standard_description: Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
problem_activity: Warm-up
The purpose of this warm-up is to elicit the idea that Venn diagrams can be used to represent events of a sample space, which will be useful when students describe events as a subset of a set of a sample space using characteristics of the outcomes, and as unions, intersections, or complements of other events in a later activity. While students may notice and wonder many things about these images, understanding how "and," "or," and "not" are used when discussing sample space and probability are the important discussion points.
This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is that traits that belong to both bats and birds are listed where the two circles overlap, all of the traits in the circles of the Venn Diagram belong to bats or birds, and the trait listed outside of both of the circles does not belong to birds or bats.
Launch
Tell students that their job is to think of at least one thing they notice and at least one thing they wonder. Display the Venn Diagram for all to see. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.
Student Facing
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What do you notice? What do you wonder?
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Student Response
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Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the Venn diagram. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification or point out contradicting information. If understanding how "and," "or," and "not" are used when discussing sample space and probability does not come up during the conversation, ask students to discuss this idea.
Tell students that when we use "or" in mathematics it is inclusive. Here are some questions for discussion.
“How many traits belong to birds or bats?” (6)
“How many traits belong to birds and bats?” (2)
“How many traits do not belong to birds or bats?” (1)
“What is an “or” question that you could ask involving something in this room? How would you answer it? Tell your partner.” (I could ask, “How many people have a pencil or pen on their desk?” I would answer it by counting the number of people with a pen or a pencil on their desk.)
“Ask a question similar to your previous one but using “and” instead of “or.” How is the question different than the previous one?” (My new question is, “How many people have a pencil and a pen on their desk?’ It is different because this would include only the students with both a pen and a pencil.)
“Is it possible for the answer to the second question to be greater than the answer to the first question? Explain your reasoning.” (No. It is not possible because the “or” question includes all of the possibilities and the second question would have to be equal to or less than the quantity of all possibilities.)
If time permits, ask students to think about this situation. You have three people with bowls of ice cream and one has chocolate, one has vanilla, and one has both chocolate and vanilla.” Here are some questions for discussion.
“How many people have vanilla ice cream?” (2)
“How many people have chocolate ice cream?” (2)
“How many people have chocolate and vanilla ice cream?” (1)
“Is the correct answer to, ‘How many people have chocolate or vanilla ice cream’ 2 or 3? Explain your reasoning.” (It is 3 because everyone has either chocolate or vanilla?)
“How is this related to the birds and bats questions?” (It is related because when you ask, ‘Which of these traits belong to birds or bats?’ it is everything but the 6 legs trait.)


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