We Are a Family!
Lesson 15 of 20
Objective: SWBAT add, subtract, multiply and divide integers by using fact families.
Students enter silently according to the daily entrance routine and work independently to answer two word problems about change and difference. Two students will have sticky notes on their papers directing them to solve each problem on the board and then go back to their seat to finish the rest of the Do Now quickly. These two students must be trustworthy in terms of behavior and urgency to complete the problem given as well as the rest of the problems on the Do Now.
We review each problem by evaluating students’ work on the board. I give students 1 minute to discuss the work displayed on the board with partners. During this time I walk around to ensure that students are completing the task and showing work, including number line models to visualize positive vs. negative change. After one minute is up, I ask students to volunteer their opinion about whether the work displayed on the board is correct (MP3). To check for understanding I ask the following questions:
- What is different about the two answers for #1? Why is one positive and the other is negative?
- Question #2 asks about difference and questions #1 asks about change. Why is the answer to #2 positive while one of the answers to #1 was negative?
If there is time left, I take questions on the Do Now or yesterday’s task.
Intro to Lesson
Students will travel to 3 stations around the room to discover the rules for dividing integers and to practice adding, subtracting, and multiplying integers. My classroom is set up in an array of 16 tables (4 by 4), with room for two students each. Each of my 3 classes includes 21 - 23 students. A room layout document included in the resource section shows how I divided up the room. Students were divided into groups according to the results on the paired problem solving task from the previous day. Students who do not score well on the paired problem solving must begin at station 1 to review adding, subtracting, and multiplying integers. Students who score 75% - 85% on the paired problem solving begin at station 2, and students who score 86% - 100% have the opportunity to sit in booths to begin at station 3. The following details the expectations at each station:
Station 1: Must begin with Part 1, then Part 2 and Part 3 at the end; Work with me to review of addition, subtraction, and multiplication using manipulatives, (i.e. counter chips)and have the option to remain in the front for the rest of class working with me ; students may not move on to station 2 u until they complete all questions within this station and have them checked by me.
Station 2: Have the option with begin with Part 1 or Part 2. Will work in groups of 3 - 4 and must complete their work and check answers provided upside down inside a box top within 10 minutes. If the group has a question they cannot find an answer to together, they may send one student to ask me while I am working with students at station 1.
Station 3: Have the option to begin with any Part. Student who score well on the paired problem solving activity from the previous day were called by highest percentage earned and given the option to sit in a booth for the remainder of class. The rest of the students in this percentage bracket are asked to sit in the tables adjacent to the booth area (shown in diagram/resource); expected to complete the entire packet within 20 minutes, including checking answers also included in box top; students who meet these expectations receive two achievement points and a seat at a booth the next day in class.
This station includes 12 fluency problems using addition, subtraction, and multiplication of integers. I will be working with students to use a visual model to conceptualize the effect of negative numbers on these operations. I ask questions to assess whether they understand the role the additive inverse has on subtraction. Students usually struggle most with problems which use multiple negative signs in one example. For example, when reviewing #6, –16 – (–5) students often don’t know how to interpret the double negative. I explain it with two examples and leave it up to them to decide which explanation they prefer. If we visualize a number line and move to –16, the double negative tells us to go the “opposite of left”. One negative symbol says “opposite” the other says “left”, and since the opposite of left is right, that means I’m adding. In this type of example I’m attempting to use vocabulary and get kids to think about the meaning of these mathematical symbols. Some students cannot follow this logic and so I try to explain using the additive inverse. The additive inverse allows us to change subtraction expressions to addition expressions by “adding the inverse/opposite”. Question 6 can thus be explained as having an equivalent expression with the use of the additive inverse: –16 – (–5) = –16 + (+5) and we can then use the number line or chip models to find the final answer (–11).
When the first 12 problems have been completed I ask students to work on the last three questions with a partner while I go check on the other stations.
At station 2 students use fact families and patterns to investigate division of rational numbers (MP8). Students complete three examples like the one below and use their answers to complete the statements along the bottom about the rules for dividing integers.
Common mistakes students commit at this station include dropping signs or writing the incorrect answer. These students are reminded that fact families use the same three numbers throughout each set. The signs of the numbers don’t change either.
–2 × –3 = _6_
–3 × –2 = _6_
6 ÷ –3 = _–2_
6 ÷ –2 = _–3_
Students working at station 3 will be able to sit in the coveted booths along the side of the room with a textbook (“Big Ideas Red”, Big Ideas Learning)
They will be asked to complete 15 problems from the chapter test and enter their answers into clickers. At the end of the limited time I will stop the assessment so that they can review their answers.
During the last 10 minutes of class, I will be asking students to copy work on the boards for problems they found most difficult. Once we begin working with multiplication and division of integers, students tend to get confused about the rules. For example, they might not understand why the sum of two integers is negative because of the rules "a negative times a negative equals a positive". This rule is sometimes mistakenly remembered as "two negatives equals a positive", omitting the operations and generalizing to all operations. I use clicker results to hold students accountable for completing the work (those who complete less than 50% of the work are kept for lunch detention) and to keep track of student mastery within each skill. When we review the answers, students will be largely responsible for explaining the answers to each other. I will also ask about the strategies students are using to solve these problems (i.e. rules for adding/subtracting integers, use of additive inverse, visual models).