Mr. Ward's Teaching Materials

024i - Substituting into Formulas

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Lesson 24 — Substituting Values into Formulas

Strand: Algebra | Descriptor: AC9M7A01 | Duration: 45 minutes

Learning Intentions

  • To substitute given values into a formula to determine an unknown.
  • To apply the order of operations correctly during substitution.

Success Criteria

I can:

  1. Replace each variable with its given value, using brackets where needed.
  2. Evaluate the result using the correct order of operations.
  3. Substitute negative values without sign errors.
  4. Set out substitution working clearly, one step per line.

Warmup

(6 minutes — order of operations retrieval, mini whiteboards)

Evaluate:

(Answers: ; ; ; ; ; .)

Bridging question: Which two pairs give different answers, and why? What does that tell you about how carefully we must substitute?

Activities

Activity 1 — Explicit Instruction: the Substitution Protocol (12 min)

The three-step protocol, modelled every time:

  1. Write the formula.
  2. Replace each letter with its value, in brackets.
  3. Evaluate using the order of operations, one step per line.

I do. Given , find when and .

Why brackets matter. Given , substituting without brackets tempts students to write . With brackets, is unambiguous. Brackets become essential with negatives, so build the habit now.

I do (a second, harder example). Given , find when .

Emphasise: multiply before adding. A student who computes then gets — wrong.

We do: Given , find when , . Given , find when , , . Given , find when , .

You do: Ten substitutions of increasing difficulty, including at least three where the order of operations is decisive.

Activity 2 — Substituting Negative Values (10 min)

This is where most errors occur. Slow, explicit modelling.

I do. Given , find when .

I do (a squared term). Given , find when .

Critical contrast — display side by side:

This is exactly why substituted values go in brackets.

We do: Given , find when . Given with , . Given when .

You do:

  1. when
  2. when
  3. when
  4. when
  5. when ,

(Answers: ; ; ; ; .)

Activity 3 — Inquiry: the Temperature Converter (8 min)

Pairs, calculators permitted.

The formula converting Celsius to Fahrenheit is

  1. Convert C, C and C to Fahrenheit.
  2. Convert C.
  3. Is there a temperature at which the two scales read the same number? Investigate.

Socratic scaffolding for Q3:

PromptPurpose
Understand: what would “the same” mean? and take the same numerical value.
Try some values. What happens?At , — is higher. At , — now higher.
Which direction should you try?The gap is growing as rises, so try going down instead.
Try .. The gap is now only . Getting closer.
Try .. They match!
Verify ✓
Looking back is the unique crossing point of the two scales — a genuinely useful fact.

Answers: 1. F, F, F; 2. F; 3. .

Checks for Understanding

(6 minutes — exit ticket, collected)

  1. Given , find when (a) (b) .
  2. Given , find when and .
  3. Given , find when .
  4. Reasoning. A student substitutes into and writes . Explain the error.
  5. Given , find the cost of a -hour job.

Answers: 1. (a) (b) ; 2. ; 3. ; 4. The student computed rather than . Squaring a negative gives a positive, so ; 5. 350$.

Common Misconceptions

MisconceptionHow to pre-empt it
.Display the side-by-side contrast in Activity 2. Require brackets around every substituted value.
Adding before multiplying, e.g. with giving .Require one operation per line. Never allow a single-line answer during this lesson.
Losing a minus sign, e.g. giving .Model the intermediate line explicitly, showing the sign change.
Substituting into the wrong variable when several appear.Have students write the value above each letter in the formula before rewriting.
Treating as “twenty-something” when is a digit.Reinforce that juxtaposition means multiplication, always.
Forgetting units in a contextual answer.Units belong with the final answer, not inside the algebra.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). If , what is when ?

Answer

E2 (AMC Junior style). Given , a rectangle has perimeter cm and length cm. Find its width.

Answer

E3 (Challenge). The formula gives the distance fallen (in metres) after seconds, where is the initial speed in m/s. Find when and .

Answer

E4 (Challenge). Given , find when and . What happens if ?

Answer

If the denominator is , and division by zero is undefined — the formula gives no value.

E5 (Challenge). If , evaluate . Then find a value of making the expression equal to .

Answer

At : . Testing small values, gives ✓

Homework

  1. Given , find when (a) (b) (c) (d) .
  2. Given , find when (a) , (b) , (c) , .
  3. Given , find when (a) , (b) , .
  4. Given , find when (a) (b) (c) (d) .
  5. Given (the cost of a job in dollars), find the cost of (a) a -hour job (b) a -hour job.
  6. Given , find when , , .
  7. Given , convert (a) C (b) C.
  8. Reasoning. Explain why and give different answers.
  9. Challenge. Given , find the value of that makes . Show your reasoning.

Answers: Q1 — (a) (b) (c) (d) . Q2 — (a) (b) (c) . Q3 — (a) (b) . Q4 — (a) (b) (c) (d) . Q5 — (a) 160$33560104^\circ23^\circ(-3)^2 = (-3)\times(-3) = 9-3^23-92x + 5 = 172x = 12x = 6$.

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