BTT Mathematics / Primary Mathematics Learning Hub / Guide 3
A fraction describes an amount using equal parts of a chosen whole. A decimal describes an amount using ones, tenths, hundredths and smaller place-value units. Both become easier to control when the learner asks: what is the whole, what is one part worth, and have the units stayed consistent?
One-half is a larger fraction than one-third. But half of eight counters is four counters, while one-third of fifteen counters is five. The first statement compares numbers using the same unit whole. The second compares actual amounts taken from different collections. Both are true, and understanding why prevents a large family of mistakes.
This guide starts with equal parts and progresses to equivalent fractions, calculations, decimal comparisons and changing wholes. The MOE syllabus, pages 33–43, places initial fractions at P2, equivalence at P3, decimals at P4 and further fraction operations at P5–P6. Later examples are for learners who have met those operations, not tasks every younger child should complete.
Name the whole · Work with fractions · Connect decimals · Track a changing whole · 24 questions · Worked answers · Teaching and next steps
1. Before the fraction, identify the whole
Imagine a strip of paper divided into eight equal lengths. Shade three of those lengths. The shaded portion is three-eighths of the strip, written 3/8. The denominator, eight, describes how many equal parts make one whole strip. The numerator, three, counts how many of those parts are selected.
The numerator and denominator do not describe two unrelated piles. They work together to name one amount. Three-eighths means three units, each of size one-eighth. Saying this aloud is useful because it connects the notation to a count of equal-sized units.
Now draw a strip with eight sections of unequal lengths. Shading three sections does not necessarily shade three-eighths of the strip’s length. Counting pieces is not enough; the pieces must represent equal shares of the relevant whole. Equality concerns the quantity being measured, not simply the existence of dividing lines.
The whole can be an object, a length or a collection
A fraction may describe part of a cake, part of a metre, or part of a set of counters. These contexts look different, but the whole must be identified in each one. For a set of forty counters, one-fifth of the set is eight counters because the set can be organised into five equal groups of eight.
The forty counters together are one whole set. One counter is not automatically the whole just because it is one physical object. This distinction becomes important when a learner sees a picture of many objects and begins counting shaded objects without asking what collection the question has named.
For a length, one whole might be one metre or the full length of a particular ribbon. Half a metre and half a ribbon are equal only when the ribbon is one metre long. The words after the fraction determine which quantity is being divided.
A fraction is also a number
On a number line from zero to one, divide the interval into eight equal steps. Three steps from zero mark 3/8. The fraction is not just a coloured region in a shape; it has a position and a magnitude.
Continue the same steps beyond one. Eight-eighths is one whole, nine-eighths is one eighth beyond one, and eleven-eighths is further still. A fraction can be greater than one. The denominator tells us the size of the unit being counted; it does not impose a rule that the numerator must always be smaller.
When a child says that fractions must be “small,” ask them to locate 5/4 on a number line. Four quarters reach one. One more quarter reaches one and a quarter. The representation explains the result without needing a special exception.
The same fraction can represent different actual amounts
Half of a small cake need not be the same amount of cake as half of a larger cake. Similarly, half of eight counters is four counters, while half of twenty counters is ten. The fraction is unchanged, but the whole is different.
When comparing the numbers 1/2 and 1/3, we use the same unit whole. When comparing portions from different objects or sets, we also need information about those wholes. A fraction comparison alone cannot settle which physical amount is larger.
This is not a reason to distrust fractions. It is a reason to finish naming the quantity. “One-third” is a number; “one-third of fifteen counters” is an instruction that produces five counters. Keeping the reference quantity visible makes the instruction usable.
2. Compare and calculate with named fractional units
Compare unit fractions by the size of a part
Divide the same whole into three equal parts, then into five equal parts. A third is larger than a fifth because fewer equal parts are used to divide the same whole. Therefore 1/3 is greater than 1/5.
