Carla Seeger AF Amistad Academy Middle School, NEW HAVEN, CT
6th Grade Math : Unit #3 - Integers and Rational Numbers : Lesson #2

Integers in the Real World

Objective: SWBAT use integers and number lines to represent quantities in real-world contexts.
Standards: 6.NS.C.5 6.NS.C.6 6.NS.C.6a MP1 MP2 MP6 MP7
Subject(s): Math
60 minutes
1 Think About It - 7 minutes

Students work in pairs on the Think About It problem.  The point of this problem is to have students thinking about integers in a real-world context.

After 2-3 minutes of partner work time, I have students share out what they came up with for the number sentences.  I also use this conversation as a way to have students articulate that integers are all positive and negative counting numbers, including zero (which they learned in the previous lesson).  

In the conversation about the problem, I want the students to conclude that even though Emari started with less money, he ends up with more money in his account than Elana.  

2 Intro to New Material - 15 minutes

In the Intro to New Material section, students will be exposed to both horizontal and vertical number lines, applied to real-world situations.  

To start this section, we work with the Think About It problem.  Students are not likely to be familiar with the word 'withdraw,' so we spend time talking about what that means with money.  We also talk about not using '+' in front of positive numbers - it is assumed that numbers are positive, so we don't need it.

For the number line in this situation, we'll place 0 ("zero") in the middle of this number line, and then count by 1s, in both directions.  I do make sure, though, that students see a variety of number lines in this unit (and beyond!) - I don't want students thinking that the origin always has to be in the middle of the visual.  I also want students in the habit of checking the scale on a number line, and not working under the assumption that the hatch marks count by ones.  

After students read the second problem, I have them paraphrase what the problem is asking us to do. This is an important step in the sense-making process.  Once students have identified that we need to represent the problem with integers, plot the numbers, and interpret the 0, I then have students turn to their neighbor and talk about why a vertical number line makes sense in this situation.  I guide students through this problem. 

 

3 Partner Practice - 15 minutes

Students work in pairs on the Partner Practice problem set. 

If, during the Intro to New Material section, it feels like students might need more support from me while working on a vertical number line, I will have pairs complete only Problem A in this problem set.  We'll come back and check in before I release them to partner practice with less support from me. 

As students work, I circulate around the room and check in with each group.  I am looking for:

  • Are students annotating/paraphrasing the question, using + and - for each integer
  • Are students drawing their number lines with hatch marks evenly spaced and correctly labeled?
  • Are students drawing vertical number lines when it fits the context?
  • Are students interpreting the given contexts correctly?
  • Are students answering in complete sentences?

 

I am asking:

  • How did you know to represent the context with that integer?
  • How did you know to draw a vertical/horizontal number line?
  • How did you know to label the number line with those integers?

 

After 10 minutes of partner work time, students complete the Check for Understanding problem independently.  I then show one student's work on the document camera.  The student explains his/her thinking.  The class then offers the student positive and constructive feedback.

 

4 Independent Practice - 15 minutes

Next I have my students work on the Independent Practice problem set.  The problems in this set expose students to many vocabulary words that they may not be familiar with.  I keep a vocabulary list on the board (or a wall) throughout this unit. Many of my students will access it as they work on these tasks.

Problem 5 is a novel problem, and students may be stumped when they first see it.  As I circulate, I'm looking to see that students are working out the different pieces of this problem and are plotting them correctly on the number line. 

Problem 9 asks students to apply what they learned previously about absolute value to a real-world situation.  

Novel Problems
Connection to Prior Knowledge

Initially, there were students who were stumped by the novelty of Problem_5.  In terms of what the problem asks students to do, the difficulty level of this problem is not that high.  Some students were initially thrown by the format of the problem.

When I saw a student was a bit stuck, I made a few moves:

  1. I asked the students to tell me what the problem was asking them to do.  The vast majority of my students were able to make sense of the problem enough to know that they needed to plot points on the number line.
  2. I asked students what they've tried already.  If they don't have an answer to that, I'll give them a chance to try something.  I'll either say that I'll stand right here for a moment while they try something, or I'll give them a concrete point at which I'll return (ex - okay, you try something.  I'm going to peek at Juan's paper, and then I'm coming right back to you).
  3. I asked students why they didn't just plot 68 ÷ 17.  Students would tell me that this isn't a singular point on a number line.  I'd push back and ask them if they knew how to find the quotient for this problem.  That'd often be enough for the light bulb moment - oh, we need to evaluate each of these, before plotting them on the number line.  

Another tactic I used was to say something along the lines of "Hmm. Well, I know that d would be plotted right here (putting my finger on 3).  How'd I know that?" before strolling away casually.

Ultimately, when my students are stuck, I do not think that my role is to tell them the answer or to tell them exactly what they should do to find the answer.  My role is to gently nudge students towards a possible direction that will lead towards the solution, in other words:

  • to help them reflect on what they know
  • to help them reconsider what they have tried
  • to help them to better understand the structure of the problem
  • to help them brainstorm a different approach
  • to help them persevere

I don't want my students relying on me, whenever they are stuck.  

 

5 Closing and Exit Ticket - 8 minutes

After independent work time, I have the class come back together.  I have three students share out their scenarios for Problem 10 and three students share out for Problem 11.  After each share, the class votes with their thumbs about whether or not the scenario represents the appropriate integer.

Students work independently on the Exit Ticket to close the lesson.