Andrea Palmer PROSPECT HILL ACADEMY CHARTER SCHOOL, SOMERVILLE, MA
6th Grade Math : Unit #3 - Integers and Rational Numbers : Lesson #5

# Adding and Subtracting Integers on a Number Line

Objective: SWBAT: • Add and subtract integers using a number line
Standards: 6.NS.C.6c 7.NS.A.1 MP1 MP3 MP4 MP7
Subject(s): Math
60 minutes
1 Do Now - 5 minutes

See my Do Now in my Strategy folder that explains my beginning of class routines.

Often, I create do nows that have problems that connect to the task that students will be working on that day.  Today I want students to review comparing integers.

I ask for a student to explain the order of the balances.  I ask, “How is 10 greater than -20?  Isn’t 20 larger than 10?”  I have students share out ideas about how Terriana could have a negative balance in her bank account.   She spent more money than she had in her account, so she owes the bank money.

If I have time I will ask the following questions:

• What do you think happened between Wednesday and Thursday?
• What do you think happened between Thursday and Friday?

I want students to recognize that her balance decreased between Wednesday and Thursday.  She could have spent more money or been charged a fee by her bank.  I want students to see that between Thursday and Friday her balanced increased from -20 to 0 dollars.  This means money was deposited into her bank.

resources
How can her balance be negative?
Real World Applications

Students were able to order the amounts of money relatively easily.  A student was able to explain that -20 is less than -5, because it is further to the left on the number line.  I asked students how Terriana could have a negative balance.  One student said that she spent all of her money.  I said that if I had \$20 and spent all of my money, I would be left with \$0.  I had students quickly participate in a Think Pair Share.  Then a few students shared that if she had a negative balance, she spent more money than she had in her bank account.  I explained that if you spend more in your account, that it is called over drafting.  The bank covers the purchase, and you owe the bank the money.  I then explained overdraft fees.  As student declared, “That should be illegal!”  I explained that banks are businesses and overdraft fees are one way they make money.  It is very important to keep track of your balance.

2 Football - 5 minutes

I ask students to share out what teams are trying to do in a football game.  Some students may say they are trying to score points and to win.  I share that we will be using football to help us add and subtract integers – it’s the Positive Yellow Team vs. the Negative Red Team.

I introduce the integer football rules.  The yard lines are a little different than regular football.  The middle of the field, where each problem starts, is the 0 yard line.  To the left of 0 are negative yards (-1, -2, etc.) and to the right of 0 are the positive yards (1, 2, 3, etc.).

Connections to Football
Developing a Conceptual Understanding

I had a student read the rules for class.  I explained that my goal for the lesson was to connect adding and subtracting integers to football, but that it wouldn’t be a perfect match.  I emphasized that my goal is to make connections to football so that students understand what is going on in the problem, instead of just learning a series of tricks.

3 Combining Drives - 10 minutes

Notes:

• For this lesson, each student will need a yellow and red writing utensil.
• The purpose of this lesson is for students to practice adding and subtracting on a number line and start to build conceptual understanding of what is actually happening.  I am avoiding telling students shortcuts and teaching them procedures because I want them to start to build their own conceptual understanding of what is going on.  This will not be easy and students will likely struggle, but I think it is important to build a conceptual foundation and have students figure out patterns.
• I would encourage you to go through the examples on your own and familiarize with how you would explain each problem in terms of the football teams.  It took me time to become comfortable in explaining problems in terms of the team and the direction.
• In the guided examples, you may want to replace some of my drawings with blank number lines so students are accountable for drawing something.

We start with combining drives.  This is when both numbers are either positive or negative and they are being added together.  Students will obviously know right away that 5 +2 = 7, but it is important for them to understand what is happening in the context of our football game.   I have students circle both numbers (not the addition sign) with the yellow marker, showing that both drives are by the Positive Yellow Team.  That means that both arrows should be yellow.  We go through problem 2 with the same language, except this time the Negative Red Team is driving towards their end zone.  If they drive -5 yards and then another -2 yards they will end at the -7 yard line.

I have students work in partners.  My students are in heterogeneous partner pairs.  It is likely that this is new material for most students.  I explain to students that the important thing they are doing right now is creating a model (MP4: Model with mathematics).  It is not good enough for students to just write an answer, they need to show what is happening with the football teams.

As students work I walk around and monitor their progress.  If students are struggling, I refer them back to problem 1 and 2 and have them follow the same steps.  If a student just writes an answer, I require them to look at my examples and draw a model of the problem.

Driving Down the Field
Modeling

As an introduction to this section, I showed a short clip of Tom Brady and the New England Patriots driving down the field.  I explained to students that when we add two numbers from the same team, they have two plays where they continue to drive down towards their end zone.

I had a student bring a football to class and we acted out problems.  I explained that unlike real football, we started at the 0 yard line at the beginning of each problem.  During the first Guided Practice problem, I picked a student to play the negative defense.  We acted out each problem in front of the number line.

During the second Guided Practice Problem, I picked two students to act it out and I asked for one student to do the “play by play”.  That student was responsible for talking through the two plays and explaining what happened, who had the ball, where they moved, and where the problem ended.  Students were expected to fill out their notes.  Students really enjoyed acted it out and playing commentator.

4 Interceptions - 15 minutes

Again we go through problems 1 and 2 together.   For problem 1 I have students circle the (-6) in red and the 2 in yellow. That means our first arrow will be red and our second arrow will be yellow.

