017i - Mixed Operations with Integers and Rational Numbers

Lesson 17 — Mixed Operations with Integers and Rational Numbers

Strand: Number | Descriptor: AC9M8N04 | Duration: 45 minutes

Learning Intentions

• To apply the order of operations correctly to expressions combining all four operations with integers, fractions and decimals.
• To combine the sign rules (Lesson 14) and rational number strategies (Lessons 15–16) fluently within a single multi-step expression.

Success Criteria

I can:

1. State the order of operations (brackets, indices, multiplication/division left to right, addition/subtraction left to right).
2. Evaluate a mixed expression involving integers, fractions and decimals, applying the correct order.
3. Use brackets correctly to preserve the sign of a negative number within a larger expression.
4. Identify and correct an order-of-operations error in someone else’s working.

Warmup

(5 minutes — spot the error, pairs)

Each of these has been solved incorrectly. Find the error and correct it.

1. (worked as )
2. (worked as )
3. (squared the sign along with the digit)

Discussion: Reinforce that multiplication and division must happen before addition and subtraction unless brackets say otherwise, and that means “the negative of ”, i.e. , while . This distinction resurfaces throughout today’s lesson.

Activities

Activity 1 — Explicit Instruction: Order of Operations with Negatives (10 min)

I do / We do / You do.

I do: Work through a mixed expression step by step, narrating each decision.

We do: Evaluate together, narrating the order used: , , .

You do: Evaluate: , , .

Activity 2 — Guided Practice: Mixed Expressions with Fractions and Decimals (11 min)

I do: Combine strategies from Lessons 15 and 16 within order of operations.

We do: Evaluate together: , , .

You do: Evaluate: , , .

Activity 3 — Applied Problem-solving: Multi-step Contexts (13 min)

Pairs. Write the full expression before evaluating.

Problem 1. A share price starts at 4.50$0.753$1.20$. Write a single expression for the final price, and evaluate it.

Problem 2. A submarine at m rises at a rate of m per minute for minutes, then a fault causes it to sink m instantly. Write a single expression for its final depth, and evaluate it.

Problem 3. A recipe requires cup of flour per batch. A baker makes fewer batches than planned (i.e. reduces the plan by 2 batches) from an original plan of batches, then doubles the reduced plan. Write a single expression for the total cups of flour needed, and evaluate it.

Socratic scaffolding for Problem 3:

Prompt	Purpose
Understand: what is the sequence of changes?	Start at 6 batches, reduce by 2, then double the result.
Devise a plan: what should happen first inside the expression?	The reduction () must be grouped in brackets so it happens before the doubling.
Write the expression for the number of batches	.
Carry out that step	batches.
Now bring in the flour rate	Multiply the batch count by cup per batch.
Carry it out	cups.
Looking back — what would go wrong without the brackets?	batches — a completely different (and wrong) answer, since multiplication would happen before subtraction.

Answers: 1 — , so 3.45-15 + 2.5 \times 4 - 3 = -15+10-3=-8-8(6-2)\times2\times\frac34 = 8 \times \frac34 = 6$ cups.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Evaluate .
2. Evaluate .
3. Evaluate .
4. Evaluate .

Answers: 1. ; 2. ; 3. ; 4. .

Common Misconceptions

Misconception	How to pre-empt it
Working strictly left to right, ignoring operation priority.	Use the warmup’s “spot the error” format regularly; always narrate which operation is done first and why.
Dropping a negative sign when it is not enclosed in brackets.	Insist negative values are written in brackets when substituted into an expression, e.g. , not .
Treating as .	Show both side by side with a numeric check every time an expression contains a leading negative before a power.
Forgetting to apply “subtract a negative = add” inside a longer expression.	Rewrite every "" as "" as an explicit intermediate step, never combine mentally until fluent.
In applied problems, evaluating operations in the order they are described in words rather than building a single correctly bracketed expression first.	Model Problem 3 explicitly: write the full expression before calculating anything.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). Evaluate .

Answer

and , so .

E2 (Kangaroo style). If , , and , evaluate .

Answer

E3 (Challenge). Insert brackets into so that the expression equals .

Answer

Brackets placed around .

E4 (Investigation). Using each of the digits exactly once, and any of and brackets, make an expression equal to .

Answer

One valid solution:

Each of is used exactly once. (Other correct arrangements exist — accept any verified example.)

Homework

1. Evaluate: (a) (b) (c) .
2. Evaluate: (a) (b) (c) .
3. Insert brackets to make each statement true: (a) (b) .
4. A lift starts at level (basement level 2). It rises levels, then descends twice as many levels as it rose. Write a single expression for its final level, and evaluate it.
5. Reasoning. Explain why and give different results. Evaluate both to support your explanation.
6. Challenge. Evaluate , showing every step in an aligned working.

Answers: Q1 — (a) (b) (c) . Q2 — (a) (b) (c) . Q3 — (a) (b) . Q4 — , level . Q5 — ; — different because squares the negative first, while squares only the 2 and keeps the negative outside. Q6 — .