Singapore School Mathematics Operating Manual · Chapter 23
Mathematics often allows many different-looking representations of the same underlying object.
1/2, 2/4 and 0.5 can represent the same number. x² − 9 and (x − 3)(x + 3) represent the same polynomial function. 2:5 and 4:10 represent the same ratio. A table, graph and equation can describe the same relationship. A fraction can be simplified without changing its value. A quadratic can be expanded, factorised or completed to square while remaining the same function.
The challenge is not merely recognising sameness. It is choosing the form that makes the next mathematical job easiest.
This chapter develops two linked ideas: equivalence, where different forms preserve the same mathematical meaning, and canonical form, where we deliberately choose one standard or especially useful representative from an equivalence class.
Equivalence · Fractions and ratios · Algebraic forms · Functions · Canonical forms · Practice · Worked answers
1. Equivalent does not mean identical in appearance
3/6 and 1/2 are not written identically.
But they represent the same rational number.
Equivalent forms belong to the same mathematical class for the relationship being considered.
The learner must distinguish visual sameness from mathematical sameness.
2. Equivalence depends on the domain
(x² − 4)/(x − 2) simplifies to x + 2 for x ≠ 2.
As formulas, the expressions agree on every input where the original rational expression is defined.
But x + 2 is defined at x = 2, while the original is not.
Therefore the two functions are not identical on all real inputs unless the domain restriction is preserved.
Equivalence statements need their conditions.
3. Reversible transformations preserve equivalence
x + 5 = 9 and x = 4 are equivalent equations because subtracting 5 is reversible.
They have the same solution set.
By contrast, squaring both sides may create extra solutions and therefore break equivalence.
The chapter on Reversible Steps and Extraneous Solutions owns the transformation-control layer.
4. Simplifying fractions selects a reduced representative
12/18 = 2/3.
Both fractions are equivalent, but 2/3 is in lowest terms.
Lowest terms act as a convenient canonical form because equivalent rational numbers can be compared and recognised more easily.
The simplification changes representation, not value.
5. Decimal and fraction forms serve different jobs
1/8 = 0.125 exactly.
The fraction may expose ratio structure. The decimal may support measurement or comparison.
Neither is universally superior.
A useful representation is the one that makes the target operation, interpretation or comparison easier while preserving exactness when needed.
6. Percentages are another equivalent representation
0.35 = 35/100 = 7/20 = 35%.
Each form emphasises something different.
35% foregrounds proportion out of 100. 7/20 may simplify exact arithmetic. 0.35 may fit calculator input or measurement conventions.
7. Ratios have canonical reduced forms
6:15 simplifies to 2:5.
Multiplying or dividing both parts by the same positive non-zero factor preserves the ratio.
The reduced ratio makes equivalence easier to see.
8. Units can create equivalent numerical forms
2.5 m = 250 cm.
The numerals differ because the units differ.
Equivalence belongs to the underlying physical length, not the raw number.
9. Expansion and factorisation reveal different structures
x² − 5x + 6 = (x − 2)(x − 3).
The expanded form makes coefficients visible.
The factorised form makes roots visible.
If solving x² − 5x + 6 = 0, factorised form is usually more useful.
If comparing coefficients with another polynomial, expanded form may be better.
10. Completed-square form reveals geometry
x² − 6x + 13 = (x − 3)² + 4.
The second form reveals vertex (3,4), minimum value 4 and axis x = 3.
The same quadratic object has become easier to interpret because its representation changed.
11. Standard form supports comparison
Writing a quadratic as ax² + bx + c creates a common template.
Once two quadratics share this standard form, coefficient comparison becomes possible.
Canonical or standard forms reduce irrelevant representational variation.
12. Common denominators create a comparison form
To add 1/(x − 1) and 1/(x + 1), a common denominator (x − 1)(x + 1) gives a shared representation.
The original fractions are not changed individually into arbitrary expressions; they are rewritten into equivalent forms with a common structural base.
13. Surd simplification can expose equivalence
√12 = √(4×3) = 2√3.
Both are exact.
The simplified surd makes common factors easier to recognise in later addition or comparison.
14. Rationalising can change form without changing value
1/√2 = √2/2.
These are equivalent exact forms.
Different curricula or conventions may prefer one presentation, but the value is the same.
15. A function can be represented by rule, table or graph
y = 2x + 1.
A table of x and y values and a straight-line graph can encode the same relationship.
Each representation exposes different information.
