Grade 3 · 3.NF.A.3C

Whole numbers written as fractions

Three pans of brownies, each cut into fourths, are all eaten. That is twelve fourths of a pan, and it is also exactly three pans. How can one amount be a fraction and a whole number at once?

You need

Fractions greater than one (improper fractions)Grade 3 What the numerator and the denominator each tell youGrade 3

Where you are 15 in Fractions

Fractions, ratio and percent3

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▴ 9 above Unit fractions2 What the numerator and the denominator each tell you

Whole numbers written as fractions

A fraction as division Dividing fractions6 ▾ Show all 15 (1 more)

Why this exists

So far whole numbers and fractions have looked like two kinds of number. Whole numbers count things: 3 pans, 6 trays. Fractions count pieces: \displaystyle \frac{3}{4} of a pan, \displaystyle \frac{5}{6} of a tray. In Fractions greater than one (improper fractions) the pieces went past one whole, and \displaystyle \frac{4}{4} turned out to be exactly 1.

That fact goes further. If three pans of brownies are each cut into fourths and every piece is eaten, then 12 fourths of a pan are gone. That is a fraction, \displaystyle \frac{12}{4}. It is also exactly 3 pans, with nothing left over. The same amount has two names.

This lesson shows that every whole number can be written as a fraction, and how to tell when a fraction is really a whole number. Once whole numbers and fractions are seen as one family of numbers, they can sit on one number line, be compared with each other, and later be added and multiplied together.

The lesson runs over two days. Part 1 writes whole numbers as fractions of one uncut whole, with denominator 1, and looks again at the many names for 1. Part 2 finds fractions such as \displaystyle \frac{12}{4} and \displaystyle \frac{6}{3} that equal whole numbers, in pictures and on the number line.

Definition

A fraction \displaystyle \frac{a}{b} names a copies of the piece you get by cutting one whole into b equal parts (see What the numerator and the denominator each tell you).

Denominator 1. Cutting a whole into 1 part leaves it as it is: the one piece is the whole. So \displaystyle \frac{1}{1} is one whole, and \displaystyle \frac{n}{1} is n whole things. For every whole number n,

n = \frac{n}{1}.

For example, \displaystyle 3 = \frac{3}{1} and \displaystyle \frac{6}{1} = 6.

Fractions equal to a whole number. A fraction \displaystyle \frac{a}{b} equals a whole number when its a pieces fill complete wholes exactly, with no piece left over. Every b pieces fill one whole, so this happens when a is a number you reach by counting by b: b, 2 \times b, 3 \times b and so on. Then the fraction is the number of wholes filled. For example, \displaystyle \frac{12}{4} = 3, because 3 \times 4 = 12: twelve fourths fill three wholes. Zero pieces fill zero wholes, so \displaystyle \frac{0}{b} = 0.

Here the whole numbers are 0, 1, 2, 3 and so on.

Intuition

Look at one pan of brownies. Leave it uncut and it is \displaystyle \frac{1}{1} of a pan. Cut it into 4 equal pieces and keep them all, and it is \displaystyle \frac{4}{4}. Cut it into 8 and keep them all, and it is \displaystyle \frac{8}{8} (Figure 1). Nothing was added or taken away. All three are the same pan, so

1 = \frac{1}{1} = \frac{4}{4} = \frac{8}{8}.

Figure 1One whole pan, named three ways. The pieces change; the pan does not.

Now take more than one pan, all the same size and cut the same way. Each full pan is one whole, so counting full pans counts whole numbers, and counting the pieces in them gives a fraction. Three full pans cut into fourths are 4 + 4 + 4 = 12 fourths, and also 3 pans. The fraction counts pieces; the whole number counts pans. They name the same amount.

On the number line the same thing happens. Cut every whole into 3 equal parts and count jumps from 0. After 3 jumps you land on 1, after 6 jumps on 2, after 9 on 3. So the point 2 has a second name, \displaystyle \frac{6}{3}. A whole number and a fraction equal to it are one point on the line with two labels.

