Arrays and Facts

Overview

In this lesson, students practise multiplication facts up to 12 × 12 and deepen understanding by linking multiplication to repeated addition, arrays, and number lines. They also apply commutativity to derive unknown facts and solve simple multiplication problems.

Learning intentions

Students will be able to:

• represent multiplication as repeated addition and as an array
• recall multiplication facts up to 12 × 12
• use commutativity (e.g., 3 × 5 = 5 × 3) to derive facts
• use known facts to find missing factors and compute missing products

Success criteria

• I can explain what × means using repeated addition.
• I can draw or identify an array for a multiplication (rows and columns).
• I can use commutativity to write the related multiplication fact.
• I can use known facts to calculate an unknown product correctly.

Curriculum links

• Number — multiplication and division (KS2): recall multiplication and division facts up to 12 × 12.
• Number — multiplication and division (KS2): recognise and use factor pairs and commutativity in mental calculations.
• Number — multiplication and division (KS2): use known and derived facts to multiply mentally, including multiplying by 0 and 1.

Key vocabulary

multiply, times, product, array, commutative, factor, repeated addition

Lesson structure (45 minutes)

1. 0–5 min · Starter retrieval. Teacher displays 8 mixed items: “4×4” (not “4+4”), arrays (e.g., 3 rows of 4 dots), and two commutativity prompts (e.g., “5×2=?”). Students answer on mini-whiteboards; teacher calls for reasoning using “It’s repeated addition because…”.
2. 5–12 min · Direct instruction: meaning of multiplication. Teacher models 3×4 as:

• repeated addition: 4+4+4
• array: 3 rows, 4 columns
• number line jumps of 4, three jumps to reach the product Teacher explicitly contrasts with addition: “If I add 4+4 only twice, that’s 2 groups, not 4 groups.”

3. 12–20 min · Direct instruction: commutativity and deriving facts. Using one worked example: start with array for 2×7. Show product = 14, then flip the array to show 7×2 also equals 14. Teacher emphasises commutative meaning: changing rows/columns doesn’t change the total. Students complete a short set: “Write the related fact” (e.g., 6×3 → 3×6).
4. 20–28 min · Guided practice: teacher-led to small groups. Teacher solves 2–3 “facts and structure” questions:

• “How many dots in this 5×3 array?”
• “What is the product of 4×6?”
• “I know 3×8=24; what is 8×3?” Then sets groups of 4 with one focused misconception task:
• Group A: repeated addition vs random addition (given 4×5 but they’re tempted to add 4+5)
• Group B: array confusion (rows/columns swapped incorrectly without using commutativity)
• Group C: commutativity/factor pair (given 9×4 and asked for 4×? )

5. 28–37 min · Independent task: differentiated three tiers. Students choose their tier (or teacher assigns based on quick checks). All tiers include arrays/number lines for support.

• Tier 1 (support): a) Draw an array and write the product for 2×6, 3×4, 5×2. b) Use commutativity to fill: 4×7= __×4. c) True/false: “4+4+4 = 4×2” (correct and explain briefly).
• Tier 2 (core): a) Calculate products: 6×7, 9×5, 8×3 (show one method: array or number line). b) Derive: “If 7×3=21, what is 3×7? What is 7×4?” c) Missing factor: “__×4=32”.
• Tier 3 (challenge): a) Find the missing factor: “6×=42”, “×9=63”. b) Reasoning: “A has 4 rows of 6 dots. B has 6 rows of __ dots. Total dots are equal. What number is missing? Explain.” c) Word problem: “There are 8 bags with 5 apples in each. How many apples?” then “If there are 40 apples total, how many apples per bag?”

6. 37–43 min · Closing plenary: exit ticket. Each student completes a single-page exit ticket (3 questions):

• Q1: Draw an array for 3×5 and write the product.
• Q2: Use commutativity: 2×9 = __×2.
• Q3: Missing factor: __×6=54 (show a quick method).

7. 43–45 min · Review misconceptions quickly. Teacher shows two anonymised common mistakes and classifies: “addition mistake” vs “array/commutativity mistake”, and restates the correct interpretation.

Resources

• Mini-whiteboards and pens
• Dot cards/counters or magnetic tiles for arrays
• Pre-printed array grids (blank and partially filled)
• Number line strips (e.g., 0–60) and highlighters
• Multiplication fact grid up to 12×12 (reference use only, not for everything)
• Tiered independent task sheets (T1/T2/T3)
• Exit ticket slips
• Group cards labelled with the misconception focus

Assessment

• Starter: teacher listens for correct language (“groups of”, “rows”, “jumps”) and checks answers on mini-whiteboards.
• Guided practice: observe group reasoning—do they use arrays/number lines or just add numbers randomly?
• Independent task: quick teacher checks with a “show me your array/number line” prompt when responses are incorrect.
• Exit ticket: mark Q1 (array/product), Q2 (commutativity), Q3 (missing factor).

Misconceptions to address

• Confusing 4+4 with 4×4 (addition used instead of repeated groups). Correction strategy: insist on “number of groups” and “number of elements per group” using arrays.
• Random addition without structure (e.g., mixing up which number is repeated). Correction: require either an array label (rows/columns) or number line jumps of the correct step size.
• Forgetting commutativity (writing only one order). Correction: always “flip the array” to generate the related fact.
• Miscounting array totals (rows×columns). Correction: point to one row, count across, then multiply by number of rows.

Differentiation

• Support: sentence starters (“I multiply because…”, “3×5 means 5 repeated 3 times”), partially completed arrays, and a number line model already drawn with jumps of the correct size.
• Support for SEN/EAL: allow counters instead of drawing; provide a factor pair word bank (“rows/columns are factors”).
• Core: expect one clear representation (array OR number line) for each product.
• Extension: missing factors and reasoning word problems requiring factor pairs/commutativity explanations, not just answers.

Homework (optional)

• Short practice (10–15 minutes):

1. Write 6 related commutativity facts (e.g., 3×4 and 4×3).
2. Choose 6 products from up to 12×12 and draw an array for any 3 of them.
3. Missing factor: ×5=35 and 9×=54.

• Brief answer guidance: products come from rows×columns; missing factor found by dividing the product by the known factor.