Grade 4 · 4.NF.A.2 · 5.NF.A.2

Benchmark fractions

A measuring tape is covered in marks of different lengths. Which few fractions are worth knowing by heart, so that any other fraction can be placed next to one of them at a glance?

You need

Fractions on the number lineGrade 3

Unlocks

Estimating with fractionsGrade 5 Comparing against a benchmarkGrade 4 Benchmark percents and mental percentMath 6

Where you are 15 in Fractions

Fractions, ratio and percent3

Fractions15

A fraction as part of a whole4 Adding and subtracting fractions5

Benchmark fractions

Comparing fractions3 Equivalent fractions4 ▾ Show all 15 (10 more)

Why this exists

Kerem, Iryna and Dragana share a measuring tape. Between any two inch numbers it has a crowd of marks: one long mark in the middle, two shorter ones, and four very short ones. Nobody reads a tape by counting every tiny mark from the start. They look for the long marks first. “Just past the half.” “Almost at the next inch.” “Right on the quarter.”

Fractions need the same kind of long marks. A fraction such as \tfrac{11}{18} is hard to picture on its own. Is it a lot? A little? Close to one whole? It becomes easy once it is set beside a fraction that everyone already knows well, such as \tfrac{1}{2}. Those well-known fractions are the benchmarks: fixed reference points, like the long marks on the tape, that every other fraction can be judged against.

Definition

A benchmark fraction is a familiar fraction used as a reference point for judging the size of other fractions. Between 0 and 1 the benchmarks are the five landmarks

0, \quad \tfrac{1}{4}, \quad \tfrac{1}{2}, \quad \tfrac{3}{4}, \quad 1

Past 1 the same landmarks repeat after every whole number: 1\tfrac{1}{4}, 1\tfrac{1}{2}, 1\tfrac{3}{4}, 2, and so on.

A fraction is a point on the number line, as in Fractions on the number line: the bottom number cuts each whole into that many equal pieces, and the top number counts the pieces. A benchmark is a point on the same line that we know without counting.

Rounding a fraction to the nearest half means replacing it with whichever of 0, \tfrac{1}{2}, 1, 1\tfrac{1}{2}, 2, and so on lies closest to it. Rounding to the nearest whole means replacing it with the closest whole number. When a fraction sits exactly halfway between two choices, this lesson rounds up to the larger one, the same rule used in Rounding whole numbers, where 45 to the nearest ten is 50.

Intuition

Part 1: the landmarks on a tape

Look at one inch of the tape. The inch marks at 0 and 1 are the longest. The half-inch mark is the next longest. The two quarter-inch marks are shorter, and the four eighth-inch marks are the shortest of all.

Figure 1The longer the mark, the more familiar the fraction. The five landmarks 0, \tfrac{1}{4}, \tfrac{1}{2}, \tfrac{3}{4} and 1 sit on the longest marks of one inch.

The lengths of the marks tell a story. The tape has eight eighth-inch spaces in each inch. Two of those spaces make a quarter inch, so the mark at \tfrac{2}{8} is also the mark at \tfrac{1}{4}. Four spaces make a half inch, so \tfrac{4}{8} is \tfrac{1}{2}. Six spaces make three quarters, so \tfrac{6}{8} is \tfrac{3}{4}. The landmarks get the longer marks because they are the points people use most.

The most useful landmark of all is \tfrac{1}{2}. It splits the stretch from 0 to 1 into two equal halves, and every fraction in that stretch falls into one of three cases: less than half, exactly half, or more than half.

Figure 2One half splits the stretch from 0 to 1 in two. Every fraction in it is less than half, exactly half, or more than half.

To tell which, look at the bottom number. It says how many pieces make one whole. Half of the whole is half that many pieces. So compare the top number with half of the bottom number.

In \tfrac{8}{17}, half of 17 is 8\tfrac{1}{2}. The top, 8, is a little less, so \tfrac{8}{17} is a little less than half.
In \tfrac{11}{18}, half of 18 is 9. The top, 11, is more, so \tfrac{11}{18} is more than half.

