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.
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This volume solves one precise learner problem: the learner memorises that squaring makes numbers larger, applies a whole-number pattern to positive proper fractions, squares only the numerator, or states the result without checking the essential condition that the fraction lies strictly between 0 and 1. 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
Squaring means multiplying a number by itself. For a positive proper fraction f, the condition is 0 < f < 1, so f² means f × f.
Multiplying a positive quantity by a factor less than 1 takes a fraction of that quantity. Therefore f × f is smaller than f when the second factor f lies between 0 and 1.
For example, (3/4)² = 9/16. Since 3/4 = 12/16, the comparison is 9/16 < 12/16.
An area model makes the same mechanism visible. Shade three quarters of a square horizontally, then take three quarters of that shaded region vertically. The overlap covers nine sixteenths of the whole.
The rule needs boundaries. At f = 0, f² = f. At f = 1, f² = f. For f > 1, squaring makes the positive number larger. A statement without conditions is unreliable.
Do not square only the numerator or only the denominator. For a/b, the square is a²/b² because the entire fraction is multiplied by itself.
The comparison can be proved without decimals: f – f² = f(1 – f). Both factors are positive for 0 < f < 1, so the difference is positive and f² < f.
A calculator can confirm examples but does not replace the mechanism. The important idea is the multiplying factor, not a list of isolated results.
In word problems, check that squaring is actually requested. Squaring a numerical scale, an area relation or a probability may have different units and interpretations even when the arithmetic is valid.
The reliable routine is state 0 < f < 1, rewrite the square as multiplication, calculate both numerator and denominator, compare with a common form, test endpoints, then label the result.
Core decision: If 0 < f < 1, then f² = f × f is smaller than f because multiplying a positive quantity by a factor less than 1 reduces it. Equivalently, f - f² = f(1 - f) > 0.
A strong argument identifies the interval before applying a pattern. It links exponent notation to multiplication, uses an exact comparison, checks boundary cases and keeps the numerical result separate from the quantity and unit represented.
Mechanism: why the rule works
1. State the domain
Confirm that f is positive and less than 1.
The conclusion changes at 0, 1 and above 1.
Diagnostic check. Which interval is guaranteed?
2. Expand the square
Rewrite f² as f × f.
The exponent is repeated multiplication, not decoration.
Diagnostic check. What factor is applied to f?
3. Use the factor effect
Recognise that the multiplying factor f is between 0 and 1.
Taking a proper fraction of a positive amount makes it smaller.
Diagnostic check. Is the factor reducing or enlarging?
4. Square the whole fraction
For a/b, calculate a²/b².
Both numerator and denominator participate in multiplication.
Diagnostic check. Which parts are multiplied by themselves?
5. Compare exactly
Use common denominators, cross-products or a difference.
Exact comparison avoids misleading decimals.
Diagnostic check. How can f² and f be written on one scale?
6. Model the overlap
Interpret f² as taking fraction f of fraction f of a whole.
The overlap area makes repeated fractional selection visible.
Diagnostic check. How much of the original whole remains?
7. Check endpoints
Test f=0 and f=1.
These produce equality and explain why strict inequalities matter.
Diagnostic check. Where does the rule stop being strict?
8. Use counterexamples
Test f=2 or f=-2 before claiming squaring always reduces.
One changed domain can reverse a pattern.
