Year 7 Mathematics · AC9M7A06

Systematic Variation in Multi-variable Formulas

We are learning to manipulate formulas involving several variables using digital tools, and describe the effect of systematic variation in the values of the variables

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Learning goalsSay it simply

We are learning to manipulate formulas involving several variables using digital tools, and describe the effect of systematic variation in the values of the variables.

A formula with several variables describes how an output depends on multiple inputs. A digital table or spreadsheet makes systematic variation visible.

To attribute an effect, hold all but one input constant; when several inputs change, combine their scale factors and expose the spreadsheet formula used.

Use controlled tables to test proportionality and sensitivity in area, volume, distance and cost models. Verify results by recalculating one row independently.

Success criteria

  • I can substitute several variables into a formula accurately.
  • I can vary one input systematically while controlling the others.
  • I can interpret and verify a digital table or spreadsheet formula.
Key conceptTeach from the board

Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.

Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.

Example 1

Vary one formula input

  1. Use A=lw while holding the controlled variable w=5.
  2. Choose l=2,4,6,8.
  3. Calculate A=10,20,30,40.
  4. Describe the systematic variation: doubling l doubles A while w remains 5.

Final answer: The areas are 10, 20, 30 and 40 square units for l=2,4,6,8.

Check: A/l=5 in every row, matching the displayed variation table.

Example 2

Compare systematic spreadsheet changes

  1. Enter l=3,6,9 and w=2,4,6.
  2. Use spreadsheet formula =A2*B2.
  3. Calculate areas 6,24,54.
  4. Explain the combined scale factors.

Final answer: Areas are 6, 24 and 54 square units.

Check: Doubling both dimensions from row 1 to 2 quadruples 6 to 24.

Example 3

Application problem 1

Problem: For A=lw, keep w=4 cm and let l take values 2,4,6,8 cm. Create the value table, describe the pattern, then predict what happens if both l and w are doubled.

  1. Plan: Represent the information first, then calculate, interpret and independently check the result.
  2. Work: Areas are 8,16,24,32 cm², so A increases by 8 cm² whenever l increases by 2 cm. Doubling both l and w multiplies area by 4.
  3. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Final answer: Areas are 8,16,24,32 cm², so A increases by 8 cm² whenever l increases by 2 cm. Doubling both l and w multiplies area by 4.

Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Application problem 2

Problem: A spreadsheet uses V=lwh. Starting from l=2, w=3 and h=4, calculate the volume when only l doubles, when only w triples, and when l doubles, w triples and h halves. Describe each scale factor.

  1. Plan: Calculate the starting row, change the named inputs only, then compare each output with 24.
  2. Work: The starting volume is 2×3×4=24 cubic units; doubling l gives 48, tripling w gives 72, and changing l by 2, w by 3 and h by ½ gives 72, so the respective volume factors are 2, 3 and 3.
  3. Interpret: The complete model for “A spreadsheet uses V=lwh. Starting from l=2, w=3 and h=4, calculate the volume when only l doubles, when only w triples, and when l doubles, w triples and h halves. Describe each scale factor.” shows The starting volume is 2×3×4=24 cubic units; doubling l gives 48, tripling w gives 72, and changing l by 2, w by 3 and h by ½ gives 72, so the respective volume factors are 2, 3 and 3.

Final answer: The starting volume is 2×3×4=24 cubic units; doubling l gives 48, tripling w gives 72, and changing l by 2, w by 3 and h by ½ gives 72, so the respective volume factors are 2, 3 and 3.

Check: The complete model for “A spreadsheet uses V=lwh. Starting from l=2, w=3 and h=4, calculate the volume when only l doubles, when only w triples, and when l doubles, w triples and h halves. Describe each scale factor.” shows The starting volume is 2×3×4=24 cubic units; doubling l gives 48, tripling w gives 72, and changing l by 2, w by 3 and h by ½ gives 72, so the respective volume factors are 2, 3 and 3.