Halving an odd number gives a half piece, which can feel awkward. Doubling the top avoids that and gives the same answer: 2 \times 8 = 16 is less than 17, and 2 \times 11 = 22 is more than 18.

Part 2: the nearest landmark

Once you know which side of one half a fraction is on, the next question is how close it sits to each landmark. Count the gap in pieces. For a fraction between 0 and 1:

the gap down to 0 is the top number of pieces,
the gap to \tfrac{1}{2} is the difference between the top and half the bottom,
the gap up to 1 is the bottom minus the top, the pieces still missing from a whole.

Every gap is counted in the same size of piece, so the smallest number wins. That is the nearest half.

Figure 3The dashed marks at \tfrac{1}{4} and \tfrac{3}{4} sit halfway between the halves, so they split the line into the stretches that round to 0, to \tfrac{1}{2} and to 1.

The picture shows where the switches happen. Halfway between 0 and \tfrac{1}{2} is \tfrac{1}{4}, and halfway between \tfrac{1}{2} and 1 is \tfrac{3}{4}. A fraction left of \tfrac{1}{4} rounds to 0; one between \tfrac{1}{4} and \tfrac{3}{4} rounds to \tfrac{1}{2}; one right of \tfrac{3}{4} rounds to 1. A fraction sitting exactly on \tfrac{1}{4} or \tfrac{3}{4} is a tie, and rounds up.

A fraction greater than 1 is handled one whole at a time. Write it as a mixed number (see Mixed numbers and improper fractions), keep the whole-number part, and round only the fraction part.

Results

Half, more than half, less than half. For a fraction \tfrac{a}{b} between 0 and 1:

\begin{aligned} 2a = b &\quad\text{means}\quad \tfrac{a}{b} = \tfrac{1}{2} \\ 2a > b &\quad\text{means}\quad \tfrac{a}{b} > \tfrac{1}{2} \\ 2a < b &\quad\text{means}\quad \tfrac{a}{b} < \tfrac{1}{2} \end{aligned}

The quarter landmarks. \tfrac{a}{b} = \tfrac{1}{4} exactly when 4a = b, and \tfrac{a}{b} = \tfrac{3}{4} exactly when 4a = 3b.

Rounding to the nearest half, between 0 and 1. Below \tfrac{1}{4} rounds to 0. From \tfrac{1}{4} up to just below \tfrac{3}{4} rounds to \tfrac{1}{2}. From \tfrac{3}{4} up rounds to 1.

Past 1. Keep the whole-number part and round the fraction part. To the nearest whole, a fraction part less than \tfrac{1}{2} leaves the whole number as it is, and a fraction part of \tfrac{1}{2} or more moves it up by one.

Why it is true

Take \tfrac{a}{b}. The whole is made of b equal pieces, so half the whole is b pieces split into two equal groups: \tfrac{b}{2} pieces. The fraction has a of those pieces. Having more pieces than half the whole means a is more than \tfrac{b}{2}, and doubling both sides keeps the order, so that is the same as 2a being more than b. The same reasoning with “equal” and “less” gives the other two lines. For a quarter, the whole splits into four equal groups of \tfrac{b}{4} pieces each, so one quarter is a pieces exactly when 4a = b, and three quarters is a pieces exactly when 4a = 3b.

For rounding, the two neighbors of a switch point are equally far from it. The point \tfrac{1}{4} is \tfrac{1}{4} away from 0 and \tfrac{1}{4} away from \tfrac{1}{2}. So any fraction to its left is closer to 0, and any fraction to its right (up to \tfrac{1}{2}) is closer to \tfrac{1}{2}. The point \tfrac{3}{4} plays the same part between \tfrac{1}{2} and 1. Past 1, moving a fraction along by a whole number moves every landmark along with it, so only the fraction part decides which landmark is nearest.

Worked examples

The lesson is taught in two parts. Examples 1 to 3 belong to the first: reading the landmarks on the tape and telling half, more than half and less than half. Examples 4 to 7 belong to the second: finding the nearest landmark and rounding, including fractions greater than 1.

