Integers, Decimals, Fractions

Overview

This 45-minute lesson is designed for a small Year 7 class of 5 students to develop fluency and confidence in applying the four operations (addition, subtraction, multiplication, division) using integers, decimals, and simple fractions (proper, improper, and mixed numbers), both positive and negative. Emphasis will be placed on formal written methods and understanding place value for working with very large or small numbers and decimals, in strict alignment with the National Curriculum for England (Mathematics programmes of study: Number – number operations and understanding number).

National Curriculum References

Key Stage 3 – Year 7 (Aged 11–12):

Number – number operations
- Use the four operations, including formal written methods, to calculate with integers, decimals, and simple fractions, both positive and negative.
- Understand and use place value when working with very large or very small numbers and decimals.
- Add and subtract fractions with different denominators and mixed numbers, using manipulation and formal written methods.
Number – multiplication and division
- Multiply and divide integers and decimals by powers of 10.

Learning Objectives

By the end of the lesson, students will be able to:

Apply formal written methods to add, subtract, multiply and divide integers, decimals, and simple fractions (proper, improper, mixed), including negative values.
Understand and use place value effectively, especially when working with very large or very small numbers and decimals.
Convert between improper fractions and mixed numbers confidently.
Recognise and apply arithmetic operations involving negative numbers in real-world contexts.

Materials Needed

Whiteboard and markers
Mini whiteboards and pens (for immediate practice)
Fraction strips or virtual fraction manipulatives (printed or on tablets)
Place value charts (for decimals and large numbers)
Worksheets tailored for each operation and format (integers, decimals, fractions)
“Operation challenge” cards (to gamify practice)
Calculator (to check answers)

Lesson Structure

1. Starter (5 minutes)

Activity: Quickfire mental arithmetic quiz on place value and signs (+/-)

Use mini whiteboards. Call out questions such as:
- “Write 3.405 in expanded form.”
- “What is 7 × (-4)?”
- “Convert 11/4 to a mixed number.”

Purpose: Activate prior knowledge on place value and operations with negative numbers and fractional forms.

2. Introduction & Modelling (10 minutes)

Teacher explains and demonstrates:

Formal written methods for addition and subtraction with:
- Integers (including negative values)
- Decimals (aligning decimal points carefully)
- Simple fractions (with unlike denominators, including mixed numbers)
Multiplication and division procedures:
- Multiplying decimals by integers (using place value understanding)
- Dividing integers and decimals (focusing on shifting decimal points)
- Multiplying and dividing fractions using formal methods

Modelling approach:

Use a step-by-step approach shown on the board.
Highlight how place value guides the alignment and calculation.
Emphasise converting improper fractions to mixed numbers after operations where needed.

Example to model:

Calculate: (-3.5) + 4.28
Calculate: 7/4 × 3/2
Calculate: (-12) ÷ 4

3. Guided Practice (15 minutes)

Structured small-group tasks:

Students work on three problem sets on their mini whiteboards with teacher circulating and providing support:

Set A: Adding and subtracting integers and decimals (including negatives)
Set B: Multiplying and dividing decimals and integers (multiplying/dividing by powers of 10, emphasising place value)
Set C: Adding, subtracting, multiplying simple fractions (including converting improper to mixed)

Peer-checking: Students swap mini whiteboards and verbally explain their reasoning to each other to reinforce understanding.

4. Problem Solving Challenge (10 minutes)

“Operation Challenge Cards” Game

Each student picks a card that presents a real-life contextual problem requiring application of the four operations on integers/decimals/fractions with negative values.
Example card:

“A submarine is 200m below sea level. It rises 75.5m, then descends another 30.25m. What is its current position?”
Students work independently, then share their approach and solution.
Teacher encourages use of written methods and place value reasoning.

5. Plenary & Assessment (5 minutes)

Class discussion with targeted questioning:

What strategies help you ensure accuracy when adding/subtracting decimals?
How does understanding place value help when multiplying or dividing decimals by powers of 10?
Why is it useful to convert improper fractions to mixed numbers?

Formative assessment:

Quick exit quiz on a mini whiteboard: One example each of addition, subtraction, multiplication, and division involving a negative number or fraction.

Review answers and provide positive, targeted feedback.

Differentiation

Support: For students struggling, provide fraction strips and place value charts; focus on operations with integers first.
Challenge: For advanced learners, extend problems to include calculations with very large numbers and decimals (e.g. 0.00456 × 1000), and mixed-number multiplication with multi-step problems.

Homework (Optional)

Worksheet consolidating all four operations on integers, decimals, and fractions including negative numbers.
Problem-solving questions involving real-life contexts (temperature changes, financial calculations).

Reflection Notes for Teacher

Observe students’ use of place value understanding especially when working with decimals and large numbers.
Monitor accuracy of formal methods and ability to switch between fraction forms.
Use peer explanation moments to gauge deep conceptual understanding.

This plan is designed to fully engage a small group through hands-on activities, real-life problems, and repeated practice, which ensures alignment with the National Curriculum while fostering confidence and enjoyment in maths.

Teaching Instructions