Results

Every whole number is a fraction with denominator 1. For every whole number n, \displaystyle n = \frac{n}{1}. In particular \displaystyle 0 = \frac{0}{1} and \displaystyle 1 = \frac{1}{1}.

Many names for one. \displaystyle \frac{1}{1} = \frac{2}{2} = \frac{3}{3} = \frac{4}{4} = \frac{6}{6} = \frac{8}{8} = 1.

When a fraction is a whole number. If a = n \times b, then \displaystyle \frac{a}{b} = n. Every b pieces make one whole, and n groups of b pieces make n wholes. For example \displaystyle \frac{12}{4} = 3 and \displaystyle \frac{6}{3} = 2. If a is not reached by counting by b, the pieces leave part of a whole over and the fraction is not a whole number.

Every whole number has a name for every denominator. \displaystyle n = \frac{n \times b}{b}. For example, \displaystyle 2 = \frac{4}{2} = \frac{6}{3} = \frac{8}{4}.

Why it is true

The denominator b says one whole is cut into b equal parts, each \displaystyle \frac{1}{b} of the whole. When b = 1 the whole is cut into one part, which is the whole itself, so \displaystyle \frac{1}{1} = 1 and n copies of it are n wholes: \displaystyle \frac{n}{1} = n.

When b is bigger than 1, b copies of \displaystyle \frac{1}{b} cover one whole with nothing missing and nothing extra, so \displaystyle \frac{b}{b} = 1. Lay out n wholes of the same size, each cut into b equal parts. Each whole holds b pieces, so the n wholes hold n \times b pieces, n equal groups of b. All of them together are \displaystyle \frac{n \times b}{b} of one whole, and also n wholes.

On the number line, each whole from one whole number to the next is b jumps of \displaystyle \frac{1}{b}. So n \times b jumps from 0 cross n wholes and land exactly on n.

Worked examples

Example 1

Part 1: six sand trays as six over one

Siosaia sets up the sand table. It holds 6 sand trays of the same size, each full of sand and none of them divided (Figure 2). Taking one tray as the whole, write the sand as a fraction of a tray, and say what whole number it equals.

Figure 2Each tray is one whole, cut into 1 part. Six of them are \displaystyle \frac{6}{1} of a tray.

Solution

The whole is one tray. A tray that is not divided is a whole cut into 1 part, so its one piece is the entire tray: \displaystyle \frac{1}{1} of a tray, which is 1 tray.

There are 6 such pieces, 6 copies of \displaystyle \frac{1}{1}. The numerator counts the pieces, 6. The denominator says how many pieces make one tray, 1. So the sand is \displaystyle \frac{6}{1} of a tray.

Every piece is a whole tray, so 6 pieces are 6 trays:

\frac{6}{1} = 6.

The whole number 6 counts trays; the fraction \displaystyle \frac{6}{1} counts pieces, each one tray in size. They say the same thing.

Example 2

Part 1: one pan, many names

Dana bakes three pans of brownies of the same size. She leaves one pan uncut, cuts one into 4 equal pieces and cuts one into 8 equal pieces (Figure 1). At a class party, every pan is finished. Write what was eaten from each pan as a fraction of a pan. Which pan gave the most brownie? Then name a finished pan cut into 6 equal pieces.

Solution

The uncut pan is 1 piece, the whole pan, and all of it was eaten: \displaystyle \frac{1}{1} of a pan.

The second pan has 4 equal pieces, each \displaystyle \frac{1}{4} of a pan, and all 4 were eaten: \displaystyle \frac{4}{4} of a pan.

The third has 8 equal pieces, each \displaystyle \frac{1}{8} of a pan, and all 8 were eaten: \displaystyle \frac{8}{8} of a pan.

Each fraction names every piece of one pan, none missing and none extra, so each is one whole pan:

\frac{1}{1} = \frac{4}{4} = \frac{8}{8} = 1.

