standard_description: Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
problem_activity: Optional activity
This activity is the same type of situation as the previous one: comparing the voting of two groups on a yes or no issue. However, the numbers make it more difficult to use "part to part" ratios. Again, students need to be thinking about how to make sense of (MP1) and quantify the class voting decisions (MP2).
Launch
Arrange students in groups of 2–4. Provide access to four-function calculators.
Student Facing
The school will be painted over the summer. Students get to vote on whether to change the color to purple (a “yes” vote), or keep it a beige color (a “no” vote).
The principal of the school decided to analyze voting results by class. The table shows some results.
In both classes, a majority voted for changing the paint color to purple. Which class was more in favor of changing?
|         | yes | no |
|---------|-----|----|
| class A |     |    |
| class B |     |    |
Student Response
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Anticipated Misconceptions
If students do not understand what comparing the two classes means, give a more extreme example, such as comparing 20 to 1 with 11 to 10. The 20 to 1 class is much more purple because almost everyone said yes. The other class had almost equal yesses and nos.
Students may be stuck with the difficult-looking numbers, expecting to be able to do calculations to create equivalent ratios mentally. Suggest that they find the total number of votes in each class.
Students may compute percentages incorrectly, forgetting that percentages are rates out of 100. So it is
not
correct to say that room A has 54% yesses. However, you can carefully make sense of this percent as a comparison: Class A had 54% as many nos as yesses; Class B had 61% as many nos as yesses. This means that Class B was more no-y, so it was less yessy than Class A.
Activity Synthesis
Invite students to show different ways of solving the problem, including using equivalent ratios and percentages. Ask students to explain their thinking. Correct ideas with incorrect calculations are still worth sharing.
Representation: Develop Language and Symbols.
Display or provide charts with symbols and meanings. Once students have determined which class was more in favor of changing, pause the class. Invite students to share their strategies for finding each fraction using percents and equivalent ratios to justify their reasoning. Create a display that includes each strategy labeled with the name and the fraction they represent of a total. Keep this display visible as students move on to the next problems.
Supports accessibility for: Conceptual processing; Memory


aligns: no