BTT Mathematics / Primary Mathematics Learning Hub / Guide 1
Place value tells us how much a digit represents. Regrouping changes the way a number is organised without changing its total value. These two ideas explain why addition and subtraction work, why a zero cannot simply be ignored, and why an answer should be checked against the quantity in the original question.
A child can write every digit in the correct column and still be unsure what the columns mean. Another child can explain tens and ones but lose track when a subtraction crosses two empty columns. This guide separates those difficulties. We will build the quantity, name its units, exchange those units, record the calculation and check that nothing has appeared or disappeared.
The route begins with tens and ones, then moves to larger whole numbers and rounding. Choose the section the learner is ready for; do not treat the whole article as one sitting. In the MOE Primary Mathematics syllabus, pages 31, 33, 35 and 37, whole-number ranges expand through P1–P4; rounding is listed at P4. The exercises here are original teaching material, not official examination questions.
Understand the number · Worked examples · Find the first wrong move · 24 practice questions · Worked answers · Teaching and next steps
1. The digit is not the value
Look at 472. The digit 4 represents four hundreds, the digit 7 represents seven tens, and the digit 2 represents two ones. The number is not a collection of three unrelated labels. It describes a single quantity assembled from different-sized units.
472 = 400 + 70 + 2.
When a question asks for the digit in the tens place, the answer is 7. When it asks for the value of that digit, the answer is 70. Confusing these requests does not necessarily mean the child cannot count. It may mean that the child has not separated a written symbol from the amount it represents.
Use two questions together: “Which digit is here?” and “How much is that digit worth here?” Then move the same digit. In 704, the 7 represents 700. In 470, it represents 70. In 407, it represents 7. The symbol remains recognisable; its position changes its contribution to the number.
Build a number before explaining a rule
Place four bundles of ten objects beside seven single objects. Count the bundles as ten, twenty, thirty, forty; then count the remaining singles. The total is 47. The four bundles and seven singles are not an optional story added after the arithmetic. They make visible what the written 4 and 7 are doing.
Now exchange one bundle for ten singles. The arrangement becomes three tens and seventeen ones. Count again. It is still 47. Nothing was added to the collection and nothing was taken away. Only the grouping changed. This is the central meaning of regrouping, and it should survive after the physical objects are put away.
A place-value chart can record the same exchange. Write 4 under tens and 7 under ones. Beneath it, write 3 under tens and 17 under ones. The second row is not the usual final form of a numeral, but it is a valid description of the quantity. Its purpose is to make a calculation possible.
One number can have several correct descriptions
The number 326 can be described as three hundreds, two tens and six ones. It can also be described as two hundreds, twelve tens and six ones, or as three hundreds, one ten and sixteen ones. Each description totals 326.
This flexibility matters because a written algorithm sometimes asks the learner to use a less familiar description temporarily. A child who believes 326 can only mean three hundreds, two tens and six ones may experience regrouping as breaking the number. A child who can rename it understands that the total remains protected.
Do not require every possible representation before moving on. Ask for one alternative and ask the learner to prove it. “Two hundreds and twelve tens make 320; six more make 326” is a sufficient explanation. The proof belongs to the quantity, not to the neatness of the handwriting.
Zero holds a place
In 305, the zero tells us there are no tens left in the standard grouping. It prevents the 3 and 5 from becoming 35. The expansion is 300 + 5, not 30 + 5. Zero contributes no tens, but its position carries information about the other digits.
Compare 4,508 and 4,580. Both have four thousands and five hundreds. The first has no tens and eight ones; the second has eight tens and no ones. Moving the zero changes the quantity because it changes where the 8 is placed. Reading each number aloud can help, but a correct reading should be connected to the units.
When a learner drops a zero while copying, ask them to compare the copied number with the original before calculating. A careful algorithm cannot repair a different starting number. The first check is whether the written quantity still matches the question.
Compare from the largest place
To compare 407 and 470, begin with hundreds. Both have four hundreds. Move to tens: the first has none, while the second has seven. That settles the comparison. The seven ones in 407 cannot outweigh the seven tens in 470.
For numbers with the same number of digits, compare from left to right until the first different place. For whole numbers written without leading zeros, a number with more digits is larger. These are not two unrelated tricks: both methods examine the largest place where the quantities differ.
