020i - Consolidation Four Operations

Lesson 20 — Consolidation and Check: Four Operations with Integers and Rational Numbers

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

Learning Intentions

• To consolidate fluent and efficient use of the four operations with integers, fractions and decimals.
• To select and justify an appropriate strategy — mental, written or digital — for a given calculation.

Success Criteria

I can:

1. Select and justify an efficient strategy for a calculation involving integers, fractions or decimals.
2. Apply the four operations accurately to integers, fractions and decimals, including combined expressions.
3. Convert between fractions and decimals when it makes a calculation more efficient.
4. Solve a substantial multi-part problem drawing on several operations, and justify the reasonableness of my answer.

Warmup

(6 minutes — “Which strategy?”, mini whiteboards)

For each calculation, decide whether you would calculate mentally, on paper, or with a digital tool, and briefly justify. You do not need to find the answer.

Discussion: There is no single “correct” answer for every student, but strong justifications refer to the size and complexity of the numbers, not just personal preference. Q2 and Q5 are natural candidates for a digital tool — the numbers are messy and a small slip is easy to make by hand; Q1, Q3 and Q4 are efficient mentally once a strategy is chosen.

Activities

Activity 1 — Diagnostic Review Circuit (9 min)

Four short, independent problems — one for each operation type from this seven-lesson block. Work through them briskly, then check as a class.

A. Integers, mixed operations. Evaluate .

B. Fractions. Evaluate .

C. Decimals. Evaluate .

D. Efficient strategy. Evaluate using a reordering strategy — do not multiply left to right.

Answers:

Discussion of D: Reordering to pair with first turns the calculation into a friendly — a much faster mental path than multiplying left to right. This is the kind of “efficient strategy” the Australian Curriculum descriptor asks for.

Activity 2 — Guided Practice: Choosing Mental, Written or Digital Strategies (8 min)

Pairs. For each calculation, agree on a strategy, carry it out, and be ready to justify your choice.

Class discussion: Q3 is the natural digital-tool candidate — two decimals with several significant figures multiply reliably on a calculator, but are slow and error-prone by hand. Every other question is efficient mentally or with brief working, once fractions are converted to a common denominator or division is rewritten as multiplication by the reciprocal.

Activity 3 — The Consolidation Check Problem: the Bakery Order (16 min)

Pairs, then whole-class share. This problem draws on integers, fractions and decimals together.

A bakery is preparing a large order. Each cake needs cups of flour. The bakery has cups of flour in stock.

1. How many whole cakes can be made from the flour in stock, and how much flour (in cups) is left over?
2. Each cake sells for 18.50$6.75$25$ applies for using the oven, regardless of how many cakes are made. Find the total profit from selling all the whole cakes made, after all costs.
3. On the collection day, the delivery van’s fridge temperature drops from by every hour for hours because of a fault, before being reset upward by . Find the final fridge temperature.
4. The safe cake-storage range is to . Is the fridge now within this range? Justify your answer.

Socratic scaffolding for Part 1:

Prompt	Purpose
Understand: what is being divided, and by what?	The total flour ( cups) is being shared into portions of cups each.
What operation finds “how many groups of size fit into “?	Division: .
Devise a plan	Convert the mixed number to an improper fraction, then divide by multiplying by the reciprocal.
Carry it out	.
Looking back — what does the fractional part of the answer mean here?	It does not mean “13.33 cakes” — a bakery cannot sell a third of a cake. It means whole cakes can be made, with some flour left over.
Looking back — how do you find the leftover flour?	cakes use cups, so cups, i.e. cup, remain.

Working for Parts 2–4:

Answers: 1. whole cakes, cup left over. 2. 127.751.5°C1.5°C-2°C6°C$, so the fridge is within the safe range.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Evaluate .
2. Evaluate .
3. Which strategy — mental, written or digital — would you use for ? Justify in one sentence.
4. Reasoning. A student calculates and gets an answer of . Explain, using the context “sharing L of juice into bottles that hold L each,” what the whole-number and decimal parts of this answer actually mean.

Answers: 1. ; 2. ; 3. Digital — the numbers have several significant figures and multiplying by hand risks a place-value or carrying error; 4. whole bottles can be completely filled, and the remaining of a bottle’s capacity is left over — it does not mean “7.5 bottles,” since a bottle cannot be partially counted as a full bottle.

Common Misconceptions

Misconception	How to pre-empt it
Dividing fractions by dividing the numerators and denominators separately instead of multiplying by the reciprocal.	Drill “keep, change, flip” explicitly every time a fraction division appears, as in Activity 3.
Rounding a division answer instead of interpreting the whole-number and remainder parts in context.	Always ask “what does the leftover part mean in this situation?” before rounding or truncating.
Treating “efficient strategy” as always meaning “use a calculator.”	Activity 1 Question D shows a reordering strategy is often faster and more reliable than any tool.
Sign errors when combining a negative integer with a decimal or fraction operation.	Insist on writing the sign as part of the number at every step, never as a separate “subtract” instruction.
Forgetting to convert a mixed number to an improper fraction before multiplying or dividing.	Model the conversion as a mandatory first line of working, as in Activity 3, Part 1.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). Evaluate .

Answer

E2 (Kangaroo style). A number is halved, then is subtracted, then the result is tripled, giving . What was the original number?

Answer

Working backwards: ; ; . The original number was .

E3 (Challenge). Without a calculator, evaluate by first rewriting as .

Answer

E4 (Investigation). A student claims that and and are three different numbers. Use a number line and a worked calculation to show they are all equal.

Answer

All three represent the same point, units to the left of zero. Algebraically, a negative sign attached to a fraction can sit in front, on the numerator, or on the denominator without changing its value, since and .

Homework

1. Evaluate: (a) (b) (c) (d) .
2. For each, state whether you would calculate mentally, on paper, or digitally, and why: (a) (b) (c) .
3. A tank holds L of water. Each bucket holds L. How many full buckets can be filled, and how much water is left over?
4. A shop’s daily profit changes were 45.50$120.25-$18.75$ over three days. Find the total change, and state whether the shop made an overall profit or loss.
5. Reasoning. Explain why but the practical answer to “how many whole cakes can be made” is , not or .
6. Challenge. A number, when increased by of itself and then decreased by , equals . Find the number. (Hint: let the number be and write a single equation.)

Answers: Q1 — (a) (b) (c) (d) . Q2 — (a) mental, friendly factors (b) digital, messy decimals (c) mental/written, common denominator of . Q3 — buckets exactly, no water left over. Q4 — 56.00\dfrac54n - 1.5 = 8 \Rightarrow \dfrac54n = 9.5 \Rightarrow n = 9.5\times\dfrac45 = 7.6$.