Clean visual examplesOne-page board

Clean one-page examples

AC9M7A06 - Systematic Variation in Multi-variable Formulas
Example 1

Example 1 Vary one formula input Use A=lw while holding the controlled variable w=5. Choose l=2,4,6,8. Calculate A=10,20,30,40. Describe the systematic variation: doubling l doubles A while w remains 5. Final answer: The areas are 10, 20, 30 and 40 square units for l=2,4,6,8. Check: A/l=5 in every row, matching the displayed variation table.

Example 2

Example 2 Compare systematic spreadsheet changes Enter l=3,6,9 and w=2,4,6. Use spreadsheet formula =A2*B2. Calculate areas 6,24,54. Explain the combined scale factors. Final answer: Areas are 6, 24 and 54 square units. Check: Doubling both dimensions from row 1 to 2 quadruples 6 to 24.

Example 3

Example 3 Application problem 1 Problem: For A=lw, keep w=4 cm and let l take values 2,4,6,8 cm. Create the value table, describe the pattern, then predict what happens if both l and w are doubled. Plan: Represent the information first, then calculate, interpret and independently check the result. Work: Areas are 8,16,24,32 cm², so A increases by 8 cm² whenever l increases by 2 cm. Doubling both l and w multiplies area by 4. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills. Final answer: Areas are 8,16,24,32 cm², so A increases by 8 cm² whenever l increases by 2 cm. Doubling both l and w multiplies area by 4. Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Example 4 Application problem 2 Problem: A spreadsheet uses V=lwh. Starting from l=2, w=3 and h=4, calculate the volume when only l doubles, when only w triples, and when l doubles, w triples and h halves. Describe each scale factor. Plan: Calculate the starting row, change the named inputs only, then compare each output with 24. Work: The starting volume is 2×3×4=24 cubic units; doubling l gives 48, tripling w gives 72, and changing l by 2, w by 3 and h by ½ gives 72, so the respective volume factors are 2, 3 and 3. Interpret: The complete model for “A spreadsheet uses V=lwh. Starting from l=2, w=3 and h=4, calculate the volume when only l doubles, when only w triples, and when l doubles, w triples and h halves. Describe each scale factor.” shows The starting volume is 2×3×4=24 cubic units; doubling l gives 48, tripling w gives 72, and changing l by 2, w by 3 and h by ½ gives 72, so the respective volume factors are 2, 3 and 3. Final answer: The starting volume is 2×3×4=24 cubic units; doubling l gives 48, tripling w gives 72, and changing l by 2, w by 3 and h by ½ gives 72, so the respective volume factors are 2, 3 and 3. Check: The complete model for “A spreadsheet uses V=lwh. Starting from l=2, w=3 and h=4, calculate the volume when only l doubles, when only w triples, and when l doubles, w triples and h halves. Describe each scale factor.” shows The starting volume is 2×3×4=24 cubic units; doubling l gives 48, tripling w gives 72, and changing l by 2, w by 3 and h by ½ gives 72, so the respective volume factors are 2, 3 and 3.

Curriculum examplesCopied content

Content description: manipulate formulas involving several variables using digital tools, and describe the effect of systematic variation in the values of the variables.

Questions and answersWith answers

Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.

  1. 1. For A=lw, find A when l=7 and w=3.

    Check answer

    Answer: 21 square units.

    Hint: Substitute both values and multiply.

    Why: Using the method “Substitute both values and multiply.” gives 21 square units.

  2. 2. In d=rt, what happens to d when t doubles and r stays fixed?

    Check answer

    Answer: d doubles.

    Hint: Hold r constant and compare scale factors.

    Why: Using the method “Hold r constant and compare scale factors.” gives d doubles.

  3. 3. Write a spreadsheet formula multiplying cells B2 and C2.

    Check answer

    Answer: =B2*C2.

    Hint: Begin a spreadsheet formula with =.

    Why: Using the method “Begin a spreadsheet formula with =.” gives =B2*C2.

