Comparing Expressions Using Reasoning
Expressions can be compared without calculating by identifying the identical parts and reasoning about the differing parts: adding or multiplying a larger number produces a greater result, while subtracting a larger number produces a smaller result. This Grade 5 math skill from Eureka Math Chapter 22 covers fraction expressions and reasoning about numerical relationships.
Key Concepts
To compare expressions without calculating, identify the parts that are identical in both expressions. Then, reason about the parts that are different to determine which expression is larger. For positive numbers $A, B, C$ where $B C$: $A + B A + C$ $A \times B A \times C$ $A C A B$.
Common Questions
How can you compare expressions without calculating?
Identify the parts that are the same in both expressions, then reason about the parts that differ. Adding or multiplying by a larger number gives a greater result, and subtracting a larger number gives a smaller result.
What is an example of comparing expressions by reasoning?
Compare (4 times 1/2) plus 10 versus (4 times 1/3) plus 10. Since 1/2 is greater than 1/3, the first product is larger, so the first expression is greater.
How does reasoning about subtraction work in expression comparison?
If you subtract a larger number, the result is smaller. So 20 minus (3 plus 5) is greater than 20 minus (3 plus 6), because subtracting a larger sum leaves less.
Why is comparison by reasoning faster than calculating?
By focusing only on the parts of expressions that differ, you can determine which is larger without doing full calculations, saving time and developing deeper mathematical understanding.