Lesson 9 — Related Operations

Summary

Learners discover that addition and subtraction undo each other, and so do multiplication and division — one fact family gives four related sentences, and a missing number is found by doing the opposite operation. Understanding the relationship is the key to checking your own work and solving for what is unknown.

Objectives

• Solve two-step word problems using the four operations and explain the relationship between addition and subtraction or multiplication and division. (D05.S2.03.02)

Connection

If you add three stones to a pile and then take three away, you are back where you started — the moves undo each other. If you make four rows of three and then share the twelve back into four groups, you are back to three again. This pairing — join and separate, group and share — is the oldest and most useful idea in arithmetic: every move has an opposite move. People use it to check a sum, to find a missing number, and to be sure an answer is fair.

Materials

• About 60 small objects
• Fact-family cards (add/subtract and multiply/divide)
• Paper or slate and a drawing tool

Preparation

• Give each learner or pair about 60 objects and a set of fact-family cards.
• Clear space to join, separate, group, and share.
• Plan a worked example (3, 4, and 12 as a fact family).

Facilitator note

This lesson is written to the learner (“you”). It completes the relationship half of the word-problem standard — the new skill is seeing the inverse pairs: add/subtract and multiply/divide. Teach with a worked example (philosophy §13): build the fact family for 3, 4, and 12 (3 × 4 = 12, 4 × 3 = 12, 12 ÷ 4 = 3, 12 ÷ 3 = 4) and show that a missing number (12 ÷ ? = 4) is found by doing the opposite. The relationship is the point, not the vocabulary “inverse” — children can say “the opposite” and still own the idea (Van de Walle et al., 2019). No one’s speed at finding a missing number measures their smartness; understanding the opposite move is worth more than racing. The pair of opposite operations is a human discovery shared across all counting peoples. Count in the home language. For learners who are Deaf or hard of hearing, show join-and-separate with signs and objects; for learners who are blind or low-vision, feel the pile grow and shrink in the hands. Learners with limited movement direct a partner. Mastery arrives at its own pace.

Procedure

1. Gather — opposite moves (5 min). Add three stones to a pile, then take three away. Where are you? Back at the start. Every arithmetic move has an opposite.
2. Worked example — a fact family (10 min). Watch and help: build 3 rows of 4. Write the four related sentences: 3 × 4 = 12, 4 × 3 = 12, 12 ÷ 4 = 3, 12 ÷ 3 = 4. They are one fact family — the same three numbers, four ways.
3. Find the missing number (10 min). Look at 12 ÷ ? = 4. What is the missing number? Do the opposite: 4 × 3 = 12, so the missing number is 3. Try 45 − ? = 27: the opposite is addition — 27 + 18 = 45, so the missing number is 18.
4. Guided practice — build families (12 min). Pick a fact-family card. Build it with objects, write all four sentences, and find one missing-number sentence by doing the opposite.
5. Check with a friend (8 min). Trade cards and read each other’s families and missing numbers. Do the opposite together, kindly, to check.
6. Close (5 min). Remember: addition and subtraction are opposites; multiplication and division are opposites. One fact family is four sentences, and the opposite move finds the missing number.

Differentiation

• Support: Work only on add/subtract families today; add multiply/divide another day.
• Support (blind/low-vision): Build families with objects in cups; say each sentence aloud and feel the opposite move.
• Extension: Write a missing-number sentence for a friend using numbers within 100, and check their answer by doing the opposite.

Assessment

• Formative (observation): Can the learner write a fact family and find a missing number by doing the opposite operation?
• Self: Can the learner explain, in their own words, why 12 ÷ 3 = 4 and 3 × 4 = 12 are the same fact seen two ways?

Home connection

At home, write a fact family for any three numbers you choose (like 6, 7, and 42) and ask a grown-up to find a missing number in it.

Resources

• Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2019). Elementary and Middle School Mathematics: Teaching Developmentally (10th ed.). Pearson. (The inverse relationship between addition and subtraction and between multiplication and division, developed through fact families.)