Visualizing the Associative Property on a Place Value Chart
This Grade 4 Eureka Math skill teaches students to visualize the associative property of multiplication on a place value chart. The key insight is that multiplying by a multiple of 10 (like 40) can be reordered: 40 times 22 = (4 times 10) times 22 = 4 times (10 times 22). On a place value chart, multiplying by 10 shifts all digits one place to the left, making the operation concrete. This strategy from Chapter 16 of Eureka Math Grade 4 supports fluency with two-digit by two-digit multiplication by breaking multiples of 10 into manageable steps.
Key Concepts
The associative property of multiplication allows us to regroup factors. When multiplying by a multiple of 10, we can change the order of operations to make the problem easier to model on a place value chart: $$(a \times 10) \times b = a \times (10 \times b)$$.
Common Questions
How does the associative property help with multiplying by a multiple of 10?
You regroup the factors: (a times 10) times b = a times (10 times b). This lets you first shift b up one place value (multiply by 10), then multiply by a, which may be easier to compute.
How do you use the associative property to compute 40 times 22?
40 times 22 = (4 times 10) times 22 = 4 times (10 times 22) = 4 times 220 = 880.
What happens on a place value chart when you multiply by 10?
Each digit shifts one column to the left, increasing its value by a factor of 10. For example, 22 becomes 220 after multiplying by 10.
Why is it easier to compute 4 times 220 than 40 times 22?
Multiplying a single digit (4) by a number is a simpler step than multiplying two-digit numbers. The associative property lets you delay the harder multiplication until after an easy place value shift.
How does this connect to the area model for two-digit multiplication?
The area model breaks two-digit factors into tens and ones, applying the distributive property. The associative property here shows the tens factor as a series of place value shifts, connecting to both the area model and standard algorithm.