Series route: This guide belongs to the complete PSLE Learner’s Guide volume directory. The original PSLE Learning Guide supplies the wider English, Mathematics and Science route, while PSLE Mathematics provide verified subject-specific continuation.
Find your route through this guide
This volume solves one precise learner problem: the learner begins long multiplication with 3375 before seeing that it is 27 groups of 125, twenty-seven eighths of 1000, 3000 plus 375 and 3500 minus 125. The goal is durable control, not a remembered slogan. A Primary 6 learner should be able to identify the trigger, explain the mechanism, perform the decision under time pressure and reject an attractive answer that exceeds the evidence.
- Start with the exact mistake and the evidence boundary.
- Build the mechanism before memorising a shortcut.
- Study worked contrasts and name why the near-miss fails.
- Repair misconceptions with one targeted check each.
- Attempt graded independent practice before reading the explanations.
- Return after a delay and prove that the method transfers.
- Use the parent and tutor notes after the learner has attempted the work.
The problem beneath the mistake
The number 3375 is 27 × 125. Since 125 = 1000 ÷ 8, multiplying by 3375 can become multiply by 27, divide by 8 and multiply by 1000 in a legal order.
For 48 × 3375, divide 48 by 8 to get 6, then calculate 6 × 27 × 1000 = 162,000. Every factor is visible, so the shortcut can be reversed and checked.
The multiplier is also 3000 + 375. Since 375 is three groups of 125, the correction uses the same eighths structure and is often quick for even multiplicands.
Another useful form is 3500 − 125. This is efficient when the 3500 product is easy and subtracting one eighth of 1000 times the multiplicand is simpler than adding from 3000.
The multiplicand decides the best route. Multiples of 8 strongly favour the twenty-seven-eighths form, while other numbers may favour distribution or doubling and halving.
A decomposition is useful only when it preserves 3375 exactly. Replacing it by 3400 without subtracting the full correction computes a different product.
Division by 8 must be justified by the factor 125 = 1000 ÷ 8. Writing the identity first prevents an unexplained cancellation or a lost factor of 27.
In money contexts, $33.75 equals 3375 cents. Whole-number cents can simplify the product, but the result must be converted back to dollars exactly once.
For positive n, 3000n < 3375n < 3500n. These bounds catch missing zeros, wrong correction signs and misplaced decimal points.
A second check should use a different structure. Repeating the same distribution can reproduce the same mistake.
The reliable routine is inspect divisibility, select an exact identity, expose every factor or correction, calculate, restore the unit and verify with bounds plus a second route.
Core decision: Use 3375 = 27×125 = (27/8)×1000 = 3000+375 = 3500−125. Choose the form that fits the multiplicand, preserve every factor and correction, and check the product between 3000n and 3500n.
Flexible calculation is exact rewriting. Associative, commutative and distributive properties allow the factors or addends to be rearranged, but the value of the multiplier is invariant. A method is trustworthy when every change is reversible.
Mechanism: why the rule works
1. Expose groups of 125
Write 3375 = 27×125 and 125 = 1000÷8.
The awkward multiplier becomes a small whole-number factor, a power of ten and a division.
Diagnostic check. Can a factor supply the required 8 exactly?
2. Use twenty-seven eighths
For n×3375, write n×27×1000÷8 and divide a convenient factor first.
Division is safe within the product only when the factor of 8 is accounted for.
Diagnostic check. Where is the 8, and is the division exact?
3. Add from 3000
Use n(3000+375)=3000n+375n.
The multiplicand must distribute to the whole correction, not merely to 3000.
Diagnostic check. Did both addends receive n?
4. Subtract from 3500
Use n(3500−125)=3500n−125n.
The product must be smaller than 3500n, which gives an immediate sign check.
Diagnostic check. Was the complete 125n correction subtracted?
5. Choose by structure
Inspect divisibility by 8, 4, 2 and useful round bases before starting.
A correct method may be inefficient for a particular multiplicand.
Diagnostic check. Which identity produces the smallest exact intermediate values?
