This retained Secondary 3 E‑Mathematics page is the Yishun modelling-and-representation guide. Its job is to connect algebra, graphs and mathematical modelling without competing with the main Secondary 3 Mathematics owner or BTT’s specialist subject teaching. It does not establish a current physical Yishun centre, class, fee, timetable or vacancy.
Current learners should follow the Mathematics subject level and school sequence actually assigned to them. For the main year-level recovery route, use Secondary 3 Mathematics in Yishun. For specialist Mathematics teaching, use BTT’s Secondary Mathematics Learning Hub.
1. A model is a relationship with assumptions
A hypothetical fixed charge of $6 plus $2.50 per kilometre can be represented as C = 6 + 2.5d under the stated assumptions. The fixed term and rate have different jobs. It is a teaching model, not a current commercial tariff.
2. Words to algebra
“Three more than twice a number is 19” becomes 2x + 3 = 19, giving x = 8. “Three times the sum of a number and two is 21” becomes 3(x + 2) = 21, giving x = 5. Grouping changes the relationship.
3. Algebra to table to graph
For y = 3x + 2, inputs 0, 1, 2 and 3 give outputs 2, 5, 8 and 11. The equation, table and plotted points describe the same linear relationship.
Direct proportion is a special case. y = 3x passes through the origin, while y = 3x + 2 does not. Constant gradient does not by itself imply direct proportion.
4. Graph to algebra
A line through (1, 4) and (5, 12) has gradient 2. Substituting into y = 2x + c gives c = 2, so the equation is y = 2x + 2.
If 0 to 40 spans eight equal graph intervals, one interval is 5. A model built from a misread scale is wrong even if the later algebra is correct.
5. Geometry can be modelled algebraically
A rectangle has width x and length x + 4. If its perimeter is 36, then 2[x + (x + 4)] = 36, giving x = 7 and length 11. If the question asks for area, the final answer is 77 square units.
6. State tracking in multi-step models
A quantity begins at 500, increases by 10% and then decreases by 40. The intermediate state is 550 and the final state is 510. Reversing the order describes a different process because the percentage base changes.
7. Fresh modelling check
Write and solve an equation for “three identical items and a $4 fee cost $25”. Find the equation of the line through (0, 5) and (4, 13). A rectangle has width w and length w + 3 with perimeter 30; find its area.
The answers are 3x + 4 = 25, x = 7; y = 2x + 5; and w = 6, length 9, area 54 square units.
8. Route onward
Use Student Diagnostics when the first modelling failure remains uncertain, Examination Performance when paper conditions disrupt otherwise sound modelling, and BTT for fuller specialist Mathematics teaching.
Extended Algebra–Graphs–Modelling System
This appended section preserves the newest live modelling-and-representation owner. It does not compete with the broader Secondary 3 recovery route. Its purpose is to make translation among words, symbols, tables, diagrams and graphs systematic, while keeping assumptions and interpretation visible.
The page does not establish a current physical centre, class, fee, timetable or vacancy. Current learners should follow the Mathematics subject level and school sequence assigned to them. The examples here are teaching models and should be read as illustrative mathematical relationships rather than current commercial or scientific claims.