Ask for a reason as well as a comparison sign. “470 is larger because it has seven tens after the same four hundreds” reveals understanding. “The 7 is nearer the front” can be a useful beginning, but the learner should eventually name why that position matters.
2. Worked examples: exchange the units, preserve the total
Example A: Why 48 + 27 is 75
Separate the units. Four tens and two tens make six tens. Eight ones and seven ones make fifteen ones. We therefore have six tens and fifteen ones. Exchange ten of those ones for one ten. The result is seven tens and five ones: 75.
48 + 27 = 60 + 15 = 75.
The small 1 often written above the tens column stands for one ten, not one ordinary unit added wherever convenient. The 5 written in the ones column stands for the five ones remaining after the exchange. Both marks should have a meaning the learner can explain.
A useful check is to reverse the operation: 75 − 27 = 48. A different check is to compare with 50 + 30 = 80. Since both original numbers are a little smaller than those convenient numbers, 75 is plausible. An estimate does not prove the answer exactly, but it can expose a result such as 615.
Example B: Why 72 − 48 is not 36
Seven tens and two ones do not contain eight loose ones in their current arrangement. Exchange one ten for ten ones. Now 72 is six tens and twelve ones. Subtract four tens and eight ones. Two tens and four ones remain, giving 24.
The invalid answer 36 can arise when a learner subtracts the smaller digit from the larger digit in each column: 8 − 2 and 7 − 4. That calculation no longer represents 72 − 48. The direction of subtraction has changed in one column but not the other.
Check the proposed result against the original relationship. If 36 were correct, then 48 + 36 would have to equal 72. It equals 84 instead. A check is most useful when it tests the story the answer claims to tell, rather than merely repeating the same mistaken procedure.
Example C: Subtracting across a zero in 503 − 278
Start with five hundreds, no tens and three ones. We need more ones to subtract eight, but there are no tens available to exchange. First exchange one hundred for ten tens. The number is now four hundreds, ten tens and three ones.
Next exchange one of those ten tens for ten ones. The number becomes four hundreds, nine tens and thirteen ones. It still totals 503: 400 + 90 + 13. This intermediate check prevents the common mistake of leaving ten tens after one of them has already been exchanged.
Subtract by matching units: thirteen ones minus eight ones leaves five ones; nine tens minus seven tens leaves two tens; four hundreds minus two hundreds leaves two hundreds. Therefore 503 − 278 = 225.
Reverse-check: 278 + 225 = 503. The zero did not turn into a mysterious 9 by itself. The 9 records ten tens received from the hundreds place, minus one ten sent to the ones place. Every change has a source and a destination.
Example D: Subtracting from 1,000
Find 1,000 − 657. Exchange the thousand for ten hundreds. Exchange one hundred for ten tens. Exchange one ten for ten ones. The useful description is now nine hundreds, nine tens and ten ones.
We have not created extra value. Nine hundreds, nine tens and ten ones total 1,000. Subtract six hundreds, five tens and seven ones. The remaining three hundreds, four tens and three ones make 343.
A second route uses counting on. From 657 to 700 is 43. From 700 to 1,000 is 300. The total difference is 343. Counting on is particularly helpful here because the upper number is a convenient boundary. It is an alternative explanation, not a reason to avoid learning the written method.
Example E: Use a nearby number without losing the adjustment
Find 398 + 207. One route is the standard addition algorithm. Another is to add two to 398 to make 400, then reduce the other addend by two. The total remains unchanged because the same two units are transferred between the addends.
398 + 207 = 400 + 205 = 605.
Another valid route is 400 + 207 − 2 = 605. Here the temporary total is two too large, so the two must be removed. Writing 400 + 207 = 607 and stopping would answer a different question.
The useful question is not “Which shortcut is fastest?” It is “What changed, and how was that change corrected?” If the learner cannot track the adjustment, a longer method with clear units is the safer choice.
Example F: Crossing a place-value boundary
Adding one to 7,099 gives 7,100. Ten ones regroup into one ten; the resulting ten tens regroup into one hundred. The thousands stay unchanged. This is why several digits can change even though the total increased by only one.
Try the reverse direction as well. One less than 7,100 is 7,099. Ask the learner to describe the boundary rather than memorise the last two digits. The same idea appears in 99 to 100, 999 to 1,000, and larger transitions.
