Linear Thinking

Overview

In this 30-minute lesson, students practise multiplication and division of whole numbers and decimals using number sentences, arrays/area models, and clear strategies for solving problems. The focus is on reasoning and communicating methods, matching the Grade 6 Ontario expectations.

Learning intentions

• Students will be able to solve problems involving multiplication and division using efficient strategies.
• Students will be able to interpret and write number sentences to represent situations.
• Students will be able to explain their thinking using math language and clear steps.
• Students will be able to check whether answers are reasonable.

Success criteria

• I can choose an operation (× or ÷) and write a matching number sentence.
• I can solve multiplication/division problems accurately with whole numbers and decimals.
• I can show my method using a model, diagram, or written steps.
• I can estimate or use an inverse operation to check my answer.

Curriculum links

• Multiplication and division of whole numbers and decimals using strategies and properties
• Number sense: estimation, reasonableness, and mental math
• Mathematical reasoning: explain and justify solution strategies

Lesson structure (30 minutes)

1. 0–3 min | Warm-up mental math

• Teacher reads two short prompts aloud (e.g., “6 × 25” and “3.6 × 0.5”).
• Student shares a quick strategy and estimate first, then computes exact answers.

2. 3–8 min | Model the situation (× then ÷)

• Present a word problem: “A package has 3.5 kg of supplies. How many kg are in 8 packages?”
• Teacher guides student to represent it as a multiplication sentence and uses an area/strip model to connect decimal × whole number.

3. 8–13 min | Guided practice: solve and explain

• Provide a second problem: “You have 28 litres of juice. You pour it evenly into 7 bottles. How much in each bottle?”
• Student writes the division sentence and solves. Teacher prompts for explanation: “How do you know your answer makes sense?”

4. 13–20 min | Flip it: inverse reasoning

• Teacher gives two connected questions:
• “If each bottle has 4 litres, how many bottles can you fill with 28 litres?”
• “If you filled 7 bottles, how can you check by using multiplication?”
• Student uses inverse operations to verify (division → multiplication check).

5. 20–25 min | Independent problem set (2 items)

• Student completes two short tasks:
• Task A: “Compute 6.4 ÷ 0.8” using a strategy (e.g., scale by 10 or use a related whole-number fact).
• Task B: “Write a number sentence for: ‘A bike costs $325. What is the total cost of 4 bikes?’ then solve.”
• Teacher circulates by asking targeted questions (in 1:1 setting, the teacher works at a steady pace and checks each step).

6. 25–28 min | Reasonableness check

• Student chooses one answer and performs a quick estimate (rounding) and/or an inverse check.
• Teacher prompts: “Is your answer larger or smaller than your estimate, and why?”

7. 28–30 min | Exit reflection

• Student completes a short response: “One strategy I used was… I checked my work by…”
• Teacher notes common errors to plan next lesson.

Resources

• Mini whiteboard or paper for number sentences and working out
• Base-ten blocks or decimal area model template (for decimals)
• Printed problem cards for the four main prompts
• Rounding/estimation reference sheet (e.g., “round to nearest whole/tenth”)
• Pencil, eraser, and ruler (optional for strip/area diagrams)
• Timer for warm-up and independent practice
• Assessment checklist (teacher-made, based on success criteria)

Assessment

• Observation during warm-up and guided practice: operation choice and strategy use
• Review of student written work for accuracy, clear number sentences, and models
• Exit reflection for reasoning about reasonableness and checking methods

Differentiation

• Support:
• Provide sentence frames: “This is multiplication because…”, “I check using the inverse: …”
• Offer a partially completed area model and ask the student to fill in missing parts.
• Allow the student to use a calculator only after attempting an estimate and showing the method.
• Targeted scaffolding for decimal division:
• Teach a single consistent approach (e.g., scale both numbers to remove decimals, then divide).
• Provide one worked example before independent Task A.
• Extension (advanced learners):
• Add a challenge: “Create a word problem that matches 6.4 ÷ 0.8 and explain how you know the correct operation.”
• Ask for two different solution methods and a comparison of efficiency (e.g., scaling vs. related fraction/fact).
• EAL/SEN:
• Use clear, repeated language and visuals; reduce reading load by highlighting key numbers.
• Provide a vocabulary prompt list for actions: “evenly,” “total,” “each,” “how many in,” “check.”
• Break tasks into smaller steps with frequent teacher check-ins.

Extension (optional)

• Create a 2-item “choice board” for next time: one multiplication-decimal problem and one division-decimal problem, both requiring an explanation and a reasonableness check.