21. Model purpose
Modelling purpose. This component helps the learner state what the model is for before choosing a form. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
22. Variable definition
Modelling purpose. This component helps the learner name each changing quantity and unit clearly. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
23. Constant terms
Modelling purpose. This component helps the learner distinguish fixed quantities from changing ones. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
24. Assumptions
Modelling purpose. This component helps the learner state the simplifications that make the model usable. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
25. Words to algebra
Modelling purpose. This component helps the learner translate verbal relationships into symbols. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
26. Algebra to words
Modelling purpose. This component helps the learner translate symbolic expressions back into meaning. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
27. Words to table
Modelling purpose. This component helps the learner organise paired values so patterns can be inspected. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
28. Table to rule
Modelling purpose. This component helps the learner infer a candidate relationship from several data pairs. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
29. Rule to table
Modelling purpose. This component helps the learner generate outputs systematically from a symbolic rule. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
30. Table to graph
Modelling purpose. This component helps the learner plot paired values with correct axes and scale. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
31. Graph to words
Modelling purpose. This component helps the learner describe numerical change without vague visual claims. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
32. Graph to equation
Modelling purpose. This component helps the learner connect intercept and constant change to a linear rule where taught. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
33. Equation to graph
Modelling purpose. This component helps the learner use a rule to generate points and interpret the resulting shape. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
34. Intercept meaning
Modelling purpose. This component helps the learner connect a fixed starting value to the context when meaningful. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
35. Gradient meaning
Modelling purpose. This component helps the learner connect rate of change to units and direction. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
36. Positive gradient
Modelling purpose. This component helps the learner interpret increasing output under a linear model. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
37. Negative gradient
Modelling purpose. This component helps the learner interpret decreasing output under a linear model. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
38. Zero gradient
Modelling purpose. This component helps the learner interpret constant output across changing input. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
39. Domain
Modelling purpose. This component helps the learner identify which input values are mathematically or contextually sensible. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
40. Range
Modelling purpose. This component helps the learner interpret the output values produced within the stated model. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
41. Linear model conditions
Modelling purpose. This component helps the learner recognise constant-rate structure rather than assume linearity. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
42. Non-linear awareness
Modelling purpose. This component helps the learner notice when constant-rate assumptions fail. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
43. Direct proportion
Modelling purpose. This component helps the learner use constant ratio as the organising relationship. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
44. Inverse proportion
Modelling purpose. This component helps the learner use constant product where that model is taught. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
45. Percentage multipliers
Modelling purpose. This component helps the learner represent repeated percentage change algebraically. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
46. Ratio tables
Modelling purpose. This component helps the learner preserve scale factors across paired quantities. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
47. Rate models
Modelling purpose. This component helps the learner keep compound units visible in relationships. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
48. Speed models
Modelling purpose. This component helps the learner use total distance and total time for average speed. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
49. Geometry to algebra
Modelling purpose. This component helps the learner convert angle or length properties into equations. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
50. Pythagorean model
Modelling purpose. This component helps the learner relate side lengths only under the right-triangle condition. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
51. Trigonometric model
Modelling purpose. This component helps the learner relate angle and side ratios after diagram labelling. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
52. Scale models
Modelling purpose. This component helps the learner connect corresponding lengths through a common factor. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
53. Area scaling
Modelling purpose. This component helps the learner distinguish area factors from linear scale factors where taught. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
54. Volume scaling
Modelling purpose. This component helps the learner distinguish cubic effects where taught. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
55. Statistics models
Modelling purpose. This component helps the learner treat summaries as compressed descriptions, not complete data. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
56. Mean model
Modelling purpose. This component helps the learner connect mean, total and count. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
57. Probability model
Modelling purpose. This component helps the learner define sample space and event before forming a probability. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
58. Sequential probability
