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 an Estimation warm-up is to practice the skill of estimating a reasonable answer based on experience and known information, and also help students develop a deeper understanding of the meaning of standard units of measure. It gives students a low-stakes opportunity to share a mathematical claim and the thinking behind it (MP3). Asking yourself, “Does this make sense?” is a component of making sense of problems (MP1), and making an estimate or a range of reasonable answers with incomplete information is a part of modeling with mathematics (MP4).
Launch
The first few times using this routine, ask students to explain the difference between a guess and an estimate. The goal is to understand that a guess is an answer without evidence, whereas an estimate is based on reasoning using the available information.
Display the image for all to see. Ask students to silently think of a number they are sure is too low, a number they are sure is too high, and a number that is about right, and write these down. Tell students that the lamp post’s height should be in relation to the person’s height. If students ask, confirm that this is the same person in the photo from the previous lesson (whose height was revealed to be 6 feet or 72 inches). Then, write a short explanation for the reasoning behind their estimate.
Student Facing
[IMG]
How tall is the lamp post?
Record an estimate that is:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "_value": "too low"}, {"_attributes": {"scope": "col"}, "_value": "about right"}, {"_attributes": {"scope": "col"}, "_value": "too high"}]}]}], "tbody": [{"tr": [{"td": [{}, {}, {}]}]}]}]}
Explain your reasoning.
Student Response
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Activity Synthesis
Ask a few students to share their estimate and their reasoning. If a student is reluctant to commit to an estimate, ask for a range of values. Display these for all to see in an ordered list or on a number line. Add the least and greatest estimate to the display by asking, “Is anyone’s estimate less than
\(\underline{\hspace{.5in}}\)
? Is anyone’s estimate greater than
\(\underline{\hspace{.5in}}\)
?” If time allows, ask students, “Based on this discussion, does anyone want to revise their estimate?” Then, reveal the actual value and add it to the display (15 feet). Ask students how accurate their estimates were, as a class. Was the actual value inside their range of values? Was it toward the middle? How variable were their estimates? What were the sources of error? Consider developing a method to record a snapshot of the estimates and actual value so students can track their progress as estimators over time.


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