Example G: Rounding is a comparison of distances
To round 3,765 to the nearest hundred, identify the neighbouring hundreds: 3,700 and 3,800. Their midpoint is 3,750. Since 3,765 lies above the midpoint, it is nearer to 3,800. Under the usual school convention, a value exactly halfway is rounded up.
The same original number rounds to 3,770 to the nearest ten and 4,000 to the nearest thousand. The requested place matters. “Round the number” is incomplete without saying the size of the step between permitted rounded values.
Use the original number for each rounding request. Rounding to tens first and then to hundreds can sometimes give a different result from rounding directly to hundreds. For example, 3,749 rounds directly to 3,700 to the nearest hundred, but rounding it first to 3,750 introduces a halfway value that was not in the original question.
Example H: Work backwards from a rounded answer
Which whole numbers round to 400 to the nearest hundred? The lower boundary is 350, which rounds up to 400. The highest whole number before the next midpoint is 449. Therefore the complete range is 350 to 449 inclusive.
Test both edges and the numbers just outside. 349 rounds to 300; 350 rounds to 400; 449 rounds to 400; 450 rounds to 500. This boundary check is stronger than naming one possible number and assuming the whole range has been found.
3. Find the first wrong move
A wrong final answer does not identify the cause by itself. Watch what happens before the final line. Did the learner copy the number correctly? Did they understand the value of each digit? Did they choose the correct operation? Did an exchange preserve the total? Did a correct method contain one arithmetic slip?
When the learner says “borrow one”
Ask, “One what?” One hundred becomes ten tens; one ten becomes ten ones. The language of borrowing can be familiar and useful, but it should not hide the change of unit. The learner does not repay a mysterious mark later. They rename an existing quantity.
A repair can be very small. Ask the child to write 600 + 0 + 8, then rename it as 500 + 100 + 8. Next write the middle 100 as ten tens. This connects a symbolic change with the value it preserves without requiring another full page of subtraction.
When the learner knows the method but loses a mark
Separate a place-value misunderstanding from a recording problem. If the child can correctly rename 503 as 400 + 90 + 13 but copies the tens as ten in the subtraction, the exchange is understood and the written record needs attention.
Try leaving more space above the columns or writing the renamed number beside the calculation before using small annotations. A larger, clearer record can be removed later. The aim is not permanently elaborate working; it is a record that allows the learner to see which units have already moved.
When counting is still the only strategy
Counting every object can produce correct answers, so do not dismiss it. Instead, invite the learner to keep a known quantity and work from it. For 38 + 27, split 27 into 2 and 25, reach 40, then add 25. Alternatively combine tens and ones. Both routes use relationships that counting can gradually make visible.
Ask which part stayed the same. The total added must still be 27. A strategy is useful only when it reduces effort without making the quantity harder to follow. An adult’s preferred mental method is not automatically the best starting method for this particular child.
Use three different kinds of checking
Unit check: Have ones been combined with ones and tens with tens? Size check: Is the answer in a sensible range? Reverse check: Does the answer recreate the original quantity when the inverse operation is used?
These checks have different jobs. An estimate can reject 6,340 as the sum of 276 and 358, but it may not distinguish 634 from 644. An inverse check can catch the ten-unit error. A place-value explanation can show why it happened. Do not ask one kind of checking to do all three jobs.
4. Practice: 24 questions, from direct meaning to transfer
Keep the answers covered until an attempt is complete. Write the question number, the working and a short reason where requested. The three groups describe increasing demand within this guide, not official school-level labels. Use objects or a place-value chart when needed, and record which help was used.