Example 1

Reading a landmark on the tape

Kerem measures a pencil from the end of the tape. The tip of the pencil lines up with the sixth eighth-inch mark past the 3-inch line. Which landmark is that mark, and how long is the pencil?

Solution

Six eighth-inch spaces past 3 is \tfrac{6}{8} of an inch past 3. Each quarter inch is two eighths, so six eighths is three quarters:

\tfrac{6}{8} = \tfrac{3}{4}

On the tape this is the quarter-length mark after the half-inch mark, the landmark \tfrac{3}{4}. The pencil is 3\tfrac{3}{4} inches long. As a check, the mark is two eighths short of 4, and 3\tfrac{3}{4} is a quarter inch short of 4.

Example 2

Half, more than half or less than half

Iryna has four fractions: \tfrac{11}{18}, \tfrac{8}{17}, \tfrac{5}{10} and \tfrac{4}{11}. Say whether each is less than half, exactly half or more than half.

Solution

For each one, double the top and compare with the bottom. Comparing the top with half the bottom gives the same answer, since doubling both numbers keeps which one is bigger.

\tfrac{11}{18}: 2 \times 11 = 22, which is more than 18. More than half. In pieces, half of 18 is 9, and 11 is two pieces past it.
\tfrac{8}{17}: 2 \times 8 = 16, which is less than 17. Less than half, but only just: half of 17 is 8\tfrac{1}{2}, and 8 is half a piece short.
\tfrac{5}{10}: 2 \times 5 = 10, equal to the bottom. Exactly half.
\tfrac{4}{11}: 2 \times 4 = 8, which is less than 11. Less than half; half of 11 is 5\tfrac{1}{2}, and 4 is a piece and a half short.

Example 3

Which landmark, if any?

Dragana finds four fractions written on a page: \tfrac{4}{16}, \tfrac{9}{12}, \tfrac{8}{16} and \tfrac{10}{16}. Which of them is exactly a landmark, and where does the one that is not sit?

Solution

Split the bottom number into four equal groups to see what one quarter is.

\tfrac{4}{16}: sixteen pieces in four equal groups is 4 in each group, so a quarter is 4 pieces. The top is 4, so \tfrac{4}{16} = \tfrac{1}{4}. Check: 4 \times 4 = 16.
\tfrac{9}{12}: twelve pieces in four groups is 3 in each group, so three quarters is 3 \times 3 = 9 pieces. So \tfrac{9}{12} = \tfrac{3}{4}. Check: 4 \times 9 = 36 and 3 \times 12 = 36.
\tfrac{8}{16}: 2 \times 8 = 16, so \tfrac{8}{16} = \tfrac{1}{2}.
\tfrac{10}{16}: a quarter is 4 sixteenths, a half is 8 and three quarters is 12. The top, 10, is none of these. It is between 8 and 12, so \tfrac{10}{16} sits between \tfrac{1}{2} and \tfrac{3}{4}, exactly in the middle of them.

Example 4

Rounding to 0, one half or 1

Round each fraction to the nearest half: \tfrac{3}{14}, \tfrac{7}{10} and \tfrac{9}{11}.

Solution

Count each gap in pieces of the fraction’s own size.

\tfrac{3}{14}: the gap to 0 is 3 fourteenths. Half of 14 is 7, so the gap to \tfrac{1}{2} is 7 - 3 = 4 fourteenths. The gap to 1 is 14 - 3 = 11 fourteenths. The smallest gap is 3, so

\tfrac{3}{14} \approx 0

\tfrac{7}{10}: the gap to \tfrac{1}{2} is 7 - 5 = 2 tenths, and the gap to 1 is 10 - 7 = 3 tenths. So \tfrac{7}{10} \approx \tfrac{1}{2}. It is more than half, but not far enough past it to be nearer 1.

\tfrac{9}{11}: half of 11 is 5\tfrac{1}{2}, so the gap to \tfrac{1}{2} is 3\tfrac{1}{2} elevenths. The gap to 1 is 11 - 9 = 2 elevenths. So \tfrac{9}{11} \approx 1.

