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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.

🔗Finding Opposites 20 minutes

Overview

Students are introduced to 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

Operators always have two inputs and, in standard mathematical notation, they always go in the middle of their inputs. You’ve created a lot of Circles of Evaluation with operators at the top.

But what if we want to do something to just one number? On which side does the operator go? Or what if we want to do something to more than two numbers? Forcing the "verb" to always be in the middle really limits what we can do!

Functions are very similar to operators…​ except that functions can have any number of inputs! If you’ve completed any of the programming explorations so far, then you are familiar with a wide variety of functions: scale, rotate, beside, etc.

Math has functions too, and they work the same way!

Incorporating functions into our Circles of Evaluation makes them more powerful! With functions, we can diagram and interpret expressions that include opposites, absolute value, exponents, square roots, Booleans…​ the list goes on.

We can even define our own functions. This allows us to tell the computer to do whatever we want, and maybe even get the computer to do the same thing over and over again but with different inputs!

Students learn to define functions in WeScheme in our Algebra curriculum, during Functions Make Life Easier!.

The function used to represent "the opposite" is called negate. When we use negate, we include just one argument inside the Circle.

(negate 2)

  • What does this Circle of Evaluation evaluate to?

    • The Circle evaluates to - 2.

(negate (negate 2))

  • What happens if we nest a negating Circle of Evaluation inside of another negating Circle, like in the above Circle?

    • The opposite of the opposite of a number is the number itself, so it evaluates to 2.

  • What do each of these Circles evaluate to?

    • 5, - 5, 5, - 5, and 5

(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? Explain.

    • 4 and - 4

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

    • 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.

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 these Circles evaluate to?

    • 20, 20, 43, 43, 43

(abs -20)

(abs 20)

(abs (abs 43))

(abs 43)

(abs -43)

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.

🔗Programming Exploration: Rotations

Overview

Students explore with the rotate function to apply what they have learned about absolute value and negation.

Launch

Today, we’re going to think deeply about the rotate function in WeScheme. In particular, what angles of rotation represent clockwise turns? Counter-clockwise? What does it mean to reverse a rotation, and how can we represent such a reversal in WeScheme?

  • Open the Negation Starter File.

  • Complete the first table on Exploring Rotations by making a prediction for each Circle of Evaluation and then testing the code in WeScheme.

  • When you are done, write a few sentences summarizing what you observed.

The opening table of the worksheet is a basic review of geometric rotations. For students who are unfamiliar with the concept of a 90-degree turn, a 180-degree half-turn, or a full 360-degree turn, this will be a necessary review. Feel free to spend as much time as needed reinforcing these concepts.

You just explored angles of rotation. Let’s review what you’ve learned.

  • What degree of rotation do we use to spin the image in a full circle, back to its starting point?

    • 360

  • Describe what a 180-degree turn looks like.

    • A 90-degree turn is a quarter turn, that rotates any image perpendicular to its starting position.

  • Think about each of the rotations you just explored. Were the angles of rotation positive or negative? Were the turns clockwise or counter-clockwise?

    • The angles of rotation in the first table are positive and the turns were all counter-clockwise.

Investigate

  • Complete the second table on Exploring Rotations, making a prediction for each Circle of Evaluation and then testing the code in WeScheme.

  • When you’re done, streamline the complicated code so that it produces an identical output without using negate or abs.

Referring to the first table on the page can be a helpful scaffold for students who are struggling.

Synthesize

  • Rotating an image 45 degrees produces a different output than rotating an image -45 degrees. Rotating an image 180 degrees, however, produces the same output as rotating that image -180 degrees. Explain why.

    • 180 degrees is exactly half of one full rotation, so no matter which direction we rotate, we end up in the same position.

  • Can you think of any other pairs of opposite angles of rotation that would produce an identical output?

    • 540, 900, 1260…​

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.

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