Mathematics · Year 5 · Additive and Multiplicative Structures · Autumn Term

Long Multiplication (2-digit by 2-digit)

Students will use long multiplication to multiply two-digit numbers by two-digit numbers.

• National Curriculum Attainment Targets
• KS2: Mathematics - Multiplication and Division

About This Topic

Long multiplication allows Year 5 students to multiply two-digit numbers by two-digit numbers efficiently, such as 23 x 45. The method involves multiplying the tens digit of the second number first, adding a zero placeholder to shift for place value, then multiplying by the units digit, and finally adding the partial products. Students justify each step, explaining how the zero maintains tens place value during the second multiplication.

This topic fits within the unit on additive and multiplicative structures, reinforcing place value understanding from earlier years while preparing for larger multiplications and division in KS2. Students construct their own problems and compare long multiplication to grid or short methods, building strategic flexibility and mathematical reasoning.

Active learning suits long multiplication well because students can use concrete tools like base-10 blocks or arrays to visualise partial products before transitioning to abstract algorithms. Collaborative problem-solving reveals errors in real time, while games comparing methods encourage discussion of efficiency, making the process engaging and deepening procedural fluency.

Key Questions

1. Justify the steps involved in long multiplication, explaining the role of the 'zero' placeholder.
2. Construct a multiplication problem that requires long multiplication and solve it.
3. Compare long multiplication with other multiplication strategies for efficiency.

Learning Objectives

• Calculate the product of two-digit numbers by two-digit numbers using the long multiplication algorithm.
• Explain the purpose of the zero placeholder in the second step of long multiplication, relating it to place value.
• Compare the efficiency of long multiplication with grid multiplication or short multiplication for solving specific two-digit by two-digit problems.
• Construct a word problem that necessitates the use of two-digit by two-digit multiplication and solve it using long multiplication.

Before You Start

Multiplication Facts to 12 x 12

Why: Students need instant recall of basic multiplication facts to perform the individual steps within long multiplication accurately.

Place Value to Thousands

Why: A strong understanding of place value is essential for correctly aligning numbers and understanding the role of the placeholder in long multiplication.

Short Multiplication (1-digit by 2-digit)

Why: This builds the foundational algorithm of multiplying digit by digit and carrying over, which is extended in long multiplication.

Key Vocabulary

partial product	A product obtained during the process of multiplication, before the final sum is calculated. For example, when multiplying 34 x 21, the partial products are 34 and 680.
placeholder	A digit, usually zero, placed in a position to maintain place value. In long multiplication, a zero is used in the tens column when multiplying by the tens digit of the second number.
algorithm	A step-by-step procedure for solving a mathematical problem. Long multiplication is an algorithm for multiplying larger numbers.
place value	The value of a digit based on its position within a number. Understanding place value is crucial for correctly aligning numbers in long multiplication.

Watch Out for These Misconceptions

Common Misconception

Forgetting the zero placeholder when multiplying by tens.

What to Teach Instead

Students often omit the zero, treating it like units multiplication and losing place value. Hands-on work with base-10 rods shows the shift clearly. Peer teaching in pairs helps as they explain the 'why' during error spotting.

Common Misconception

Adding partial products incorrectly by ignoring alignment.

What to Teach Instead

Misalignment leads to wrong totals since columns mix place values. Station activities with place value mats reinforce vertical setup. Group discussions during rotations allow students to correct each other visually.

Common Misconception

Believing long multiplication is just repeated addition without structure.

What to Teach Instead

This overlooks the efficiency of partial products. Array-building tasks connect to grouping, while strategy comparisons in games highlight why the algorithm scales better for larger numbers.

Active Learning Ideas

Stations Rotation: Multiplication Steps

Set up stations for each step: one for units multiplication with counters, one for tens with place value charts, one for adding partial products, and one for self-created problems. Groups rotate every 10 minutes, recording justifications at each. End with whole-class share-out.

45 min·Small Groups

Pairs: Problem Construction Challenge

Pairs create a two-digit by two-digit problem, solve using long multiplication, and swap with another pair to check and justify steps. Discuss the zero placehold's role. Teacher circulates to prompt comparisons with grid method.

30 min·Pairs

Whole Class: Strategy Race

Divide class into teams. Project problems; teams solve using long multiplication, grid, or short method and vote on most efficient. Debrief on when each works best, with students explaining choices.

35 min·Whole Class

Individual: Array Visualiser

Students draw arrays for given multiplications, then convert to long method on paper. Use coloured pencils to show partial products and zero shift. Share one with partner for peer feedback.

25 min·Individual

Real-World Connections

• Retail buyers at a large department store might calculate the total cost of ordering 45 units of a product that costs $32 per unit, using long multiplication to determine the overall expense.
• Event planners calculating the seating capacity for a conference might need to multiply the number of rows (e.g., 24) by the number of chairs per row (e.g., 18) to find the total number of seats, using long multiplication.
• Construction site managers estimating material needs might multiply the number of wall sections (e.g., 36) by the number of bricks needed per section (e.g., 15) to order the correct quantity of bricks.

Assessment Ideas

Exit Ticket

Provide students with the calculation 57 x 34. Ask them to solve it using long multiplication. On the back, have them write one sentence explaining why they placed a zero in the second line of their calculation.

Quick Check

Display the problem: 'A school is buying new library books. Each book costs $16, and they need to buy 25 books. How much will the books cost in total?' Ask students to show their working using long multiplication and circle their final answer.

Discussion Prompt

Pose the question: 'Imagine you need to calculate 42 x 53. Would you choose long multiplication, grid multiplication, or short multiplication? Explain why your chosen method is the most efficient for this specific problem and justify your steps.'

Frequently Asked Questions

How do I teach the role of the zero placeholder in long multiplication?

Use base-10 blocks to model: show multiplying by units, then regroup tens into hundreds for the tens step, adding the zero on paper to track. Students justify by drawing the shift. Practice with scaffolded worksheets fading to independent work builds confidence over sessions.

What are common errors in two-digit by two-digit long multiplication?

Errors include skipping the zero shift, misaligning columns, or calculation slips in partial products. Address with visual aids like expanded notation first. Regular low-stakes quizzes and error analysis discussions help students self-correct patterns.

How can active learning help students master long multiplication?

Active approaches like manipulatives and partner problem-swapping make abstract steps concrete and social. Students manipulate arrays to see partial products, discuss justifications in groups, and race strategies for efficiency. This boosts retention, reduces anxiety, and develops reasoning as peers challenge misconceptions collaboratively.

How to differentiate long multiplication for Year 5?

Provide concrete supports like grids for visual learners, challenge advanced students with three-digit extensions or efficiency proofs. Pair mixed abilities for construction tasks. Use success criteria checklists so all justify steps at their level, ensuring progress across the ability range.