Modelling purpose. This component helps the learner use trees or tables where stages affect later outcomes. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
59. Unit consistency
Modelling purpose. This component helps the learner align quantities before combining or comparing them. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
60. Dimensional checking
Modelling purpose. This component helps the learner use units to test whether a model output is plausible. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
61. Parameter interpretation
Modelling purpose. This component helps the learner explain what coefficients and constants mean. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
62. Parameter change
Modelling purpose. This component helps the learner predict how the graph or output changes when a parameter changes. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
63. What-if analysis
Modelling purpose. This component helps the learner change one assumption or input and predict before calculating. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
64. Reverse modelling
Modelling purpose. This component helps the learner infer a parameter from input-output evidence. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
65. Model validation
Modelling purpose. This component helps the learner compare predictions with data or given constraints. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
66. Model revision
Modelling purpose. This component helps the learner update the relationship when new evidence contradicts it. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
67. Model comparison
Modelling purpose. This component helps the learner compare simplicity, fit, interpretability and assumptions. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
68. Counterexample
Modelling purpose. This component helps the learner construct a case where an overgeneralised model fails. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
69. Calculator-supported modelling
Modelling purpose. This component helps the learner write the model and estimate before using technology. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
70. Error tracing
Modelling purpose. This component helps the learner locate whether failure arose in assumptions, representation, algebra or arithmetic. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
71. Communication
Modelling purpose. This component helps the learner present variables, relationship, assumptions and interpretation clearly. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
72. Independent modelling
Modelling purpose. This component helps the learner build a defensible model without a supplied template. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
73. Delayed modelling transfer
Modelling purpose. This component helps the learner rebuild the same structure in a new context after time. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
74. Mixed-representation transfer
Modelling purpose. This component helps the learner switch representations deliberately inside one problem. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
75. Examination modelling
Modelling purpose. This component helps the learner use modelling habits to enter unfamiliar mixed questions. A mathematical model is useful because it makes a relationship inspectable: quantities are named, assumptions are visible, and algebra, tables, graphs or diagrams can be checked against one another.
Modelling Practice Bank
The learner is ready when they can define variables, select and translate representations, state important assumptions, carry out the mathematics, check the result and explain what the model does and does not claim. The representation should support judgement, not replace it.
That is the distinct owner role of this page: connect algebra, graphs and mathematical modelling so unfamiliar questions can be entered through relationships rather than template matching.
Worked Modelling Casebook
These original cases add concrete demonstrations to the modelling system. Each case begins with a relationship, makes its assumptions visible and then changes one feature so the learner has to interpret rather than copy.
Case 1 · Fixed charge plus usage
A teaching model charges a fixed amount of 8 units plus 3 units for each unit of usage. Let u be usage and C be total cost, so C=8+3u. The constant 8 is the intercept; the coefficient 3 is the rate. A table for u=0,1,2 gives C=8,11,14, and the graph is linear. The model is only valid under the stated assumption that the rate remains constant.
Change the fixed charge to 5 and predict the graph before recalculating. The line should shift downward without changing its gradient. This separates parameter meaning from arithmetic.
Case 2 · Direct proportion
A quantity y is directly proportional to x and y=18 when x=6. The constant of proportionality is 3, so y=3x. The model passes through the origin because zero input gives zero output under the stated relationship.
Now add a fixed term: y=3x+4. The gradient remains 3, but the relationship is no longer direct proportion. This contrast helps the learner distinguish constant rate from direct proportionality.
Case 3 · Inverse proportion
Under an idealised inverse model, xy=48. If x=6, y=8; if x=12, y=4. Doubling x halves y because the product is fixed. The model is useful only when the situation genuinely supports inverse behaviour.
Ask what happens if x becomes 3. Then y=16. A table makes the constant product visible and prevents the student from treating inverse proportion as a memorised ‘flip’ rule.
Case 4 · Percentage model
A quantity begins at 200, increases by 10% and then decreases by 5%. The multiplier model gives 200×1.10×0.95=209. The overall factor is 1.045, or a 4.5% increase.
A student who adds 10% and subtracts 5% directly would predict a 5% increase and obtain 210. The one-unit difference reveals why changing reference values matter in repeated percentage models.
Case 5 · Geometry model
A rectangle has width x and length x+4. If its perimeter is 40, then 2[x+(x+4)]=40, giving 4x+8=40 and x=8. The length is 12 and the area is 96 square units.
The equation is not the geometry itself; it is a representation of the perimeter condition. Change the perimeter to 52 and rebuild the equation rather than modifying the previous answer mechanically.
Case 6 · Right-triangle model
A right triangle has legs 5 and 12, so the hypotenuse is 13 by Pythagoras. The theorem models a relationship that is valid because the triangle is right-angled. The condition is as important as the formula.