No pan gave more than another. The cuts changed how many pieces there were and how big each piece was, but not how much brownie was in the pan. A finished pan cut into 6 equal pieces is \displaystyle \frac{6}{6} of a pan, which is 1 too.

Example 3

Part 1: whole numbers on a number line

A number line runs from 0 to 5 with a tick at each whole number and no ticks in between. Name each tick with a fraction whose denominator is 1. Which point is \displaystyle \frac{0}{1}, and which is \displaystyle \frac{5}{1}?

Solution

From 0 to 1 is one whole, and it is not cut, so it is 1 part. One jump of that size is \displaystyle \frac{1}{1}, and every tick is a count of those jumps from 0.

Point	0	1	2	3	4	5
Jumps of \displaystyle \frac{1}{1}	0	1	2	3	4	5
Fraction	\displaystyle \frac{0}{1}	\displaystyle \frac{1}{1}	\displaystyle \frac{2}{1}	\displaystyle \frac{3}{1}	\displaystyle \frac{4}{1}	\displaystyle \frac{5}{1}

\displaystyle \frac{0}{1} is no jumps at all, the point 0. \displaystyle \frac{5}{1} is 5 jumps, the point 5, the right end of the line. Every whole number n sits at \displaystyle \frac{n}{1}, so every whole number is a fraction with denominator 1.

Example 4

Part 2: three pans of brownies in fourths

Losaline brings three pans of brownies of the same size to a bake sale, each cut into 4 equal pieces. Every piece is sold (Figure 3). Write the brownies sold as a fraction of a pan. Then say how many pans that is.

Figure 3Twelve fourths of a pan, grouped into pans. Every 4 fourths fill one pan.

Solution

Each piece is \displaystyle \frac{1}{4} of a pan. Three pans of 4 pieces hold 3 \times 4 = 12 pieces, and all 12 were sold. So the brownies sold are \displaystyle \frac{12}{4} of a pan, twelve fourths.

Now group the pieces into pans. Every 4 fourths fill one pan, \displaystyle \frac{4}{4} = 1. Twelve pieces make 4 + 4 + 4 = 12, three full groups of 4 with none left over. So

\frac{12}{4} = 3.

Losaline sold 3 pans, which is also \displaystyle \frac{3}{1} of a pan. Asking how many groups of 4 are in 12 is the division 12 \div 4 = 3, and it gives the same answer.

Example 5

Part 2: the point 2 in thirds

Dana draws a number line from 0 to 3 and cuts each whole into 3 equal parts (Figure 4). Name the point 2 as a fraction in thirds. Name 0, 1 and 3 the same way.

Figure 4Each whole number on the line also has a name in thirds. The point 2 is \displaystyle \frac{6}{3}.

Solution

Each part is \displaystyle \frac{1}{3} of a whole, so each jump from one tick to the next is \displaystyle \frac{1}{3}. Count jumps from 0.

To reach 1 takes 3 jumps, so 1 is \displaystyle \frac{3}{3}. To reach 2 takes 3 more, 3 + 3 = 6 jumps, so

2 = \frac{6}{3}.

To reach 3 takes 6 + 3 = 9 jumps, so \displaystyle 3 = \frac{9}{3}. The point 0 is no jumps, \displaystyle \frac{0}{3}.

The thirds that land on whole numbers are \displaystyle \frac{0}{3}, \displaystyle \frac{3}{3}, \displaystyle \frac{6}{3} and \displaystyle \frac{9}{3}, the numerators you reach counting by 3. A point such as \displaystyle \frac{7}{3} lands between 2 and 3, so it is not a whole number.

Example 6

Part 2: the number 2 with five denominators

Write 2 as a fraction with denominator 2, 3, 4, 6 and 8.

Solution

Picture 2 pans of brownies of the same size, cut the same way. The denominator says how many equal pieces each pan is cut into, and the numerator counts all the pieces in the 2 pans, which is 2 groups of that many.