The equation is compact. The table provides selected values. The graph reveals global shape and intercepts visually.
16. Representation switching is not automatic equivalence unless information is preserved
A graph drawn from only three points may suggest a line, but the three points alone do not prove the relationship is globally linear.
A table sampled from a curve loses information between the sampled x-values.
Representation conversion must distinguish exact translation from partial sampling.
17. A graph transformation can preserve the underlying family while changing parameters
y = x² and y = (x − 2)² + 3 are not the same function.
They belong to the same quadratic family but represent different objects.
This is an important boundary: similarity of form is not equivalence of value.
18. Canonical form reduces many equivalent descriptions to one preferred representative
For rational numbers, lowest terms can play this role.
For ratios, simplest integer ratio can play this role.
For lines, y = mx + c may be a convenient standard form when the line is non-vertical.
For quadratics, ax² + bx + c may be standard for coefficient work, while completed-square form is canonical for vertex analysis.
The “best” canonical form depends on the job.
19. Canonical forms make equality testing easier
Are 4/6 and 10/15 equal?
Reduce both: 2/3 and 2/3.
The equivalence becomes immediate.
Canonicalisation converts a difficult comparison into an ordinary equality check.
20. Normalisation can remove irrelevant scale
The vectors (2,4) and (1,2) point in the same direction but have different magnitudes.
If only direction matters, normalising to a unit vector removes scale and reveals the shared direction.
The operation changes the vector itself, so it is not equality in the ordinary vector sense. It creates equivalence under a chosen relation: same direction.
This demonstrates that equivalence must specify what features are being ignored.
21. Congruence is an equivalence relation on geometric shapes
Two congruent triangles may lie in different positions or orientations.
Translation, rotation or reflection can change placement while preserving size and shape.
Under congruence, location is irrelevant.
22. Similarity uses a coarser equivalence relation
Similar triangles may have different sizes.
They preserve angle structure and proportional side relationships.
Congruence implies similarity, but similarity does not imply congruence.
The equivalence relation determines which differences are ignored.
23. Modular arithmetic creates equivalence classes
17 and 5 are congruent modulo 12 because they leave the same remainder.
We write 17 ≡ 5 (mod 12).
Under this relation, numbers differing by a multiple of 12 belong to the same class.
This is enrichment beyond some school routes, but it gives a clean example of equivalence classes.
24. Time-of-day arithmetic uses modular equivalence intuitively
On a 12-hour clock, 15 o’clock corresponds to 3 o’clock after reducing modulo 12.
The raw numbers differ while the clock position is equivalent.
25. Equivalent equations can have different visual difficulty
2(x + 3) = 14 and x + 3 = 7 are equivalent.
The second is simpler to solve.
Good algebra often consists of moving toward an equivalent but more transparent form.
26. A useful form exposes the target feature
To find roots, factorise.
To find a vertex, complete the square.
To compare fractions, use common denominators or reduced form.
To interpret a linear model, use gradient-intercept form.
To inspect scale, use scientific notation.
Representation choice is problem-solving strategy.
27. Canonicalisation can support error detection
If two independently derived expressions should be equal, simplify both to a common form.
If the canonical forms differ, at least one derivation is wrong.
This is especially useful in algebraic proof and checking.
28. Equivalent probabilities can be expressed as fractions, decimals or percentages
1/4 = 0.25 = 25%.
The form should match context.
A probability tree may work naturally with fractions. A report may prefer percentages. A calculator may display decimals.
29. Approximation can destroy exact equivalence
√2 and 1.414 are not exactly equal.
They are approximately equal.
Once an exact form is replaced by a rounded decimal, later algebra may no longer preserve exact identities.
Use ≈ rather than = when appropriate.
30. Rounding can merge distinct values
1.2344 and 1.23449 both round to 1.234 to three decimal places.
Equal rounded displays do not imply equal exact values.
Canonical forms built from approximation have weaker equality meaning.
31. Data standardisation creates comparable scale but changes raw units
Converting values to z-scores subtracts a mean and divides by standard deviation.
The transformed scores express position relative to a distribution rather than original units.
This is not ordinary equivalence of raw values; it is a standardised representation for comparison.
32. A canonical form should not erase necessary information
Simplifying (x² − 4)/(x − 2) to x + 2 without recording x ≠ 2 loses domain information.
A useful standard form must preserve every constraint relevant to the original object.
33. Representation equivalence can support transfer
A student who recognises 0.25, 1/4, 25% and ratio 1:4 as related structures can transfer reasoning across topics.