Diagnostic check. Does the claim survive outside 0 Attach units and interpret what was squared. A correct number can answer the wrong quantity. Diagnostic check. What real quantity does f² represent? 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. Task. Compare (3/4)² with 3/4. Reasoning. (3/4)²=9/16 and 3/4=12/16. Answer. 9/16<12/16. Independent check. Both numerator and denominator are squared. Contrast-and-transfer pass. Keep the governing principle from “Three quarters”, but change one surface detail in the task: Compare (3/4)² with 3/4. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—9/16<12/16.—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. Task. Compare (2/3)² with 2/3. Reasoning. 4/9<6/9. Answer. The second factor 2/3 reduces the first. Independent check. No decimal is needed. Contrast-and-transfer pass. Keep the governing principle from “Two thirds”, but change one surface detail in the task: Compare (2/3)² with 2/3. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The second factor 2/3 reduces the first.—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. Task. (1/10)²=1/100. Reasoning. One tenth of one tenth is one hundredth. Answer. The square is much smaller. Independent check. The area model gives a 10-by-10 grid. Contrast-and-transfer pass. Keep the governing principle from “One tenth”, but change one surface detail in the task: (1/10)²=1/100. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The square is much smaller.—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. Task. Compare 0² and 0. Reasoning. They are equal. Answer. The strict condition excludes 0. Independent check. Do not write 0²<0. Contrast-and-transfer pass. Keep the governing principle from “Endpoint zero”, but change one surface detail in the task: Compare 0² and 0. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The strict condition excludes 0.—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. Task. Compare 1² and 1. Reasoning. They are equal. Answer. The strict condition excludes 1. Independent check. This is the second fixed point. Contrast-and-transfer pass. Keep the governing principle from “Endpoint one”, but change one surface detail in the task: Compare 1² and 1. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The strict condition excludes 1.—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. Task. Compare (5/4)² and 5/4. Reasoning. 25/16>20/16. Answer. A factor greater than 1 enlarges the positive amount. Independent check. The proper-fraction condition matters. Contrast-and-transfer pass. Keep the governing principle from “Improper fraction”, but change one surface detail in the task: Compare (5/4)² and 5/4. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—A factor greater than 1 enlarges the positive amount.—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. Task. A pupil writes (3/5)²=9/5. Reasoning. The denominator must also be squared. Answer. Correct result: 9/25. Independent check. The exponent applies to the whole fraction. Contrast-and-transfer pass. Keep the governing principle from “Square-only-numerator error”, but change one surface detail in the task: A pupil writes (3/5)²=9/5. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—Correct result: 9/25.—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. Task. 0.6²=0.36. Reasoning. Since 0.6 is a factor less than 1, the result is smaller. Answer. Convert 3/5 if an exact fraction helps. Independent check. Decimal and fraction forms agree. Contrast-and-transfer pass. Keep the governing principle from “Decimal form”, but change one surface detail in the task: 0.6²=0.36. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—Convert 3/5 if an exact fraction helps.—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. Task. Let f=7/8. Compute f-f². Reasoning. 7/8-49/64=56/64-49/64=7/64>0. Answer. A positive difference proves f>f². Independent check. The method generalises. Contrast-and-transfer pass. Keep the governing principle from “Difference proof”, but change one surface detail in the task: Let f=7/8. Compute f-f². Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—A positive difference proves f>f².—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. Task. An event has probability 1/2 and two independent repetitions are modelled by (1/2)². Reasoning. The product is 1/4. Answer. The arithmetic is valid only because the situation justifies multiplication. Independent check. Do not square a probability merely because two events appear. Contrast-and-transfer pass. Keep the governing principle from “Probability”, but change one surface detail in the task: An event has probability 1/2 and two independent repetitions are modelled by (1/2)². Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—The arithmetic is valid only because the situation justifies multiplication.—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. Task. A square’s side is scaled by 3/4. Reasoning. Its area scale is (3/4)²=9/16. Answer. Area shrinks more than the side-length factor. Independent check. Label linear and area scales separately. Contrast-and-transfer pass. Keep the governing principle from “Area scale”, but change one surface detail in the task: A square’s side is scaled by 3/4. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—Area shrinks more than the side-length factor.—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. Task. Compare (99/100)² with 99/100 without long division. Reasoning. (99/100)² is 99/100 of 99/100, so it is smaller. Answer. Exactly, 9801/10000 is less than 9900/10000. Independent check. A number can be very close to 1 and still satisfy the strict rule. Contrast-and-transfer pass. Keep the governing principle from “Near one”, but change one surface detail in the task: Compare (99/100)² with 99/100 without long division. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—Exactly, 9801/10000 is less than 9900/10000.—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. Task. Three quarters of a garden is planted, and two thirds of the planted part has flowers. Reasoning. The flower part is (3/4)×(2/3)=1/2 of the garden. Answer. Repeated fractional selection reduces the original positive part. Independent check. The factors need not be equal for the factor effect to work. Contrast-and-transfer pass. Keep the governing principle from “Fraction of a fraction”, but change one surface detail in the task: Three quarters of a garden is planted, and two thirds of the planted part has flowers. Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—Repeated fractional selection reduces the original positive part.—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. Task. Compare (3/5)² and (4/5)². Reasoning. 9/25<16/25. Answer. Squaring preserves the order of positive fractions even though each square is smaller than its own base. Independent check. Do not confuse within-number change with between-number order. Contrast-and-transfer pass. Keep the governing principle from “Order after squaring”, but change one surface detail in the task: Compare (3/5)² and (4/5)². Before recalculating or rewriting, name which evidence remains controlling and which detail has changed. The original conclusion—Squaring preserves the order of positive fractions even though each square is smaller than its own base.—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. Write every square as repeated multiplication first. Sort values into below 0, 0, between 0 and 1, 1 and above 1. Square one value from each region and compare. Use a 4-by-4 grid to model (3/4)². Use a 10-by-10 grid to model (7/10)². Convert the original fraction to the squared denominator for comparison. Practise cross-product comparisons without decimal rounding. Repair examples that square only one fraction part. Find f-f² and factor it as f(1-f). Explain why both factors are positive in the target interval. Test f=0 and f=1 as boundary cases. Find a counterexample to “squaring always makes larger”. Distinguish squaring a number from finding a square’s area. After a delay, compare (11/12)² with 11/12 mentally. Finish with an audit: domain, exponent, whole fraction, comparison, endpoints, units and context. What the learner may think. The learner overgeneralises whole-number examples. Why it fails. A positive factor below 1 reduces. Targeted repair. Check the interval first. What the learner may think. The learner ignores the fraction bar. Why it fails. The exponent applies to numerator and denominator. Targeted repair. Rewrite as repeated multiplication. What the learner may think. The learner confuses smaller with below zero. Why it fails. A positive fraction squared stays positive. Targeted repair. Keep order and sign separate. What the learner may think. The learner ignores equality at 0 and 1. Why it fails. Both are fixed by squaring. Targeted repair. Test boundaries. What the learner may think. The learner lists examples without mechanism. Why it fails. Examples support; factor reasoning explains. Targeted repair. Name the factor less than 1. What the learner may think. The learner applies notation without a model. Why it fails. The situation must justify repeated multiplication. Targeted repair. Label the represented quantity. 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. Question. What does f² mean? Explained answer. f × f. Why this earns trust. The exponent shows two equal factors. 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 “f × f.” from memory, but to recreate a justified answer from the new evidence. Question. What condition makes the square strictly smaller? Explained answer. 0 Why this earns trust. Endpoints give equality. 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 “0 Question. Find (4/5)². Explained answer. 16/25. Why this earns trust. Square numerator and denominator. 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 “16/25.” from memory, but to recreate a justified answer from the new evidence. Question. Compare 16/25 and 4/5. Explained answer. 16/25<20/25. Why this earns trust. Use a common denominator. 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 “16/25<20/25.” from memory, but to recreate a justified answer from the new evidence. Question. Why is f×f Explained answer. The second factor is positive and less than 1. Why this earns trust. It takes only a fraction of f. 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 second factor is positive and less than 1.” from memory, but to recreate a justified answer from the new evidence. Question. What is 1²? Explained answer. 