6. Protect units
Name whether 3375 represents a pure number, cents, grams or another quantity.
Correct arithmetic can still give a wrong reported answer when a conversion is missed.
Diagnostic check. What unit enters and what unit must leave?
7. Check independently
Use 3000n and 3500n as bounds, then recompute by a different identity.
Magnitude and structural checks reveal different classes of error.
Diagnostic check. Does the product fit both the bound and the second route?
Worked examples: make the decision visible
Pause before each answer. State the controlling evidence in one line, then complete the operation. The explanations deliberately contrast a valid route with a plausible but invalid one, because transfer grows when the learner can say why a tempting route does not apply.
Worked example 1: Multiple of eight
Task. Calculate 48×3375.
Reasoning. 48÷8×27×1000 = 6×27×1000.
Answer. 162,000.
Independent check. 3000×48=144,000 and 3500×48=168,000.
Contrast-and-transfer pass. Keep the governing principle from “Multiple of eight”, but change one surface detail in the task: Calculate 48×3375. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—162,000.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 2: Twenty-four groups
Task. Calculate 24×3375.
Reasoning. 24÷8=3, so 3×27×1000=81,000.
Answer. 81,000.
Independent check. 3000+375 gives 72,000+9,000.
Contrast-and-transfer pass. Keep the governing principle from “Twenty-four groups”, but change one surface detail in the task: Calculate 24×3375. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—81,000.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 3: Add from 3000
Task. Calculate 32×3375.
Reasoning. 32×3000=96,000 and 32×375=12,000.
Answer. 108,000.
Independent check. The eighths route gives 4×27×1000.
Contrast-and-transfer pass. Keep the governing principle from “Add from 3000”, but change one surface detail in the task: Calculate 32×3375. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—108,000.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 4: Subtract from 3500
Task. Calculate 16×3375.
Reasoning. 16×3500=56,000 and 16×125=2,000.
Answer. 54,000.
Independent check. 16÷8×27×1000 also gives 54,000.
Contrast-and-transfer pass. Keep the governing principle from “Subtract from 3500”, but change one surface detail in the task: Calculate 16×3375. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—54,000.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 5: Non-multiple of eight
Task. Calculate 14×3375.
Reasoning. Use 14×3000 + 14×375 = 42,000+5,250.
Answer. 47,250.
Independent check. It lies between 42,000 and 49,000.
Contrast-and-transfer pass. Keep the governing principle from “Non-multiple of eight”, but change one surface detail in the task: Calculate 14×3375. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—47,250.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 6: Money
Task. Find the cost of 48 items at $33.75 each.
Reasoning. Work in cents: 48×3375=162,000 cents.
Answer. $1,620.00.
Independent check. Estimate 48×about $34, which is about $1,632.
Contrast-and-transfer pass. Keep the governing principle from “Money”, but change one surface detail in the task: Find the cost of 48 items at $33.75 each. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—$1,620.00.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 7: Reverse problem
Task. How many groups of 3375 make 108,000?
Reasoning. From the 32-group result or by 108,000÷125=864, then 864÷27.
Answer. 32 groups.
Independent check. 32×3375 restores 108,000.
Contrast-and-transfer pass. Keep the governing principle from “Reverse problem”, but change one surface detail in the task: How many groups of 3375 make 108,000? Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—32 groups.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 8: Missing factor
Task. Solve n×3375=81,000.
Reasoning. Using 3375=(27/8)×1000, n=81×8÷27.
Answer. n=24.
Independent check. 24×3375=81,000.
Contrast-and-transfer pass. Keep the governing principle from “Missing factor”, but change one surface detail in the task: Solve n×3375=81,000. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—n=24.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 9: Distribution error
Task. A pupil writes 20×3375=20×3000+375.
Reasoning. The correction 375 must also be multiplied by 20.
Answer. 60,000+7,500=67,500.
Independent check. Factor 20 from 20×3000+20×375 to restore 20×3375.
Contrast-and-transfer pass. Keep the governing principle from “Distribution error”, but change one surface detail in the task: A pupil writes 20×3375=20×3000+375. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—60,000+7,500=67,500.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 10: Decimal factor
Task. Calculate 2.4×3375.
