This activity allows students to make a stronger connection between the process of finding in a situation in which of is and representing the situation with an equation of the form and finding the value of . To reach this generalization, students perform repeated calculations, observe regularity in their calculations, and represent the regularity with algebraic expressions (MP8).

The purpose of this discussion is for students to see how writing and solving an equation can be an efficient way to solve a problem about percentages. 

Invite students to share their responses and strategies for the last question. If any students used tape diagrams or double number line diagrams to reason about a situation, consider displaying the diagrams alongside the equations. Doing so would help students make connections between strategies they understand well and the more abstract strategy of writing and solving an equation. 

Highlight that every percentage problem in this activity is presented as “ of a number, , equals another number .” or:

which can be thought of as:

 

Ask students:

• “In all the problems in this activity, which of the three quantities corresponds to 100%?” ()
• “Which quantity was unknown?” (the value of 100% or )
• “Do we know the other two values?” (Yes)
• “How can we solve equations that have this form?” (Divide both sides of the equation by or its decimal equivalent.)

This activity gives students an opportunity to determine and request the information needed to solve percentage problems. Each problem involves finding the value that corresponds to 100% in a situation. While students have the opportunity to write equations of the form and then solve it to find the unknown value, they may also choose to reason in other ways. 

The Information Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Here are some questions for discussion:

• “For the first problem, what question did you ask first? Why?” (How much time did Elena spend doing something active? It’s one of the two pieces of information that were needed.)
• “At what point did you realize that the recommended amount of time corresponds to 100%?” (From the beginning, because the problem said Elena did more than 100%. When hearing that Elena did physical activity for 125% of the recommended time.)
• “For each problem, what strategy did you use to solve the problem once you had all the information?”

If writing and solving an equation is not mentioned as a strategy for answering either question, ask students to think about an equation that can be written to represent each problem. 

Highlight for students that each problem can be described by an equation of the form , where is the percentage, is the recommended number of minutes of physical activity, and is the value that corresponds to the given percentage. To solve it, we can reason about what value of makes the equation true, or divide each side of the equation by (or its decimal equivalent).