Question

1. a. draw a group of 2 circles in the first box. draw six times as many circles in the boxes to the right. b. use the model to complete the sentence and multiplication equation. is 6 times as many as 2. = × 2 4. complete the descriptions and equations for each model. a. 4 circles times as many is times as many as 4. = × 4 b. 3 circles times as many is times as many as 3. = × 3 5. how are the multiplication equations you wrote in problem 4 alike? how are they different?

Explanation:

Step1: Analyze problem 3b

We know that we need to find a number that is 6 times as many as 2. Multiplication is used for such 'times - as - many' problems.
$6\times2 = 12$
So the number is 12 and the equation is $12=6\times2$.

Step2: Analyze problem 4a

There are 4 circles in the first group and there are 3 groups of 4 circles in the second part. So the number of circles in the second part is 3 times as many as 4. The equation is $12 = 3\times4$.

Step3: Analyze problem 4b

There are 3 circles in the first group and there are 4 groups of 3 circles in the second part. So the number of circles in the second part is 4 times as many as 3. The equation is $12=4\times3$.

Step4: Analyze problem 5

Alike: In problems 4a and 4b, the product in both multiplication equations ($12 = 3\times4$ and $12=4\times3$) is the same. This is because of the commutative property of multiplication which states that for any two numbers $a$ and $b$, $a\times b=b\times a$.
Different: The factors are switched. In $12 = 3\times4$, the first factor is 3 and the second is 4, while in $12=4\times3$, the first factor is 4 and the second is 3.

Answer:

3b: 12 is 6 times as many as 2. $12 = 6\times2$
4a: 3 times as many as 4. $12=3\times4$
4b: 4 times as many as 3. $12=4\times3$
5: Alike - The product is the same due to the commutative property of multiplication. Different - The factors are switched.