Rotate the triangle or place it inside a larger diagram. The side labels change position on the page, but the hypotenuse remains opposite the right angle. The model survives orientation.
Case 7 · Trigonometric model
Where trigonometry is in the student’s current course, a right triangle with hypotenuse 10 and angle 30° has opposite side 10 sin30°=5. The model connects an angle to a ratio of side lengths.
Use a different reference angle and ask the learner to relabel opposite and adjacent before choosing the ratio. This prevents SOHCAHTOA from becoming a purely verbal mnemonic detached from the diagram.
Case 8 · Linear graph from data
Suppose a table contains points (0,2), (1,5), (2,8), (3,11). The output increases by 3 for every unit increase in input, giving the rule y=3x+2. The graph provides a visual representation of the same constant-rate relationship.
Ask the learner to predict y at x=10 without extending the table row by row. The equation gives 32. Translation between table and equation creates efficiency.
Case 9 · Model validation
A proposed model y=2x+1 is tested against data points (1,3), (2,5) and (3,8). The first two fit, but the third does not because the model predicts 7. The learner should not ignore the mismatch simply because the first examples worked.
This creates a decision: perhaps the third data point contains error, perhaps the relationship is not perfectly linear, or perhaps a different model is needed. Modelling requires judgement about evidence.
Case 10 · Mean as a model of centre
The data 4,5,5,6,30 have mean 10. The mean compresses the dataset into one number but does not communicate the extreme value clearly. The median is 5, showing a very different centre.
Ask what conclusion each measure supports and what it hides. This is modelling in a statistical sense: summary measures simplify reality and therefore have limitations.
Case 11 · Probability model
A bag contains 3 red and 2 blue counters, and each counter is equally likely to be selected. The model gives P(red)=3/5. If one red counter is removed without replacement, the state changes and the next red probability becomes 2/4=1/2.
The change is not a new formula. It follows from rebuilding the sample space after the first event. Modelling remains tied to the process.
Case 12 · Examination transfer
A mixed question describes a fixed starting value plus a constant increase per unit, but never says ‘linear’. The learner defines variables, writes a table for two or three values, recognises constant change and builds a linear equation.
The examination skill is not memorising a template. It is recognising that words, table and equation are representations of one relationship. The model gives the learner a route into unfamiliar wording.
Casebook handoff
After studying a case, close it and rebuild the same structure with a different context or representation. The learner should be able to define variables, explain the role of each parameter, identify assumptions and test the model without relying on the original wording.
Final Modelling Transfer Studio
These original transfer cases deepen the page without changing its owner role. Each case asks the learner to move between representations, assumptions and interpretation rather than merely manipulate symbols.
Transfer Studio 1 · Same rule, three representations
Take the relationship y=2x+5. Build a table for x=0,1,2,3, then plot the four points, then write one sentence describing the change. The equation, table and graph are not three separate topics; they are three representations of one relationship. Ask which form makes prediction easiest and which makes visual change easiest to see.
Transfer Studio 2 · Model from two observations
A teaching scenario records paired values (1,7) and (4,16) and assumes a linear relationship. The change in y is 9 while the change in x is 3, so the gradient is 3. Substituting (1,7) into y=3x+c gives c=4, hence y=3x+4. Test the second point to validate the rule before using it for prediction.
Transfer Studio 3 · Change the assumption
Suppose a model assumes constant speed of 60 km/h. Under that assumption, distance d after t hours is d=60t. If the journey includes stops or changing speed, that simple rule no longer describes the whole trip. The learner should identify which assumption failed rather than treating the equation as universally true.
Transfer Studio 4 · Reverse modelling
A line has rule y=4x+c and passes through (3,17). Substitute to obtain 17=12+c, so c=5. The unknown is a model parameter rather than an output. This reverse direction matters because examinations may ask students to infer a rule from evidence rather than only evaluate a known rule.
