Fact Families Inverse

Overview

Lesson 5 of 8 in the unit “Multiplication and Division Facts”. Students use arrays, counters and known multiplication facts to build related division facts for 3, 4, 5 and 10. The lesson develops accurate mathematical language and understanding that multiplication and division are inverse operations.

Learning intentions

• Students will connect multiplication and division using arrays and equal groups.
• Students will create fact families for multiplication facts involving 3, 4, 5 and 10.
• Students will use a known multiplication fact to solve a related division fact.
• Students will explain division as both sharing and grouping.

Success criteria

• I can create a fact family from an array.
• I can write two multiplication and two division equations.
• I can use a known multiplication fact to solve a division problem.
• I can explain the roles of the dividend, divisor and quotient.

Curriculum links

• Number and algebra: recall, represent and apply multiplication facts for 3, 4, 5 and 10.
• Number and algebra: use the inverse relationship between multiplication and division.
• Mathematical reasoning: explain strategies, make connections and justify answers.
• Mathematical proficiency: develop fluency, problem-solving and precise mathematical communication.

Lesson structure (50 minutes)

1. 0–5 minutes – Hook and activate prior knowledge Display an array of 3 rows of 4 using counters. Ask: “What multiplication fact can you see? What division facts might belong to the same numbers?” Open with the array hook and discussion prompt. Invite students to turn and talk, then collect responses without correcting them yet.

2. 5–17 minutes – Explicit teaching and modelling Build the 3-by-4 array with counters. Model and record:

• 3 × 4 = 12
• 4 × 3 = 12
• 12 ÷ 3 = 4
• 12 ÷ 4 = 3 Explain that these four equations form a fact family because they use the same three numbers. Model 12 shared into 3 equal groups and 12 grouped in groups of 3, emphasising that division can mean sharing or grouping. Use and repeat the terms inverse, divide, dividend, divisor and quotient. Refer to the worked array and inverse-operation slides. Ask: “Which number is the total? Which number tells us the groups?”

3. 17–32 minutes – Guided group practice Place students in groups of four, with roles of builder, recorder, checker and explainer. Give each group counters and a set of fact-family triangles. Groups select facts involving 3s, 4s, 5s or 10s, build an array, and write the two multiplication and two division equations. Then partners use the division facts matching cards to match related division equations and answers, checking each match with counters or an array. Circulate and ask:

• “How do you know these equations belong together?”
• “Which multiplication fact helps you solve this division fact?”
• “Is this division showing sharing, grouping or both?” Display the guided-practice instructions using the group-work instructions and discussion prompts.

4. 32–45 minutes – Independent practice Distribute the fact-family and missing-number worksheet. Students complete fact-family sets and missing-number equations, including:

• 20 ÷ 5 = ___
• 40 ÷ 4 = ___
• ___ ÷ 5 = 6
• 4 × ___ = 28 Students should draw an array or use a related multiplication fact when needed. Prompt students to explain how knowing 4 × 7 helps solve 28 ÷ 4. Early finishers create a fact family for a chosen product involving 3, 4, 5 or 10.

5. 45–50 minutes – Conference and exit assessment Confer briefly with selected students, asking them to explain one completed fact family and identify the dividend, divisor and quotient. Finish with the division facts exit ticket slips. Students complete the family for 5 × 6 = 30 by writing two multiplication and two division equations, then answer: “How does knowing 4 × 7 help you solve 28 ÷ 4?” Collect slips as students leave.

Resources

• Counters or linking cubes, enough for groups of four
• Array mats or hoops
• Fact-family triangles
• Mini-whiteboards and markers
• Equation cards
• the fact families teaching and activity deck
• the fact-family and missing-number worksheet
• the division facts matching cards
• the division facts exit ticket slips

Assessment

• Observe group work for accurate arrays, related equations and correct use of mathematical vocabulary.
• Check the worksheet for correct fact families, missing-number equations and evidence of using inverse operations.
• Use conferences and exit tickets to identify misconceptions, particularly reversing the dividend and divisor, treating division only as sharing, or writing unrelated equations.

Differentiation

• Support students with concrete sharing using counters, smaller products, fact-family templates and a teacher-led group.
• Provide sentence stems: “I know ___ × ___ = ___, so ___ ÷ ___ = ___.” Allow students to explain orally before recording.
• Support EAL learners with labelled examples, gesture and visual models for dividend, divisor and quotient; pair them with a supportive partner.
• Extend confident students by explaining both sharing and grouping interpretations of one division equation and generating a fact family from a chosen product.