Fractions as parts, division and numbers

Decide what counts as one whole

Half of what? Half of one loaf, half of a metre and half of a collection of twelve stones describe different quantities. Before using a fraction, name the reference whole: the complete quantity you are treating as one. It may be one object, a length, an area, a volume, an amount of time or an entire collection.

The fraction 1/2 names the same number in each case, but its application depends on the whole. Half of a 20 cm ribbon is 10 cm; half of a 30 cm ribbon is 15 cm. Both pieces are half their own ribbon, yet their lengths differ. Keep the whole fixed when comparing amounts by their fractions.

Equal parts must contain equal amounts

Divide a 12 cm ribbon into four lengths of 3 cm. Each part is one quarter of the ribbon because the four lengths are equal. Cutting it into lengths of 1 cm, 2 cm, 3 cm and 6 cm creates four pieces, but does not make each piece one quarter. The number of pieces alone cannot establish their fractional size.

What must be equal depends on the question. For a fraction of an area, the parts must have equal area; they need not have identical shapes. For a fraction of volume, compare volumes. For a fraction of the number of objects in a collection, compare object counts, even if the objects have different masses. Equal counts do not automatically mean equal mass. State the quantity before deciding whether the partition is equal.

One number names the part size; the other counts parts

Write a fraction as a/b or with a horizontal fraction bar. The number a is the numerator; b is the denominator. In our equal-parts model, b is a positive whole number: it tells us how many equal parts make one whole. One such part has size 1/b, called a unit fraction. The numerator counts how many of these parts we take. Thus 3/4 means three parts, each of size 1/4: three quarters.

For a nonnegative whole-number numerator a, a/b counts a copies of 1/b. Zero copies give 0/b = 0, while b copies give b/b = 1. For example, 0/4 = 0 and 4/4 = 1. With denominator 1, each part is an entire unit, so 3/1 = 3. A fraction’s denominator cannot be zero: a/0 would require division by zero. We will use positive denominators and place any negative sign in front of the fraction or in its numerator.

A whole can be a collection

Let the whole be a collection of twelve stones, and ask for 3/4 of its object count. Divide the collection into four groups of three stones. Each group is 1/4 of the collection; three groups contain nine stones. We have taken three of the four equal groups, not three individual stones. The denominator specifies the number of equal groups, while the group’s count depends on the total collection.

The context can restrict what a share means. Half of three whole tickets has the numerical value 3/2 tickets, but if the tickets must remain intact, three tickets cannot be split into two equal shares of whole tickets. We could give one ticket to each person and leave one, or change the arrangement. Half of a 3 m ribbon can be a 3/2 m length. A valid fractional number does not guarantee that every physical task permits that share.

A fraction can be the answer to division

Share three same-size loaves equally among four people. Cut every loaf into four equal portions. Each person receives one quarter from each of the three loaves: three quarters of one loaf. The division 3 ÷ 4 therefore gives 3/4 loaf per person. Multiplying the share by four recovers the original three loaves. A fraction bar represents division: a/b = a ÷ b, with b ≠ 0.

Notice the reference whole. Each person has 3/4 of one loaf but 1/4 of the entire batch of three loaves. These describe the same amount of bread relative to different wholes. The denominator 4 in the sharing calculation counts recipients; in the part-of-one-loaf representation it also corresponds to four equal portions per loaf. The connection comes from the equal sharing, not from treating the two numbers as unrelated counts.

Count equal intervals from zero

On a number line, first fix zero, the positive direction and the distance from 0 to 1. That distance is one numerical unit. To locate 3/4, divide the interval from 0 to 1 into four equal intervals and move three of them to the right from zero. The point reached is 3/4. There are five boundary marks when both endpoints are included, but only four intervals: the denominator counts the intervals in one unit, not all the marks.

Fractions are numbers even when there is no shaded shape or physical collection. The point 3/4 can represent a number by itself, or become a measurement such as 3/4 m when we attach a unit. Keep the spacing of whole-number marks equal. To continue beyond 1, subdivide every further unit interval into the same-sized fractional intervals; do not shrink the unit because the numerator becomes larger.

