Questions: Multiplication with Arrays

Question 1

You arrange 24 tiles into a 4-by-6 array. Then you rotate the entire arrangement 90 degrees. What is true about the new arrangement?

• A. The product changes to 4 + 6 = 10 because the rows and columns switched
• B. The total is now 6 × 4 = 24 — the same product, but rows and columns have swapped
• C. You now have a 4-by-6 array again, because rotation doesn't change anything
• D. The array no longer works for multiplication because the shape changed

Answer: (B)

Rotating the array swaps the rows and columns: a 4-by-6 array becomes a 6-by-4 array. But the total number of tiles hasn't changed — there are still 24. This is the visual proof of the commutative property: 4 × 6 = 6 × 4. Arrays make this insight concrete and undeniable — you can see with your own eyes that the total doesn't change when you swap the factors, because you're just looking at the same objects from a different orientation.

Question 2

What does a 3-by-4 array drawn on grid paper have in common with a rectangle that is 3 units tall and 4 units wide?

• A. They are the same shape drawn in different ways — no mathematical connection
• B. Both contain exactly 12 unit squares, connecting multiplication to area
• C. Both can be used to practice addition but not multiplication
• D. The rectangle shows subtraction while the array shows multiplication

Answer: (B)

The 3-by-4 array has 12 objects arranged in 3 rows of 4. The 3-by-4 rectangle on grid paper has 12 unit squares, with 3 rows of 4 squares each. They are mathematically identical. This connection is profound: every multiplication problem can be visualized as a rectangle, and every rectangle's area can be found by multiplication. This is why area is taught using multiplication — the concepts are the same thing expressed differently.

Question 3

In an array, 'rows' refers to the vertical columns running up and down.

• T. True
• F. False

Answer: False

Rows run horizontally — left to right — like the rows of seats in a theater. Columns run vertically — up and down — like the columns of a building. In a 3-by-4 array, the '3' counts the rows (there are 3 horizontal lines of objects) and the '4' counts the columns (each row has 4 objects). Mixing up rows and columns is the most common error with arrays, and it matters when connecting arrays to area problems with labeled length and width.

Question 4

A 3-by-4 array and a 4-by-3 array both contain 12 objects, which visually demonstrates the commutative property of multiplication.

• T. True
• F. False

Answer: True

This is exactly the insight arrays are designed to reveal. Both arrangements use the same 12 objects — they're just organized differently. The 3-by-4 array has 3 rows of 4; the 4-by-3 array has 4 rows of 3. Rotating one gives you the other. With equal groups alone, it's hard to see why 3 groups of 4 should equal 4 groups of 3. With arrays, it's visually obvious — the same total, just oriented differently.

Question 5

How do arrays show that 3 × 4 = 4 × 3? Describe what happens visually.

Model answer: A 3-by-4 array has 3 rows and 4 columns, for 12 objects total. If you rotate the array 90 degrees, you get a 4-by-3 array — 4 rows and 3 columns — which also has 12 objects. The same objects are arranged differently, but the total count stays the same. This shows that swapping the order of the factors (3 × 4 vs. 4 × 3) doesn't change the product.

This is the power of arrays over equal-groups thinking. With equal groups, 3 groups of 4 and 4 groups of 3 look very different and require separate reasoning. With an array, rotating the grid turns one into the other instantly, making the commutative property obvious. The visual proof is more convincing than any abstract rule.