Weak transfer often appears when each representation is learned as an isolated chapter symbol.
34. A practical equivalence audit
Ask:
What mathematical object is being represented? Which differences are merely representational? Which conditions must be preserved? Is the transformation reversible? What standard form would make comparison easiest? Have I replaced exact equality with approximation? Did I accidentally enlarge or shrink the domain?
35. Independent practice
1. Reduce 18/24 to lowest terms.
2. Express 0.375 as a fraction in lowest terms.
3. Simplify ratio 14:35.
4. Factorise x² − 7x + 12.
5. Write x² − 8x + 20 in completed-square form.
6. Are (x² − 1)/(x − 1) and x + 1 identical functions over all real x? Explain.
7. Convert 3/5 to decimal and percentage form.
8. Explain why y = x² and y = (x − 1)² are not equivalent functions.
9. Why can lowest terms be treated as a canonical form for rational numbers?
10. Give two different exact forms of 1/√3.
11. Explain why √2 = 1.414 is false as an exact equality.
12. Name one target where factorised form is more useful than expanded form.
36. Worked answers
1. 18/24 = 3/4.
2. 0.375 = 375/1000 = 3/8.
3. 14:35 = 2:5.
4. (x − 3)(x − 4).
5. (x − 4)² + 4.
6. No. They agree for x ≠ 1, but the rational expression is undefined at x = 1 while x + 1 is defined there.
7. 0.6 and 60%.
8. Their outputs differ for most inputs; the second is a horizontal translation of the first.
9. Every rational number has a unique reduced numerator-denominator pair up to the usual positive-denominator convention, making equality comparison easy.
10. 1/√3 and √3/3.
11. 1.414 is only a rounded approximation to the irrational number √2.
12. Finding roots of a polynomial equation.
37. Continue through the operating manual
Use Assumptions and Model Validity to preserve meaning when a representation returns to context, Scale and Order of Magnitude when representation should expose size, and Monotonicity and Ordering when a standard form makes comparison decisive.
Return to the BTT Mathematics Hub for Batch 06.
Mixed representation challenge: choose the form that makes the target easiest to see
Do not announce the topic. The learner must decide whether a fraction, decimal, percentage, ratio, unit rate or another equivalent form is most useful for the question being asked. More than one form may be correct; the explanation must show why one is efficient.
| Situation | Target | Useful representation and why |
|---|---|---|
| A 24-litre tank currently contains 18 litres. | Communicate how full the tank is to someone comparing several tanks of different capacities. | 75% is efficient because percentage normalises different capacities to the same 100% scale. The equivalent fraction is 18/24 = 3/4. |
| The same 24-litre tank contains 18 litres. | Compare the filled amount with the empty amount. | 3:1 is efficient because 18 litres filled and 6 litres empty reduce to a filled:empty ratio of 3:1. The target changed, so the best representation changed. |
| A drink mixture uses 300 g of ingredient A for every 180 g of ingredient B. | Find how much B is needed when A becomes 500 g. | Ratio or unit rate exposes scale. A:B = 5:3, so B/A = 3/5 = 0.6 and 500 × 0.6 = 300 g. |
| A price rises from $80 to $92. | Compare the size of the increase with the original price. | Percentage change matches the target: the increase is 12/80 = 0.15 = 15%. Reporting only “12 dollars” does not normalise the change to the starting value. |
| 12 of 30 students choose option X. | Compare this class with another class of a different size. | 2/5, 0.4 or 40% are equivalent. Percentage is usually easiest for a direct cross-class comparison; the fraction may be better when exact part-whole structure matters. |
Transfer test: change both the numbers and the target. A learner has not transferred the skill merely by converting among forms. They should be able to say, “I chose this representation because it makes this comparison, scale or part-whole relationship visible.”
Repair underlying rational-number meaning in Fractions, Decimals and Percentages. Repair multiplicative comparison and scale in Ratio, Rate and Proportion. Return here when the learner can use both and needs to choose between them independently.
Representation Decision Casebook: Choose the Form That Exposes the Target
Equivalent forms are valuable only when the learner knows why one form is more useful for the current question. The goal is not to convert everything into one preferred format. It is to choose a representation that makes the target feature easier to see while preserving the relevant information.
1. Fraction, decimal or percentage?
Consider:
[ rac38=0.375=37.5%. ]
The forms are equivalent as exact values.