Change the part size and the number of parts

In the explorer, each complete bar represents one unit. Choose a numerator from 0 to 8 and a denominator from 2 to 8. The denominator divides every bar into equal parts, and the numerator counts the selected parts across the bars. The number line keeps the same scale from 0 to 4. Begin with 3/4: find three selected quarter-parts and the point between 0 and 1. Then try 0/4, 4/4 and 7/4. Describe how many whole units and leftover quarter-parts each value contains.

Keep the denominator at 4 and change only the numerator: each additional part moves the point by 1/4. Next compare 3/4 with 3/8, keeping the numerator at 3. The same count of smaller parts produces a smaller value. Finally compare 3/4 with 6/8: the number of parts and their size both change, yet the selected amount and number-line position agree. This tool displays nonnegative values up to 4; the number line itself continues in both directions.

A fraction can reach or pass one

A fraction is not limited to less than one. Seven quarter-parts need more than one whole: four quarters make the first whole and three quarters remain. Thus 7/4 = 1 + 3/4. We can write this as the mixed number 1 3/4, read “one and three quarters”. Here the space expresses a whole-number part plus a fractional part; it does not mean 1 × 3/4.

For nonnegative numerators and positive denominators, a/b is less than one when a < b, equals one when a = b, and is greater than one when a > b. A positive fraction with numerator smaller than its denominator is called a proper fraction; one whose numerator is at least its denominator is called an improper fraction. “Improper” is a conventional name, not an error. For example, 8/4 = 2 is a whole number written as a fraction, and 7/4 lies between 1 and 2.

A negative fraction is the opposite number

The opposite of 3/4 is −3/4. It lies three quarter-unit intervals to the left of zero, at the same distance from zero as 3/4. A sign can indicate a change or position: a change of −3/4 m might mean moving three quarters of a metre in the chosen negative direction. It does not describe a negative amount of shading or a negative physical length.

Division’s sign rules still apply: (−3)/4 = 3/(−4) = −3/4, while (−3)/(−4) = 3/4. Keeping the denominator positive makes the unit-part description easier to read; any sign goes in the numerator or before the fraction. Zero remains zero, so (−0)/4 = 0/4 = 0. A denominator of zero stays undefined regardless of the numerator’s sign.

Compare the part size as well as its count

For the same whole, one quarter is smaller than one half: dividing that whole into more equal parts makes each part smaller. The same reasoning works with a fixed positive numerator. Three eighths are smaller than three quarters because we take three smaller parts: 3/8 < 3/4. If the positive denominator increases while the positive numerator stays fixed, the numerical value decreases. With numerator zero, every allowed denominator still gives zero.

With the same positive denominator, a larger numerator counts more of the same-sized parts: 2/5 < 4/5. When applying fractions to physical amounts, also compare the wholes. Half of a 20 cm ribbon is 10 cm, while one quarter of a 60 cm ribbon is 15 cm. Although 1/2 > 1/4 as numbers, that second piece is longer because its reference ribbon is larger. Comparisons with two changing denominators and numerators need further methods, which we will develop next.

Different partitions can name the same value

Take three quarters of a fixed bar. Now split each quarter into two equal pieces without changing the bar or the selected amount. The whole contains eight eighth-parts, and the selected region contains six. Therefore 3/4 = 6/8. These are equivalent fractions: different fraction expressions that represent the same number and occupy the same point on the number line.

Both counts changed because every original part was split in two. Changing just the numerator to make 6/4 would double the selected amount; changing just the denominator to make 3/8 would halve it. Equality requires preserving the value. This observation prepares us for simplifying fractions and finding common denominators; their general rules and calculations belong to the following lesson.

Check what the answer says about the original task

Ask four questions: What is the whole or unit? Are the parts equal in the relevant quantity? How many such parts does the numerator count? Where should the value lie relative to 0, 1 and other whole numbers? A drawing, an equal-sharing explanation and a number-line position can check one another. Include the unit when the answer is a measurement or physical share.

Distinguish a whole-number quotient with a remainder from an exact fractional quotient. Sharing seven same-size loaves among four people gives one whole loaf each with three loaves left at first. Sharing those three remaining loaves gives each person another 3/4 loaf, so 7 ÷ 4 = 7/4 = 1 3/4. If instead we are distributing seven indivisible tickets, one ticket each with three left may be the appropriate practical answer. The arithmetic and the permitted action must both fit the question.

Worked examples

Try it yourself

1. A 12 cm ribbon is cut into pieces of 1 cm, 2 cm, 3 cm and 6 cm. Are all four pieces one quarter of the ribbon? What is the reference whole, and what must be equal?

