(Also available in WeScheme)
Students use properties to determine when equations with and without variables are true.
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Student-facing Lesson Goals |
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🔗Seeing Equivalence 25 minutes
Overview
Students review what they have learned about computation and equivalence to lay a foundation for work with variables.
Launch
As we’ve observed, Circles and laws of arithmetic can help us understand whether an equation - a mathematical statement of equality - is true or false.
Look at Equation 1, represented by these two Circles of Evaluation:
(- (* 22 3) 4) |
= |
(- (* 3 22) 4) |
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What do you Notice? What do you Wonder?
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Possible responses: The outer circle subtracts 4, the inner circle multiplies 22 and 3. The difference between the two Circles of Evaluation is that 22 and 3 swap positions.
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Does this equation appear to be true or false?
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True. We see the Commutative Property of multiplication being applied when 22 × 3 is transformed to 3 × 22.
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We can also use Computation to verify that both sides evaluate to 62.
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Look at Equation 2, represented by these two Circles of Evaluation:
(- (* 22 3) 4) |
= |
(- 4 (* 22 3)) |
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What do you Notice? What do you Wonder?
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Possible responses: The same numbers and operations are included. This time, the difference between the Circles of Evaluation is that the 4 moves from the beginning to the end.
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Does this equation appear to be true or false?
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False. This appears to be a misguided attempted to apply the Commutative Property. The Commutative Property does not hold for subtraction.
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The equation’s left side is 62, but its right side is - 62.
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As students observe and share out, urge them to justify their ideas about equivalence.
Investigate
We have learned that computation is an effective strategy for determining whether or not two expressions are equivalent. Pretty soon we’ll be looking at expressions with variables, where our computation strategy won’t work as well. Let’s see if we can figure out how to determine whether or not two expressions are equivalent without computation! For now, while we’re still looking at expressions with numbers, we can double check our thinking with computation.
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Complete True or False? Commutative and Associative Properties using your knowledge of the Commutative Property and the Associative Property. Try not to rely on computation! Be sure to explain your responses.
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Finally, examine the Circles of Evaluation to decide Which One Doesn’t Belong? Again, try to use properties rather than computation. Be sure to explain your thinking.
Synthesize
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Summarize the Associative Property in your own words. When does it apply?
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Summarize the Commutative Property in your own words. When does it apply?
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How can we use computation to verify equivalence?
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Can you think of any expressions involving subtraction or division when the Associative Property or the Commutative Property would apply?
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What strategies did you use to locate equations that were false, or expressions that did not belong?
🔗Equivalence with Variables 25 minutes
Overview
Students examine Circles of Evaluation to determine if expressions with variables are equivalent.
Launch
In this class, sometimes we’ll work with variables - symbols that stand for values we don’t know yet.
The equation c + a = r could be true or false.
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What are some values that make the equation true?
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Sample answer: c = 5, a = 10, r = 15
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What are some values that make the equation false?
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Sample answer: c = 1, a = 2, r = 900
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We can represent the Commutative Property like this:
(* 5 g) |
= |
(* g 5) |
5 × g |
= |
g × 5 |
Or like this:
(+ 12 h) |
= |
(+ h 12) |
12 + h |
= |
h + 12 |
It doesn’t matter what values we use for g or for h - the equations above are always true.
The Commutative Property will hold no matter what values we substitute in for c or for m.
The same goes for the Associative Property! We can represent it with variables, like this:
(* 2 (* 3 c)) |
= |
(* (* 2 3) c) |
2 × (3 × c) |
= |
(2 × 3) × c |
Or like this:
(+ 5 (+ 6 m)) |
= |
(+ (+ 5 6) m) |
5 + (6 + m) |
= |
(5 + 6) + m |
These examples above show us that the Commutative and Associative properties are in fact more powerful than sheer computation. We can’t use computation to prove that 12 + h and h + 12 are equivalent – because without knowing what h is, we can’t evaluate the expression! Commutativity lets us prove that these expressions are equivalent even with variables.
Investigate
The activities below invite students to apply their knowledge of the Commutative and Associative Properties in equations and Circles of Evaluation that include variables. For some students, this cognitive leap can be a challenging one. These pages parallel those in the previous section to make this transition a bit smoother.
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Complete True or False? Variables using your knowledge of the Associative Property and the Commutative Property.
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Done early? Substitute in numbers to confirm your response. The ability to make substitutions to check equivalence is an extremely valuable skill!
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Decide Which One Doesn’t Belong? Variables. Be sure to explain your thinking.
If students get stuck, ask: What do you notice, at first glance? What makes the Circles of Evaluation alike? What makes them different?
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On Label the Arrows, each arrow represents a transformation from an expression to an equivalent expression.
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Label each arrow with the type of transformation that you observe: Associative Property ("AP"), Commutative Property ("CP"), or Computation ("Comp").
Synthesize
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Are equations with variables always true?
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No. If we see a representation of the Associative Property or the Commutative Property, then yes. An equation such as j - y = y - j is only sometimes true. And an equation like g × 0 = 600 is never true.
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What does it mean for two expressions with variables to be equivalent?
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When two expressions with variables are equivalent, that means that no matter what number we substitute in, we will still get the same result.
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🔗Programming Exploration: Variables
Overview
Students learn about examples in Pyret, and use their new knowledge to think explore variables, commutativity, and associativity.
Launch
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On Variables and Code (Commutative Property), record what you Notice and what you Wonder about the starter file.
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Let’s share out some of our Noticings and Wonderings.
Here are some common Noticings. Is there anything on this list that you didn’t notice?
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The first section includes variable definitions. When we define variables in Pyret, we use an equal sign.
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The second section includes four examples. When we want to represent equality but not a definition, we use
==. -
Some words are in bold.
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Some words are in different colors and have hashes in front of them.
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In the starter file, all four examples are labeled as
true. -
The examples end with
end.
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Return to your starter file and click "Run".
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With your partner, respond to questions 2 through Q10 on Variables and Code (Commutative Property).
As students work, encourage them to interact with the starter file! They are free to change definitions, edit examples, and hit "Run" as many times as needed. If they break something, they can always open a fresh copy of the file.
Reading the messages that appear encourages students to think deeply about the Commutative Property and 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.
Investigate
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On Variables and Code (Associative Property), record what you Notice and what you Wonder about the starter file.
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Let’s share out some of our Noticings and Wonderings.
As students share, encourage them to draw on discoveries made during Variables and Code (Commutative Property). You can prompt them by inviting them to consider how this starter file is different from the first one that they explored.
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Predict what will happen when you hit "Run".
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Record your prediction by circling one of the options listed in Q2 on Variables and Code (Associative Property).
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Let’s share and discuss our predictions.
Students have a tendency to skip over the "prediction" part of these worksheets, but talking about predictions before diving into the code is really valuable! Discuss student predictions as a whole class, or have students talk with their partner.
Complete the rest of the page with your partner.
Synthesize
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Using Pyret allows us to test variables in algebraic expressions easily and efficiently. In your own words, describe how you tested different variables - and how you interpreted the results that Pyret produced.
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In Pyret, what does it mean for a test to pass? What does it mean for a test to fail?
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What did this programming exploration teach you about the Commutative and Associative Properties?
These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, 1738598, 2031479, and 1501927). 
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