Reasoning. Use 24×3375=81,000, then divide by 10.
Answer. 8,100.
Independent check. It lies between 7,200 and 8,400.
Contrast-and-transfer pass. Keep the governing principle from “Decimal factor”, but change one surface detail in the task: Calculate 2.4×3375. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—8,100.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 11: Fractional group
Task. Calculate three quarters of 3375×16 without first finding the full product.
Reasoning. Three quarters of 16 groups is 12 groups, so calculate 12×3375 as 12×3000+12×375.
Answer. 40,500.
Independent check. The full product is 54,000 and three quarters of 54,000 is 40,500.
Contrast-and-transfer pass. Keep the governing principle from “Fractional group”, but change one surface detail in the task: Calculate three quarters of 3375×16 without first finding the full product. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—40,500.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
Worked example 12: Choosing between exact routes
Task. Calculate 25×3375 and explain why the eighths route is not automatically the neatest first move.
Reasoning. Use 3375×(100÷4): first 3375×100=337,500, then divide by 4, or use 3000+375.
Answer. 84,375.
Independent check. 25×3000+25×375=75,000+9,375=84,375. The non-integer 25÷8 warns against forcing division first.
Contrast-and-transfer pass. Keep the governing principle from “Choosing between exact routes”, but change one surface detail in the task: Calculate 25×3375 and explain why the eighths route is not automatically the neatest first move. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—84,375.—may remain valid only if the changed detail does not alter the trigger, relationship or scope. Use the independent check again rather than trusting visual similarity. This second pass trains selection: the learner must decide whether the same method still applies, adapt one step when a condition changes, and refuse the method when its trigger disappears.
The method lab: change one condition at a time
Build an identity web for 3375 with branches to 27×125, 27/8×1000, 3000+375, 3500−125 and 2500+875. Verify each branch before using it.
Test multiplicands 14, 16, 24 and 40. Choose the route before calculating and justify the choice by divisibility or correction size. Compare intermediate values, not just final speed.
Place 3375 in three contexts: cents, grams and millilitres. Keep the numerical structure but record how the required final unit changes the reporting step.
After a delay, reconstruct three identities, one illegal cancellation, one distribution trap and one magnitude bound from a blank page.
Compare 3375×25, 3375×32 and 3375×48. For each, write two correct routes and circle the route with the smaller or more transparent intermediate values. Explain why a route can be valid without being the best choice.
Create a mini error clinic containing one missing factor of 27, one undistributed correction, one wrong sign after 3500 and one cents-to-dollars mistake. Repair each line and add a different independent check.
Common misconceptions and targeted repairs
Misconception 1: Unjustified division
What the learner may think. The learner divides by 8 because 3375 looks related to 125.
Why it fails. The division is legal only through the exact factor 125=1000/8.
Targeted repair. Write the full product identity first.
Misconception 2: Drop 27
What the learner may think. The learner uses 125 but forgets there are 27 groups.
Why it fails. That changes the multiplier from 3375 to 125.
Targeted repair. Recombine 27×125 before calculating.
Misconception 3: Add 375 once
What the learner may think. The learner writes n×3000+375.
Why it fails. Distribution requires n×375.
Targeted repair. Factor n back out as a check.
Misconception 4: Wrong correction sign
What the learner may think. The learner adds 125 after using 3500.
Why it fails. 3375 is smaller than 3500, so the correction must be subtracted.
Targeted repair. Use the magnitude relation before arithmetic.
Misconception 5: Money decimal
What the learner may think. Cents are reported as dollars.
Why it fails. 3375 cents is $33.75, so divide the final cent total by 100 once.
Targeted repair. Write the unit beside every conversion.
Misconception 6: Repeated check
What the learner may think. The learner repeats the same long multiplication.
Why it fails. The same structural error can survive.
Targeted repair. Use a different identity and bounds.