The whole-number habit “five is larger than three” does not decide the size of these fractions. Five describes the number of equal pieces in a complete whole, not the size of one selected piece. More equal pieces in that same whole mean smaller pieces.
For fractions with the same positive denominator, the unit size is already equal. Five sevenths is greater than two sevenths because five copies of one-seventh exceed two copies. For fractions with the same positive numerator, compare the unit sizes: two-fifths is greater than two-sevenths of the same whole.
Equivalent fractions rename the same amount
Take two-thirds of a strip. Split each third into four equal smaller pieces. The whole strip now contains twelve equal pieces, and the selected two thirds contain eight. Therefore 2/3 = 8/12.
The quantity did not grow. The selected region stayed fixed while the counting unit became smaller. Multiplying both numerator and denominator by four records that finer partition. If only the numerator changed, we would select more of the old-sized units; if only the denominator changed, we would change the unit size without correcting the count.
Simplifying a fraction reverses this renaming. Six ninths can be grouped into pairs of three ninths. Three ninths make one third, so six ninths make two thirds. Dividing both numerator and denominator by three gives 6/9 = 2/3.
The goal is not to force every fraction into its simplest form before any thinking occurs. Equivalent forms serve different jobs. Twelfths may help an addition, while thirds may make the final amount easier to recognise. Choose the form that preserves the amount and makes the next step clearer.
Example A: Add fractions with the same denominator
Find 2/7 + 3/7. Both amounts are measured in sevenths. Two sevenths plus three sevenths make five sevenths, so the answer is 5/7.
Adding the denominators would give 5/14, which describes five units of a different size. We did not cut the whole into fourteen equal pieces merely by combining two groups of sevenths. The unit remains one-seventh, and only the number of those units changes.
A useful language check is to replace “sevenths” with another unit. Two metres plus three metres gives five metres, not five units of twice the denominator. The analogy does not replace a fraction model, but it makes the role of a shared unit explicit.
Example B: Add unlike fractions by renaming
Find 2/3 + 1/6. Thirds and sixths are different-sized units. Rename two-thirds as four-sixths, because each third contains two sixths. Then add four-sixths and one-sixth to obtain 5/6.
The denominator six is useful because both original quantities can be expressed exactly in sixths. We did not change either amount. We changed their descriptions so that the unit counted by each numerator was the same.
For 2/3 + 1/4, use twelfths: 2/3 = 8/12 and 1/4 = 3/12. Add to obtain 11/12. A size check helps: two-thirds plus one-quarter is more than two-thirds but less than one, so eleven-twelfths is plausible.
Example C: Subtract by preserving the whole
Find 5/6 − 1/3. Rename one-third as two-sixths. Five-sixths minus two-sixths leaves three-sixths, which simplifies to 1/2.
Check by addition: 1/2 + 1/3 = 3/6 + 2/6 = 5/6. This reconstruction tests whether the remaining part and the removed part account for the starting amount. The answer is not secure merely because it has a small numerator and denominator.
Example D: A whole can be renamed into fractional units
Find 2 1/4 − 3/4. The starting amount is two wholes and one quarter. To subtract three quarters, rename one of the wholes as four quarters. The amount becomes one whole and five quarters.
Subtract three quarters to leave one whole and two quarters, or 1 1/2. This is the same kind of value-preserving exchange used in whole-number subtraction. One whole becomes four quarters because quarters are the chosen unit here.
A mixed number combines a whole-number part and a fractional part. The improper fraction 11/4 contains eight quarters for two wholes and three quarters more, so 11/4 = 2 3/4. Both forms describe the same point on a number line.
Example E: Find a fraction of a set
Find 3/5 of forty counters. Divide the whole set into five equal groups: 40 ÷ 5 = 8 counters in one fifth. Take three groups: 8 × 3 = 24 counters.
The denominator tells us the number of equal groups in the whole; the numerator tells us how many groups are wanted. The order “divide by five, then multiply by three” is not arbitrary. It follows the structure of the fraction.