Students work with their partners on the Interception Practice.  I am looking that students are drawing the models.  A common mistake is that students ignore the signs and add the numbers as if they were both positive or both negative.  If I see students doing this I ask them to color the positive number yellow and the negative number red.  I guide them through the problem the same way as the examples.

When most partners have finished the Interception practice, we come back together as a class. For each example I have one student come to the document camera to show and explain their thinking.  If students just list steps I push them to explain the why they did each step.  Then I ask a student if he/she agrees or disagrees with the student and why.  Students are engaging with MP3: Construct viable arguments and critique the reasoning of others.

Last, I ask them to look at problem 1 and problem 3.  I ask them what they notice.  How can those two problems have the same answer?  I want students to recognize that even though the problems started differently they both included a drive of 6 yards by the Negative Red Team and a drive of 2 yards by the Positive Yellow Team.  Students are engaging with MP7: Look for and make use of structure.

Clarifying Mistakes
Checks for Understanding

I asked for a student to explain what happens when there is an interception in football.  Then I showed a quick video clip of an interception.  I explained that we can tell there is an interception by seeing that one value is positive and one value is negative.  I continued to mention that since the sign was addition, each team was gaining yards and moving towards their own end zone.

I continued to pick volunteers to act out the problems and do the “play by play”.  Students worked on the practice problems independently.  Problem 5 was problematic for some students.  Some students first drew a red line from 0 to -6 and then drew a yellow line from 0 to 6, instead of a yellow line from -6 to 0.  I called on students to act this out.  I asked students, “At which yard line does the yellow team get possession of the ball?”  Students were able to tell me that the yellow team got the ball at the -6 yard line.  Then the yellow team gains 6 yards towards their own endzone.

5 Getting Pushed Back - 15 minutes

Using football, if you subtract an integer it means that that color team gets pushed back, or pushed away from their end zone a particular number of yards.  I ask a volunteer to act out this motion.  I am on the Positive Yellow Team and I have the ball.  The Negative Red Team overpowers me and pushes me back, away from my end zone.  I don’t use the words “loss” or “gain” here because I don’t want students get confused.

We go through the example problems in a similar manner.  I have students circle -3 in red and circle 2 in yellow.  That means that the Negative Red Team will have a drive and then the Positive Yellow Team will intercept it.  Because we are subtracting a positive 2, the Positive Yellow Team is getting pushed back 2 yards.  The result is that the problem ends at negative 7.

We go through the other examples in a similar fashion.  If needed, I have a student come up and help me act out what is going on, then we review it on the number line.

This section may prove to be the most confusing for students.  Take it slowly and act it out.  Emphasize that when we are subtracting integers, that team is getting push back, the opposite of adding integers.

Loss of Yards
Modeling

For this section, I showed a clip of the patriots getting a gain of yards on one play and then Brady getting sacked on the next play.  I asked students how a team can lose yards, or get pushed back from their end zone, in football.  Students mentioned penalties, sacks, and getting pushed back by the defense.

I explained that when there is a subtraction sign, the team of the second number is losing yards, or getting pushed back.  I explained that in problem one, the negative team drives -3 yards for the first play.  Then during the second play, the yellow team intercepts the ball.  But since it is subtracting 2, that means that the yellow team gets pushed back 2 yards, away from their end zone.

For problem 2, I asked students who had possession of the ball on the first and second play.  A common mistake was for students to think that the second play involved the red team.  I called for volunteers to help me act out the problem.  They started at the 0 yard line and gained 3 yards towards their end zone.  Then in the second play, their quarterback was sacked and they lost five yards.  The play ended at the -2 yard line.

6 Closure and Ticket to Go - 10 minutes

For Closure I ask students these questions:

• What happens when we add two positive numbers together?
• What happens when we add two negative numbers together?

I want students to realize that if we add two positive numbers together, the result will always be a positive number and that if we add two negative numbers together, the result will always be a negative number.  Students will look for more patterns in the next lesson using integer chips to model problems.

I pass out the Ticket to Go for students to complete independently.  At the end of class I pass out the HW Adding and Subtracting Integers on a Number Line.

Ticket to Go Work
Exit Tickets

I collected the tickets to go to see what students understood about adding and subtracting integers and what gaps in understanding they had.  I corrected the tickets to go and grouped them in the following way:

Novice:  These students struggled to connect the problem, model, and answer.  This particular student thinks that (-2)+(-2) is -4, but her model involves yellow and red, and does not end at -4.  For problem 2, she is able to create an accurate model, but says the answer is -9.  She is not using her markers to color the integers in the problem by their team color.  She has some instinctive answers, but needs more practice with creating model and using it to support her answer.

Approaching Mastery: These students were able to identify which team possessed the ball during each problem.  They were able to create accurate models for 1 to 2 of the problems.  A common problem was counting the yards on the number line.  Some of the mistakes involved counting too many or too few yards.  In class in the next lesson I will model counting yards on the number line by making hops.

Proficient:  These students were able to accurately identify which team had possession of the ball and they were able to accurately model the problems.  If there were any mistakes, they were small ones.  For instance, this particular student for problem one drew the first drive from 0 to -2 and then the second drive from -3 to -5.  He then realized his mistake, and adjusted this model leaving a gap.  These students understood for problem 3 that the yellow team possessed the ball for each play.  They were able to show that the yellow team lost 6 yards during the second play.

For this situation, I did not include an “advanced” category because the content did not require students to explain or analyze their work.   Most students were able to identify which team possessed the ball and could accurately model 2 to 3 problems.  Students will learn how to use counters to represent adding and subtracting integers.  I want students to be able to use both strategies and see the connections between the methods.