More Expressions and Equations

15 min

Teacher Prep
Setup
Familiarize students with school plays. 5 minutes of quiet work time followed by whole-class discussion.

Narrative

This activity walks students through the process of defining a variable, writing an expression, writing the expression with fewer terms, estimating a reasonable solution, computing a solution, and finally checking that the solution makes sense and is correct. Through this process, students reason abstractly and quantitatively (MP2). Note that there are two unknown quantities (prices for student and adult tickets) and students are guided to express one in terms of the other.

Monitor for students who write the expressions in different ways in the first three questions and invite them to share during the discussion. There are many possible correct answers, but some forms will lead to equations that are easier to solve (such as px+qpx+q).

Launch

If your school or a nearby school has recently performed a play, consider asking if any students went to see it and have them briefly describe the experience. Alternatively, display photos from any school play, including images of the tickets or ticket booth. Invite students to share what they notice and what they wonder.

Give students 5 minutes of quiet work time followed by a whole-class discussion.

Student Task

Student tickets for the school play cost $2 less than adult tickets.

  1. If aa represents the price of 1 adult ticket, write an expression for the price of 1 student ticket.

  2. Write an expression that represents the amount of money they collected each night:

    1. The first night, the school sold 60 adult tickets and 94 student tickets.
    2. The second night, the school sold 83 adult tickets and 127 student tickets.
  3. Write an expression that represents the total amount of money collected from ticket sales on both nights.

  4. Over these two nights, they collected a total of $1,651 in ticket sales.

    1. Write an equation that represents this situation.

    2. What was the cost of each type of ticket?

  5. Is your solution reasonable? Explain how you know.

Sample Response

  1. a−2a−2
    1. 60a+94(a−2)60a+94(a−2) (or 154a−188154a−188)
    2. 83a+127(a−2)83a+127(a−2) (or 210a−254210a−254)
  3. 364a−442364a−442 or equivalent
    1. 364a−442=1651364a−442=1651
    2. $5.75 adult, $3.75 student
  5. Sample response: Yes, it's reasonable that tickets to the school play would cost somewhere between $0 and $10 and also that the adult tickets would cost more than the students tickets. I can show it is correct using that 5.75(60+83)+3.75(94+127)=822.25+828.75=1,6515.75(60+83)+3.75(94+127)=822.25+828.75=1,651.
Activity Synthesis (Teacher Notes)

The purpose of the discussion is for students to reflect on the problem solving process. Consider asking these questions:

  • “Why was the price of an adult ticket chosen as the variable?” (This was an arbitrary choice.)
  • “Could the problem be worked by choosing the price of a student ticket as the variable? How would the expressions be different? The equation? The solution?” (Yes, the expressions would be 60(s+2)+94s60(s+2)+94s and 83(s+2)+127s83(s+2)+127s, where s represents the price of 1 adult ticket. The equation would become 364s+286=1,651364s+286=1,651. The solution to this equation is 5.75. The price of the tickets do not change.)
  • “What does each term in the expressions you wrote represent?” (The numbers represent the numbers of each type of ticket sold. The variable aa represents the price of one adult ticket.)
  • “How did you find the price of a student ticket?” (First, I solved the equation for the price of an adult ticket. Then I used the first expression a−2a-2, to solve for the price of a student ticket because student tickets cost $2 less than adult tickets.)
  • “How did you check that your solution is correct?” (I checked using the number of tickets sold and the amount of money earned.)
Anticipated Misconceptions

If students are struggling with writing and solving an equation, consider asking:

  • “Can you explain how you wrote your expressions?”
  • “How can your expressions be combined to represent both nights?”
  • “How can you use what you know about the adult ticket price to find out the student ticket price?”

If students represent the price of a student ticket as “2−a2-a” instead of “a−2a-2,” consider asking:

  • “Tell me more about this expression.”
  • “How could you use test values to make sure your expression is correct?”
     
Standards
Addressing
  • 7.EE.1·Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
  • 7.EE.4.a·Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. <em>For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?</em>
  • 7.EE.A.1·Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
  • 7.EE.B.4.a·Solve word problems leading to equations of the form <span class="math">\(px + q = r\)</span> and <span class="math">\(p(x + q) = r\)</span>, where <span class="math">\(p\)</span>, <span class="math">\(q\)</span>, and <span class="math">\(r\)</span> are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. <span>For example, the perimeter of a rectangle is <span class="math">\(54\)</span> cm. Its length is <span class="math">\(6\)</span> cm. What is its width?</span>

15 min

15 min