Graded independent practice with explained answers
Attempt each item without looking ahead. Write the evidence or relationship before the final response. Early items check recognition; middle items require explanation; later items change the surface form so that copying a worked example is insufficient.
Practice 1: Identity
Question. Express 3375 using 125.
Explained answer. 3375=27×125.
Why this earns trust. Twenty-seven groups are retained.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “3375=27×125.” from memory, but to recreate a justified answer from the new evidence.
Practice 2: Eighths
Question. Express 3375 using 1000.
Explained answer. 3375=(27/8)×1000.
Why this earns trust. Each 125 is one eighth of 1000.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “3375=(27/8)×1000.” from memory, but to recreate a justified answer from the new evidence.
Practice 3: Multiple of eight
Question. Calculate 56×3375.
Explained answer. 56÷8×27×1000=7×27×1000=189,000.
Why this earns trust. The division is exact.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “56÷8×27×1000=7×27×1000=189,000.” from memory, but to recreate a justified answer from the new evidence.
Practice 4: Add route
Question. Calculate 8×3375 using 3000+375.
Explained answer. 24,000+3,000=27,000.
Why this earns trust. Both addends receive 8.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “24,000+3,000=27,000.” from memory, but to recreate a justified answer from the new evidence.
Practice 5: Subtract route
Question. Calculate 40×3375 using 3500−125.
Explained answer. 140,000−5,000=135,000.
Why this earns trust. The complete correction is subtracted.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “140,000−5,000=135,000.” from memory, but to recreate a justified answer from the new evidence.
Practice 6: Non-multiple
Question. Calculate 18×3375.
Explained answer. 18×3000+18×375=54,000+6,750=60,750.
Why this earns trust. Distribution avoids an early fraction.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “18×3000+18×375=54,000+6,750=60,750.” from memory, but to recreate a justified answer from the new evidence.
Practice 7: Money
Question. Find 16 items at $33.75 each.
Explained answer. $540.00.
Why this earns trust. 16×3375=54,000 cents.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “$540.00.” from memory, but to recreate a justified answer from the new evidence.
Practice 8: Inverse
Question. How many groups of 3375 make 135,000?
Explained answer. 40 groups.
Why this earns trust. 40×3375 confirms the quotient.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “40 groups.” from memory, but to recreate a justified answer from the new evidence.
Practice 9: Repair
Question. Correct 24×3375=24×3000+375.
Explained answer. Use 72,000+9,000=81,000.
Why this earns trust. The correction is 24×375.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “Use 72,000+9,000=81,000.” from memory, but to recreate a justified answer from the new evidence.
Practice 10: Bound
Question. Give bounds for 36×3375.
Explained answer. The answer lies between 108,000 and 126,000.
Why this earns trust. 3375 lies between 3000 and 3500.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “The answer lies between 108,000 and 126,000.” from memory, but to recreate a justified answer from the new evidence.
Practice 11: Decimal
Question. Calculate 0.8×3375.
Explained answer. 2,700.
Why this earns trust. Eight tenths is 8×3375÷10.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “2,700.” from memory, but to recreate a justified answer from the new evidence.
Practice 12: Choice
Question. Which route suits 64×3375?
Explained answer. The eighths route: 64÷8×27×1000=216,000.
Why this earns trust. Divisibility by 8 gives an immediate exact simplification.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “The eighths route: 64÷8×27×1000=216,000.” from memory, but to recreate a justified answer from the new evidence.
Practice 13: Quarter of a product
Question. Find one quarter of 32×3375.
Explained answer. One quarter of 32 groups is 8 groups, so the result is 8×3375=27,000.
Why this earns trust. Regrouping is exact and avoids calculating a larger product first.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “One quarter of 32 groups is 8 groups, so the result is 8×3375=27,000.” from memory, but to recreate a justified answer from the new evidence.
Practice 14: Method restraint
Question. Why might 25×3375 be clearer as 25×3000+25×375 than as 25÷8×27×1000?
Explained answer. The distributive route keeps whole numbers immediately, while dividing 25 by 8 first creates a fraction.
Why this earns trust. Method choice responds to the multiplicand rather than forcing one memorised shortcut.