Each pan cut into	Pieces in 2 pans	Fraction
2	2 \times 2 = 4	\displaystyle \frac{4}{2}
3	2 \times 3 = 6	\displaystyle \frac{6}{3}
4	2 \times 4 = 8	\displaystyle \frac{8}{4}
6	2 \times 6 = 12	\displaystyle \frac{12}{6}
8	2 \times 8 = 16	\displaystyle \frac{16}{8}

So \displaystyle 2 = \frac{4}{2} = \frac{6}{3} = \frac{8}{4} = \frac{12}{6} = \frac{16}{8}, and also \displaystyle \frac{2}{1}. On a number line, all six fractions mark the same point, 2. The more pieces each whole is cut into, the bigger both numbers get, but the amount stays 2 pans.

Example 7

Part 2: which labels are whole pans?

Losaline packs leftover brownie pieces in boxes and labels each box with how much of a pan is inside. All the pans were the same size. Which boxes hold a whole number of pans?

\frac{8}{4}, \quad \frac{9}{4}, \quad \frac{18}{6}, \quad \frac{5}{3}, \quad \frac{0}{2}.

Solution

A label is a whole number of pans when its pieces fill whole pans with none left over: the numerator must be reached by counting by the denominator.

\displaystyle \frac{8}{4}: counting by 4 goes 4, 8. So 8 = 4 + 4 fills 2 pans exactly: \displaystyle \frac{8}{4} = 2.

\displaystyle \frac{9}{4}: counting by 4 goes 4, 8, 12 and skips 9. Nine fourths fill 2 pans and leave 9 - 8 = 1 fourth over, so the box holds two whole pans and one fourth more. Not a whole number.

\displaystyle \frac{18}{6}: counting by 6 goes 6, 12, 18. So 3 \times 6 = 18 and \displaystyle \frac{18}{6} = 3.

\displaystyle \frac{5}{3}: counting by 3 goes 3, 6 and skips 5. Five thirds is one whole pan and two thirds more. Not a whole number.

\displaystyle \frac{0}{2}: no pieces at all, an empty box. Zero pieces fill zero pans, so \displaystyle \frac{0}{2} = 0, and 0 is a whole number.

The boxes labeled \displaystyle \frac{8}{4}, \displaystyle \frac{18}{6} and \displaystyle \frac{0}{2} hold whole numbers of pans: 2, 3 and 0.

Boundaries

The denominator is never zero. A whole cannot be cut into 0 parts, so a fraction such as \displaystyle \frac{3}{0} names no piece and no number. Denominator 1 is fine: it means the whole is not cut.

Not every fraction is a whole number. \displaystyle \frac{a}{b} is a whole number only when a pieces fill complete wholes exactly. \displaystyle \frac{9}{4} fills 2 pans and leaves a fourth over, so it lies between 2 and 3.

The wholes must match. \displaystyle \frac{12}{4} = 3 pans only if every pan is the same size and cut into 4 equal pieces. A fourth of a small pan is not a fourth of a big one, and counting it gives no fraction of either.

Say what the whole is. \displaystyle \frac{12}{4} of a pan is 3 pans. If the whole is taken to be all three pans together, the same 12 pieces are \displaystyle \frac{12}{12} of the three pans, which is 1. The number changes because the whole changed.

Where this goes next

Since a whole number like 2 has a name in fourths, \displaystyle \frac{8}{4}, it can be compared with or added to other fourths. Naming one number with different fractions is the idea of equivalent fractions in pictures, which also renames fractions that are not whole numbers, and later of Equivalent fractions in general.

The grouping in \displaystyle \frac{12}{4} = 3 is the division 12 \div 4 = 3. That link, that the bar in a fraction means divide, is A fraction as division. A fraction past a whole number, like \displaystyle \frac{9}{4}, is written as wholes and a part in Mixed numbers and improper fractions: rewriting and comparing, and taking several copies of a fraction, such as \displaystyle 3 \times \frac{4}{4}, is A fraction times a whole number.

