1. Auditorium Problem
6.RP - Understand ratio concepts and use ratio reasoning to solve problems. 7.RP - Analyze proportional relationships and use them to solve real-world and mathematical problems. 8.EE - Understand the connections between proportional relationships, lines, and linear equations.

2. 5/7 of the students seated in an auditorium were girls
5/7 of the students seated in an auditorium were girls. There are 48 more girls than boys. How many students are seated in the auditorium?
Provide your students with this problem. Allow them to work in groups to solve. Mathematical Practice #3: Construct viable arguments and critique the reasoning of others,
states that mathematically proficient student justify their conclusions, communicate them to others, and respond to the arguments of others. Have your students present their solutions to the class and justify their reasoning. During presentations, encourage other students to ask questions and give feedback to the presenters to clarify or improve the arguments.

3. Use Proportional Reasoning- Method 1
5/7 of the students seated in an auditorium were girls.
There are 48 more girls than boys.
How many students are seated in the auditorium?
What is the ratio of girls to boys?
What does x represent?
What does x + 48 represent?
What does 32 represent?

4. Use Proportional Reasoning - Method 1
5/7 of the students seated in an auditorium were girls.
There are 48 more girls than boys.
How many students are seated in the auditorium?

5. Use Proportional Reasoning – Method 2
5/7 of the students seated in an auditorium were girls.
There are 48 more girls than boys.
How many students are seated in the auditorium?
What is the ratio of
girls to total students?
What does x represent?
Have students compare method 1 to method 2 and explain why they both work to solve the problem.
Why does 2x+48 represent
the total # of students?

6. Use Proportional Reasoning – Method 2
5/7 of the students seated in an auditorium were girls.
There are 48 more girls than boys.
How many students are seated in the auditorium?

7. Use logical Reasoning- Method 3
5/7 of the students seated in an auditorium were girls.
There are 48 more girls than boys.
How many students are seated in the auditorium?
If 5/7 of the students are girls, then 2/7 of the students have to be boys. Therefore the difference between girls and boys is 3/7 of the students and since there are 48 more girls than boys, then 3/7 of the students must be equal to 48.

8. Use logical Reasoning- Method 3
If 5/7 of the students are girls, then 2/7 of the students have to be boys. Therefore the difference between girls and boys is 3/7 of the students and since there are 48 more girls than boys,
then 3/7 of the students must be equal to 48.
What does x represent?

9. Use Model Drawings – method 4
5/7 of the students seated in an auditorium were girls.
There are 48 more girls than boys. How many students are seated in the auditorium?
Girls 16 16 16 16 16
Each box represents 16 48
Boys 16 16

10. Use Systems of Equations – Method 5
5/7 of the students seated in an auditorium were girls.
There are 48 more girls than boys. How many students are seated in the auditorium?
Let g = # of girls
Let b = # of boys
Total # of students = b + g

11. Use Systems of Equations – Method 5
Use substitution

12. Use Systems of Equations – Method 5
How many students are seated in the auditorium?

13. Use Systems of Equations – Method 6
7[ ]
-5g
-5g
Use substitution

14. Use Systems of Equations- Method 6
5/7 of the students seated in an auditorium were girls.
There are 48 more girls than boys. How many students are seated in the auditorium?
Let g = # of girls
Let b = # of boys
Total # of students = b + g

15. Discussion How are the solution methods similar?
How are the solution methods different?
Identify correspondences between different solution methods.
Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.

16. What is the Error?
Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. The next few slides show solutions with errors. Present these to your students and have them explain the errors in reasoning.

17. What is the Error?

18. What is the Error?