Question

look at the expressions. explain how you can use the factors you see to compare without computing.
the same 3 factors are on each side of the circle i know that the same factors grouped in a different way have the same product. so, the expressions are equal
④×(6×2) 〇 (6×2)×④
patterns and structure
wrote the expression at the left of the circle below. riaz wrote expression at the right of the circle below. without computing, find which of the symbols >, <, or = should be placed in the circle.
maria
4×8×3 〇 riaz
2×3×8

1. tell how you can use the structure of mathematics to complete the ta
2. look at the expressions. explain how you can use the factors you see to compare without computing.

Explanation:

Part 1: Determine the symbol between \( 4\times8\times3 \) and \( 2\times3\times8 \)

Step1: Identify common factors

Both expressions have factors \( 8 \) and \( 3 \). Maria's expression has an additional factor of \( 4 \), Riaz's has an additional factor of \( 2 \).

Step2: Compare the unique factors

Compare \( 4 \) (from Maria) and \( 2 \) (from Riaz). Since \( 4>2 \), and the other factors (\( 8\times3 \)) are the same, Maria's product is larger.

Step1: Rearrange Riaz's expression

Using the commutative property, \( 2\times3\times8 = 3\times8\times2 \).

Step2: Compare with Maria's expression

Maria's expression: \( 4\times8\times3 = 8\times3\times4 \) (commutative property). Now compare the last factors: \( 4 \) (Maria) vs \( 2 \) (Riaz). Since \( 4 > 2 \), \( 4\times8\times3>2\times3\times8 \).

1. Identify Common Factors: Both expressions share \( 8 \) and \( 3 \).
2. Compare Unique Factors: Maria’s expression has \( 4 \) as the remaining factor; Riaz’s has \( 2 \).
3. Determine Relationship: Since \( 4 > 2 \) and the common factors (\( 8\times3 \)) are identical, the product with the larger unique factor (\( 4 \)) is greater.

Alternatively, using the commutative property: Rearrange both to \( (8\times3)\times4 \) (Maria) and \( (8\times3)\times2 \) (Riaz). The first term \( (8\times3) \) is equal, so compare \( 4 \) and \( 2 \); \( 4 > 2 \), so \( 4\times8\times3 > 2\times3\times8 \).

Answer:

\( 4\times8\times3 \boldsymbol{>} 2\times3\times8 \)

Part 1 (Alternative using Commutative/Associative Properties):