Exercises

Part 1: one whole and denominator one

Write each whole number as a fraction with denominator 1: 3, 9, 11.
Write each fraction as a whole number: \displaystyle \frac{8}{1}, \displaystyle \frac{12}{1}, \displaystyle \frac{15}{1}.
Siosaia puts 4 undivided sand trays of the same size on the sand table. Write the sand as a fraction of a tray with denominator 1, and as a whole number.
Losaline cuts a pan of brownies into 2 equal pieces and serves both. What fraction of the pan was served? Write it as a whole number too.
Which of these is the same point on the number line as \displaystyle \frac{1}{1}: \displaystyle \frac{3}{3}, \displaystyle \frac{1}{3} or \displaystyle \frac{3}{1}?
Fill in each box. (a) \displaystyle \frac{\square}{6} = 1. (b) \displaystyle 10 = \frac{\square}{1}. (c) \displaystyle 1 = \frac{1}{\square}.
A number line runs from 0 to 8 with a tick at each whole number and none in between. Name the point 7 with a fraction whose denominator is 1. Where is \displaystyle \frac{8}{1}?
Which of these equal 1: \displaystyle \frac{1}{1}, \displaystyle \frac{2}{1}, \displaystyle \frac{2}{2}, \displaystyle \frac{1}{2}? Say whether each of the others is more or less than 1.

Part 2: fractions that are whole numbers

Dana cuts 5 pans of brownies of the same size into halves, and the class eats every piece. How many halves were eaten? Write the amount as a fraction of a pan and as a whole number.
Siosaia divides each sand tray into 6 equal sections. He smooths 24 sections, and every tray he works on ends up fully smoothed. As a fraction of a tray, how much did he smooth? How many trays is that?
Write each as a whole number: \displaystyle \frac{16}{2}, \displaystyle \frac{28}{4}, \displaystyle \frac{48}{8}, \displaystyle \frac{27}{3}.
A number line from 0 to 4 is cut into thirds. Name the point 4 as a fraction in thirds. Which whole number is 3 jumps before it, and what is its name in thirds?
A number line from 0 to 5 is cut into fourths. Which fraction in fourths sits at 5?
Write 9 as a fraction with denominator 2, with denominator 4 and with denominator 6.
Fill in each box. (a) \displaystyle \frac{\square}{3} = 5. (b) \displaystyle \frac{30}{\square} = 5, using a denominator from 2, 3, 4, 6, 8. (c) \displaystyle 0 = \frac{\square}{4}.
Which of these are whole numbers? For each that is, say which: \displaystyle \frac{12}{2}, \displaystyle \frac{10}{4}, \displaystyle \frac{24}{3}, \displaystyle \frac{14}{6}, \displaystyle \frac{40}{8}.
Losaline cuts 4 pans of brownies of the same size into 8 equal pieces each and sells every piece. Write the brownies sold as a fraction of a pan and as a whole number.
Which is more, \displaystyle \frac{16}{4} of a pan or \displaystyle \frac{3}{1} of a pan? By how much?
On a number line cut into halves, which whole number is at \displaystyle \frac{22}{2}? Is \displaystyle \frac{23}{2} a whole number?

Reasoning

Use a picture of brownie pans to explain why \displaystyle \frac{5}{1} = 5.
Explain why \displaystyle \frac{0}{6} = 0.
Can every whole number be written as a fraction with denominator 4? Explain, and give an example.
Without drawing, explain why \displaystyle \frac{21}{3} and \displaystyle \frac{14}{2} sit at the same point on the number line.

