Massachusetts Curriculum Framework for Mathematics:
3.NBT.A.3 - Number and Operations in Base Ten
STANDARD: Use place value understanding and properties of operations to perform multi-digit arithmetic. [1]
OBJECTIVE: 3. Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 x 80, 5 x 60) using strategies based on place value and properties of operations.
[1] A range of algorithms may be used
So what exactly does the standard mean?!
Place value is an extremely important concept that we will be emphasizing throughout the year. Place value is the idea that each digit in a number represents a certain amount, depending on the position that it is in (a “digit” is a certain number within a bigger number; the “number” refers to the entire group of digits). For example, a number like 365 has a 3 in the hundreds place, a 6 in the tens place, and a 5 in the ones place. The digit 3, in the hundreds place, does not represent 3, it represents 300. Students understand this concept by using base ten blocks (refer to the image below):
3.NBT.A.3 - Number and Operations in Base Ten
STANDARD: Use place value understanding and properties of operations to perform multi-digit arithmetic. [1]
OBJECTIVE: 3. Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 x 80, 5 x 60) using strategies based on place value and properties of operations.
[1] A range of algorithms may be used
So what exactly does the standard mean?!
Place value is an extremely important concept that we will be emphasizing throughout the year. Place value is the idea that each digit in a number represents a certain amount, depending on the position that it is in (a “digit” is a certain number within a bigger number; the “number” refers to the entire group of digits). For example, a number like 365 has a 3 in the hundreds place, a 6 in the tens place, and a 5 in the ones place. The digit 3, in the hundreds place, does not represent 3, it represents 300. Students understand this concept by using base ten blocks (refer to the image below):
This idea is to help students manipulate numbers and solve problems. If a child understands that 365 is actually 300+60+5, he or she can play around with this number more easily. Understanding this concept will make it easier for students to add or subtract numbers, as well as multiple numbers, which is what we are working on! For example, by understanding place value a student can break down 365 x 3 = (300 x 3) + (60 x 3) + (5 x 3). This method simplifies the traditional multiplication process, which usually causes many students to make mistakes.
The idea that numbers can be taken apart and then put back together gives the student a more solid understanding of how the different operations work. Not only that, but the student can also figure out how to solve problems more independently. Once a child has a good understanding of place value, he or she will have an easier time with addition, subtraction, multiplication, and division. Place value helps teach the logic behind the mathematics.
Examples of How the Different Properties of Operations Work:
1. Commutative Property
2 x 6 = 12 OR 6 x 2 = 12
2. Associative Property
2 x 4 x 3 = 24 which can be found by multiplying: 2 x 4 = 8 then 8 x 3 = 24 OR 4 x 3 = 12 then 12 x 2 = 24
3. Distributive Property
8 x 6 is the same thing as 8 x (4 + 2) = (8 x 4) + (8 x 2) = 32 + 16 = 48
How Will the Student Master the Objective?!
When students master this content they will be able to answer questions using strategies based on place value and properties of operation.
EXAMPLE: Mrs. Thompson buys 8 boxes of crayons. Each box contains 30 crayons. How many crayons did she buy in all?
The students will learn how to multiply by a multiple of ten by using the base ten block strategy based on place value. Base ten blocks are organized by 100's, 10's and 1's. When looking at the number 60 the student will understand that there are 0 1's (60) and 6 10's (60). Therefore, in order to create 60 with base ten blocks the student will need 6 groups of ten. That tells us that 6 x 10 = 60. A common misunderstanding students have with base then blocks is that they use 10 tens for 100. The students should exchange 10 tens for a hundred block.
When the students look at the word problem above, they will know it is a multiplication problem. Based off the information they will know they are multiplying 8 x 30. When they look at the numbers they will know the 8 is eight 1's and 30 is zero 1's and three 10's. In order to multiple one-digit whole numbers by multiples of 10 using the base ten block strategy the student will gather 8 groups of 3 tens using the 10's from the base blocks. The 8 groups of 3 tens means that there is a total of 24 tens. Successful student work will show 24 tens (8 groups of 3 tens) of base ten blocks (refer to image below).
The idea that numbers can be taken apart and then put back together gives the student a more solid understanding of how the different operations work. Not only that, but the student can also figure out how to solve problems more independently. Once a child has a good understanding of place value, he or she will have an easier time with addition, subtraction, multiplication, and division. Place value helps teach the logic behind the mathematics.
