035i - Rearranging Expressions and Formulas

Lesson 35 — Rearranging Linear Expressions and Formulas

Strand: Algebra | Descriptor: AC9M8A01 | Duration: 45 minutes

Learning Intentions

• To rearrange a formula to make a different variable the subject.
• To apply the inverse property to “undo” operations in the correct order.

Success Criteria

I can:

1. Identify the subject of a formula.
2. Apply an inverse operation to both sides of a formula to isolate a variable.
3. Rearrange a two-step formula to make a chosen variable the subject.
4. Rearrange a formula involving a bracket or a fraction.
5. Verify a rearrangement by substituting numbers into both the original and rearranged formulas.

Warmup

(6 minutes — retrieval, mini whiteboards)

Find the additive or multiplicative inverse used to solve each equation, and solve.

Answers: (subtract ); (add ); (divide by ); (multiply by ).

Bridging question: Every one of these used an inverse operation to “undo” what was done to . Formulas work exactly the same way — except the “answer” is another variable, not a number.

Activities

Activity 1 — Explicit Instruction: Making a Variable the Subject (16 min)

Definition. The subject of a formula is the variable by itself on one side, e.g. in , the subject is .

The protocol — identical to solving an equation:

1. Identify what is done to the variable you want as the new subject.
2. Apply the inverse operation to both sides, undoing operations in reverse order (last done, first undone).
3. The rearranged formula is complete once the target variable stands alone.

I do — one-step rearrangement. Make the subject of (area of a rectangle).

I do — two-step rearrangement. Make the subject of (velocity formula).

Check with numbers. If : original gives . Substitute back into : ✓

I do — subject appears inside a bracket. Make the subject of (perimeter of a rectangle).

We do: Make the subject of ; make the subject of .

You do: Rearrange to make the bracketed variable the subject.

1.   (make the subject)
2.   (make the subject)
3.   (make the subject)
4.   (make the subject)
5.   (make the subject)
6.   (make the subject)

(Answers: ; ; ; ; ; .)

Activity 2 — Independent Practice: Harder Rearrangements (11 min)

You do: Rearrange each formula to make the bracketed variable the subject. Show every inverse step.

1.   (make the subject)
2.   (make the subject)
3. — skip; too advanced. Replace with:   (make the subject)
4.   (make the subject)
5.   (make the subject; treat as a single quantity)

(Answers: ; ; ; ; .)

Activity 3 — Inquiry: Which Formula is Easier to Rearrange? (9 min)

Pairs, then whole-class share.

A phone plan’s monthly cost is , where is the number of data GB used.

A second plan is written as .

1. Show that both formulas are equivalent by expanding the second.
2. Rearrange each formula to make the subject.
3. Which version was easier to rearrange, and why?
4. Use your rearranged formula to find how much data was used if the monthly bill was 45$.

Socratic scaffolding for Q3–4:

Prompt	Purpose
Understand the problem.	Two equivalent formulas may not be equally convenient to rearrange.
Devise a plan for Q3.	Count the inverse steps needed for each version, and compare.
Carry out the plan.	Version 1 () needs two inverse steps: subtract , then divide by . Version 2 () needs two different steps: divide by , then subtract — same number of steps, different order.
Looking back — is one “easier”?	Both take two steps; the factorised version avoids a decimal division until the very end, which some students find cleaner.
Carry out Q4.	Using with : GB.
Looking back — check.	Substitute into the original: ✓

Answers: 1. ✓; 2. from version 1: ; from version 2: (both equivalent); 3. either is reasonable — discuss as a class; 4. GB.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Make the subject of .
2. Make the subject of .
3. Make the subject of .
4. Reasoning. A student rearranges to make the subject and gets . Verify this is correct using .
5. Make the subject of .

Answers: 1. ; 2. ; 3. ; 4. Original: ; check in : ✓; 5. .

Common Misconceptions

Misconception	How to pre-empt it
Applying the inverse operation to only one side of the formula.	Insist “whatever you do, do to both sides” — physically draw a line down the middle of the equals sign.
Undoing operations in the wrong order (e.g. dividing before subtracting in ).	Use the “last done, first undone” rule — identify the order of operations on the original subject first.
Forgetting brackets are needed when a subtracted or divided term contains more than one term.	Model carefully — show why (an equally valid alternative form) also works, but does not.
Treating a rearranged formula as a completely different, unrelated formula.	Always verify with the same numbers substituted into both the original and rearranged versions.
Believing rearranging changes the relationship between the variables.	Emphasise: rearranging is just applying inverse operations — the formula still describes the exact same relationship.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Make the subject of .

Answer

E2 (AMC Junior style). The formula for the perimeter of a rectangle is . If and , use a rearranged formula to find .

Answer

E3 (Challenge). Make the subject of .

Answer

E4 (Challenge). The formula converts Celsius to Fahrenheit. Rearrange it to make the subject, then find when .

Answer

At : .

E5 (Investigation). Two students rearrange to make the subject. One gets ; the other gets . Are both correct?

Answer

Yes — both are algebraically identical: . This uses the distributive property on the fraction.

Homework

1. Make the bracketed variable the subject: (a) (d) (b) (x) (c) (u) (d) (l).
2. Rearrange to make the subject.
3. Rearrange to make the subject, then find when and .
4. Rearrange to make the subject, then verify using (find , then use your formula to recover ).
5. The formula for simple interest is . Rearrange to make the subject.
6. Reasoning. Explain why the “last done, first undone” rule matters, using as your example.
7. Reasoning. A student says rearranging a formula changes what it means. Explain why this is false, using a numerical check.
8. Challenge. Make the subject of .
9. Challenge. The formula converts Celsius to Fahrenheit. Rearrange to make the subject, then find the Celsius temperature equivalent to . What do you notice about your answer?

Answers: Q1 — (a) (b) (c) (d) . Q2 — . Q3 — ; at : . Q4 — ; at : ; check ✓. Q5 — . Q6 — in , is multiplied after and is added; to undo, you must subtract first (undoing the last-applied addition) before dividing by — reversing the order would mix operations incorrectly. Q7 — e.g. with gives ; rearranged ✓ — the same relationship, just solved for a different letter. Q8 — . Q9 — ; at : — the freezing point of water is C, matching F exactly.