Use the fraction when exact arithmetic matters, the decimal when calculator or measurement work is natural, and the percentage when the question concerns a part of one hundred or comparison of rates.
2. Comparison can determine the best form
Which is larger:
[ rac7{12}quad ext{or}quad0.6? ]
Convert one representation:
[ rac7{12}approx0.5833. ]
So 0.6 is larger.
Alternatively convert 0.6 to (3/5) and compare exact fractions.
3. Common denominators expose order
Compare:
[ rac56quad ext{and}quad rac78. ]
Common denominator 24:
[ rac{20}{24}quad ext{and}quad rac{21}{24}. ]
The second is larger.
The common denominator is useful because it creates directly comparable units.
4. Percentages expose multiplicative change
A price rises from 80 to 92.
Absolute increase:
[ 12. ]
Percentage increase:
[ 12/80=15%. ]
The percentage form answers a different question from the absolute difference.
5. Ratio can remove irrelevant scale
Groups 6:9 and 10:15 have the same reduced ratio:
[ 2:3. ]
The reduced form preserves relative proportion while discarding common scale.
That makes it useful for comparison.
6. Ratio is not a fraction of the whole unless the whole is reconstructed
For red:blue = 2:3, the fraction that is red is:
[ rac{2}{2+3}= rac25. ]
Writing (2/3) would compare red to blue, not red to total.
7. Unit rate is a canonical comparison form
Pack A costs 5.40 for 900 g.
Pack B costs 4.20 for 650 g.
Convert both to cost per kilogram or cost per gram before comparing.
The common unit creates a fair basis.
8. Standard form exposes order of magnitude
Compare:
[ 3.2 imes10^7 ]
and:
[ 9.5 imes10^6. ]
The exponent already shows that the first is larger.
Standard form compresses scale without changing value.
9. Factorised form exposes zeros
For:
[ x^2-7x+10, ]
factor:
[ (x-2)(x-5). ]
The roots (2) and (5) become visible immediately.
10. Expanded form exposes coefficients
For:
[ (x-2)(x-5), ]
expansion gives:
[ x^2-7x+10. ]
If the task asks for the coefficient of (x), expanded form is more useful.
11. Completed-square form exposes the turning point
For:
[ x^2-6x+5, ]
write:
[ (x-3)^2-4. ]
The minimum point ((3,-4)) is visible.
Different forms expose different properties of the same quadratic.
12. The best representation depends on the target
| Target | Useful form |
|---|---|
| roots | factorised |
| turning point | completed square |
| coefficient comparison | expanded |
| rough graph | form revealing roots/turning point |
13. Algebraic fractions should simplify without erasing restrictions
Consider:
[ rac{x^2-9}{x-3}. ]
Factor:
[ rac{(x-3)(x+3)}{x-3}=x+3, ]
but the original domain still excludes:
[ x=3. ]
Equivalent simplified appearance does not restore a value that the original expression never had.
14. Approximate decimal form is not exactly equivalent to an irrational value
[ sqrt2approx1.414. ]
The decimal 1.414 is an approximation, not the exact same number.
Use the surd when exactness matters.
15. Rounding can merge distinct values
1.44 and 1.46 both round to 1.5 to one decimal place? More carefully:
[ 1.44 o1.4,qquad1.46 o1.5. ]
But 1.46 and 1.54 both round to 1.5 to one decimal place.
The rounded representation has lost information.
16. A graph is not automatically equivalent to the full function rule
A plotted graph on a limited window may hide:
- roots outside the window;
- asymptotic behaviour;
- holes;
- fine-scale oscillation;
- domain restrictions.
The graph is a representation of selected information, not necessarily the entire symbolic object.
17. A table samples a function
For:
[ y=x^2, ]
a table at (x=-2,-1,0,1,2) gives:
[ 4,1,0,1,4. ]
The table suggests symmetry but does not by itself prove the rule for all real (x).
18. Equation, table, graph and verbal description should cross-check one another
For:
[ y=2x+3, ]
the representations include:
- equation: (y=2x+3);
- table: values increasing by 2 when (x) increases by 1;
- graph: straight line;
- verbal: constant rate of change 2 with starting value 3 at (x=0).
If one representation contradicts the others, inspect the translation.
19. Unit conversion changes the number but preserves the quantity
[ 1.5 ext{ m}=150 ext{ cm}. ]
The numerical value changes because the unit changes.
The physical length is the same.