Check the unused part. Two fifths remain, containing sixteen counters. The selected twenty-four and remaining sixteen total forty. This check also confirms that the fraction was applied to the full original set.
Example F: Work backwards from a fractional part
Two-fifths of a collection contains eighteen counters. Find the whole collection. The eighteen counters represent two equal fifths, so one fifth contains 18 ÷ 2 = 9 counters. Five fifths contain 9 × 5 = 45 counters.
Do not calculate two-fifths of eighteen: eighteen is already the given part, not the whole. Label what it represents before choosing an operation. A bar with five equal units, two of them labelled eighteen together, makes the missing whole visible.
Check forward: 45 ÷ 5 × 2 = 18. A backwards solution should return successfully through the original relationship. This forward check is especially useful when the question gives a part but asks for a larger starting amount.
3. Decimals are place value extended beyond ones
One whole contains ten tenths. One tenth contains ten hundredths. Therefore one whole contains one hundred hundredths. Decimal notation records these units in places to the right of the ones place.
In 0.47, the 4 represents four tenths and the 7 represents seven hundredths. The amount is 4/10 + 7/100 = 47/100. It is not a whole number forty-seven with a decorative dot placed in front.
Why 0.7 is greater than 0.65
Rename seven tenths as seventy hundredths. Now compare 0.70 with 0.65: seventy hundredths is greater than sixty-five hundredths. The fact that 65 has more digits than 7 is irrelevant because the written digits occupy different decimal places.
Adding a zero at the end of a decimal’s fractional part does not change its value: 0.7 = 0.70. But inserting a zero between the decimal point and the 7 does change its place: 0.07 is seven hundredths, not seven tenths.
Use the names of the units before relying on the phrase “add a zero.” The same visible action can have different effects depending on where the zero goes. Place value, rather than the appearance of a string of digits, controls the amount.
Example G: Add 0.4 and 0.35
Rename four tenths as forty hundredths. Then forty hundredths plus thirty-five hundredths gives seventy-five hundredths: 0.4 + 0.35 = 0.75.
When written vertically, align decimal points so that tenths are combined with tenths and hundredths with hundredths. Writing 0.40 can make the matching places visible. The alignment is a consequence of using the same unit, not a purely visual rule.
An answer of 0.39 might result from treating 4 and 35 as whole-number labels and then putting a decimal point in front. A quick size check rejects it: adding a positive amount to 0.4 must produce more than 0.4, not less.
Example H: Subtract from a whole number
Find 2 − 0.68. Write two as 2.00, or think of two hundred hundredths. Subtract sixty-eight hundredths to leave one hundred and thirty-two hundredths, which is 1.32.
The written algorithm can rename 2.00 as one whole, nine tenths and ten hundredths. This is the decimal version of regrouping across zero. Check by addition: 1.32 + 0.68 = 2.00.
The amount being subtracted is less than one, so the result must lie between one and two. That range check is useful before or after the exact calculation. It does not provide the final digits, but it gives the learner a reason to question an answer such as 0.32.
Connect fractions and decimals through equal amounts
The decimal 0.06 means six hundredths, so 0.06 = 6/100 = 3/50. The denominator comes from the last occupied decimal place. A decimal with two places is not automatically a fraction over ten.
The decimal 0.375 equals 375/1,000, which simplifies to 3/8. A learner ready for this extension can also check by dividing three by eight. Fractions and decimals are different notations for the same number when the values match exactly.
Some fractions do not have a terminating decimal. For example, one-third has a repeating decimal expansion, so 0.33 is an approximation rather than an exact equality with 1/3. This distinction matters whenever a task asks for an exact answer or a specified rounding. Do not replace an exact fraction with a rounded decimal without recording that change.
Units still belong to the answer
A length of 1.25 metres is 125 centimetres because each metre contains one hundred centimetres. The number becomes larger when a smaller measuring unit is used, but the length is unchanged.
