## The Water Fountain Problem – Real World Math

### Using Linear Relations and Proportional Reasoning to Model Water Flow

The following lesson resource material provides Real World Math Problems that were created by the mathematics department at Herman Secondary School from the Greater Essex County District School Board. I had the pleasure of working with Mr. Fabris, Mrs. Austen, Mr. Loebach and Mr. Marusic to create their first Real World Problem or 3 Act Math Task as Dan Meyer would call it. What a success! We will be getting together in a couple weeks to decide on some learning goals, suggested prompts, and other useful information to go along with these videos and photos. The goal here is to allow their classes to begin thinking about how long it would take to get their Recommended 8 Cups of Water Each Day from the Herman Secondary School Water Fountain. Use as you see fit in your classroom!

### Minds On:

Watch the Video Below:

Question that will likely come up through class discussion:

How fast is the water flowing from the fountain?

### Action:

Another question to explore:

How many Herman Griffins Water Bottles will it take to fill a 2L Bottle of Coca-Cola?

## The Drive to Work – Real World Math

### Using Linear Relations to Model a Car Commute

The following lesson resource material provides Real World Math Problems that were created with the Grade 9 Ontario Mathematics Curriculum in mind. A video and series of screenshots from a smartphone were taken by a passenger as we attempted to best capture The Drive to Work. On this particular day, we were running late and thought we might be able to inspire some deep thinking with the questions that could be posed to our grade 9 students.

### Minds On:

Students will watch a 1-minute real world math video The Drive to Work. In the video, students view a section of the drive to work from the perspective of passengers in the car.

Watch the Video Below:

Question that will likely come up through class discussion:

How fast is the car going?

In Dan Meyer “3 Act” fashion, a great start to the discussion might be to have students make predictions:

• What is a number that is too low?
• What is a number that is too high?
• What number do you feel best represents the speed?

Discussion about whether the car is going at a consistent speed, what road we are travelling on and other interesting pieces of information may be asked. For the record, cruise control was used where possible and the road was County Road 42 outside of Windsor, Ontario, Canada.

### Action:

Students can now analyse this photo and have a discussion with their table group to see what questions they have and what information they will need to answer their question(s):

Some questions students may have:

• Where did the car start?
• Was the car travelling at a consistent speed?
• What is the speed limit of that section of road?

Other questions are also possible.

In this particular situation, cruise control was being used at the starting point on the map to the end point.

Assuming students determine that the time and distance travelled at the start and end point of the map are important, you can reveal the following:

In my classroom, I have students use their own method to find the distance travelled at the end point on the map. Since we have a paperless iPad classroom, I have students use Google Maps or similar to find the distance travelled at the end point.

Scaffolding:

If your students are having trouble getting started, or if you want to promote the use of finding a linear equation using two points, then you may want to offer the following visual as a starting point. In my classroom, students are able to use any method they see fit and we then share out over the Apple TV via AirPlay on our iPads to compare solutions and attempt to find the most convenient solution for our bag of math tricks.

Something else students might not consider is the fact that the timer shows minutes and seconds travelled. This could provide for a great teachable moment when student answers differ and may provide for a great classroom discussion about units and conversion.

### Consolidation:

Students may then work together to form their solution to the problem.

In some cases, if students have had experience working with different representations of linear equations, you can ask them to show their solution in multiple ways such as the following:

Interested in a Solution to Share:

## Real World Percentages – Estimating and Calculating Discounts

### Sharpening Your Shopping Math Skills – Calculating Discounts

The following lesson resource material provides Real World Math Problems that were created with the Grade 6 Ontario Mathematics Curriculum course in mind. Teachers from Ontario schools as well as schools world-wide are welcome to use this lesson and series of videos/photos in their classroom to engage their students with a real world application to strengthen the estimation and calculation skills of percentages as well as apply their understanding of discounts as a percentage. Throughout the lesson, students are asked to estimate the cost of each item of clothing after a discount is applied. Students will then be able to take their estimate and complete the calculation with a calculator to compare their estimate with the actual result. Teachers in grade 5 may also find this activity useful for an extension and teachers from grade 7 to 10 may also find this useful for a quick and fun activity for recalling prior knowledge of percentages and discounts.

### Success Criteria:

After our Real World Percentages Math Lesson, I will be able to:

• estimate quantities using benchmarks of 10%, 25%, 50%, 75%, and 100%,
• calculate percentage quanitites,
• and, apply percentage discounts to find the sale price of an item.

### Minds On:

Students will watch a 1-minute video called Real World Percentages – Estimating and Calculating Discounts (1/3). In the video, Mr. Pearce notices that infant clothes for his daughter are on sale for a 75% discount.

Watch the Video and Pause When Video Indicates:

Estimate the sale price of the item.

The teacher can then ask students to lead a discussion at their table groups to determine ways that they can go about estimating the sale price after the 75% discount. Some guiding questions:

• What are some friendly percentages we can use to help us get a close approximation?
• Would you feel more comfortable rounding the original price to a more friendly number?

After the discussion, students can share out via Apple TV or using chart paper in your classroom to model as many creative solutions as possible. Students can then check their estimates using a calculator.

At this point, the teacher may resume the video to see the answer in the video.

### Action:

Students will then watch the next 1-minute video clip called Real World Percentages – Estimating and Calculating Discounts (2/3). In the video, Mr. Pearce is in the Calvin Klein outlet store and shopping for a new suit jacket. Luckily, the suit jackets are discounted.

Watch the Video and Pause When Video Indicates:

Estimate the sale price of the item.

Teacher can have students lead discussions in their table groups as they did in the Minds On section and share out in a similar fashion.