Answers

\displaystyle \frac{3}{1}, \displaystyle \frac{9}{1}, \displaystyle \frac{11}{1}. Each whole is one uncut piece, so n wholes are n pieces of size one whole.
8, 12, 15. A denominator of 1 means each piece is a whole.
\displaystyle \frac{4}{1} of a tray, which is 4 trays.
\displaystyle \frac{2}{2} of the pan, which is 1: both pieces are the whole pan.
\displaystyle \frac{3}{3}. Both it and \displaystyle \frac{1}{1} are one whole. \displaystyle \frac{1}{3} is less than 1 and \displaystyle \frac{3}{1} is 3.
(a) 6, since \displaystyle \frac{6}{6} = 1. (b) 10, since \displaystyle 10 = \frac{10}{1}. (c) 1, since \displaystyle \frac{1}{1} = 1.
\displaystyle \frac{7}{1}, seven jumps of one whole from 0. \displaystyle \frac{8}{1} is 8, the right end of the line.
\displaystyle \frac{1}{1} and \displaystyle \frac{2}{2} equal 1. \displaystyle \frac{2}{1} = 2 is more than 1, and \displaystyle \frac{1}{2} is less than 1.
5 \times 2 = 10 halves, so \displaystyle \frac{10}{2} of a pan, which is 5 pans.
\displaystyle \frac{24}{6} of a tray. Counting by 6: 6, 12, 18, 24, four full trays, so \displaystyle \frac{24}{6} = 4 trays.
8, 7, 6, 9, since 8 \times 2 = 16, 7 \times 4 = 28, 6 \times 8 = 48 and 9 \times 3 = 27.
\displaystyle 4 = \frac{12}{3}, since 4 \times 3 = 12 jumps. Three jumps back is one whole back, the point 3, which is \displaystyle \frac{9}{3}.
\displaystyle \frac{20}{4}, since 5 wholes of 4 jumps each make 5 \times 4 = 20 jumps.
\displaystyle \frac{18}{2}, \displaystyle \frac{36}{4}, \displaystyle \frac{54}{6}, since 9 \times 2 = 18, 9 \times 4 = 36 and 9 \times 6 = 54.
(a) 15, since 5 \times 3 = 15. (b) 6, since 5 \times 6 = 30. (c) 0, since \displaystyle \frac{0}{4} = 0.
\displaystyle \frac{12}{2} = 6, \displaystyle \frac{24}{3} = 8 and \displaystyle \frac{40}{8} = 5 are whole numbers. \displaystyle \frac{10}{4} is two wholes and two fourths more, and \displaystyle \frac{14}{6} is two wholes and two sixths more, so they are not.
4 \times 8 = 32 pieces, so \displaystyle \frac{32}{8} of a pan, which is 4 pans.
\displaystyle \frac{16}{4} of a pan is more. \displaystyle \frac{16}{4} = 4 pans, since 4 \times 4 = 16, and \displaystyle \frac{3}{1} = 3 pans, so it is 4 - 3 = 1 pan more.
11, since 11 \times 2 = 22. \displaystyle \frac{23}{2} is not a whole number: it is one half past 11, halfway between 11 and 12.
Draw 5 pans of the same size and do not cut them. Each pan is a whole cut into 1 part, so each is \displaystyle \frac{1}{1} of a pan. Five of them are \displaystyle \frac{5}{1} of a pan, and they are 5 pans.
\displaystyle \frac{0}{6} means a whole cut into 6 equal pieces, and 0 of those pieces. No pieces is no amount at all, so it is 0, the point where the number line starts.
Yes. Cut each whole into 4 equal pieces; n wholes hold n \times 4 pieces, so \displaystyle n = \frac{n \times 4}{4}. For example, \displaystyle 6 = \frac{24}{4}, since 6 \times 4 = 24.
Both equal 7. Counting by 3 reaches 21 after 7 counts, since 7 \times 3 = 21, so \displaystyle \frac{21}{3} = 7. Counting by 2 reaches 14 after 7 counts, since 7 \times 2 = 14, so \displaystyle \frac{14}{2} = 7. Equal numbers are the same point on the line.