Getting Positive About Integer Arithmetic

  • Getting Positive About Integer Arithmetic

    Grades: 6th to 8th
    Periods: 1
    Author: Jessica Bishop, Lisa Lamb, Ian Whitacre, Randolph Philipp, and Bonnie Schappelle


    Instructional Plan

    This instructional plan is based on the MTLT article, “Beyond the Sign Rules " by Jessica Bishop, Lisa Lamb, Ian Whitacre, Randolph Philipp, and Bonnie Schappelle. In this lesson, we share a series of tasks that can support students to order and compare signed numbers and develop multiple ways to reason productively about integer addition and subtraction. The lesson is designed to engage students with integers and integer operations through interaction with peers, independent reflection time, whole-class conversation, and relevant representations/tools. 

    0 ICON PDF  Integers Student Handout


    Begin the lesson by asking students to compare pairs of integers on individual whiteboards or use the provided student handout. The purposes of the initial task are to (re)introduce negative numbers, elicit the idea of magnitude (for example, count a number of “things”) and compare it to order (for example, recognize the ordered nature of numbers, often by using a number line), and collectively generate and locate numbers on a number line. After writing each pair of integers on their whiteboard, students should circle which is larger, write the equals sign if equivalent, or “?” if they don't know. We suggest having students initially work independently on the comparisons. 

    -5      5

     0     -5

    -5     -6

    To encourage student interaction and engagement, we suggest discussing each pair of numbers (i.e., -5 vs. 5) one at a time, using a think-pair-share format. After providing time for independent solutions and partner discussion, lead a brief whole-class discussion of the given number pair. During partner talk, each student can share which number they think is larger and why. If they disagree, each person should be able to explain their partner's reasoning and consider whether they want to revise their thinking after hearing their partner's thinking. 

    0 ICON PDF  Vertical Number Line

    0 ICON PDF  Horizontal Number Line

    When discussing the number comparisons as a whole class, consider having a student draw a number line at the board and record the location of each pair of numbers on the number line. You may also consider providing students with a copy of the vertical or horizontal number line provided. Key ideas you may choose to highlight in your discussion include magnitude (e.g., there are more negatives in -6 than in -5), distance from zero, opposites, interpreting the negative symbol as negation (e.g., reading -8 as “the opposite of 8” and - 8 as “the opposite of negative 8”), less than/greater than, and differentiating order and magnitude. Probing questions you might ask during whole-class discussion or for think-pair-share conversations include the following:

    • How are -5 and 5 both similar and different?
    • I have heard some students say -5 is larger than 0 since 0 is nothing and -5 is 5 negatives-what do you think about that reasoning?
    • In what way is -6 less than -5?
    • Is there any way that -6 can be seen as larger than -5?

    Conclude the introduction by asking students to draw and complete a number line from -10 to 10 (for only integer values) on their whiteboards or paper. They can refer to their number line during the next part of the lesson. 

    Open Number Sentences:

    Pose the following six open number sentences one at a time (see student handout). Students can solve each on a piece of paper, use the premade handout, or use whiteboards and markers if available. If you are teaching older students who may already be familiar with procedures, encourage them to solve the problem in a way that does not involve a rule or procedure and that would be understandable to an elementary school child. Students can generate a second way to solve the given open number sentence if they have time.

    The open number sentences have been selected purposefully to encourage students to solve problems using different ways of reasoning, to discuss those ways of reasoning, and to consider when a way of reasoning might be more or less helpful for a given problem. The three ways of reasoning we expect students to use across these problems are (1) order-based reasoning, (2) analogy-based reasoning, and (3) formal reasoning

    To increase student participation, consider using a think-pair-share format by asking students to first independently solve the given open number sentence, followed by a conversation with a partner during which they share their ideas. During partner talk, students can determine if they have the same answer and share their strategies. Did they use the same strategy or a different strategy than their partner? Do they understand their partner's strategy? Can they restate their partner's strategy? During the subsequent whole-class discussion of the open number sentence, consider recording all possible answers for a given problem and asking one student per answer to share their reasoning. If more than one possible answer is shared, ask the class if both answers can be correct and then ask how they can determine which is correct, moving between partner talk and whole-class discussion as needed.