### Consolidation:

Students will then watch the next 40-second video clip called Real World Percentages – Estimating and Calculating Discounts (3/3). In the video, Mr. Pearce continues shopping at the Calvin Klein outlet store and comes across another sale table.

Watch the Video and Pause When Video Indicates:

Estimate the sale price of the item.

Teacher can have students lead discussions in their table groups as they did in the Minds On section and share out in a similar fashion.

## Real World Trigonometry – Primary Trigonometric Ratios

### Finding Height Given a Side and Angle

The following lesson provides Real World Math Problems that was created with the Foundations of Mathematics (MFM2P) Grade 10 Applied course in mind. Teachers from Ontario schools as well as schools world-wide are welcome to use this lesson and series of videos in their classroom to engage their students with a real world application using the Primary Trigonometric Ratios (SOH CAH TOA) to find height given a side and an angle. Throughout the lesson, students are asked to determine how tall a student is, to estimate the height of a support pole in the main hallway of Tecumseh Vista Academy K-12 School and then finally, calculate the actual height using prior knowledge from our Similar Triangles and Trigonometry unit.

### Success Criteria:

After our Real World Math Trigonometry Lesson, I will be able to:

• analyse a real-world situation and make a connection to the prior knowledge I have learned throughout this math course,
• deconstruct the problem and use prior knowledge of the primary trigonometric ratios acquired throughout this unit to create a solution to the problem,
• and, show adequate steps to clearly demonstrate my understanding.

### Minds On:

Students will watch a 30 second video called Real World Math – Trigonometry (1/3) – Using Primary Trig Ratios to Find Height. In the video, Dylan (a student) is standing against a wall with a metre stick on the floor touching his shoe.

Watch the Video:

How Can We Find Dylan’s Height?

After the video, the teacher can have the students share out ideas regarding how we can go about finding the height of Dylan in the video. Some guiding questions include:

• What information would be useful to help us solve the problem?
• Why do you think the metre stick is on the ground next to Dylan?
• Is the metre stick the only piece of information required to determine a solution?

After the discussion, prompt students to determine Dylan’s height using the method of their choice. Sharing out via Apple TV would be ideal to see if there are any creative solutions other than the Primary Trigonometric Ratios.

### Action:

Students will then watch the next 30-second video clip called Real World Math – Trigonometry (2/3) – Estimate the Height of the Support Pole. In the video, Dylan stands next to one of the building support poles in the main hallway at our school, Tecumseh Vista Academy. You can pause the video or use the image to the right to help them engage in a discussion with their table groups.

Estimate the Height of the Support Pole.

Working in table groups, the students can try to come up with they believe to be their best estimates.

Some guiding questions:

• Estimate heights that you believe to be too low, too high and the actual height (Dan Meyer’s 3Acts).
• What are some estimates you can make using the height of objects you know in the picture?

You may find that some groups decide to make random guesses, while others might use objects from the photo as a reference to make their estimates. Encourage groups to make estimates based on any information they can relate to the scenario.

Groups Having Difficulty?

If you find that some table groups are having difficulty, or this is the first time you are attempting a problem outside of your regular routine, you might want to give them some additional guidance.

After discussing student estimates and why they believe their estimate would be close to the real height, you can then show them some more detailed information like the following:

### Consolidation:

Students will then watch the last 30-second video clip called Real World Math – Trigonometry (3/3) – Find the Height of the Support Pole. In the video, students are provided with a quick view of the supprt pole as well as a triangle with the angle of elevation and length of the adjacent side (the metre stick).

How Tall is the Support Pole?

Feel free to pause the video to give the required information, or bring up the following images for students to use as they work independently or with an elbow partner:

Encourage students to share their results over the Apple TV or by simply explaining from their seat. Using a single iPad as a document camera can also be useful to quickly share student work over a projector or Interactive Whiteboard.

## Exploring Linear Relationships – The Detention Buy-Out

### Using iPads in Math as a Tool for Problem Solving

#### Success Criteria:

After our exploring linear relationships lesson, I will be able to:

• analyse a real-world situation and make a connection to the prior knowledge I have learned throughout this math course,
• deconstruct the problem and use prior knowledge to create a solution to the problem,
• and, show adequate steps to clearly demonstrate my understanding.

### Minds On:

Students will watch a video called The Detention Buy-Out. In the video, three administrators from Tecumseh Vista Academy K-12 School are interviewed and propose individual options for students to avoid serving detentions by paying the administrators according to their buy-out offers.

See the Exploring Linear Relationships – Detention Buy-Out video below:

### Action:

Students will then be split into groups of 2 or 3, arranged by the teacher, to determine which administrator should each student buy-out from.

Encourage students to show their solution in any way they would like or you can assign certain methods to particular groups.

The exploring linear relationships problem can be solved in a number of ways, including:

• Trial and error / guess and check (grades 6-10)
• Table of values (grades 6-9)
• Graphing to find point of intersection (grades 9-10)
• Creating equations and substitute different values of x (grades 9-10)
• Solving a system of equations using elimination (grade 10)
• Solving the system of equations using substitution (grade 10)

Students can solve the problem using iPads in neu.Annotate+ or GoodNotes – Notes & PDF.

### Consolidation:

Once groups have solved the problem, they will take screenshots of different parts of their solution in the PDF annotation app they used and import the photos into Explain Everything to create a video that allows the students to communicate their understanding. Once they have completed their video, they will share over Apple TV with the rest of the class.

Alternatively, students can share their solutions directly from an annotation app like neu.Annotate+ PDF or GoodNotes – Notes & PDF over the Apple TV.

If you do not have technology in the classroom, you can also have students solve the problems on chart paper and present them from the front of the room to explore the different methods of solving the problem.

Additionally, the teacher can give another problem involving linear relationships as a consolidation activity for students to complete in the last portion of class or on their own time.