Questions 1–8: Build and calculate
1. What is the value of the digit 7 in 472?
2. Write the number represented by six tens and four ones.
3. Write 305 in expanded form using its non-zero place values.
4. Are nine tens and two ones equal to eight tens and twelve ones? Explain.
5. Which is larger, 407 or 470? Name the place that decides.
6. Complete the sequence: 598, 599, __, 601.
7. Calculate 38 + 27. Explain one regrouping or adjustment.
8. Calculate 72 − 48. Show how you rename 72.
Questions 9–16: Control the representation
9. Write four hundreds, thirteen tens and seven ones as a standard numeral.
10. Rename 608 after exchanging exactly one hundred for tens. State the hundreds, tens and ones.
11. Calculate 1,000 − 486.
12. Calculate 276 + 358.
13. Calculate 402 − 178.
14. What is the value of the digit 5 in 4,508?
15. Find one more than 7,099.
16. Find one hundred less than 9,604.
Questions 17–24: Explain, reverse and check
17. Round 3,746 separately to the nearest ten, hundred and thousand.
18. Find the smallest and largest whole numbers that round to 2,400 to the nearest hundred.
19. How much greater is 24,570 than 24,507?
20. A school has 8,000 sheets of paper and uses 2,746. How many sheets remain?
21. Write nineteen hundreds, fifteen tens and twenty-four ones as a standard numeral.
22. Calculate 499 + 286 using a nearby convenient number. Include a rough estimate.
23. You know 658 + 297 = 955. Find 658 + 307 without starting the addition again.
24. A learner writes 503 − 287 = 384 by subtracting the smaller digit from the larger in each column. Explain the mistake, find the correct answer and check it.
5. Worked answers
Compare the reasoning, not just the final numeral. A different valid method is welcome. Where an answer was obtained with help, return to a similar question later without the same prompt; do not relabel a supported answer as independent work.
Answers 1–8
1. 70. The 7 is in the tens place, so it represents seven tens. The digit is 7; its value in this number is 7 × 10 = 70. This distinction is the whole purpose of the question.
2. 64. Six tens make 60 and four ones make 4. Combine the values: 60 + 4 = 64. Writing 604 would place the 6 in the hundreds column and describe a different quantity.
3. 300 + 5. There are three hundreds, no tens and five ones. It is also correct to explain 300 + 0 + 5, but the requested non-zero expansion omits the zero term. The zero remains necessary in the numeral 305.
4. Yes; both equal 92. Nine tens and two ones give 90 + 2. Eight tens and twelve ones give 80 + 12. Exchange ten of the twelve ones for one ten to obtain nine tens and two ones again.
5. 470. The hundreds are equal. In the tens place, 470 has seven tens while 407 has none. The comparison is already decided before the ones place is considered. The larger ones digit in 407 does not reverse that result.
6. 600. One more than 599 crosses a boundary. The ones become ten ones, regrouping through the tens to the next hundred. Continue with 601 to check that the sequence still increases by one at each step.
7. 65. Three tens plus two tens make five tens. Eight ones plus seven ones make fifteen ones. Exchange ten ones for one ten to obtain six tens and five ones. A reverse check gives 65 − 27 = 38.
8. 24. Rename 72 as six tens and twelve ones. Subtract four tens and eight ones to leave two tens and four ones. Check by addition: 48 + 24 = 72. The subtraction must keep the same direction in both places.
Answers 9–16
9. 537. Four hundreds, thirteen tens and seven ones total 400 + 130 + 7. Exchange ten of the thirteen tens for one hundred. Five hundreds, three tens and seven ones give the standard numeral 537.
10. Five hundreds, ten tens and eight ones. The hundred that was exchanged contributes ten tens. The value is 500 + 100 + 8 = 608. The question asks for one exchange only, so the eight ones are unchanged.
11. 514. Rename 1,000 as nine hundreds, nine tens and ten ones. Subtract four hundreds, eight tens and six ones. Five hundreds, one ten and four ones remain. Check: 486 + 514 = 1,000.
12. 634. Six ones plus eight ones make fourteen ones; write four ones and regroup one ten. Seven tens plus five tens plus that ten make thirteen tens. Regroup one hundred. The hundreds total six. Check: 634 − 358 = 276.
13. 224. Rename 402 as three hundreds, nine tens and twelve ones. Subtract one hundred, seven tens and eight ones. The result is two hundreds, two tens and four ones. Its reverse check is 178 + 224 = 402.
14. 500. The digit 5 occupies the hundreds place. The number expands as 4,000 + 500 + 8. Do not confuse the empty tens place with the value of the 5 beside it.
15. 7,100. Adding one to the nine ones creates a new ten; that new ten completes a hundred with the existing nine tens. The thousands stay at seven. Subtracting one from 7,100 returns 7,099.
16. 9,504. Subtract one hundred from the six hundreds in 9,604. The thousands, tens and ones remain unchanged. A difference check gives 9,604 − 9,504 = 100, exactly the amount requested.
Answers 17–24
17. 3,750; 3,700; 4,000. For tens, 3,746 is above the midpoint 3,745. For hundreds, it is below 3,750. For thousands, it is above 3,500. Use 3,746 afresh for each comparison rather than successively rounding previous answers.
