069i - Consolidation Four Operations

Lesson 69 — Consolidation and Check: Four Operations with Rational Numbers

Strand: Number | Descriptor: AC9M7N06 | Duration: 45 minutes

Learning Intentions

• To consolidate all four operations across fractions, decimals and percentages.
• To apply the order of operations to rational number calculations.

Success Criteria

I can:

1. Perform all four operations with fractions, decimals and percentages accurately.
2. Apply the order of operations with rational numbers.
3. Select efficient strategies and check with estimates.
4. Solve mixed practical problems reliably.

Warmup

(6 minutes — error clinic, pairs)

Each contains one error. Find and fix it.

5. of 45 = $4.05$

Answers: 1. Needs a common denominator: ; 2. Multiply straight across: ; 3. Estimate — less than half of : ; 4. How many halves in ? ; 5. Divide by : 4.50$.

Note: all five errors were caught by estimation or meaning, not by redoing the algorithm. That is the checking standard for today.

Activities

Activity 1 — Order of Operations with Rationals (12 min)

The conventions carry over unchanged: brackets, then indices, then multiplication and division (left to right), then addition and subtraction (left to right).

I do.

(Multiplication before addition — resist reading left to right.)

(Brackets change everything — compare the two results.)

You do:

(Answers: ; ; ; ; ; .)

Activity 2 — Mixed Skills Circuit (14 min)

Stations or a graded worksheet covering the whole block (Lessons 62–68).

Station A — fractions.

Station B — decimals.

Station C — percentages.

9. of 450$
10. 6815%$
11. 927.5%$
12. of a number is — find the number.

Station D — mixed applications.

13. A kg bag of nuts costs 10.80$. Find the price per kilogram.
14. Ana reads of a -page book on Saturday and of the book on Sunday. How many pages remain?
15. A 140020%10%$ off the sale price. Find the final price.

(Answers: 1. ; 2. ; 3. ; 4. ; 5. ; 6. ; 7. ; 8. ; 9. 126$57.80$98.9014010.80 \div \tfrac34 = $14.40\tfrac25 + 0.3 = 70%1081400 \times 0.8 \times 0.9 = $1008$.)

Activity 3 — Inquiry: Make the Target (8 min)

Pairs. A construction challenge.

Using each of the numbers , , and exactly once, with any operations and brackets:

1. Make the largest value you can.
2. Make the smallest positive value you can.
3. Make exactly .

Socratic scaffolding:

Prompt	Purpose
For “largest”: which operations grow numbers?	Dividing by small numbers; multiplying by numbers over .
Which of our numbers is the big lever?	The — and dividing by multiplies by .
Try a build.	; then doubles it: ; then : .
For “smallest positive”: invert the logic.	Multiply the small numbers together, divide by the big one: .
For “exactly ”: what pairs combine to friendly values?	, and — then lands exactly on .
Verify.	✓
Looking back	Other builds exist. The hunt itself — trying, estimating, adjusting — is the practice.

Verified solutions (teacher reference):

• Largest: .
• Smallest positive: .
• Exactly : .

Accept any correct construction — students may find different routes, and comparing them is valuable.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Calculate: (a) (b) .
2. Calculate: (a) (b) .
3. A 24035%$. Find the sale price.
4. of a number is . Find the number.
5. Reasoning. Explain how estimation caught the error in "".

Answers: 1. (a) (b) ; 2. (a) (b) ; 3. 156720.20.40.8$ is bigger. The size alone exposes it.

Common Misconceptions

Misconception	How to pre-empt it
Working left to right regardless of operation.	The paired bracket/no-bracket items in Activity 1 make the difference visible.
Reverting to “add tops and bottoms” under pressure.	The error clinic reopens the misconception deliberately before the circuit.
Percentage of the wrong base in two-stage discounts.	Station D Q15; ask “of what?” at each stage.
Answering the intermediate step (the discount, not the price).	Sentence answers required for all applied items.
Treating as “make smaller”.	Station A Q4 and the target inquiry both confront it.
Abandoning the estimate under time pressure.	Estimates stay compulsory in the circuit.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). What is ?

Answer

Telescoping again — everything cancels but the ends.

E2 (AMC Junior style). A price is increased by and the result is decreased by . What fraction of the original is the final price?

Answer

of the original.

E3 (Challenge). Evaluate .

Answer

E4 (Challenge). Ana, Ben and Cara share a prize. Ana gets , Ben gets , and Cara gets the remaining 76$. Find the prize.

Answer

E5 (Challenge). Using each of , , exactly once, make (a) the largest possible value (b) exactly .

Answer

(a) . (b) ✓

Homework

1. Calculate: (a) (b) (c) (d) .
2. Calculate: (a) (b) (c) (d) .
3. Calculate: (a) (b) (c) (d) .
4. Calculate: (a) of 320$7622.5%$5415%$.
5. of a number is . Find the number.
6. of a number is . Find the number.
7. A kg block of cheese costs 8.10$. Find the cost per kilogram.
8. Leo spends of his money, then of the remainder, and has 63$ left. How much did he start with?
9. A 240015%15%$. Find the final price and the overall percentage change.
10. Challenge. Using each of , , exactly once, make (a) the largest value possible (b) exactly .

Answers: Q1 — (a) (b) (c) (d) . Q2 — (a) (b) (c) (d) . Q3 — (a) (b) (c) (d) . Q4 — (a) 144$58.90$62.1072120$13.50\tfrac34 \times 0.7 = 0.5250.525m = 63$1202400 \times 0.85 \times 1.15 = $23462.25%10 \div \tfrac14 \div 0.8 = 5010 \times \tfrac14 \times 0.8 = 2$ ✓