Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 3 chapters and video availability01 · Read and understand
What you will learn
- Use a rotated array to justify commutative multiplication.
- Explain your method and check what the result means in the stated situation.
Before you start
Place value through hundreds, addition and subtraction, equal groups, and equal shares.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Explain why 4 × 7 and 7 × 4 have the same value.
Why this math matters
Rotate a rectangular arrangement to choose the easier counting direction. The worked example shows how to connect the situation, its mathematical representation, and a check on the result.
Set up the model
A useful answer starts with clear assumptions:
- Use the units, grouping rules, and reference whole stated in the question.
- Treat the supplied counts and measurements as exact within the classroom model, unless an estimate or a random outcome is requested.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Explain why factor order can change
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Explain why 4 × 7 and 7 × 4 have the same value.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Represent the situation
4 rows of 7 = 28
Draw a rectangle of equal-spaced dots.
Connect the parts
Turn it: 7 rows of 4
The original columns become rows.
State and check the result
4 × 7 = 7 × 4 = 28
No dot was added or removed by turning the picture.
The result
4 × 7 = 7 × 4 = 28
No dot was added or removed by turning the picture.
Common mistakes to catch
- Changing factor order does not justify changing division order.
- Story meanings of factors may differ even when products match.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Which known fact helps with 8 × 3: 3 × 8 or 3 + 8?
Show a hint
Swap the factors, not the operation.
Reveal answer and explanation
3 × 8
Both multiplication facts count the same array.
Practice 2
Does swapping 12 ÷ 3 to 3 ÷ 12 keep the answer?
Show a hint
Commutativity here concerns multiplication.
Reveal answer and explanation
No
12 ÷ 3 = 4, while 3 ÷ 12 is one quarter.
Take the idea with you
Rotate a rectangular arrangement to choose the easier counting direction.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Break a product into useful parts
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.