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Students examine expressions and Circles of Evaluation that use both opposite and absolute value.
Lesson Goals |
Students will be able to…
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Student-facing Lesson Goals |
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Materials |
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Preparation |
This activity involves a card sort. The easiest way to prepare for this is to give each pair of students an envelope containing the three sets of cards. Keep each set together with rubber bands or paperclips. |
🔗Translating Absolute Value & Opposite
Overview
Students match Circles of Evaluation to expressions in words with absolute value and opposite, and then they consider how arithmetic expressions correspond to those two representations.
Launch
During this activity, students sort three separate sets of cards: Circles of Evaluation, Verbal Expressions, and Arithmetic Expressions. Give each pair of students an envelope containing the three unique sets of cards. Keep each set together with rubber bands or paperclips.
Explain to students that they are going to receive three sets of cards.
The first set of cards includes Circles of Evaluation that use the negate and abs functions.
The second set of cards translates those Circles of Evaluation into Verbal Expressions.
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With your partner, match each Circles of Evaluation card with the corresponding Verbal Expressions card.
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Lay them out on the table in front of you so you can clearly see both the Circle of Evaluation and the Words.
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Some Circles of Evaluation have more than one translation into words.
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Discuss each match that you make.
Circulate as students sort their cards, ensuring that they carefully examine each card in order to see the connection between the Circle of Evaluation and its translation into words. Students will need to consider if translating the Circle of Evaluation into words involves reading from the outside-in or the inside-out.
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You’re going to receive a third set of cards - Arithmetic Expressions.
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One at a time, look at each of these cards. After examining a card closely, place it with the appropriate Circle of Evaluation / Verbal Expression pairing.
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Explain to your partner how and why you placed each card.
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You and your partner must agree on each card’s placement before advancing to the next.
Again, circulate during the activity. As students match Verbal Expressions to Arithmetic Expressions and Circles of Evaluation, encourage them to use the Circle of Evaluation as a tool to help them see the structure of the math.
Investigate
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Each row on Translating (Absolute Value & Opposite) represents a single arithmetic expression, written in three different forms. Sometimes, you will be provided with a Circle to translate, other times you will start with a verbal or arithmetic expression.
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Fill in the empty spaces so that all three forms represent the same expression.
Here are some key takeaways:
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We can find the opposite of an absolute value: - |x|
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We can find the absolute value of an opposite: | - x|
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Thinking about the structure of the expression can help us understand if it is positive or negative.
Synthesize
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We did lots of different translations between Circles of Evaluation, verbal expressions, and arithmetic expressions. Was there any type of translation that was more challenging for you?
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Was there any particular representation (Circle, verbal expression, or arithmetic expression) that you found yourself returning to often? Why?
🔗Evaluating with Absolute Value & Opposite
Overview
Students evaluate expressions that include both abs and negate.
Launch
Circles can be extremely useful in helping students to navigate problems with abs and negate, which can be quite complicated, otherwise; after all, "negative" and "absolute value" are not typically delegated in the order of operations.
Consider this equation: - |4| = | - 4|
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Is it true or false?
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Jenny says that the equation is true because absolute value makes everything positive, so both sides of the equation evaluate to 4. Is she correct?
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Jenny is not correct - although students commonly assume that the presence of absolute value indicates a positive outcome.
Representing the equation with Circles of Evaluation highlights the structure of these expressions, and it becomes clearer that they are not equivalent.
(negate (abs 4)) |
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(abs (negate 4)) |
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Turn to True or False? Absolute Value & Negate. Are the expressions on either side of the equal sign equivalent?
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Evaluate each side of the equation to confirm your response.
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Complete the page. The first one is done for you.
Investigate
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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 of Evaluation that meets the condition stated on the left.
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Next, try Matching Circles and Expressions, where you will match expressions with their corresponding Circles of Evaluation. Note: some expressions can be matched to more than one correct Circle of Evaluation!
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At the bottom of the page, respond to the open response questions, thinking carefully about each Circle of Evaluation that you encountered.
Synthesize
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How do you decide the order in which to apply absolute value and opposite when an expression includes both?
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Did you work from the inside-out or the outside-in when evaluating Circles of Evaluation with absolute value and opposite? Did your strategy change, depending on the Circle of Evaluation? Explain.
🔗Programming Exploration
Overview
Students apply their knowledge of examples in WeScheme to think about abs and negate.
Launch
We are going to complete an activity that involves (1) making predictions about equations with absolute value and negation, and (2) running tests in WeScheme to see if our predictions are correct, and then (3) reflecting on what we learned.
On Programming with Absolute Value and Opposite, complete the table at the top of the page. If you get stuck, translate the code into Circles of Evaluation. Discuss your predictions with your partner.
Students may need you to demonstrate your thought process for one or two of the examples.
When finished, invite students to share their predictions.
Investigate
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With your partner, complete the second section of Programming with Absolute Value and Opposite using Negation Starter File (2).
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If you discover that one of your predictions was wrong, revise the table at the top of the page.
As you circulate, ensure that students are looking at the messages that appear in WeScheme. This activity not only provides practice thinking about the absolute value and opposite; it also gives students exposure to tests - bits of code used to verify that code is working as we would expect. Examples and tests are widely used in programming! We explore examples in greater depth in Functions: Contracts, Examples & Definitions.
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Complete the final section of Programming with Absolute Value and Opposite.
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How can you use WeScheme to help you decide if an equation is true?
Two algebraic expressions are equivalent if they produce the same outcome, no matter what value we substitute in for the variable(s). In this activity, none of the equations that tested always passed… which means that none of them were true.
Can you write an example/equation in WeScheme that includes both abs and negate which is always true?
Synthesize
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How is WeScheme similar to having a handheld calculator available to use? How is it different?
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Did you like having WeScheme available to run tests? Why or why not?
These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, 1738598, 2031479, and 1501927). 
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