The rounding-zones figure shows all three: \tfrac{3}{14} left of the dashed mark at \tfrac{1}{4}, \tfrac{7}{10} just left of the dashed mark at \tfrac{3}{4}, and \tfrac{9}{11} to its right.

Example 5

Fractions greater than one

Iryna’s seedling is 2\tfrac{3}{8} inches tall, and a strip of paper is \tfrac{19}{6} inches long. Round each length to the nearest whole inch and to the nearest half inch.

Solution

The seedling first. The whole number 2 stays; only \tfrac{3}{8} is rounded.

Nearest whole: half of 8 is 4, and 3 is less than 4, so \tfrac{3}{8} is less than half. The length stays at 2 inches.
Nearest half: \tfrac{3}{8} is 3 eighths above 0 and 4 - 3 = 1 eighth below \tfrac{1}{2}. The half is closer, so the seedling is about 2\tfrac{1}{2} inches.

The two answers differ, and both are right. A whole inch is a coarse step and a half inch is a finer one.

The strip next. First write \tfrac{19}{6} as a mixed number. Six sixths make one whole, and 3 \times 6 = 18, so 19 sixths is 3 wholes and 1 sixth left over:

\tfrac{19}{6} = 3\tfrac{1}{6}

Nearest whole: half of 6 is 3, and 1 is less than 3, so the strip is about 3 inches.
Nearest half: \tfrac{1}{6} is 1 sixth above 0 and 3 - 1 = 2 sixths below \tfrac{1}{2}. Zero is closer, so to the nearest half inch the strip is also about 3 inches.

Example 6

Exactly halfway

Kerem cuts two ribbons, one 6\tfrac{1}{4} inches long and one 9\tfrac{3}{4} inches long. Round each to the nearest half inch.

Solution

6\tfrac{1}{4} inches sits on the quarter mark after 6. In eighths, \tfrac{1}{4} = \tfrac{2}{8}, which is 2 eighths above 6 and 2 eighths below 6\tfrac{1}{2}. The gaps are equal. Neither half is closer, so a rule has to decide, and the rule of this lesson is to round up:

6\tfrac{1}{4} \approx 6\tfrac{1}{2}

9\tfrac{3}{4} inches is \tfrac{6}{8} past 9: 2 eighths above 9\tfrac{1}{2} and 2 eighths below 10. Again a tie, so it rounds up to 10 inches.

This is the same choice as rounding 45 to the nearest ten. Any fixed rule would work; what matters is that the same rule is used every time, so two people rounding the same length get the same answer. If Kerem had been asked for the nearest whole inch instead, 6\tfrac{1}{4} would be no tie at all: \tfrac{1}{4} is less than half, so the answer is 6.

Example 7

Every mark on one inch

Dragana looks at the nine marks from 3 inches to 4 inches on her tape, one at every eighth inch. Which marks round to 3, which to 3\tfrac{1}{2} and which to 4, to the nearest half inch?

Solution

The switch points are halfway between the halves: 3\tfrac{1}{4}, which is 3\tfrac{2}{8}, and 3\tfrac{3}{4}, which is 3\tfrac{6}{8}.

3 and 3\tfrac{1}{8} are left of 3\tfrac{2}{8}, so they round to 3.
3\tfrac{2}{8} is a tie between 3 and 3\tfrac{1}{2} and rounds up to 3\tfrac{1}{2}.
3\tfrac{3}{8}, 3\tfrac{4}{8} and 3\tfrac{5}{8} are each at most 1 eighth from 3\tfrac{4}{8}, so they round to 3\tfrac{1}{2}.
3\tfrac{6}{8} is a tie between 3\tfrac{1}{2} and 4 and rounds up to 4.
3\tfrac{7}{8} and 4 round to 4.

Figure 4Every eighth mark from 3 to 4 inches, grouped by the nearest half inch. The marks 3\tfrac{2}{8} and 3\tfrac{6}{8} sit exactly halfway and round up.

So 2 marks round to 3, 4 marks round to 3\tfrac{1}{2} and 3 marks round to 4. The groups differ in size for two reasons. The marks 3 and 4 sit at the ends of the inch, so only one side of the stretch around each of them is on this inch of tape. And each tie joins the group above it.