   Show the answer and reasoning

   No. The whole is the 12 cm ribbon, and quarter-parts must each have length 3 cm.

   Four pieces do not establish four equal parts. The relevant quantity is length. Only the 3 cm piece is one quarter of this ribbon; for area fractions, equal area would be the corresponding requirement, even if the parts had different shapes.

2. Explain 5/8 in unit-fraction language. What are the numerator and denominator? Find 0/8, 8/8 and 5/1. Can 5/0 be used as a numerical fraction here?

   Show the answer and reasoning

   Five parts of size 1/8; numerator 5 and denominator 8. The values are 0, 1 and 5. The denominator 0 is not allowed.

   The denominator says that eight equal eighth-parts make one unit, and the numerator counts five of them. Zero parts give zero, eight give a whole, and a part of size 1/1 is a whole unit. A fraction bar means division, so 5/0 would require undefined division by zero.

3. Find 2/3 of a collection of fifteen beads by count. Can three indivisible tickets be shared equally between two people as whole tickets? Is the number 3/2 nevertheless valid?

   Show the answer and reasoning

   10 beads. Three indivisible tickets cannot give two equal shares of whole tickets, but 3/2 is a valid number.

   Partition the fifteen beads into three equal groups of five; two groups contain ten. With tickets, 3 ÷ 2 = 3/2 numerically, but a share of one and a half tickets violates the requirement that they stay intact. The context restricts the distribution, not the existence of the fractional number.

4. Five same-size loaves are shared equally among six people, with cutting allowed. State each share in loaf units and as a fraction of the entire batch. Explain why the two fraction expressions differ.

   Show the answer and reasoning

   Each person receives 5/6 loaf, which is 1/6 of the five-loaf batch.

   Cut each loaf into six equal pieces and give everyone one piece from each loaf: five sixth-loaf pieces each. The whole batch is also partitioned into six equal shares. One loaf and the five-loaf batch are different reference wholes, so 5/6 and 1/6 describe the same physical share relative to those different wholes.

5. The interval from 0 to 1 has five boundary marks including both endpoints, creating four equal spaces. Which fraction is at the third step from zero? Where is 9/4, and how do you write it as a mixed number?

   Show the answer and reasoning

   3/4 is at the third step. 9/4 = 2 + 1/4 = 2 1/4, between 2 and 3.

   Four equal intervals make one unit, so each step has size 1/4. For 9/4, eight quarter-unit steps make two whole units, leaving one further quarter-unit step. Five boundary marks do not mean the unit was divided into five parts.

6. Place −3/4 relative to −1 and 0. Rewrite 3/(−4) and (−3)/(−4) with positive denominators. What does the sign describe on the number line?

   Show the answer and reasoning

   −1 < −3/4 < 0. The rewritten values are −3/4 and 3/4. The negative sign selects the position to the left of zero.

   The point −3/4 is three quarter-unit steps left of zero, at distance 3/4 from it. A quotient with one negative input is negative; two negative inputs give a positive quotient. The sign expresses direction or numerical position, not a negative physical size.

7. For the same whole, which is larger: 3/4 or 3/8? Why does 6/8 equal 3/4? Is half of a 20 cm ribbon longer than one quarter of a 60 cm ribbon?

   Show the answer and reasoning

   3/4 is larger than 3/8; 6/8 = 3/4. The ribbon lengths are 10 cm and 15 cm, so the quarter-piece of the longer ribbon is longer.

   Three quarter-parts are larger than three eighth-parts of the same whole. Splitting each of three selected quarters in two gives six selected eighths while preserving the amount, establishing the equality. Fractions applied to different reference wholes can give amounts ordered differently from the numerical fractions.

What to carry forward

• Name the reference whole or measurement unit. Equal fractional parts must be equal in the quantity the question uses.
• With a positive denominator b, 1/b names one fractional unit; a nonnegative numerator a counts a such units. The denominator cannot be zero.
• A fraction can express a part of a whole, a share of a collection, the quotient of division or a number-line position.
• Fractions include zero, values at or beyond one, and negative values. A mixed number combines a whole-number part and a fractional part by addition.
• Compare counts and part sizes while keeping the unit clear. Equivalent fractions name the same value even though their numerator and denominator differ.