Inverse Operations: Solving Single-Step Mult & Div Equations

Title: Balancing Equations: Solve for the Unknown Number using Inverse Operations (Multiplication & Division)

Materials Needed

• Pencil or pen
• Notebook or whiteboard
• 12-20 small, identical objects (e.g., counters, marbles, cubes) for grouping
• Small plastic cups or circles drawn on paper (to represent the unknown "groups")
• Review materials (notes from the previous lesson on Add/Sub inverses)

Learning Objectives

Building on the skill of balancing equations, by the end of this lesson, you will be able to:

• Identify the inverse relationship between multiplication and division.
• Apply division as the inverse operation to solve single-step multiplication equations for the unknown.
• Apply multiplication as the inverse operation to solve single-step division equations for the unknown.

Success Criteria

You know you've succeeded when you can:

• Accurately determine the required inverse operation for multiplication and division problems.
• Show the step where you perform the inverse operation on both sides of the equation.
• Check your final answer to verify the equation is balanced and true.

Part 1: Introduction (10 minutes)

Review & Connection (The Continuing Case)

Educator (E): Welcome back, Detectives! Last time, we solved cases where the mystery number (X) was being added or subtracted. Quick review: If we have $X + 5 = 12$, how do we isolate X?

(Learner response: Subtract 5 from both sides.)

E: Exactly! We use the inverse operation—subtraction undoes addition—to keep the equation balanced. Today, we're leveling up. We are tackling multiplication and division puzzles, but the core rule of balancing the seesaw never changes!

Hook: Sharing the Loot

E: Imagine you and a friend found a stash of 14 gold coins. You know that when you divide them equally between the 2 of you, you each get the unknown number of coins (X). The equation is $2 \times X = 14$. We need to undo the multiplication to find out how many coins X represents.

Vocabulary Reinforcement

• Inverse Operation: The opposite action. The inverse of multiplication ($\times$) is division ($\div$). The inverse of division ($\div$) is multiplication ($\times$).
• Isolating the Unknown: Getting the variable (X) alone on one side of the equation.

Part 2: Body – Teaching the Concept (Teach it)

I Do: Modeling Multiplication (15 minutes)

Step 1: Introducing the concept of "Groups" (Visual/Kinesthetic)

E: Let's use our physical objects to model the first case: $$3X = 15$$

E: Remember, $3X$ means 3 multiplied by X, or X groups of 3, or 3 groups of X. We have 15 total objects on the right side. On the left, we have 3 empty cups (or circles) representing the 3 groups, and we need to find how many objects (X) go into each cup.

Step 2: Using the Inverse Operation (Division)

E: Since X is being multiplied by 3, we must use the inverse: division. We must divide the left side by 3 and the right side by 3. This is the same as taking the 15 objects and physically splitting them into 3 equal piles.

E: The $3 \div 3$ cancels out, leaving X alone. When we divide 15 objects into 3 equal groups, how many objects are in each group? (5). So, $X = 5$.

Check: Does $3 \times 5 = 15$? Yes, it balances!

We Do: Guided Practice (20 minutes)

Activity: The Detective's Grouping Case (Collaborative Writing/Discussion)

Learners practice solving one multiplication and one division equation, focusing on identifying the inverse operation.

Problem 1 (Multiplication): $$4 \cdot Y = 28$$

• Identify the Operation: Y is being multiplied by 4.
• Apply the Inverse: We must divide both sides by 4. (Write down: $28 \div 4$).
• Solve: $Y = 7$.
• Check: Does $4 \times 7 = 28$? Yes!

Problem 2 (Division): $$\frac{Z}{5} = 4$$

E: This equation means the unknown number (Z) was divided into 5 pieces, and each piece was 4. To get Z alone, we must undo the division. How do we undo division?

(Learner response: Multiplication.)

• Identify the Operation: Z is being divided by 5.
• Apply the Inverse: We must multiply both sides by 5. (Write down: $4 \times 5$).
• Solve: $Z = 20$.
• Check: Does $20 \div 5 = 4$? Yes!

Formative Assessment Check: Ask the learner: "In the second problem, why did multiplying by 5 balance the equation? How is this balancing act similar to using subtraction in our previous lesson?" (Expected answer: Multiplying by 5 undoes dividing by 5, just like the previous lesson where subtracting 5 undid adding 5. We always do the opposite to keep the balance.)

You Do: Independent Application (15 minutes)

Activity: Isolating the Factor Challenge Learners solve three problems independently. They must show the inverse operation applied to both sides and check their final answer. Encourage them to state the inverse operation aloud before writing it down.

1. $$6 \times x = 42$$ (Show your work and check)
2. $$\frac{y}{2} = 9$$ (Show your work and check)
3. $$50 = 10a$$ (Show your work and check. Remember, 10a means $10 \times a$)

Success Criteria Review: Learners should underline the division/multiplication step on both sides of the equation to confirm they balanced it correctly.

Part 3: Conclusion (Tell them what you taught) (5 minutes)

Recap and Reinforcement of Progression

E: Excellent work! We have now mastered two pairs of inverse operations to solve for the unknown. Let’s review our tool kit:

• If you ADD: You use SUBTRACTION to balance. (Previous Lesson)
• If you SUBTRACT: You use ADDITION to balance. (Previous Lesson)
• If you MULTIPLY: You use DIVISION to balance. (Today)
• If you DIVIDE: You use MULTIPLICATION to balance. (Today)

E: The most important rule we learned over these two lessons is that equations must always stay balanced by performing the same inverse action on both sides.

Summative Assessment: The Cumulative Case File

Ask the learner to solve one problem that requires them to choose the correct inverse operation from the four options they now know, and explain their choice.

Final Challenge: $$45 = 9 \times M$$

E: "M is being multiplied by 9. How do you get M alone? What is the inverse operation, and what step must you take to balance the equation?" (Assess both the answer and the explanation of the inverse process.)

Next Steps & Extension

E: We have solved single-step puzzles. Next time, we will put our full toolkit together and solve tricky, multi-step equations where we might have to use addition inverse AND division inverse in the same problem!

Differentiation and Adaptations

Scaffolding (For learners needing more support or visualization)

• Concrete Manipulation: For multiplication problems ($3X=12$), allow the learner to physically count out the total (12 counters) and sort them into the correct number of groups (3 cups) to see the answer directly before writing the formal division step.
• Repeated Subtraction: Explain division as repeated subtraction (e.g., $15 \div 3$ is how many times you can subtract 3 from 15), linking back to the previous lesson’s inverse skills.

Extension (For learners ready for a challenge)

• Introduction to Two-Step Equations (Early Look): Provide a problem that combines the skills learned across both lessons: $$\frac{x}{2} + 3 = 7$$ (The learner must first subtract 3, then multiply by 2).
• Complex Word Problems: Challenge the learner to create a word problem involving sharing/grouping that translates to a division equation with an unknown variable.