Boundaries

The test “compare the top with half the bottom” works only when the bottom number counts the pieces in one whole and every piece is the same size. A fraction of an uneven split has no bottom number to halve.
The rounding rules for 0, \tfrac{1}{2} and 1 are for fractions between 0 and 1. A fraction greater than 1 must first be written as a mixed number (see Fractions greater than one (improper fractions)); then only the fraction part is rounded.
A tie, a fraction exactly halfway between two choices, needs a rule because neither choice is closer. This lesson rounds up. Another book may choose differently; say which rule you used.
Rounding throws information away. Different fractions can round to the same landmark: both \tfrac{7}{10} and \tfrac{3}{8} round to \tfrac{1}{2}. A rounded fraction says roughly where a number is, never exactly.
A benchmark is a fraction of the same whole. Half an inch and half a foot are both \tfrac{1}{2}, but they are not the same length.

Where this goes next

With the landmarks in place, two fractions can be put side by side by asking which landmark each one is near, the subject of Comparing against a benchmark. Rounding each fraction to 0, \tfrac{1}{2} or 1 makes it possible to guess a sum or difference before working it out, in Estimating with fractions, and choosing nearby friendly numbers for quick mental work is Friendly numbers and deliberate over and under estimates. The same idea comes back with percents: 50 percent is one half and 25 percent is one quarter, in Benchmark percents and mental percent.

Exercises

Exercises 1 to 12 go with the first part of the lesson (Examples 1 to 3), Exercises 13 to 22 with the second (Examples 4 to 7), and the reasoning group draws on both.

Part 1: landmarks on the tape

The tape is marked in eighths of an inch. How many eighth-inch spaces make \tfrac{1}{2} inch? How many make \tfrac{1}{4} inch? How many make \tfrac{3}{4} inch?
Kerem’s crayon ends at the second eighth-inch mark past the 5-inch line. Which landmark is that, and how long is the crayon?
Iryna’s paper strip ends at the fourth eighth-inch mark past the 2-inch line. How long is the strip?
A key is 2\tfrac{3}{4} inches long. How many eighth-inch marks past the 2-inch line is its end?

Part 1: half, more or less

Is \tfrac{5}{9} less than half, exactly half or more than half?
Is \tfrac{8}{19} less than half, exactly half or more than half?
Is \tfrac{6}{12} less than half, exactly half or more than half?
Is \tfrac{9}{20} less than half, exactly half or more than half?
With a bottom number of 21, find the largest top number that makes a fraction less than half and the smallest that makes one more than half.

Part 1: exactly a landmark?

Which landmark is \tfrac{6}{24}?
Which landmark is \tfrac{15}{20}?
Is \tfrac{10}{12} a landmark? If not, between which two landmarks does it sit?

Part 2: rounding to the nearest half

Round \tfrac{2}{13} to the nearest half.
Round \tfrac{9}{16} to the nearest half.
Round \tfrac{15}{17} to the nearest half.
Round \tfrac{3}{11} to the nearest half.

Part 2: lengths past one inch

Iryna’s seedling grows to 4\tfrac{1}{8} inches. Round its height to the nearest whole inch and to the nearest half inch.
Round 7\tfrac{4}{5} to the nearest whole and to the nearest half.
Write \tfrac{15}{4} as a mixed number, then round it to the nearest whole and to the nearest half.
Write \tfrac{23}{10} as a mixed number, then round it to the nearest whole and to the nearest half.
Dragana’s ribbon is 7\tfrac{4}{8} inches long. Round it to the nearest half inch and to the nearest whole inch.
Kerem reads 5\tfrac{6}{8} inches on the tape and says that to the nearest half inch it is 5\tfrac{1}{2} inches. Using this lesson’s rule for ties, what is the right answer?

Reasoning

Explain why doubling the top number and comparing it with the bottom number tells you whether a fraction is more than half.
Explain why a fraction whose bottom number is odd can never be exactly \tfrac{1}{2}.
Iryna says: “If two fractions round to the same benchmark, they must be equal.” Give two fractions that show she is wrong, and explain.

