064i - Adding and Subtracting Decimals

Lesson 64 — Operations with Decimals: Addition and Subtraction

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

Learning Intentions

• To add and subtract decimals using place value alignment.
• To use estimation to check decimal calculations.

Success Criteria

I can:

1. Line up decimal points to align place values before adding or subtracting.
2. Use zeros as placeholders where decimal lengths differ.
3. Subtract with regrouping across the decimal point.
4. Estimate first and use the estimate to check.

Warmup

(6 minutes — place value recall, mini whiteboards)

1. In , what does the represent? The ?
2. Which is larger: or ? Why do many people get this wrong?
3. Write with two decimal places without changing its value.
4. What is ?
5. What is ?

Answers: 1. tenths; hundredths; 2. — people wrongly compare with as whole numbers; 3. ; 4. (not ); 5. .

Activities

Activity 1 — Explicit Instruction: Alignment and Addition (12 min)

The one rule that matters:

Line up the decimal points. Then every column holds matching place values, and the addition works exactly as for whole numbers.

I do.

Note the placeholder zero: writing makes both numbers two decimal places long and prevents misalignment. Estimate first: , so is plausible ✓

The error this prevents. Right-aligning the digits (as with whole numbers) gives the nonsense treated as . Show this wrong layout once, cross it out, and name the rule again.

I do — money context. 25.95 + $8.50 + $0.85$

Estimate: ✓. Note the final zero in is kept for money.

We do: ; ; .

(Answers: ; ; .)

Activity 2 — Subtraction with Regrouping (12 min)

I do.

Model the regrouping aloud: ” hundredths take — regroup a tenth into ten hundredths…” The placeholder zero in is what makes the regrouping visible.

Estimate: ✓

I do — subtracting from a whole number.

This is the highest-error item type in the topic: students must first rewrite as .

We do: ; ; .

(Answers: ; ; .)

You do:

7. 50 - $32.45$

(Answers: ; ; ; ; ; ; 17.551.191$.)

Activity 3 — Inquiry: the Decimal Magic Square (10 min)

Pairs. The full square (teacher reference) has magic sum in every row, column and diagonal:

Give students the version with the four corners and the centre blanked out:

Every row, column and diagonal of this magic square has the same sum. Fill the five gaps.

?		?
	?
?		?

Socratic scaffolding:

Prompt	Purpose
What must you find first?	The magic sum.
Which line is complete enough to use?	The middle row once the centre is known — or use the middle column and middle row together.
Both middle lines pass through the centre. Write their sums.	Row: . Column: .
These are equal. Solve.	— no help. Try the property of the centre instead.
In a magic square, the magic sum is three times the centre. And the middle column gives sum .	Set , so and . Magic sum .
Now fill the corners.	Middle row: ✓. Each corner comes from its row or column, e.g. top-left — use the diagonal through the centre.
Verify everything.	All eight lines total ✓

Extension: design your own decimal magic square with magic sum . (The centre must be ; a scaled-down copy of any known square works.)

Checks for Understanding

(5 minutes — exit ticket)

1. Calculate: (a) (b) (c) .
2. Calculate: (a) (b) (c) .
3. A meal costs 18.75$4.60$30$?
4. Reasoning. A student calculates as by right-aligning the digits. Explain the error and give the correct answer.
5. Estimate, then calculate: .

Answers: 1. (a) (b) (c) ; 2. (a) (b) (c) ; 3. 6.65655.60 + 2.15 = 7.7520 + 30 = 5050.07$.

Common Misconceptions

Misconception	How to pre-empt it
Right-aligning digits instead of aligning decimal points.	Show the wrong layout once, cross it out, and reinforce placeholder zeros.
Treating as having no decimal form, failing at .	Rewrite whole numbers as explicitly, every time.
Believing longer decimals are larger: .	Warmup Q2. Compare tenths first.
Writing .	Ten tenths make one whole. Use a place value chart.
Dropping the final zero in money answers: 35.3$.	Money always shows two decimal places.
Skipping the estimate.	Estimate-first remains compulsory from Lesson 60 onwards.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). What is ?

Answer

— each term occupies its own decimal place, so the digits simply line up.

E2 (AMC Junior style). Three parcels weigh kg, kg and kg. The courier’s limit is kg. Are they under, and by how much?

Answer

E3 (Challenge). Find the missing digits: .

Answer

Hundredths: ends in , so (carry 1). Tenths: ends in , so (carry 1). Check: ✓

E4 (Challenge). A number line shows at and at . Point is exactly halfway. Where is ?

Answer

E5 (Challenge). In a m relay, four runners clock s, s, s and s. The school record is s. Did they break it, and by how much?

Answer

No — they missed the record by s.

Homework

1. Calculate: (a) (b) (c) (d) .
2. Calculate: (a) (b) (c) (d) .
3. Find the change from 50$12.85$7.60$21.95$.
4. A jug holds L. After pouring out L and then L, how much remains?
5. Three sides of a quadrilateral measure cm, cm and cm. The perimeter is cm. Find the fourth side.
6. Balance the books: a wallet starts with 87.35$23.90$15.75$40.50$. Find the final amount.
7. Find the missing digits: .
8. Reasoning. Explain why placeholder zeros make decimal subtraction safer, using as your example.
9. Challenge. The sum of two decimals is , and their difference is . Find both numbers.

Answers: Q1 — (a) (b) (c) (d) . Q2 — (a) (b) (c) (d) . Q3 — 7.602.5 - 1.655 = 0.84520 - 16.425 = 3.57587.35 - 39.65 + 40.50 = $88.203 + \square = 96\square + 4 = 776.73 + 1.46 = 8.197.00(10 + 2.46) \div 2 = 6.233.77$.