20. Area and volume conversion require squared and cubed unit relationships
[ 1 ext{ m}=100 ext{ cm}. ]
Therefore:
[ 1 ext{ m}^2=10,000 ext{ cm}^2. ]
And:
[ 1 ext{ m}^3=1,000,000 ext{ cm}^3. ]
A learner who converts area using only a factor of 100 has preserved neither value nor quantity.
21. Probability forms serve different communication jobs
[ rac14=0.25=25%. ]
A fraction may show exact structure, a decimal may support calculator comparison, and a percentage may be easier for a general audience.
The underlying probability is unchanged.
22. Odds and probability are not the same representation
If probability of success is:
[ rac34, ]
odds in favour are:
[ 3:1. ]
These are related but not identical expressions. Converting between them requires reconstructing successes and failures.
23. Geometry diagrams can preserve or distort information
A sketch can preserve incidence and relative arrangement while not preserving length or angle size.
A scale drawing adds metric information.
A coordinate representation adds algebraic access.
Choose the representation according to the question.
24. Congruence preserves more structure than similarity
Congruent shapes preserve lengths and angles.
Similar shapes preserve angles and proportional side relationships but not absolute size.
These are different equivalence ideas.
25. Coordinates can turn geometry into algebra
For points (A(1,2)) and (B(5,5)):
[ overrightarrow{AB}= egin{pmatrix} 4\3 end{pmatrix}. ]
Distance:
[ 5. ]
The coordinate representation makes displacement and distance calculable.
26. A verbal rule can hide mathematical ambiguity
“Increase by 10” means additive change.
“Increase by 10%” means multiplicative change.
Converting both into symbolic form prevents the wording from being treated as equivalent.
27. Canonical form can support comparison across answers
Two students may produce:
[ rac{6}{8} ]
and:
[ rac34. ]
Reducing fractions to canonical form makes equivalence easier to recognise.
28. Canonicalisation should not be used blindly
The expanded form of a quadratic may be less useful than factorised form for roots. The decimal form of a fraction may be less useful than exact fraction form for later cancellation.
A canonical form is useful for a particular job, not universally superior.
29. Representation choice can reduce working-memory load
Consider:
[ 49^2-1. ]
Expanding 49² creates large arithmetic.
Rewriting as:
[ (49-1)(49+1)=48cdot50 ]
reduces the load dramatically.
The value is unchanged; the form is better suited to calculation.
30. Representation choice can expose invariants
A ratio written as (2:3) makes relative scale visible.
A fraction written as (2/5) makes part-to-whole visible.
A bar model may make a preserved total visible.
An equation may make an unknown relation visible.
31. A representation decision routine
Before converting, ask:
- What is the target?
- Which feature must be visible?
- Which form preserves exactness?
- Which form reduces unnecessary work?
- What information might be lost?
32. Casebook: best form for comparing discounts
Shop A gives 20% off.
Shop B reduces 15 dollars from a 60-dollar item.
Convert Shop B reduction to a percentage:
[ 15/60=25%. ]
Now the reductions are comparable on the same scale.
33. Casebook: best form for sharing a quantity
Share 84 in ratio 3:4.
Total ratio units = 7.
One unit = 12.
Shares:
[ 36, 48. ]
The ratio-unit representation is more efficient than converting to percentages first.
34. Casebook: best form for repeated percentage growth
A quantity grows by 5% each period.
Use multiplier:
[ 1.05. ]
After (n) periods:
[ A_n=A_0(1.05)^n. ]
The multiplier form exposes repeated composition.
35. Casebook: best form for roots
For:
[ x^2-11x+24, ]
factor:
[ (x-3)(x-8). ]
Do not complete the square if the target is only the roots.
36. Casebook: best form for minimum value
For:
[ x^2-8x+5, ]
complete square:
[ (x-4)^2-11. ]
Minimum value = -11 at (x=4).
37. Casebook: best form for line comparison
Lines:
[ 2x-y=5 ]
and:
[ y=2x+1. ]
Rearrange the first:
[ y=2x-5. ]
Both now show gradient 2, so they are parallel.
38. Casebook: best form for probability comparison
Compare:
[ 0.42,quad rac25,quad39%. ]
Convert to decimals:
[ 0.42,quad0.40,quad0.39. ]
Now order is immediate.
39. Casebook: best form for unit conversion
Compare 1.8 km and 1750 m.
Convert 1.8 km:
[ 1800 ext{ m}. ]
Then 1.8 km is longer by 50 m.