This is another form of renaming. The statement is about one length measured in two units, not about making the ribbon longer. Before adding measurements, put them into compatible units and make clear which unit the final answer uses.
4. When the reference whole changes
Example I: A fraction of a fraction
A ribbon is three-quarters of a metre long. Two-thirds of that ribbon is used. How much is used? The phrase “of that ribbon” makes the three-quarter-metre length the reference quantity for the two-thirds.
Divide the ribbon into three equal lengths. Since the ribbon contains three quarter-metres, each third of it is one quarter-metre. Two such parts measure two quarters of a metre, or 1/2 metre.
The calculation is 2/3 × 3/4 = 1/2. The interpretation explains why the answer is smaller than three-quarters of a metre. It also explains why taking two-thirds of a full metre would answer a different question.
Example J: How many fractional pieces fit?
A three-quarter-metre ribbon is cut into pieces one-eighth of a metre long. Assume no length is lost in cutting. How many pieces can be made? Rename three-quarters as six-eighths. Six pieces of one-eighth metre fit exactly.
3/4 ÷ 1/8 = 6 pieces. The quotient counts pieces; it does not measure the length of each piece. This is the grouping meaning of division, with a fractional group size instead of a whole-number group size.
The answer six is larger than the numerical value three-quarters, which is not a contradiction. We changed what was being counted: a length measured in metres became a count of smaller pieces. The labels make the result sensible.
Example K: Remaining money is not the original whole
A child spends three-fifths of their money and has twenty-four dollars left. The remaining twenty-four dollars represents two-fifths of the original amount. One fifth is twelve dollars, so the original five fifths total sixty dollars.
Multiplying twenty-four by five and dividing by three would treat the remainder as the spent part. The calculation might look like a familiar fraction method, but it would be attached to the wrong label. State what the known amount represents before operating on it.
Example L: Half of the remainder
Start with forty-eight counters. Remove one-third, which is sixteen, leaving thirty-two. Then remove half of the remainder, which is sixteen. Sixteen counters are left, equal to one-third of the original forty-eight.
In fractional terms, two-thirds of the original amount remained after the first removal. Keeping half of those two-thirds leaves one-third. Adding one-third and one-half as though both removals referred to the original whole would misread the second instruction.
A before-and-after record helps: original forty-eight; after first removal thirty-two; after second removal sixteen. At each stage, write the quantity to which the next fraction applies. This is often more useful than drawing additional bars without naming their wholes.
5. Practice: 24 questions
Attempt only the operations already introduced to the learner. Questions 1–8 focus on direct fraction meaning and simple calculations. Questions 9–16 develop renaming, sets and decimals. Questions 17–24 require more interpretation, including upper-primary operations. Keep the solutions covered while working.
Questions 1–8: Identify and rename
1. Three of eight equal parts of a strip are shaded. What fraction is shaded?
2. A strip is divided into four unequal lengths. One length is shaded. Can we conclude that one-quarter of the strip’s length is shaded? Explain.
3. Which is greater, 1/3 or 1/5, when compared as numbers?
4. Calculate 2/7 + 3/7.
5. Express 6/9 in its simplest form.
6. Complete 3/4 = __/12.
7. Calculate 1/2 + 1/4.
8. Calculate 5/6 − 1/3.
Questions 9–16: Match the units
9. Calculate 2/3 + 1/4.
10. Calculate 7/8 − 1/4.
11. Write 11/4 as a mixed number.
12. Write 2 1/3 as an improper fraction.
13. Find 3/5 of forty counters.
14. Three-quarters of a collection contains twenty-seven counters. How many counters are in the whole collection?
15. Which is greater, 0.7 or 0.65? Explain using hundredths.
16. Calculate 0.4 + 0.35.
Questions 17–24: Interpret and transfer
17. Calculate 2 − 0.68.
18. Express 0.06 as a fraction in simplest form.
19. Is 0.375 equal to 3/8? Show a calculation that decides.
20. Find two-thirds of a ribbon that is three-quarters of a metre long.
21. A three-quarter-metre ribbon is cut into one-eighth-metre pieces, with no cutting loss. How many pieces are made?
22. A child spends three-fifths of their money and has $24 left. How much did the child have at first?
23. One-third of forty-eight counters is removed. Then half of the remainder is removed. How many counters remain, and what fraction of the original set is that?
