This learning module is the first unit in a senior level course designed as a 4th year math course for students. The prerequisite is Algebra 2. Is it designed to help students see the real world applications of the mathematics they have learned throughout their schooling. Students generally learn about ratios and means in middle school, but may not have been ready to learn some of their more advanced applications. This unit’s goal is to provide real world examples of these ideas.

Not only will students model and apply the ideas of ratio and weighted averages, but they will do so in a collaborative environment. Students will often work with one another to brainstorm ideas, present their ideas, critique the ideas of others, and showcase their learning in a project to end the unit.

These lessons are part of a larger unit that takes 4 weeks of instructional time.

The following materials will be needed:

• Tennis balls
• Boxes
• Objects with UPC codes

Background: Students will be collecting data on how many people fit in a specified space and then using that data to esimate the size of a large crowd. To prep for today's lesson. Use painter's tape or masking tape to create a 5 ft by 5ft square somewhere in the classroom. 

Opener: Display the picture of the large crowd. Ask them to share their experience of being in a large crowd. Explain that today, you will be estimating crowd size. 

Notes: Begin question 1 together as a class, and explain that you will be collecting data as a class to estimate the size of crowds. Before calling up volunteers, you may have a discussion about how close together the "crowd" you are making will be - mosh pit close? comfortably close? Make a decision as a class about this. This will assist in a future discussion about making assumptions to solve problems. Ask for volunteers to stand in the 5 by 5 foot square you have made. 

Questions 1-3 are designed to be done in groups. Emphasize the importance of perseverance and collaboration. Student may want to you tell them the answer or how to do it, but it will be more useful for them if they have to reason the problem with their group. 

Circulate the room asking probing questions about the work that students are doing. Take note of the different methods that students are doing, and ask some groups to present their methods to the class. Emphasize the importance of listening to the contributions of others. 

Closure: End class with the video of the protest in Hong Kong. Lead a short discussion about the discrepancy in crowd size that is shown in the video.  

Background: Students will be estimating how many tennis balls it takes to fill the classroom. They will need knowledge of volume for this task, which they have enountered in middle school. You will need tennis balls and boxes (preferably enough for each group to make an estimate).  One method is to fill a box and use that to extrapolate how many tennis balls it would take to fill the room. If you do not have that many tennis balls, use what you have to collect data as a class. Then let students make their estimates in their groups. It is not advisable to make a class estimation, as that may allow students to rely on the thinking of too many other students. Working in a group towards a common answer will help all group members to contribute. You will also then be able to compare the solutions that all groups came up with. 

Opener: Ask students to make a quick estimate for how many tennis balls it takes to fill the classroom. Give them 60 seconds. 

Notes: Direct students to the notesheet and have them begin to collect data to determine how many tennis balls would be needed to fill the room. If they ask you about assumptions, let the group make their own assumptions (are they assuming the room is empty? is the classroom a perfect rectangle? , etc.)and have them document any assumptions they made. 

As students work, circulate and take note of different methods that students are using. Have groups present their solutions to the class. Encourage the class to ask the presenters questions about their work and methods. Ask the class about similarities and differences between methods that groups use. 

Closure: Ask students to individually write about their preferred method for solving these types of problems. 

Background: Students will be using aspect ratio to calculate the length and width of different types of screens. They will need to recall Pythagorean Theorem for this calculation, as the diagonal of a screen creates the hypotenuse of a right triangle. Pythagoerean Theorem is a major topic of geometry and middle school, so students should have some knowledge of it. 

Opener: Ask students to name the different types of screens that they deal with (phones, tv, chromebooks, calculators, etc.). If there is a screen in the classroom, measure it as a class to show students that the measurement comes from the diagonal, not the length and width of the screen (e.g. a 48 inch television is called that because of the length of its diagonal.). Ask students if they have ever noticed the black bars when watching a movie or dvd on television. Indicate that all of this has to do with aspect ratio. 

Notes: Direct students to the notesheet. Students will be able to figure out question 2 because of its nicer numbers, but may struggle for a method on number 3. Encourage them to share and present ideas to come up with a class method. Students will then use aspect ratio to answer a series of questions. Ask probing questions and listen to students discussions to highlight some things for closure. 

Closure: Highlight some of the good conversations you heard as you circulated the room. Praise the collaboration and praise the perseverance students showed on question 3.  Praise the effort you are seeing in your students.  

Background: This section is about calculating grade averages. The next 2 lessons use the idea of weights to calculate different numbers. Calculating grade averages is something that many students may appreciate knowing how to do, especially for any post-secondary schooling they may do. Although the idea of averages may seem like a simple one to many students, its applications can be far reaching and complex. 

Opener: Ask students if they know about the grade breakdown of this course (i.e. what percentage of the grade is made up of tests? projects? homework? classrwork? etc.). 

Notes: Direct students to the note sheet. Circulate the room and encourage students to check answers with one another to ensure accuracy. Students will have to either use algebra, work backwards, or guess and check on question 5. Encourage multiple methods and have students present their methods to showcase the different ways to solve these problems. 

Closure: Ask students to think about what their ideal grade weighting system would be. Have them share their ideas with a partner.

Background: Students will be calculating the check digit on UPC codes using weighted averages. You will need several products with UPCs on them. Encourage students to take out products with barcodes they might have with them (pop bottles, single use water bottles, binders, folders, etc.). They can verify the check digit on these items. 

Opener: Ask students if they know anything about the barcodes listed on items they purchase. Some students might have experience with scanning barcodes because of jobs they have. Ask them to share their experience about this (i.e. what happens if the bar code isn't scanning, or if computers are down, etc.).

Notes: Direct students to the note sheet. Ensure that they understand the alternating nature of the weights (i.e. that every other number will be multiplied by 3).  Encourage students to check their work with one another. 

Closure: Give students a UPC code and ask them to calculate the correct check digit. Have them write their work and answer on an index card and turn it in. 

Background: This is the project for the unit on Fermi Questions. Students will create a Fermi question to answer, create assumptions to help answer their question, research any necessary information, and calculate their answer. 

Opener: Ask students to share with a partner what some of the main ideas were for the crowd size calculations or the tennis ball challenge. They should come up with some ideas around ratios and/or proportional reasoning. 

Notes: Have students watch the Fermi question video. Explain the project and go through the rubric with students. Encourage students to be creative with their Fermi questions. (Note: they do not have to use the method in the video to answer their Fermi questions.) Give students time to work on their projects and assist as needed. Student may need assistance creating a quetion and making assumptions to help answer that question. 

Closure: Remind students that their project will be peer reviewed based on the rubric provided. Encouarge them to read through each piece of the rubric before submitted their work.