Transfer variation. Close the answer, wait several minutes, and rebuild the reasoning with different names, values or wording while preserving the same decision. Start by stating the evidence that activates the method; then produce the response and a separate check. If the new surface detail changes the governing condition, explain why the earlier answer pattern must not be copied. The aim is not to repeat “The distributive route keeps whole numbers immediately, while dividing 25 by 8 first creates a fraction.” from memory, but to recreate a justified answer from the new evidence.
A four-session mastery cycle
Session 1: construct. Read the mechanism once, cover it and write a three-line model containing the trigger, the operation and the check. Add one example and one near-miss. Compare only after the model is complete. If the operation is right but the trigger is missing, the knowledge is unsafe because it may be applied in the wrong place.
Session 2: discriminate. Mix six items: two that clearly require this method, two that tempt the method but violate a condition, and two that require another skill. For every item, write “use” or “do not use” before answering. Accuracy in method selection matters more than speed at this stage.
Session 3: explain. Give a complete answer to an imagined classmate. Include the smallest sufficient evidence, the mechanism and the independent check. Remove any sentence that merely repeats the question or adds an unsupported claim. A concise causal explanation is stronger than a longer collection of familiar words.
Session 4: perform. Attempt a timed mixed set without notes. Mark uncertain decisions, but continue. During review, classify each error as trigger, evidence, operation, scope, unit or check. Reattempt one changed version after a delay. Mastery is shown when the learner chooses and adapts the method without needing the original example beside it.
Delayed transfer: prove that the learning lasts
Transfer test 1: Multiplier family
Derive exact methods for 3125, 3250, 3375 and 3500.
Success criterion. Every identity recombines to the stated multiplier.
Transfer test 2: Mixed multiplicands
Choose methods for 14, 24, 33 and 64 without a route prompt.
Success criterion. Choice is justified before calculation.
Transfer test 3: Context switch
Solve money, mass and rate problems containing 3375.
Success criterion. Units and conversions remain correct.
Transfer test 4: Reverse transfer
Create a division problem recovered through 27×125.
Success criterion. Multiplication restores the dividend.
Examination control
1. Inspect divisibility. Look for 8, 4, 2 and convenient round bases.
2. Write the exact identity. Make every factor and correction visible.
3. Calculate in a safe order. Divide only within a justified product.
4. Protect the unit. Convert cents or other units exactly once.
5. Check differently. Use bounds and a second decomposition.
Under time pressure, compress the method to 3375 = 27×125 = 27/8×1000 = 3000+375 = 3500−125. The compressed routine is useful only after the full reasoning is understood. If it produces uncertainty, expand the step instead of guessing.
A compact error log
Classify errors as identity wrong, factor dropped, division unjustified, correction undistributed, sign reversed, unit lost or check repeated.
Retest with new multiplicands so the learner must select a route rather than reproduce one worked line.
Notes for parents and tutors
Ask for the identity before the answer.
Praise legal cancellation and clear units, not just speed.
Use estimation to catch place-value errors immediately.
Mix numbers divisible and not divisible by 8 so method choice stays active.
Frequently asked questions
Why is 3375 related to 125?
It equals 27 groups of 125.
Why divide by 8?
Because 125 equals 1000 divided by 8.
Must the multiplicand be divisible by 8?
No, but exact cancellation must occur somewhere in the product.
Which route is best?
The route with the cleanest exact first step for the given multiplicand.
Can I use long multiplication?
Yes, but the structural routes may be faster and easier to check.
How do I check the size?
For positive n, the product lies between 3000n and 3500n.
What about $33.75?
Use 3375 cents and convert the final total back to dollars once.
What is the common distribution trap?
Writing n×3000+375 instead of n×3000+n×375.
Next routes
Return to the complete volume directory to locate the series, use the original PSLE Learning Guide for the foundation route, and continue through PSLE Mathematics for verified subject guidance. The next useful task is a mixed set in which this idea is sometimes relevant and sometimes irrelevant; that is where the learner must choose a method rather than imitate one.