    1.   3 − 5 = _____

    We refer to this open number sentence as a change-positive problem because the change value (or subtrahend) of 5 is positive. Many students will likely solve this by either moving left on a number line, subtracting 3 to reach zero and then subtracting 2 more, or extending a counting down strategy into the negatives. These strategies are all order-based reasoning because they involve students using the ordered and sequential nature of numbers to move up or down the number sequence when adding and subtracting. Possible questions to pose during whole-class conversation or in additional partner talk are these:

    • How did you think about this problem?
    • How did you know where to start, which direction to move, when to stop?

    Some students may suggest answers of 0 or 2; ask them to explain their reasoning. In this task, the starting value is a positive number, but what happens when the starting value is a negative number? Pose question 2 to find out what your students will do.

    2.  -4 + _____ = 2

    This is also a change-positive problem (the unknown addend of 6 is positive). Again, students are most likely to solve this using order-based reasoning by using motion on a number line, jumping from -4 to 0 and then to 2, or counting up from -4 to 2. If so, you might pose some of the following questions: 

    • How did you think about this problem?
    • How did you know where to start, which direction to move, when to stop?

    However, some students may choose to use analogy-based reasoning and compare -4 to an object, such as owing money or being 4 feet below the ground, and use that context to solve the problem. If so, consider posing the following question:

    • I noticed that you compared negatives to _________ (positives, owing, etc.). Why did you make this comparison?

    We also note this open number sentence involves crossing zero when moving from -4 to 2. Consequently, students may not have to confront which way to move or count when the starting value is negative for this problem. But what happens when the starting and ending values are not on opposite sides of zero? Pose question 3 to find out how students reason about these problems.

    3.  -6 − 2 = ______

    For this change-positive problem, students may struggle with which way to move or count since the starting and ending values (-6 and the unknown of -8, respectively) are both negative. Is the answer to this problem -4 or -8? After all, 6 − 2 is 4, so does subtraction work the same way with negatives? You might ask students to discuss with a partner whether -4 or -8 is correct and why. Or, if nobody has suggested -4 as a possible answer, ask them what a student who thinks -6 − 2 is -4 might be thinking. To resolve this dilemma, you might suggest that your students compare the current open number sentence to the previous problem and see if that could help them decide which way to move or count. In fact, we have seen many students do just that and use formal reasoning to successfully engage with this problem. Students might conjecture that If I moved right (or counted up) to add when solving question 2, then I should move the opposite direction when subtracting for this problem since addition and subtraction are inverses. That means the answer would have to be -8. Consider asking these questions:

    • Is there a related problem you can solve that can help you think about -6 − 2?
    • How does knowing the answer to that problem help you to answer -6 − 2?

    This task can also support a conversation about order and magnitude. If we sort numbers by order, then -8 < -6. But if we sort numbers by magnitude, then -8 can be thought of as more than -6, which we can write using absolute value: |-8| > |-6|. We point out that in each of these first three open number sentences, students add or subtract a positive number. What happens when students are asked to add a negative number? Pose question 4 to find out. 

    4.  -5 + -1 = ______ 

    This open number sentence is what we call an all-negatives problem because all quantities (including the unknown) are negative. Many students will solve this problem by using analogy-based reasoning either by treating -5 as 5 negative ones and adding an additional negative one for a total of 6 negative ones, or perhaps by comparing -5 and -1 to an object like sad thoughts or debt and reasoning about the problem in that context (adding more debt or sad thoughts). Or some may treat negatives like positives and compare 
    -5 + -1 to 5 + 1, solve the related problem involving positives, and then add the negative back at the end. You may notice that using order-based reasoning is challenging for this problem unless students have an understanding of negative numbers as negation or additive inverses. Possible questions to pose during whole-class discussion or during partner talk include the following:

    • I noticed that you compared negatives to ________ (positives, owing, etc.). Why did you make this comparison?
    • Did anyone try other approaches (i.e., order-based) and decide they would be challenging to use? Which ones?
    • Why do you think your approach was challenging to use for this problem?
    • Is there a related problem that might help you solve this?

