Being able to count fluently up and back is a critical mathematical skill. Second grade students are expected to be able to add and subtract numbers to 20 using mental strategies by the end of second grade. One way students begin to do this is by counting forward and backward on a number line. The number line serves as a "mental model" for computation.
Today I ask students to take out their Math Journal and find the next empty half page. I put the number 18 on the board and ask students to start at 18 and count to 30, writing the counts in their journal. I circulate around to assess who can do this and who can not. I've included a photo example of my recording sheet. When most of the class is done, we read the numbers aloud together.
I purposely do not start at 1 each time when I ask students to count because, later on when they are adding and subtracting, they start with the larger number and count on or back to find their answer.
Next I put the number 25 on the board. I ask students to start at 25 and count back, writing the counts in their journals. I circulate around to assess how students are doing. When most of the class is done, we read the numbers aloud together.
For this activity I use sentence strips marked out into equal parts, far enough apart for students to stand on the marks. Working with one group of students, I hand 10 students a number and ask those students to arrange themselves on the line with 8 at one end and 18 at the other without talking. They are to show each other the number and move along the line until the numbers are in order from 8 to 18.
Next, one-by-one, I ask a child to stand next to a number. For example, the first child is asked to stand next to 11, holding a red dot. This indicates the starting point, where the second child begins their "hops" on the number line. For example, I give the first child a red dot and ask her to go stand next to 11 on the number line. The second child, who is also given a red dot, is asked to go to the first red dot (11), and hop up 6 and hold their red dot where they stop. I ask the rest of the class what addition sentence we just made. I pick one child to write 11 + 6 =17 on the board.
The idea of hopping goes along with how many people mark a number line when they count up or down. The use of the number line is shown in the number line video which I show to students after they have practiced walking the number line.
I repeat the activity several times and then represent a new set of numbers on the number line. (Several students will still be part of the line during this second round.)
With this new group I ask them to do the same thing but counting back. What is different about the number sentence? (It is now subtraction).
Counting backwards is a skill that is often neglected, but if students are to understand subtraction, they need to understand and be able to use the process of counting backwards. (If I am counting to 10 for students to finish cleaning up, etc, I count down instead of up to reinforce the process of counting backwards.)
I watch carefully for the students to become restless to decide when to stop the activity.
Students have now practiced the number line using a 3D model. It is time for independent work using a practice page: numberline problems. I put the page on the SmartBoard and model the first problem with the students. This is an opportunity for students to do guided practice before working on their own. After that I invite anyone who would like to work with me to come to the rug with their papers. I tell the other students that they may work on their own to complete the page. I know that student experience with a number line varies. Some students used number lines extensively in first grade while others never did. I want those students who are unfamiliar with the tool to join me as we work together using the number line. I want students to model with mathematics using the number line as a tool (MP4). My job working with the students who are struggling is to help them to make sense of the problems (MP1). I want them to look at the problem and think of how the number line might be a model for trying to get an answer.(MP4)
I tell students that I will work with them, so if they are not sure what to do, they can come work with me. Usually students are aware that they do not know what to do and welcome the opportunity to be a part of a group working together. If students are struggling, I may invite them to join us even after we have started our work together. I also encourage students that I feel may have difficulty to come with me.
The group that has come to me may be having some difficulty grasping counting up or back. We work the problems together. I post the page on the SmartBoard, we talk about where to start and whether to count up or back. I show the jumps with swooped lines above the numbers. I observe how the children complete each problem using the number line tool.
I have all students hand in papers at the end of this time so I can assess their understanding of counting up and back (adding on and taking away with numbers within 20 which is a Common Core skill 2.OA.B.2.)
I always take a few minutes to wrap up the lesson before the end of the math time. Today I ask students to tell me what they know about the number line. Is it a math tool? Do adults use anything like a number line? How might the number line help us when we want to add things on or take things away?
Can number lines do numbers above 20? Students can share their thinking using materials and drawings. Allow students to critique the reasoning of others as they consider this question.