Note the Arrows: Modeling Addition and Subtraction of Integers on Number Lines

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Objective

SWBAT add and subtract integers and represent these operations on a number line.

Big Idea

Students will grapple with integer subtraction concepts including double negatives by exploring direction on number lines.

Do Now + HW Check

15 minutes

Students enter silently, following the Daily Entrance Routine. Assignments are already on their table. A large timer is displayed on the SMART Board giving students 3 minutes to complete 5 integer addition/subtraction problems. If they finish early they are to raise their hands for me to come check their work. If a student gets any number of problems incorrect, I ask them to draw the number line model if they haven’t already done so. If they use the incorrect sign (+/-) in an answer, I ask them to “reconsider” the sign and raise their hand when they’re ready for me to check. When I go back to this student’s seat, I ask them why they changed the sign. At the end of 4 minutes, I call out the answers and ask students to check and ask any clarifying questions.

As I walk around I stop to narrate student work. For example, "Martin is doing a great job drawing positive and negative chips for #2", or "Interesting, Laura has decided to draw a number line to visualize #3". By narrating these examples, other students may take and use them as strategies of their own if they're stuck.

After 4 minutes, we stop and I read off the answers for students to check. I allow 2 minutes for questions on the Do Now. Before students ask questions, I take the opportunity to set the expectation for an appropriate math question:

  • specific and direct
  • instead of "i don't get it" or "I don't understand the whole thing", ask about meanings of words
  • understand what is happening in the problem
  • if students disagree with one of my calculations on the board they should let me know
  • "I don't understand why the answer is negative" 
  • "I did get a negative answer, but not the same number"
  • "Could you show me how you calculated #___, the ___ step?"

 

After we check the Do Now, I ask students to take out their homework and I read off the answers. I stop and take questions for the last 3 minutes of this section. I make sure to fully explain #10 and to alert students that these types of questions should always be done with the use of a vertical number line which is useful for visualizing the distance traveled. I also make sure to explain the meaning of the word "change" and the fact that in a problem that asks about change, the sign of an answer will indicate the direction in which the object is traveling. 

Class Notes + Intro to Lesson

10 minutes

In this lesson students will continue exploring integer addition and will focus on subtraction. We begin our Cornell Notes by filling out the heading and copying the aim off the board. I summarize our work with positive and negative numbers over the last week. I explain that visualization is important for these types of problems because it will help us build rules for more difficult problems. The two visualization tips that are given for the cornel notes are counters/chips and number lines.  Counters/chips are helpful when adding integers. I ask students to imagine blue and red chips on the table. For each example, I ask students to tell me how many of each color counters they are visualizing. I also give them a third example and ask them to write it down and solve (example C: -13 + (-25)

I go on and explain that number lines are helpful when subtracting integers, especially when we see double negatives. Example A in this section can be read as “go left 4 units and then go left 7 units:” Example B has 2 negatives next to each other. This problem could be read as “go right 9 units, then go the opposite of left 5 units”. Since the opposite of left is right, this problem could also be read as “go right 9 units, then go right 5 units.” This idea was also reviewed in the lesson titled “Lines Around the World: Combining and Graphing Integers on a Number Line.

Students are given 3 minutes to complete examples c and d with their neighbor and then we check together. I ask at least 3 students to tell me again, how we could read a problem like example d: -20-(-5). I ask them to use directional words (left, right) and remind them to also use the word opposite. I ask them to tell me how the number line can be useful when determining the sign and numerical value of an answer. Example d can be read as “move 20 units to the left of zero. Then, move the opposite of left 5 units.” I ask students to consider if they will pass 0 again. What does it mean if they won’t? (that we will remain on the negative side of the number line and the answer will be negative)”. Once questions have been answered I transition students to the task by giving them 1 minute to copy their homework and clear their desks of everything except for a pencil.

Task

20 minutes

Students work in partner pairs or independently to complete the Task. MP5 (using appropriate tools strategically) will be in use during this class as students explore rigorous subtraction problems on their number lines (i.e. -17-18 and -17-(-18)). They will need to draw the arrow annotations for each integer addition or subtraction problem. These arrows will visually demonstrate the process of adding and subtracting integers and most importantly, the sign of the answer.

Students must raise their hands to have their work checked by myself or a “chelper” once they are finished with two problems at a time. A “chelper” is a highly coveted job in my classroom. The “chelper” checks for correct answers and helps students with incorrect answers. This individual also gets to use one of my “super fancy, awesome LePens” to chelp. I choose the student(s) who finish first, show all work neatly, and have followed all directions in class. This is a great motivator.

I also choose at least 3 students who display their drawing neatly, to show their answers and work on the SMART board. 

Closing

10 minutes

For the closing, I ask students to retrieve “red” books from the back of the room and turn to a page that includes negative number combinations or complicated subtraction problems. I ask them to choose a problem they feel would be difficult for them. I choose a low, middle, and high achieving student to share their example and we all copy their problem in one of the three blank cells provided at the end of their task paper. I ask student volunteers to read the problem using directional language so that I can draw the arrow annotations on the SMART Board while all students copy the work. At the end, I stop for questions. We close out the lesson by distributing Homework and lining up silently for the next class.