40. Casebook: best form for checking an equation solution
After solving:
[ 3x+7=25, ]
with (x=6), substitute:
[ 3(6)+7=25. ]
The original equation is the best checking representation because it tests the exact constraint.
41. Casebook: best form for a word problem
“Lina has 7 more stickers than Kai. Together they have 31.”
Let Kai = (x).
Lina = (x+7).
[ x+(x+7)=31. ]
Equation form compresses the relationship after the verbal roles are understood.
42. Casebook: best form for a changing ratio
Red:blue changes from 2:3 to 4:3 after red items are added.
Keep blue at 3 matching ratio units in both stages.
The matched ratio representation exposes that red increased by 2 units.
43. Casebook: best form for a histogram comparison
Raw frequencies cannot be compared by bar height if class widths differ.
Convert to frequency density:
[ ext{density}= rac{ ext{frequency}}{ ext{class width}}. ]
The representation matches histogram area to frequency.
44. Casebook: best form for average speed
Do not average two speeds unless the weighting supports it.
Use:
[ ext{average speed}= rac{ ext{total distance}}{ ext{total time}}. ]
The distance-time representation is canonical for the reader job.
45. Representation failure: converting too early
Turning:
[ rac13 ]
into 0.333 too early introduces approximation.
If later multiplied by 3, the decimal gives 0.999 rather than exact 1.
46. Representation failure: keeping an awkward form too long
Conversely, refusing to convert can add unnecessary work.
Comparing 37.5% with (3/8) is easiest once one is converted.
47. Representation failure: losing the reference whole
Ratio 2:3 is not automatically (2/3) of the whole.
The whole has five ratio units.
The correct fraction for the first part is (2/5).
48. Representation failure: losing units
1.5 and 150 can be equivalent measurements only if the units are 1.5 m and 150 cm.
Numbers without units can hide whether the quantities are comparable.
49. Representation failure: graphing before defining the domain
A real-life model may only make sense for:
[ tge0 ]
or within a finite time interval.
A graph drawn across all real values can display mathematically defined but contextually meaningless regions.
50. Representation failure: simplifying an equation with a one-way operation
Squaring both sides can introduce extra solutions.
Example:
[ sqrt{x+2}=x. ]
Squaring gives:
[ x+2=x^2, ]
but any resulting candidate must be checked in the original equation.
The transformed equation is not automatically equivalent in both directions.
51. A representation-choice worksheet should not name the representation
Give mixed problems and ask:
“Which form makes the target easiest to see?”
Do not title the page “Fractions to Decimals” if the point is to choose among representations.
52. A tutor should ask for two valid forms
For a quadratic, ask for expanded and factorised forms.
For a probability, ask for fraction and percentage.
For a function, ask for equation and graph.
Then ask what each form reveals.
53. A parent can support representation choice without solving
Ask:
- Could you draw this?
- Would a fraction or percentage be easier?
- Can you put both quantities in the same unit?
- Which form shows the roots?
- What information might be lost if you round?
54. Fresh representation transfer 1
Compare:
[ rac{11}{20},quad0.52,quad54%. ]
Convert to decimals:
[ 0.55, 0.52, 0.54. ]
Order:
[ 0.52<54%<11/20. ]
55. Fresh representation transfer 2
Rewrite:
[ x^2+10x+21 ]
in factorised form:
[ (x+3)(x+7). ]
Then state what feature becomes visible: roots (-3,-7).
56. Fresh representation transfer 3
Rewrite:
[ x^2+10x+21 ]
in completed-square form:
[ (x+5)^2-4. ]
Now the turning point ((-5,-4)) is visible.
57. Fresh representation transfer 4
Convert 72 km/h to m/s:
[ 72 imes rac{1000}{3600}=20 ext{ m/s}. ]
The quantity is the same speed; the numerical value changes because the unit changes.
58. The final MA-S1-040 release test
Give six unlabeled problems:
- compare fraction/decimal/percentage;
- reduce a ratio for comparison;
- choose quadratic form for roots;
- choose quadratic form for turning point;
- convert units before comparison;
- choose equation/table/graph for a function question.
Require the learner to state:
- chosen representation;
- why it is useful;
- what information is preserved;
- what information could be lost.
59. The durable representation model
The complete loop is:
identify target → inspect current form → choose a useful equivalent form → preserve domain/units/exactness → solve or compare → check against another representation.
The student has learned representation choice when they can explain why a conversion helps, not merely perform the conversion because a worksheet told them to.
