Decimal and integer methods

Overview

This 55-minute session builds Year 11 fluency with standard column methods for addition/subtraction of integers and decimals, focusing on place value accuracy and the additive/multiplicative properties that support checking. Students practise written procedures and use reasoning to explain errors.

Prerequisite Knowledge

• The ability to order numbers
• Appreciation of place value
• Some experience of the 4 operations using whole numbers

Key words

• Place value, sum, difference, product, number, numeral, digit, decimal

Key Concepts

• Students must be very secure applying the place value table to identify the value of any digit (including within decimals).
• Additive/multiplicative properties support mental and written checking: 2 + 3 = 3 + 2 and 2 × 3 = 3 × 2.

Common Misconceptions

• When subtracting, students may “borrow” incorrectly, subtracting the smaller digit from the larger digit (e.g. 43 − 25 = 22).
• Misaligning correct digits in column addition/subtraction; encourage using a place value table and lining up decimal points.

4-Lesson Sequence (use these as your mini unit)

Topic: Standard written methods with place value (integers + decimals)

Learning objectives (shared across the 4 lessons)

• Use standard column procedures for addition and subtraction of integers and decimals.
• Use place value to write and compare decimals in order of size.
• Multiply/divide integers and decimals using known facts about place value (including powers of 10 and × with numbers between 0 and 1 in future lessons if needed).

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Lesson 1 (55 minutes) — Place value for decimals, then addition/subtraction

Success criteria

• I can identify the value of each digit using the place value table (including after the decimal point).
• I can order decimals correctly using digit value.
• I can add and subtract decimals using correct alignment and decimal points.

Curriculum links

• Mathematics Appendix 1: formal written methods for addition and subtraction
• Students apply place value knowledge to decimals and whole number operations
• Number — additive properties used to check results (e.g. commutativity)

Prerequisite Knowledge (from above)

• Ordering numbers, place value, some whole-number 4-operation experience

Careers/Real life

• Money calculations (pounds and pence): adding costs, finding change, comparing decimal values.

PSHRE Links

• Budgeting responsibly: checking accuracy before committing to spending decisions.

Learning Objective / activities / feedback & assessment (55 mins)

1. 0–6 min · Drop. Teacher shows 3 decimals (e.g. 3.07, 3.7, 3.007). Students write the value of one chosen digit (e.g. “the 0”).

• Teacher listens for correct interpretation of digit value; quick oral checks.

2. 6–18 min · Flow. Direct teach: place value table for decimals; model how each digit maps to value. Show ordering reasoning: compare tenths, then hundredths.

• Students complete “digit value” cards and order them on mini-whiteboards.

3. 18–35 min · Mix. Column addition of decimals with emphasis on alignment: teacher models 2.36 + 1.04, then 0.7 + 3.25.

• Students practise 6 questions in pairs; each answer must include a one-line check (e.g. estimating or using inverse subtraction).

4. 35–48 min · Deep. Column subtraction of decimals: teacher models 5.03 − 2.7 (with explicit place value mapping before column work).

• Students complete 5 mixed subtraction questions; teacher circulates to prevent decimal-point and digit misalignment.

5. 48–55 min · Calm. Exit ticket (2 questions):

• (a) identify digit value (e.g. value of the 4 in 0.42)
• (b) compute 3.6 − 1.27
• Mark for correct place value and correct column procedure.

Extended writing opportunities

• One sentence: “I knew the answer was reasonable because…” using estimate/inverse reasoning.

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Lesson 2 (55 minutes) — Integers + additive checking

Success criteria

• I can add and subtract integers using correct column procedures.
• I can explain what a negative sign means in context and in calculation.
• I can use inverse operations to check a result.

Curriculum links

• Mathematics Appendix 1: formal written methods for addition and subtraction
• Number properties supporting checking (additive inverse)

Learning Objective / activities / feedback & assessment

1. 0–8 min · Drop. Quick retrieval: write the value of −3, 0, and 5 on a number line; order 4 integers.
2. 8–20 min · Flow. Teach sign rules through place value and inverse: add integers then use subtraction to confirm.
3. 20–38 min · Mix. Practise column addition/subtraction of integers (including mixed signs). Students must circle alignment and place-value columns before calculating.
4. 38–48 min · Deep. Error analysis: teacher shows an incorrect subtraction (a “borrowing” style error). Students diagnose which digit value was mishandled and rewrite correctly.
5. 48–55 min · Calm. Exit ticket: one addition, one subtraction; include a required check step.

Common misconception focus: borrowing confusion—students must use place value table to justify each digit.

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Lesson 3 (55 minutes) — Decimal multiplication by whole numbers (place value)

Success criteria

• I can multiply a decimal by a whole number using column method with correct placement of the decimal point.
• I can estimate to check whether an answer is sensible.

Curriculum links

• Mathematics Appendix 1: examples of formal written methods for multiplication (including decimals by whole numbers)
• Place value precision after multiplication

Learning Objective / activities / feedback & assessment

1. 0–8 min · Drop. Estimate-first warm-up: 0.6 × 8, 2.4 × 3 (range answers acceptable, then compute).
2. 8–20 min · Flow. Teach decimal × whole number: count decimal places; link to multiplication by powers of 10 conceptually.
3. 20–40 min · Mix. Students practise 8 multiplications with varied tenths/hundredths; after each, they do a quick estimate check.
4. 40–50 min · Deep. Reasoning task: “How many decimal places should the answer have?” Students justify using place value table.
5. 50–55 min · Calm. Exit ticket: 2 calculations + one estimate statement.

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Lesson 4 (55 minutes) — Decimal order + multi-step integer/decimal problem solving

Success criteria

• I can order decimals correctly and use written methods in a combined problem.
• I can choose a sensible method and check using inverse/estimation.

Curriculum links

• Mathematics Appendix 1: formal written methods for addition/subtraction/multiplication/division
• Place value for ordering decimals

Learning Objective / activities / feedback & assessment

1. 0–10 min · Drop. Ordering decimals tournament (individual then pair justify).
2. 10–25 min · Flow. Model a realistic multi-step problem: e.g. money scenario requiring decimal addition/subtraction then integer multiplication for totals.
3. 25–45 min · Mix. Students solve 2 multi-step written questions, showing alignment and place value working.
4. 45–52 min · Deep. Peer marking using a checklist: (i) decimal points aligned (ii) digit values correct (iii) check step present.
5. 52–55 min · Calm. Exit ticket: one short problem + one “check statement”.

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Assessment (across the 4 lessons)

• Daily exit tickets focused on place value accuracy and column correctness.
• Teacher formative checks during pair practice: alignment, decimal-point placement, and use of checks.
• Error analysis in Lesson 2 and 4 to confirm misconceptions are addressed.

Differentiation

• Support: provide a place value table template beside the working; sentence starters for checks (“I estimated…” “I checked by…”).
• Support: pre-drawn column grids with decimal point markers.
• Extension: include reasoning-only prompts (“Which digit value is affected by the decimal point and why?”).
• EAL/SEN: reduce cognitive load by doing one operation per question at first; encourage verbal explanation of each step before writing.

Final note for the teacher (how it matches UK curriculum expectations)

Across the sequence, students repeatedly practise standard written methods for addition/subtraction and use place value to control decimals, aligning with the intent of the Mathematics Appendix 1 examples for formal written procedures and the need for correct digit value and column alignment.