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Lessons

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    (Also available in Pyret)

    Students use Circles of Evaluation to explore the effect of negation and absolute value on arithmetic and algebraic expressions.

    Lesson Goals

    Students will be able to…​

    • Define absolute value as a number’s distance from zero on the number line.

    • Recognize that two numbers are referred to as "opposites" if they are the same distance from zero.

    • Use Circles of Evaluation to analyze expressions with opposite and absolute value.

    Student-facing Lesson Goals

    • Let’s use Circles of Evaluation to think about opposites.

    • Let’s use Circles of Evaluation to think about absolute value.

    Materials

    Preparation

    • Draw or project a number line of the integers from -10 to 10. This will be the focal point of a lot of the conversation for today’s lesson.

    Key Points For The Facilitator

    • The difference between absolute value and opposite can be subtle and confusing for students. Sometimes, given an input, they produce the same output - but at other times, their outputs differ.
    Glossary
    absolute value

    the (positive) distance of a number from zero, annotated | x |

    algebraic expression

    a mathematical expression that consists of numbers, operators, and variables. The variables' values are sometimes unknown. If we define variables, algebraic expressions can be simplified to a single numeric value.

    arithmetic expression

    a mathematical expression that consists of numbers and operators, which does not have an equal sign. When we apply the laws of arithmetic, arithmetic expressions can be simplified to a single numeric value.

    equation

    a statement that two expressions are equal

    function

    a relation from a set of inputs to a set of possible outputs, where each input is related to exactly one output

    operator

    a symbol that manipulates two Numbers and produces a result

    opposite

    Two numbers are opposites when they are the same distance from zero on the number line.

    variable

    a name or symbol that stands for some value or expression, often a value or expression that changes

    🔗Finding Opposites 20 minutes

    Overview

    Students are introduced to their first non-operator function - negate. Using this function, they create and evaluate Circles of Evaluation to explore the notion of opposites.

    Launch

    Instruct students to draw a number line that spans from −10 to 10, or draw/project one at the front of the room.

    • What number is the same distance from zero as 8?

      • −8

    • What number is the same distance from zero as −3?

      • 3

    • The pairs of numbers we’ve just identified are called opposites - two numbers that are the same distance from zero on the number line, with one negative and one positive.

    • What are some other pairs of opposites that you can identify?

      • Possible responses: 6.5 and −6.5; 4 and −4; 7/3 and − 7/3.

    Investigate

    Today is an exciting day! Rather than using an operator at the top of each Circle of Evaluation, we are going to use…​ functions!

    Operators always have two inputs and, in standard mathematical notation, they always go in the middle of their inputs. Functions, however, can have any amount of inputs - and all kinds of inputs, not just numbers!

    Incorporating functions into our Circles of Evaluation will expand their utility. With functions, we can diagram and interpret expressions that include opposites, absolute value, exponents, square roots, Booleans…​ the list goes on.

    The function used to represent "the opposite" is called negate. It can be used like any other operator that we put at the top of a Circle of Evaluation - with one important difference: when we use negate, we include just one argument inside the Circle.

    • What does the Circle of Evaluation below evaluate to?

      • The Circle below evaluates to −2.

    (negate 2)

    • What happens if we nest a negating Circle of Evaluation inside of another negating Circle?

      • The opposite of the opposite of a number is the number itself.

    (negate (negate 2))

    • Try negating a few more times, to make sure you’ve got the hang of it. What do each of these Circles evaluate to?

    (negate -5)

    (negate (negate -5))

    (negate (negate 5))

    (negate (negate (negate 5)))

    (negate (negate (negate -5)))

    • Practice applying negate on True or False? Negate. Evaluate each side of the equation to confirm if it is true or false.

    • On True or False? Negate (2), explore Circles of Evaluation with Variables, and review previously explored properties (Identity, Commutative, Associative, and Distributive).

    When we want to show negation of a number or an expression, we just use a negative sign! So, while we can read −45 by simply saying "negative 45," another way to read that value’s name is by saying, "the opposite of 45." Similarly, −(𝑥 + 𝑚) can be translated as "the opposite of the sum of 𝑥 and 𝑚."