Answers

4, 2 and 6. Half of 8 eighths is 4, a quarter of 8 is 2, and three quarters is 3 \times 2 = 6.
\tfrac{2}{8} = \tfrac{1}{4}, so the mark is the quarter inch and the crayon is 5\tfrac{1}{4} inches long.
\tfrac{4}{8} = \tfrac{1}{2}, so the strip is 2\tfrac{1}{2} inches long.
6 marks: \tfrac{3}{4} = \tfrac{6}{8}, since each quarter is 2 eighths.
More than half: 2 \times 5 = 10, which is more than 9.
Less than half: 2 \times 8 = 16, which is less than 19.
Exactly half: 2 \times 6 = 12.
Less than half: 2 \times 9 = 18, which is less than 20.
10 and 11: half of 21 is 10\tfrac{1}{2}, so \tfrac{10}{21} is less than half and \tfrac{11}{21} is more.
\tfrac{1}{4}: 24 pieces in four equal groups is 6 each, and 4 \times 6 = 24.
\tfrac{3}{4}: 20 in four groups is 5 each, three groups is 15; check 4 \times 15 = 60 and 3 \times 20 = 60.
Not a landmark. Three quarters of 12 is 9, so \tfrac{3}{4} = \tfrac{9}{12} and \tfrac{10}{12} sits between \tfrac{3}{4} and 1.
0: the gap to 0 is 2 thirteenths, the gap to \tfrac{1}{2} is 6\tfrac{1}{2} - 2 = 4\tfrac{1}{2} thirteenths.
\tfrac{1}{2}: half of 16 is 8, so the gap to \tfrac{1}{2} is 1 sixteenth, against 7 sixteenths to 1.
1: the gap to 1 is 2 seventeenths, the gap to \tfrac{1}{2} is 15 - 8\tfrac{1}{2} = 6\tfrac{1}{2} seventeenths.
\tfrac{1}{2}: the gap to 0 is 3 elevenths, the gap to \tfrac{1}{2} is 5\tfrac{1}{2} - 3 = 2\tfrac{1}{2} elevenths. It is just past \tfrac{1}{4}, since 4 \times 3 = 12 is more than 11.
4 inches both ways. \tfrac{1}{8} is less than half, and it is 1 eighth from 0 against 3 eighths from \tfrac{1}{2}.
8 both ways. 2 \times 4 = 8 is more than 5, so the whole rounds up; to the nearest half, \tfrac{4}{5} is 1 fifth from 1 and 1\tfrac{1}{2} fifths from \tfrac{1}{2}.
\tfrac{15}{4} = 3\tfrac{3}{4}, since 3 \times 4 = 12 with 3 left. Nearest whole: 4. Nearest half: \tfrac{3}{4} is a tie between \tfrac{1}{2} and 1, so it rounds up to 4.
\tfrac{23}{10} = 2\tfrac{3}{10}. Nearest whole: 2, since 3 is less than 5. Nearest half: 3 tenths from 0 against 2 tenths from \tfrac{1}{2}, so 2\tfrac{1}{2}.
7\tfrac{4}{8} = 7\tfrac{1}{2}, so to the nearest half inch it stays 7\tfrac{1}{2} inches. To the nearest whole inch it is a tie between 7 and 8, so it rounds up to 8 inches.
6 inches. \tfrac{6}{8} is 2 eighths from \tfrac{1}{2} and 2 eighths from 1, a tie, and ties round up.
The bottom number is how many pieces make the whole, so half the whole is half that many pieces. The top is more than half the bottom exactly when double the top is more than the bottom, because doubling two numbers keeps which one is bigger.
To be exactly half, the top doubled must equal the bottom. A number doubled is always even, so an odd bottom number can never equal it. The half sits in the middle of a piece.
For example \tfrac{4}{10} and \tfrac{6}{10}. Both round to \tfrac{1}{2}, since each is 1 tenth from it, but one is 2 tenths more than the other. Rounding keeps only the nearest landmark, not the exact place.