle \frac{4}{6} of the strip, 4 \times 2 = 8 centimeters.
- 3 feet and 6 feet from the left handle, since 9 \div 3 = 3. The second tape has 2 parts behind it, so it is \displaystyle \frac{2}{3} of the way along.
- \displaystyle \frac{5}{6} is left, 15 centimeters. Each length is 18 \div 6 = 3 centimeters, and 5 \times 3 = 15.
- \displaystyle \frac{1}{3} of the way. Three lengths of 2 feet make the rope, 2 + 2 + 2 = 6, so 2 feet is one of 3 equal parts.
- 4 small squares in \displaystyle \frac{1}{4}, since 16 \div 4 = 4, and 3 \times 4 = 12 in \displaystyle \frac{3}{4}.
- Pihla eats 6 centimeters and Kaarlo 8. Pihla’s thirds are 9 \div 3 = 3 centimeters and 2 \times 3 = 6; Kaarlo’s are 12 \div 3 = 4 centimeters and 2 \times 4 = 8.
- \displaystyle \frac{1}{2} of the 8 centimeter strip, which is 4 centimeters. \displaystyle \frac{3}{4} of the 4 centimeter strip is 3 \times 1 = 3 centimeters, since each fourth is 4 \div 4 = 1.
- 4 small squares in each piece, since 24 \div 6 = 4, and 5 \times 4 = 20 small squares in \displaystyle \frac{5}{6}.
- Yes. Each piece covers 4 of the 16 small squares, so each is one fourth, although the pieces are not all the same shape (a column, a square and two L shapes). Two of the fourths are shaded, so \displaystyle \frac{2}{4} of the cake is shaded.
- 18 centimeters. The piece is 4 sixths, so one sixth is 12 \div 4 = 3 centimeters, and the strip is 6 \times 3 = 18 centimeters.
- 12 centimeters. The piece is 3 fourths, so one fourth is 9 \div 3 = 3 centimeters, and the strip is 4 \times 3 = 12 centimeters.
- 6 pieces. The 4 pieces are 2 thirds, so one third is 4 \div 2 = 2 pieces, and the whole is 3 \times 2 = 6 pieces.
- 12 feet. The tape has 5 sixths behind it, so one sixth is 10 \div 5 = 2 feet, and the rope is 6 \times 2 = 12 feet.
- 8 small squares, since 4 \times 2 = 8. For example, 2 rows of 4, or one row of 8, or an L of 8 squares; any shape of 8 small squares will do.
- No. The 4 pieces must be the same size. Cut a rectangle into one big piece and three thin slivers, shade a sliver, and the shaded piece is far less than one fourth.
- Only if the two quesadillas are the same size. Each \displaystyle \frac{2}{3} is two thirds of its own quesadilla, and two thirds of a bigger quesadilla is more.
- Each strip is one of 4 equal parts of its square, and so is each triangle: the four triangles are the same triangle turned around. Both are one fourth of squares of the same size, so they are the same amount, though the shapes differ.
- Yes, when the wholes differ. One fourth of a cake of 24 small squares is 6 squares, and three fourths of a cake of 4 small squares is 3 squares; 6 is more.
- The piece is 2 thirds, so half of it is one third, and three thirds put together with no gaps or overlaps make the whole. Lengths can only go end to end, so the piece and one more third make one whole strip. A flat third can be fitted against the piece in different places, so the cake’s shape can differ. Every such cake has the same area: three thirds, the piece and half as much again.