24. Jar A contains sixty counters, of which two-fifths are red. Jar B contains ninety counters, of which one-third are red. Which jar has more red counters, and by how many?
6. Worked answers
Answers 1–8
1. 3/8. The whole is the complete strip. Eight equal parts make that whole, and three are selected. The denominator names the unit size; the numerator counts the selected units.
2. No. One out of four pieces does not necessarily represent one-quarter of the total length when the pieces are unequal. The shaded piece could happen to measure one-quarter, but the information given does not establish that. We need equal-part information or actual lengths.
3. 1/3. A unit whole divided into three equal parts has larger individual parts than the same whole divided into five. The comparison is between one third-sized unit and one fifth-sized unit, not between the whole numbers three and five.
4. 5/7. Both fractions count sevenths. Add the counts: two sevenths plus three sevenths gives five sevenths. The size of one unit remains a seventh, so the denominator stays seven.
5. 2/3. Divide the numerator and denominator by three: 6 ÷ 3 = 2 and 9 ÷ 3 = 3. Both parts of the fraction are renamed together, preserving the value.
6. 9. Each quarter contains three twelfths. Three quarters therefore contain nine twelfths. Multiplying both numerator and denominator by three gives 3/4 = 9/12.
7. 3/4. Rename one-half as two-quarters. Then 2/4 + 1/4 = 3/4. The result is greater than one-half and less than one whole, consistent with the amounts being added.
8. 1/2. One-third equals two-sixths. Subtract: 5/6 − 2/6 = 3/6 = 1/2. Add the removed one-third back to one-half to check that five-sixths is recovered.
Answers 9–16
9. 11/12. Rename two-thirds as eight-twelfths and one-quarter as three-twelfths. Add the equal-sized units: 8/12 + 3/12 = 11/12. Adding the denominators would change the unit instead of matching it.
10. 5/8. One-quarter equals two-eighths. Seven-eighths minus two-eighths leaves five-eighths. Check: 5/8 + 1/4 = 5/8 + 2/8 = 7/8.
11. 2 3/4. Four quarters make one whole, so eight of the eleven quarters make two wholes. Three quarters remain. The mixed number and improper fraction describe the same amount.
12. 7/3. Two wholes contain six thirds. Add the existing one-third to obtain seven thirds. The denominator stays three because all seven pieces are still third-sized units.
13. 24 counters. One fifth of forty is 40 ÷ 5 = 8. Three fifths contain 8 × 3 = 24. The remaining sixteen counters form the other two fifths, accounting for the full set of forty.
14. 36 counters. Twenty-seven represents three quarters. Divide by three to find one quarter: nine counters. Four quarters contain 9 × 4 = 36. Check forward: three-quarters of thirty-six is twenty-seven.
15. 0.7. Seven tenths is seventy hundredths, so 0.7 = 0.70. Compare seventy hundredths with sixty-five hundredths. The first is larger by five hundredths.
16. 0.75. Rewrite 0.4 as 0.40. Forty hundredths plus thirty-five hundredths gives seventy-five hundredths. The answer is larger than either positive addend and smaller than one.
Answers 17–24
17. 1.32. Rewrite two as two hundred hundredths. Subtract sixty-eight hundredths to obtain one hundred and thirty-two hundredths. The reverse check 1.32 + 0.68 = 2 confirms the result.
18. 3/50. The decimal is six hundredths: 6/100. Divide both numerator and denominator by two to obtain 3/50. It is not 6/10, which would equal 0.6.