    5.   -5 − -3 = ______

    To solve this all-negatives problem, students may engage in analogy-based reasoning similar to problem 4 or they may use formal reasoning. We encourage you to elicit both ways of reasoning in the discussion of this open number sentence. For example, we have seen students use formal reasoning to compare -5 - -3 to the previous problem, -5 + -1, and conjecture that if adding a negative one resulted in a “more negative” value of -6, that subtracting a negative value must result in a “less negative” value. Possible questions during whole-class discussion or partner talk include these:

    • I noticed that you compared negatives to _______ (positives, owing, etc.). Why did you make this comparison?
    • Can someone use the same kind of reasoning we used to solve -5 + -1 by treating negatives like _______ to solve -5 − -3? (The goal here is to elicit analogy-based reasoning.) 
    • Some students used the previous problem, -5 + -1 = -6 to help them think about this problem-how does knowing that adding -5 and -1 is -6 help you think about subtracting -3 from -5? (The goal here is to elicit formal reasoning.)

    6.  6 + -4 = ______

    This counterintuitive problem may be particularly challenging for students to reason about without a rule or procedure. Consider asking students to compare this open number sentence to the second problem -4 + 6 = 2 or the fourth problem -5 + -1 = -6 to see if either problem might help them develop a meaning for what it means to add a negative number. Perhaps they will use formal reasoning by invoking the commutative property and arguing that -4 + 6 = 6 + -4 or observing that adding a negative number in task 4 resulted in moving to the left on the number line and conjecture that adding a negative moves in the opposite direction as adding a positive number. Possible questions to ask include the following:

    • Let's compare 6 + -4 to a related problem, -4 + 6. Work with your partner to solve -4 + 6 and see if knowing that answer can help you reason about 6 + -4.
    • This problem seems challenging for us/you-what features of the problem make it challenging? 

    Assessments and Extensions

    Assessment Options:

    Encourage students to solve each of the following four problems using a strategy other than a rule or procedure. Their solutions and ways of reasoning will give you insight into their fluency with integer arithmetic and their flexibility in reasoning.

    1.   -7 + ____ = -2
    2.   ____ − (-2) = -8
    3.   -3 + 6 = ____
    4.   6 + (-3) = ____


    We suggest the following tasks to extend students' understanding of integer addition and subtraction. 

    1. Solve -2 + ____ = 3 in as many different ways as you can.
    2. 6 + ____ = 4 
    3. 5 − ____ = 8
    4. -7 − (-9) = ____

    The first task (and others like it) can help students to develop proficiency with multiple Ways of Reasoning (WoR) -what we call flexibility. The last three tasks are more challenging open number sentences for some students. In fact, some students argue that these problems are not possible to solve. Do you understand why a student might say that 6 + __ = 4 is impossible? What about the others? Encourage students to verbalize what makes these problems tricky and consider pairing each of these tasks with a more familiar “partner” to encourage sense making. For example, can students solve (6 + -2) or (5 − -3) or (-7 − 9)?  


    Questions and Reflections


    The following questions can be posed during whole-class discussion of the open number sentences or during partner or small-group conversations.

    1. How did you think about this problem? Why did you decide to use this strategy? Was there a feature of the problem or numbers that encouraged you to use your strategy? (General questions that can be asked regardless of problem type or Ways of Reasoning (WoR).)
    2. How did you know which way to count/move? Where to start/end? Is -8 (or -4) greater than or smaller than -6? In what way could you see -8 as larger than -6?
    3. I noticed that you compared negatives to _________ (positives, owing, etc.). Why did you make this comparison?
    4. Did anyone try other approaches and decide they would be challenging to use? Which ones? Why do you think your approach was challenging to use for this problem?
    5. You shared that you knew that the answer had to be negative. What features of this problem helped you to realize that the answer had to be negative?
    6. Is there a related problem that might help you solve this? When sharing pairs of problems such as -3 + 6 and 6 + -3 or 6 − 2 and 6 − -2, ask, “How does knowing the answer to the first problem help you to answer the second problem?”
    7. If a younger child asked you why this rule works, what would you say?


    Objectives and Standards

    Students will:

    • Order and compare negative numbers.
    • Construct a number line extending into the negative numbers.
    • Understand numbers, and negative numbers in particular, as being both ordinal and cardinal. In other words, numbers have a location or place in the counting sequence (-2 is the number before -1 and after -3) and numbers have magnitude or cardinality (-2 means 2 negative ones).
    • Use multiple ways of reasoning to solve additive integer open number sentences including number lines/counting, magnitude/analogies, and formal mathematical properties and principles (e.g., commutativity, inverses).


    Related Resources

    Beyond the Sign Rules

    Journal Article