    Pedagogy Note: Can we call it "minus 45"?! Precise language is crucial when dealing with this symbol: -.

    In math we want students to read for comprehension, not just mechanically sounding out each symbol in order. We read the arithmetic expression 12 − 9 as "12 minus 9" because subtraction is the "verb", not because the minus sign happens to be in the middle!

    When we encounter a value such as −20, however, it is confusing and misleading to name that value "minus 5." Subtraction is not happening here - negation is! But if the teacher uses the "symbols in order" wording, it reinforces the lower level of reading comprehension for students.

    Synthesize

    • Two opposites are 8 units apart from each other. What are they?

      • 4 and −4

    • Two opposites are 42.5 units apart from each other. What are they?

      • 21.5 and −21.5

    • Cia says that opposites are always negative. Are they right? Why?

      • No, opposites are not always negative. Positive numbers always have negative opposites - but negative numbers have positive opposites!

    • We have learned that opposites are two numbers that are the same distance from zero on the number line, with one negative and one positive. Knowing this, what do you think is the opposite of zero?

      • Zero is its own opposite!

    🔗Absolute Value 30 minutes

    Overview

    Students consider the meaning of absolute value, and apply the concept to Circles of Evaluation using abs.

    Launch

    What is the distance between these two points on the number line: −8 and 5?

    Give students a minute to contemplate, and then invite them to verbally share their strategies. Record students' thinking on the board, annotating the number line. All strategies are welcome, with a special interest in any discussion that hones in on the idea of the distance of a number from zero.

    Explain to students that we have a term for the distance of a number from zero - it’s absolute value.

    Absolute value is the (positive) distance of a number from zero.

    We annotate absolute value like this: |𝑥|, with 𝑥 being any given number. When we encounter an expression like |𝑥|, we say "the absolute value of 𝑥."

    Because opposites are the same distance away from zero, they will always have the same absolute value. So, |4| = 4 and |−4| = 4.

    Investigate

    The function that we will use to represent absolute value is abs. It can be used like any other operator that we put at the top of a Circle of Evaluation. As with negate, when we use abs, we include just one argument inside the Circle of Evaluation.

    • Let’s try evaluating some Circles of Evaluation with abs! What does each of the Circles below evaluate to?

      • 20, 20, 43, 43, 43

    (abs -20)

    (abs 20)

    (abs (abs 43))

    (abs 43)

    (abs -43)

    • On True or False? Absolute Value & Negate, you will compare expressions with abs to expressions with negate

    • On the bottom half of the page, determine whether variable equations featuring negate and abs are always, sometimes, or never true. Be sure to explain your response.

    • Examine the Circles of Evaluation to determine Which One Doesn’t Belong? Absolute Value & Negate. The page starts with numeric values and then integrates variables. Place a check mark by each Circle that meets the condition stated on the left.

    Check in with students to ensure that they have a solid understanding of absolute value before moving forward.

    Synthesize

    Think about the algebraic expressions |ℎ| and −ℎ.

    • What do we know about the outcomes of each of these expressions?

      • |ℎ| is always positive or zero, while −ℎ can be negative, zero, or positive.

    • When do they produce the same outcome?

      • −ℎ is positive when ℎ is negative, and −ℎ is negative when ℎ is positive. As a result, |ℎ| and −ℎ produce the same outcome only when ℎ is negative or zero.

    • When do they produce different outcomes?

      • |ℎ| and −ℎ produce different outcomes when ℎ is positive.

    These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, 1738598, 2031479, and 1501927). CCbadge Bootstrap by the Bootstrap Community is licensed under a Creative Commons 4.0 Unported License. This license does not grant permission to run training or professional development. Offering training or professional development with materials substantially derived from Bootstrap must be approved in writing by a Bootstrap Director. Permissions beyond the scope of this license, such as to run training, may be available by contacting contact@BootstrapWorld.org.