60. Continue through the BTT Mathematics system
Use Fractions, Decimals and Percentages for the detailed fraction/decimal/percentage object. Use Ratio, Rate and Proportion for ratio and rate structures. Use Functions and Graphs for equation/table/graph switching. Return to the World Mathematics Atlas for the wider route.
Final Representation Lab: One Quantity, Several Useful Forms
The final MA-S1-040 check should present one mathematical object in several forms and ask the learner to choose the form that best exposes the target.
Lab 1: fraction, decimal and percentage
Given:
[ rac7{20}=0.35=35%. ]
Ask:
- Which form is easiest for comparing with 32%?
- Which is best for exact addition with (3/10)?
- Which communicates the result most naturally in a survey?
No single representation wins every task.
Lab 2: ratio and fraction of a whole
If red:blue = 3:5:
red as a fraction of total:
[ rac{3}{8}. ]
blue as a fraction of total:
[ rac{5}{8}. ]
The ratio compares parts; the fractions compare each part with the whole.
Lab 3: quadratic forms
For:
[ x^2-8x+15, ]
expanded form shows coefficients.
Factorised form:
[ (x-3)(x-5) ]
shows roots.
Completed-square form:
[ (x-4)^2-1 ]
shows the turning point.
The learner should choose by target rather than habit.
Lab 4: equation and graph
For:
[ y=-3x+6, ]
the equation shows exact gradient and intercept. The graph makes the decreasing trend and intercept locations visible.
Ask the learner which representation is better for finding the x-intercept and which is better for explaining the overall direction.
Lab 5: exact and approximate values
[ sqrt5 ]
is exact.
[ 2.236 ]
is approximate.
If a later expression allows cancellation with (sqrt5), exact form is preferable. If a physical measurement needs a decimal, approximation may be appropriate.
Lab 6: unit conversions
[ 2.4 ext{ m}=240 ext{ cm}. ]
The numerical value changes while the length stays the same.
For area:
[ 2.4 ext{ m}^2=24,000 ext{ cm}^2. ]
The conversion factor is squared because the unit is squared.
Lab 7: probability forms
[ rac58=0.625=62.5%. ]
Ask which is most useful for exact probability arithmetic, which is easiest to compare on a calculator, and which is most natural for a general report.
Lab 8: standard form
[ 0.00072=7.2 imes10^{-4}. ]
Standard form makes scale and multiplication/division with powers of ten easier to manage.
Lab 9: word problem to equation
“A value is 9 more than twice another value.”
Represent as:
[ y=2x+9. ]
The equation compresses the verbal relationship while preserving its structure.
Lab 10: equation to table to graph
For:
[ y=x^2-1, ]
a table can sample points, the graph can show symmetry and zeros, and the equation gives the exact rule.
Ask what each representation can and cannot prove.
Lab 11: bar model to equation
“Mina has 12 more than Sol. Together they have 48.”
A bar model can show one equal base bar plus an extra 12.
Equation:
[ x+(x+12)=48. ]
The equation is more compact once the relationship is understood.
Lab 12: changing ratio
Red:blue changes from 2:5 to 4:5 because red increases.
Keeping blue at five matched ratio units exposes that red increased by two ratio units.
A fraction-of-total representation would make the before-and-after comparison less direct.
Lab 13: histogram frequency density
When class widths differ, frequency density creates a representation in which bar area corresponds to frequency:
[ ext{frequency density}= rac{ ext{frequency}}{ ext{class width}}. ]
Using raw frequency as height would distort the data.
Lab 14: average speed
For journeys at changing speeds, the useful representation is:
[ ext{average speed}= rac{ ext{total distance}}{ ext{total time}}. ]
The arithmetic mean of individual speeds may answer the wrong question.
Lab 15: equation transformation with a one-way step
From:
[ sqrt{x+1}=x-1, ]
squaring gives:
[ x+1=(x-1)^2. ]
The squared equation can contain extra candidates. Every candidate must be checked in the original equation.
This is a case where the transformed form is useful but not automatically equivalent in both directions.
A final decision checklist
Before converting a representation, ask:
- What do I need to see?
- Do I need exactness?
- Do I need easy comparison?
- Do units need normalising?
- Will the conversion lose a domain restriction?
- Will rounding lose information?
- Could another representation provide an independent check?
Delayed transfer set
After several days, give:
- a percentage that should become a multiplier;
- a ratio that should become a fraction of the whole;
- a quadratic whose roots are easiest in factorised form;
- a quadratic whose minimum is easiest in completed-square form;
- a function question where a graph is useful but the exact rule is still needed;
- a unit-conversion comparison.