18. Smallest 2,350; largest 2,449. The lower midpoint rounds up to 2,400. The next midpoint, 2,450, rounds to 2,500, so it is excluded. Testing 2,349 and 2,450 confirms that the stated endpoints are the limits for whole numbers.
19. 63. Subtract 24,507 from 24,570, or compare the final part after the common 24,500: 70 − 7 = 63. The repeated prefix can be held fixed. The question concerns a difference, not which numeral looks longer.
20. 5,254 sheets. Calculate 8,000 − 2,746. A useful renamed form is seven thousands, nine hundreds, nine tens and ten ones. Subtract matching units to obtain 5,254. Check that used plus remaining equals the starting amount: 2,746 + 5,254 = 8,000.
21. 2,074. Nineteen hundreds contribute 1,900, fifteen tens contribute 150, and twenty-four ones contribute 24. Their sum is 2,074. The zero in the hundreds place belongs to the final standard grouping; it does not mean no hundreds were involved in forming the number.
22. 785. Increase 499 to 500, giving 500 + 286 = 786. Remove the extra one: 786 − 1 = 785. A rough estimate is 500 + 300 = 800, so 785 is plausible. The exact check is 785 − 286 = 499.
23. 965. Only the second addend changed, from 297 to 307, an increase of ten. Increase the known sum by ten: 955 + 10 = 965. This method uses the relationship between the two questions and avoids repeating unchanged work.
24. 216. Renaming is required; reversing subtraction inside selected columns changes the question. Rename 503 as four hundreds, nine tens and thirteen ones. Subtract two hundreds, eight tens and seven ones to obtain 216. Check: 216 + 287 = 503. The proposed 384 fails because 384 + 287 = 671.
6. Turn the result into a useful next lesson
Choose a small cluster of responses to discuss. If questions 1–5 are uncertain, return to value, position and comparison before increasing the size of calculations. If those are secure but questions 11 and 13 fail, focus on exchanges across zero. If the arithmetic works but questions 17 and 18 do not, use neighbouring multiples and midpoints rather than more subtraction practice.
Try a short teaching sequence: the adult models one exchange and names its units; the learner completes a second with a place-value chart; the learner attempts a third without the chart; later, the learner explains a changed example. This is a suggested sequence, not a guarantee or a fixed timetable. Stop and adjust when the explanation reveals a different difficulty.
A conversation that reveals more than “Do you understand?”
Ask, “What amount did we start with? Which unit did we exchange? What did it become? How can we show the total stayed the same?” For 503, the reply 400 + 90 + 13 = 503 connects the written method to a checkable statement.
Avoid completing every sentence for the child. A pause may be needed to translate a familiar action into language. Accept a clear explanation with objects or pointing before demanding polished vocabulary. The mathematical relationship matters more than performing an adult script.
Keep a small repair record
Write one line for the original difficulty, one for the corrected idea and one for a later independent attempt. For example: “I reversed the ones subtraction. I need to exchange a ten instead. Today I solved a new subtraction and checked it by addition.” The record should describe something observable, not label the child as careless or weak.
Correctness, explanation and independence are separate observations. A correct answer obtained after a parent names every exchange is useful practice, but it is not the same evidence as a correct answer produced and checked alone. Keeping that distinction makes the next task easier to choose.
Use the number in the world again
Give the learner a simple stock problem: a box contains 600 counters and 248 are used. Before calculating, ask whether more or fewer than 400 should remain. Then find 352 and check 248 + 352 = 600. The subtraction is not finished until the answer has returned to its meaning: counters remaining.
The durable idea is that numbers can change form without changing value. That idea will support equal groups, fractions and later algebra. We do not need to rush towards those topics to make this work worthwhile. A learner who can explain and check an exchange already has a stronger way of controlling the mathematics in front of them.
Continue through the Primary Mathematics series
For multiplication and division, continue to Equal Groups, Division and Remainders. To extend the meaning of a unit, use Fractions, Decimals and the Same Whole. To choose a calculation from a story, use Word Problems, Bar Models and Checking.
Return to the BTT Primary Mathematics Learning Hub to choose a different starting point. For the broader concept map, see Number and Place Value. Families considering guided support can use the separate Primary Mathematics Tuition route.
Original learning guide. Curriculum reference checked 6 September 2026. Worked examples and practice questions are educational illustrations; they are not reproduced from school papers and do not predict an examination.