  4. 4. For P=2l+2w with w=5, compare l=4 and l=8.

    Check answer

    Answer: P changes from 18 to 26.

    Hint: Calculate both cases rather than assuming doubling.

    Why: Using the method “Calculate both cases rather than assuming doubling.” gives P changes from 18 to 26.

  5. 5. For V=lwh, double h only. What happens to V?

    Check answer

    Answer: V doubles.

    Hint: Hold l and w fixed.

    Why: Using the method “Hold l and w fixed.” gives V doubles.

  6. 6. A student changes l and w together and credits l alone for the area change. Evaluate.

    Check answer

    Answer: The claim is invalid because both l and w changed, so their effects are confounded; vary l while holding w fixed to isolate the effect of l.

    Hint: Change only one input to attribute an effect.

    Why: A controlled variable is held fixed so systematic variation can isolate the effect of the input being changed.

  7. 7. For C=3n+10, list C for n=0,5,10.

    Check answer

    Answer: 10, 25, 40.

    Hint: Apply the formula to each input.

    Why: Using the method “Apply the formula to each input.” gives 10, 25, 40.

  8. 8. In V=lwh, l doubles, w triples and h halves. Find the volume factor.

    Check answer

    Answer: The volume scale factor is 2×3×½=3, so the new volume is three times the original even though one dimension was halved.

    Hint: Multiply 2×3×1/2.

    Why: The complete model for “In V=lwh, l doubles, w triples and h halves. Find the volume factor.” shows The volume scale factor is 2×3×½=3, so the new volume is three times the original even though one dimension was halved.

  9. 9. A spreadsheet uses V=lwh. Starting from l=2, w=3 and h=4, calculate the volume when only l doubles, when only w triples, and when l doubles, w triples and h halves. Describe each scale factor.

    Check answer

    Answer: The starting volume is 2×3×4=24 cubic units; doubling l gives 48, tripling w gives 72, and changing l by 2, w by 3 and h by ½ gives 72, so the respective volume factors are 2, 3 and 3.

    Hint: Calculate the starting row, change the named inputs only, then compare each output with 24.

    Why: The complete model for “A spreadsheet uses V=lwh. Starting from l=2, w=3 and h=4, calculate the volume when only l doubles, when only w triples, and when l doubles, w triples and h halves. Describe each scale factor.” shows The starting volume is 2×3×4=24 cubic units; doubling l gives 48, tripling w gives 72, and changing l by 2, w by 3 and h by ½ gives 72, so the respective volume factors are 2, 3 and 3.

  10. 10. For A=lw, keep w=4 cm and let l take values 2,4,6,8 cm. Create the value table, describe the pattern, then predict what happens if both l and w are doubled.

    Check answer

    Answer: Areas are 8,16,24,32 cm², so A increases by 8 cm² whenever l increases by 2 cm. Doubling both l and w multiplies area by 4.

    Hint: Represent the information first, then calculate, interpret and independently check the result.

    Why: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Practice and reviewReady for practice

Common mistake: Several inputs varied together without plan.

Correction: Isolate effects before combining them.

Common mistake: Spreadsheet values accepted without formula audit.

Correction: Check references and units.

Common mistake: Additive language used for multiplicative effect.

Correction: Say doubles, triples or scales by a factor.

Curriculum alignmentStart here

We are learning to manipulate formulas involving several variables using digital tools, and describe the effect of systematic variation in the values of the variables.

A formula with several variables describes how an output depends on multiple inputs. A digital table or spreadsheet makes systematic variation visible.

To attribute an effect, hold all but one input constant; when several inputs change, combine their scale factors and expose the spreadsheet formula used.

Use controlled tables to test proportionality and sensitivity in area, volume, distance and cost models. Verify results by recalculating one row independently.

Success criteria

  • I can substitute several variables into a formula accurately.
  • I can vary one input systematically while controlling the others.
  • I can interpret and verify a digital table or spreadsheet formula.
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