Do not name the required representation.
Final MA-S1-040 standard
The learner is ready when they can choose a useful representation voluntarily, explain what it reveals, preserve domain and units, avoid unnecessary approximation, and confirm the result through a second equivalent form.
Release principle: do not convert because conversion is possible; convert because the new form exposes the job.
Final Floor: Representation Choice Under Pressure
For the final representation check, use six short questions and forbid the worksheet from naming the desired form.
1. Compare 0.46, 45% and 7/15. Convert to a common form. Since (7/15approx0.4667), the order is (45%<0.46<7/15).
2. Find the roots of (x^2-9x+20). Factorised form ((x-4)(x-5)) exposes the roots immediately.
3. Find the minimum of (x^2-6x+1). Completed-square form ((x-3)^2-8) exposes the minimum value -8.
4. Compare 1.8 km and 1750 m. Convert to a common unit: 1800 m is 50 m longer.
5. Interpret (y=4x+7). Equation form exposes exact gradient and intercept; graph form exposes overall behaviour. The useful representation depends on the question.
6. Simplify ((x^2-16)/(x-4)). Factorisation gives (x+4) while preserving the original restriction (x e4).
After every conversion, ask what information was gained and what might have been lost. Decimal approximation can lose exactness. Graph windows can hide behaviour. Simplification can hide excluded values. Unit conversion changes the numerical value while preserving the physical quantity.
Final MA-S1-040 standard: choose a representation because it exposes the target, preserve exactness, domain and units, and use a second representation as an independent check when useful.
Delayed Transfer Floor: Choose the Form Without Being Told
After a short delay, give the learner five unrelated questions and require the representation choice before calculation.
Compare (0.58) and (7/12). Converting (7/12approx0.5833) makes the comparison visible.
Find the roots of (x^2-13x+40). Factorised form ((x-5)(x-8)) exposes the roots immediately.
Find the minimum of (x^2-10x+18). Completed-square form ((x-5)^2-7) exposes the minimum.
Compare 2.05 km and 1980 m. Convert both to metres: 2050 m is 70 m longer.
Interpret (y=-2x+9). The equation gives exact gradient and intercept; the graph makes the decreasing trend visible.
For every answer, ask what information the chosen form reveals and what could be lost by conversion. Decimal approximation can lose exactness, a graph window can hide behaviour, and algebraic simplification can hide original restrictions if they are not recorded.
Final MA-S1-040 release standard: representation is chosen deliberately, not by habit; exactness, domain and units are preserved; and a second form can be used as an independent check.
Final floor: complete one delayed representation-choice set using a fraction/percentage comparison, a quadratic, a unit conversion, a function, a probability and an algebraic fraction. The learner should name the target feature first, choose the form that reveals it most efficiently, preserve exactness and original restrictions, and then use another representation to check the result. A conversion is successful only when it makes the job clearer without silently changing the mathematical object. This final delayed check confirms that representation choice has become an independent decision rather than a classroom instruction.
Release verification: the final task should force a representation change that is useful rather than decorative: convert to a common unit for comparison, factor to expose roots, complete the square to expose a minimum, use a percentage to compare relative change, and keep an exact surd when a decimal would lose information. The learner should explain what became easier to see after each conversion and what information had to be preserved. This confirms that equivalence and canonical form have become tools for reasoning rather than isolated conversion exercises.
Final completion note: the learner should complete one fresh task where two equivalent forms are both correct but only one exposes the target efficiently. They should explain why the chosen form reduces work, what conditions must remain attached, and how another representation can verify the result. This confirms that representation choice is a reasoning skill rather than a conversion routine.
Safety-margin release note: a final learner explanation should compare two valid equivalent forms and state which is more useful for the immediate target, why the other remains correct, and what restrictions, units or exactness conditions must survive the conversion. This short explanation is the clearest evidence that canonical forms are being selected for mathematical reasons rather than produced mechanically.
Final verification: representation choice, exactness, domain, units and independent cross-checking remain stable on unseen problems.
Safe margin: unseen representation choices remain mathematically justified, exact where needed and independently checked.
Final: representation selection remains deliberate, exact and independently checked on fresh work.
Verified: equivalent forms are selected deliberately, with domain, units, exactness and independent checking preserved.
Complete: delayed representation transfer remains stable on fresh work.