standard_description: (+)Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
problem_activity: Activity
In this activity, students interpret equations that represent situations (MP2). The purpose is for students to see that equations of the form
\(x + p = q\)
can be solved by adding the opposite of
\(p\)
to the equation, regardless of whether
\(p\)
is positive or negative. Students also see that equations of the form
\(px = q\)
can be solved by multiplying the equation by the reciprocal of
\(p\)
. Through this work, students see that the structure of equations can be used to reason about a path to a solution (MP7) even when negative values are included or when a variable can represent a negative number.
Launch
Give students 5–6 minutes of quiet work time followed by whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention
Reading: MLR6 Three Reads.
Use this routine to help students interpret the representations of the situation in this activity. Use the first read to help students comprehend the different stages of situation (e.g., hiking a mountain, temperature falls, cost of the trip) without using numbers. Use the second read to identify the important quantities by asking students what can be counted or measured (e.g., increased elevation 290 feet, now at 450 feet; temperature fell 4 degrees, now is at 32 degrees; 3 times as many students this year, now 42 students; cost is
\(\frac23\)
of last year’s, cost
$
32 this year). For the third read, ask students to brainstorm possible strategies to answer the given questions.
Design Principle(s): Support sense-making
Student Facing
The Hiking Club is on a trip to hike up a mountain.
The members increased their elevation 290 feet during their hike this morning. Now they are at an elevation of 450 feet.
Explain how to find their elevation before the hike.
Han says the equation
\(e + 290 = 450\)
describes the situation. What does the variable
\(e\)
represent?
Han says that he can rewrite his equation as
\(e=450 + \text-290\)
to solve for
\(e\)
. Compare Han's strategy to your strategy for finding the beginning elevation.
The temperature fell 4 degrees in the last hour. Now it is 21 degrees. Write and solve an equation to find the temperature it was 1 hour ago.
There are 3 times as many students participating in the hiking trip this year than last year. There are 42 students on the trip this year.
Explain how to find the number of students that came on the hiking trip last year.
Mai says the equation
\(3s=42\)
describes the situation. What does the variable
\(s\)
represent?
Mai says that she can rewrite her equation as
\(s=\frac13 \boldcdot 42\)
to solve for
\(s\)
. Compare Mai's strategy to your strategy for finding the number of students on last year’s trip.
The cost of the hiking trip this year is
\(\frac23\)
of the cost of last year's trip. This year's trip cost
$
32. Write and solve an equation to find the cost of last year's trip.
Student Response
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Student Facing
Are you ready for more?
A number line is shown below. The numbers 0 and 1 are marked on the line, as are two other rational numbers
\(a\)
and
\(b\)
.
[IMG]
Decide which of the following numbers are positive and which are negative.
\(a-1\)
\(a-2\)
\(\text-b\)
\(a+b\)
\(a-b\)
\(ab+1\)
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
Students may be misled by words to add or multiply (or subtract or divide) by the wrong numbers. For example, the word "increased" in question 1 may lead students to simply add the numbers they see, while the words "three times as many" in question 4 may lead students to multiply the numbers in the problem. Encourage students to make sense of the situations by acting them out or using visual diagrams, which will help them understand the actions and relationships in the stories.
Activity Synthesis
Tell students: “We learned four things about the hiking trip in this activity: the students were climbing, the temperature was falling, there were more students this year than last, and the cost of the trip was less this year than last.” Then ask them:
“Think about how you knew what operation described the rise in elevation, fall in temperature, rise in number of students, and fall in the cost. How did you know whether the situation used adding or multiplying?”  (This conversation can highlight the problem with relying on “key words” . For example, when students see “times as many” , they might want to multiply the numbers they see in the problem. Encourage students to make sense of the situations by acting them out or drawing diagrams.)
“How did you decide how to solve for the unknown quantity?”
“What are some ways to know that a situation involves negative values?”


aligns: no