Transfer Studio 5 · Model comparison
Two candidate rules are y=2x+1 and y=2.5x for data points (2,5), (4,9), (6,13). The first rule fits all three exactly: 5,9,13. The second gives 5,10,15. The comparison should be based on evidence and purpose, not on which formula looks simpler.
Transfer Studio 6 · Geometry as a model
A rectangle has perimeter 50 and length three units more than width. Let width be w and length w+3. Then 2[w+(w+3)]=50, so 4w+6=50 and w=11. The model translates a geometric condition into algebra. A different perimeter requires rebuilding the equation, not adjusting the old answer.
Transfer Studio 7 · Statistical model limits
A mean can summarise a dataset but can hide extreme values. Compare 4,5,5,6,30 with 8,9,10,11,12; both have mean 10. A model based only on mean would treat them as similar even though their spread is very different. The learner should state what the summary captures and what it omits.
Transfer Studio 8 · Probability as a state model
A bag begins with 4 red and 6 blue counters. Under equal selection, P(red)=4/10. If a red counter is removed without replacement, the state becomes 3 red and 6 blue, so the next red probability is 3/9. The model changes because the system state changed.
Transfer Studio 9 · Unit-aware modelling
A rate model uses kilometres and hours. If one part of a problem gives metres and seconds, convert units before combining values. Dimensional consistency is not decorative notation; it is evidence that the model is combining comparable quantities. A numerical answer with incompatible units is a sign that the representation may be wrong.
Transfer Studio 10 · Independent model construction
Give a fresh context involving a fixed start plus a constant per-unit increase. The learner defines variables, states assumptions, creates two or three test values, writes an equation, sketches the graph and explains the meaning of the constant and coefficient. The final check is whether a new reader can understand the model without the tutor’s explanation.
Final modelling standard
The learner is ready when words, algebra, tables, graphs and diagrams feel like interchangeable tools for representing relationships. They can state assumptions, test a model, interpret parameters and revise the model when the evidence changes.
Final Modelling Validation Workshop
This final workshop checks whether algebra, tables, graphs and modelling have become one connected system. Give the learner a fresh relationship without naming the preferred representation. The learner should define variables and units, state the important assumption, choose a useful form, calculate at least two test values and explain what the model predicts.
Workshop A · Build, test and revise
A hypothetical process begins with a fixed value of 6 and increases by 4 for each unit of input. Let x be the input and y the output. The linear model is y = 4x + 6. A table for x = 0, 1, 2 and 3 gives y = 6, 10, 14 and 18. Plotting those points gives the same relationship visually. The coefficient 4 represents the constant increase per input unit; the constant 6 represents the modelled output when the input is zero.
Now introduce a new observation: when x = 4, the observed output is 23 rather than the modelled 22. The learner should not force the old model to fit by ignoring the mismatch. They should state that the simple exact linear rule no longer matches every observation. Depending on the purpose, the observation could represent measurement variation, an error, or evidence that a more complicated relationship is needed. Modelling includes deciding when a representation is no longer adequate.
Workshop B · Translate in reverse
Provide only the points (0,3), (2,9), (4,15) and (6,21). The output rises by 6 whenever the input rises by 2, so the rate is 3 per input unit. Since the output is 3 when x = 0, the rule is y = 3x + 3. Ask the learner to reconstruct the equation, then explain the constant and coefficient in words. Reverse translation from data to algebra is stronger evidence than substituting into a rule already provided.
Workshop C · State the boundary of the model
Ask whether the same rule should be trusted for every possible input. A classroom model may be intended only for non-negative inputs, a geometric model may require positive lengths, and a rate model may assume a constant process that is valid only over a stated interval. The learner should distinguish mathematical manipulation from contextual validity.
The final standard is independence: choose a representation for a new relationship, justify the choice, validate it against evidence and revise it when the assumptions or observations change. When those decisions can be made after earlier examples are hidden, the modelling system has transferred.
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