1. Why this earns trust. No strict decrease occurs. 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 “1.” from memory, but to recreate a justified answer from the new evidence. Question. Give a positive value whose square is larger. Explained answer. 2, because 2²=4. Why this earns trust. The value exceeds 1. 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, because 2²=4.” from memory, but to recreate a justified answer from the new evidence. Question. Repair (2/7)²=4/7. Explained answer. 4/49. Why this earns trust. Both parts are squared. 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 “4/49.” from memory, but to recreate a justified answer from the new evidence. Question. Why is f-f² positive? Explained answer. It equals f(1-f), whose factors are positive. Why this earns trust. This proves the order. 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 “It equals f(1-f), whose factors are positive.” from memory, but to recreate a justified answer from the new evidence. Question. A side scale is 1/2. What is the area scale? Explained answer. 1/4. Why this earns trust. Area uses two linear factors. 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 “1/4.” from memory, but to recreate a justified answer from the new evidence. Question. Can the square of a positive proper fraction be negative? Explained answer. No. Why this earns trust. A positive times a positive is positive. 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 “No.” from memory, but to recreate a justified answer from the new evidence. Question. State the checks. Explained answer. Domain, expand square, square whole fraction, compare exactly, test endpoints and label context. Why this earns trust. Each check protects the conclusion. 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 “Domain, expand square, square whole fraction, compare exactly, test endpoints and label context.” from memory, but to recreate a justified answer from the new evidence. 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. Interpret multiplication by a proper fraction as taking part of a quantity. Success criterion. The factor effect is explicit. Translate notation into repeated multiplication. Success criterion. The whole base is identified. Connect linear scale to area scale. Success criterion. Two dimensions create a squared factor. Use a positive difference and boundary cases. Success criterion. The claim is general and properly limited. 1. Name the target. State exactly what the question asks before choosing a rule or explanation. 2. Expose the relationship. Write the grammatical, numerical, causal or measurement relationship. 3. Use decisive evidence. Choose the smallest fact, calculation or observation that controls the answer. 4. Check the boundary. Reject a nearby claim that the evidence does not support. 5. Present precisely. Keep units, conditions, certainty and scope visible. Under time pressure, compress the method to check 0 Label errors as domain omitted, exponent misread, numerator-only square, denominator-only square, sign confusion, endpoint error, example-only claim or wrong contextual quantity. Repair by writing f-f²=f(1-f) beside one exact numerical example. Shade fraction-of-a-fraction grids. Compare values near 0 and near 1. Ask for boundary cases. Use scale drawings and area factors. Require an explanation beyond calculator output. The number is multiplied by a factor below 1. No. Yes. Equality. Equality. Their positive squares are larger. No; use the factor or difference argument. Locate the base relative to 0 and 1. 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.9. Return to context
Worked examples: make the decision visible
Worked example 1: Three quarters
Worked example 2: Two thirds
Worked example 3: One tenth
Worked example 4: Endpoint zero
Worked example 5: Endpoint one
Worked example 6: Improper fraction
Worked example 7: Square-only-numerator error
Worked example 8: Decimal form
Worked example 9: Difference proof
Worked example 10: Probability
Worked example 11: Area scale
Worked example 12: Near one
Worked example 13: Fraction of a fraction
Worked example 14: Order after squaring
The method lab: change one condition at a time
Common misconceptions and targeted repairs
Misconception 1: Squaring always increases
Misconception 2: Square only numerator
Misconception 3: The result is negative
Misconception 4: Endpoints are smaller
Misconception 5: A calculator proves the rule
Misconception 6: Every context requires a square
Graded independent practice with explained answers
Practice 1: Definition
Practice 2: Domain
Practice 3: Calculation
Practice 4: Comparison
Practice 5: Mechanism
Practice 6: Endpoint
Practice 7: Counterexample
Practice 8: Error repair
Practice 9: Difference
Practice 10: Area
Practice 11: Sign
Practice 12: Routine
A four-session mastery cycle
Delayed transfer: prove that the learning lasts
Transfer test 1: Fractions
Transfer test 2: Exponents
Transfer test 3: Geometry
Transfer test 4: Proof
Examination control
A compact error log
Notes for parents and tutors
Frequently asked questions
Why does the square get smaller?
Is the square negative?
Do both fraction parts get squared?
What happens at 0?
What happens at 1?
What about values above 1?
Is a decimal example enough proof?
Fastest check?
Next routes