Mathematics

1.NBT.C.5 Transcript

This is Common Core State Standards Support Video in Mathematics; the standard is 1.NBT.C.5. This standard states: Given a two-digit number, mentally find 10 more or 10 less than the number without having to count and explain the reasoning used.

This implies the necessity of a solid foundation in understanding place value. Back in kindergarten with standard K.NBT.A.1, students were introduced to this. The standard states: Compose and decompose numbers from 11 to 19 into 10 ones and some further ones, for example, by using objects or drawings and record each composition or decomposition by a drawing or equation such as 18 equals to 10 plus 8. Within the same grade level, we have this other standard 1.NBT.B.2. It connects back to standard 1.NBT.C.5. This additional standard states: Understand that the two digits of a two-digit number represent amounts of tens and ones and understand the following as special cases. So, we do have some pieces to this. Part 2a states: Ten can be thought of as a bundle of 10 ones called a ten; 2b states the numbers from 11 to 19 are composed of a 10 and 1, 2, 3, 4, 5, 6, 7, 8, or 9 ones. And then the third component here, 2c, states: the numbers 10, 20, 30, 40, 50, 60, 70, 80, and 90 refer to 1, 2, 3, 4, 5, 6, 7, 8, or 9 tens and 0 ones.

So, there’s a big emphasis here on place value. Within this same cluster, we have standard 1.NBT.C.6 that deals with subtracting multiples of 10 in the range 10 through 90 from multiples of 10 in the range of 10 through 90, but you need to have positive or zero differences. And you have to use concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. Standard 1.NBT.C.5 lays the foundation for a second grade standard, 2.NBT.B.8, that states: mentally add 10 or 100 to a given number 100 through 900, and mentally subtract 10 or 100 from a given number 100 through 900.

The key idea here is that students have to mentally find 10 more or 10 less than a number: no calculators, no pencil and paper. They have to do this mentally. So let’s take the number 28. If we go to the right, that would be adding 10 in that direction. If we go to the left, subtract 10. If we think of this in terms of a number line, this is what we would have. We start with 28, add 10 it’s 38; subtract 10, it’s 18. However, this is first grade. There are no number line expectations at this grade level. Number lines don’t appear in the Common Core until second grade. So let’s table this right now because, again, this is not an expectation.

So what else can we do mentally? Well, let’s say we start with 37, and we’re going to add 10. Students have worked with manipulatives dealing with tens and ones. So if they can visualize that here, I would have 3 tens and 7 ones. And if I’m adding 10, I would throw in an additional 10. So now I have 4 tens and 7 ones, which would be 47. A key observation here would be that nothing happened with the ones since I’m adding zero. Let’s take 37 again, but this time, we’ll subtract 10. And again visualizing our 3 tens and our 7 ones, all we have to do is take 1 ten away. So we have 2 tens and 7 ones, which is 27. Again, key observation; nothing happened with the ones.

Prior to the expectation of students doing this mentally, well they need experience actually doing it in writing. Now when they do these written examples, they’ll start seeing the patterns when you’re adding and subtracting 10. For example, if I start with 24 and I add 10, I get 34, or when I subtract 10 from 46, I get 36, and they start seeing the pattern that nothing changed with the ones place. The change happened with the tens place. By doing additional examples, students start to get this idea really cemented in their minds that again, when you’re subtracting 10 or adding 10, nothing changed with the ones place. It’s only the tens places that’s getting affected.

Now, here’s a little bit of a different example. Let’s say we start with a number in the teens like 16. If I subtract 10, in this case I would get six. We typically don’t put a 0 in front of the 6, but it’s just a little bit of a difference from the other numbers that they’re dealing with. Now what if you have a number in the nineties? What about something like 93? We know that we’re going to get a three-digit solution. There is standard 1.NBT.A.1 in this same grade level where students are expected to count to 120. So they are expected to do something, know some things about three-digit numbers. So having them add 10 to a number in the nineties would be an expectation here. So for example, here 93 plus 10 is 103. It should be a problem that the students could handle under this standard.

Let’s take 53 plus 10, which is 63. To really be able to do this mentally, students have to transition from seeing this is as 10 ones to seeing this as 1 ten. Now that’s critical. It’s almost like okay, we’re not even dealing with the ones place. It’s just the tens. So mentally, it’s almost like the ones place isn’t even there. I’m only going to be dealing with the tens place. So I would take 53. I’m only concerned with the five, with the tens place. I’ll add 1 ten to that, which gives me 6 tens. So again, it’s only the tens that we’re dealing with. I’m adding 1 ten to that. With the ones place nothing happens. There’s no change because I’m adding nothing. I’m adding 0 ones. So when students see problems like these where they’re adding or subtracting 10 mentally, it’s almost like they eliminate the 0. We’re only dealing with the tens place. So they do the computation, either add one or subtract one depending on the context. And then, they can visualize the 0 being there. But again, there’s going to be no change with the ones place.

So with enough experience, students should be able to start with a number, say like 53, add 10 and get 63 and keep going. Add another 10, that’s 73. And then 73 plus 10 would be 83. Of course, the expectation would be that they should be able to take 10 less than a number also. So if I started with 53 and I wanted 10 less, that would be 43; subtract 10 from that, 33; subtract 10 from that, 23. Again the key is simply to have a lot of practice but focus on the idea of place value. That’s the key.

If we look at our standards for mathematical practice, if you look at the first four, we did apply two, three, and four. Students would reason abstractly and quantitatively doing the activities in this standard. They would construct viable arguments and critique the reasoning of others, and they would model with mathematics. If we look at the last four standards for mathematical practice, students, by doing this standard, would need to attend to precision. They would look for and make use of structure, and they would also look for and express regularity in repeated reasoning.