19. Yes. 0.375 = 375/1,000. Divide numerator and denominator by 125 to obtain 3/8. This is an exact equivalence, not a rounded approximation.
20. 1/2 metre. Divide the three-quarter-metre ribbon into three equal lengths of one-quarter metre each. Take two of them to obtain two-quarters, or one-half metre. The fraction two-thirds applies to the ribbon, not to a full metre.
21. 6 pieces. Three-quarters of a metre equals six-eighths of a metre. Six pieces of one-eighth metre fit exactly. Check: six pieces multiplied by one-eighth metre per piece total three-quarters of a metre.
22. $60. The unspent part is two-fifths, and it equals $24. One fifth is $12; five fifths total $60. Check that spending three-fifths, or $36, leaves the stated $24.
23. 16 counters; 1/3 of the original set. Remove 48 ÷ 3 = 16, leaving thirty-two. Half of thirty-two is sixteen, so the second removal leaves sixteen. Compare with the original forty-eight: 16/48 = 1/3.
24. Jar B, by 6 red counters. Jar A has 60 ÷ 5 × 2 = 24 red counters. Jar B has 90 ÷ 3 = 30 red counters. The difference is six. The smaller fraction in Jar B produces more red counters because it is applied to a larger whole.
7. Use the errors to choose the next representation
If a child says the shaded part in question 2 must be one-quarter, return to equal shares before teaching additional fraction procedures. Make two strips with the same total length: one split equally, one unequally. Ask what information is needed before a fraction of the length can be named.
If comparison is secure but addition fails, focus on unit matching. Ask the learner to say “two thirds” and “one sixth” and identify that the named units differ. Then show a common partition. The repair is not merely remembering which numbers to multiply; it is seeing why both quantities must be described in compatible units.
If the operations work but questions 22–24 fail, keep the arithmetic simple and examine the reference whole. Label the original set, the remaining set and any compared set separately. A correct fraction rule applied to the wrong whole still gives the wrong amount.
Give the learner something to verify
Ask for a prediction before calculation. A proper fraction of a positive quantity is smaller than that quantity. Adding a positive fraction increases the starting number. Removing less than one from two leaves more than one. These statements give a range within which an exact answer must sit.
Then use a different check where possible. Recombine a removed and remaining part. Multiply a backwards-found whole by the stated fraction. Change both quantities to hundredths before comparing decimals. A check should challenge the answer, not merely repeat the same steps more quickly.
Do not remove every support at once
A learner may initially use a strip to compare thirds and fifths, a bar to find the whole, or a place-value chart to align decimals. Ask for an explanation while that representation is available. On a later example, invite the learner to decide whether it is still needed.
The goal is flexible control, not a rule that strong students never draw. A quick diagram can be an efficient mathematical tool. Equally, a diagram that the child cannot label is not evidence of understanding just because it resembles a familiar classroom model.
A small, useful record of progress
Record three things: the unit or whole that was confused, the explanation that repaired it, and a later question completed with less help. For example: “I treated 24 dollars as three-fifths. It is the two-fifths left. I found one fifth first and checked the original money by spending the given fraction.”
This record describes an observable change in reasoning. It is more informative than writing only “fractions revised.” It also makes it easier to decide whether the next task should extend the calculation or revisit the same relationship in a different context.
Continue through the Primary Mathematics series
For decimal place-value difficulties, return to Place Value and Regrouping. For the sharing and grouping meanings used throughout this guide, use Equal Groups, Division and Remainders. For longer stories and before-and-after models, continue to Word Problems, Bar Models and Checking.
Return to the BTT Primary Mathematics Learning Hub for the first four-guide route. The wider topic map remains at Fractions, Decimals and Percentages. For guided support, use the separate Primary Mathematics Tuition route.
Original learning guide. Curriculum reference checked 6 September 2026. These questions and examples are educational illustrations, not reproduced school questions or examination predictions.