Examples of How the Different Properties of Operations Work:
1. Commutative Property
2 x 6 = 12 OR 6 x 2 = 12
2. Associative Property
2 x 4 x 3 = 24 which can be found by multiplying: 2 x 4 = 8 then 8 x 3 = 24 OR 4 x 3 = 12 then 12 x 2 = 24
3. Distributive Property
8 x 6 is the same thing as 8 x (4 + 2) = (8 x 4) + (8 x 2) = 32 + 16 = 48
How Will the Student Master the Objective?!
When students master this content they will be able to answer questions using strategies based on place value and properties of operation.
EXAMPLE: Mrs. Thompson buys 8 boxes of crayons. Each box contains 30 crayons. How many crayons did she buy in all?
The students will learn how to multiply by a multiple of ten by using the base ten block strategy based on place value. Base ten blocks are organized by 100's, 10's and 1's. When looking at the number 60 the student will understand that there are 0 1's (60) and 6 10's (60). Therefore, in order to create 60 with base ten blocks the student will need 6 groups of ten. That tells us that 6 x 10 = 60. A common misunderstanding students have with base then blocks is that they use 10 tens for 100. The students should exchange 10 tens for a hundred block.
When the students look at the word problem above, they will know it is a multiplication problem. Based off the information they will know they are multiplying 8 x 30. When they look at the numbers they will know the 8 is eight 1's and 30 is zero 1's and three 10's. In order to multiple one-digit whole numbers by multiples of 10 using the base ten block strategy the student will gather 8 groups of 3 tens using the 10's from the base blocks. The 8 groups of 3 tens means that there is a total of 24 tens. Successful student work will show 24 tens (8 groups of 3 tens) of base ten blocks (refer to image below).
Since there are 24 tens, the student will count by ten. When the student counts by ten 24 times, they will get 240. Since counting by ten to 240 could be a lot and the student might lose count, the student can break it up. They can break it off by 100. If the student does it that way, the will have 2 groups of 100 and then 40 remaining. It would look like this:
So from here the student can add 100 + 100 + 40 = 240. This means that the student has successful solved 8 x 30 using base ten blocks to master the objective!
How Will the Student Master the Standard?!
By the student grasping the objective, they are able to master the standard.
EXAMPLE: A parking garage has 4 levels. There are 29 parking spots on each level. How many parking spots are there in all?
In order for the students to solve this problem, they will need to use their understanding of place value and the properties of operations. The first step is for the students to identify that the problem is a multiplication problem. The students will write 4 x 29 and by knowing the commutative property, they will know that 29 x 4 is the same thing as 4 x 29. Next the students will use their understanding of place value to break down 29 into (20 + 9). By breaking down 29 into (20 + 9), it makes the multiplication problem easier for the students to solve! Since the students know how to master the objective (multiplying one-digit whole numbers by multiples of 10 in the range 10–90) the students will be able to solve 4 x 20 in this problem! The students will use the distributive property to solve the problem so they will multiple 4 x 20 and get 80. The students know that they can use base ten blocks to help them solve 4 x 20 like they would in the example above for 8 x 30. The students will then multiple 4 x 9 and since the students know their basic multiplication facts they will know 4 x 9 = 36. Finally, the students will add 80 + 36 = 116. By solving this problem using this concept the student successfully solves 4 x 29.
How Will the Student Master the Standard?!
By the student grasping the objective, they are able to master the standard.
EXAMPLE: A parking garage has 4 levels. There are 29 parking spots on each level. How many parking spots are there in all?
In order for the students to solve this problem, they will need to use their understanding of place value and the properties of operations. The first step is for the students to identify that the problem is a multiplication problem. The students will write 4 x 29 and by knowing the commutative property, they will know that 29 x 4 is the same thing as 4 x 29. Next the students will use their understanding of place value to break down 29 into (20 + 9). By breaking down 29 into (20 + 9), it makes the multiplication problem easier for the students to solve! Since the students know how to master the objective (multiplying one-digit whole numbers by multiples of 10 in the range 10–90) the students will be able to solve 4 x 20 in this problem! The students will use the distributive property to solve the problem so they will multiple 4 x 20 and get 80. The students know that they can use base ten blocks to help them solve 4 x 20 like they would in the example above for 8 x 30. The students will then multiple 4 x 9 and since the students know their basic multiplication facts they will know 4 x 9 = 36. Finally, the students will add 80 + 36 = 116. By solving this problem using this concept the student successfully solves 4 x 29.