​I am always trying to create or find projects where each student’s project will be different than every other student’s in the class because it is more authentic this way. And of course, I want it to be applicable in the real world! This is not always easy to make because if every student’s project is different, or even has an unlimited number of solutions, checking each one is extremely difficult and time-consuming for the teacher. Well, guess what?!...the project I’m about to tell you about is authentic AND I was able figure out how each student could have a different solution. And at the same time, have an answer key for the teacher with all the possible solutions….I know, fabulous, right!! And it covers one of the most popular math topics…Linear Systems of Equations!

Oh wait, one more cool thing about it….it’s not quite STEM because there isn’t any engineering, but I’d say it’s STAM (Science, Technology, Art, and Math)…fun little twist…so check it out:

Every project needs to have a focus and goal. In this project, students are to…

• Research the growth of a tree and write an equation to represent the growth
• Determine when two trees will be the same height (algebraically & graphically)
• Analyze the results
• Create visuals to represent information about their tree and data results

​Students select a tree from the Arbor Day site and find the growth rate range and mature size. Students write two equations – 1 to represent their tree’s growth over time and 2 - their teacher’s tree growth over time. Students use the substitution or elimination method to solve the system and explain the solution. Included is a spreadsheet where the teacher can record the student name, tree, and growth rate to ensure each student has a different growth rate.

Student’s us graph paper or Desmos to re-create their system of linear equations by graphing both lines. Students also draw, paint, or find a picture of their tree online and label it by finding specific characteristics of their tree. I learn a lot of neat things about different trees from each project. And students love the artsy aspect!

Students find at what year the trees are the same height and determine if it’s realistic based on the year each tree will reach their mature size. This helps students understand that not every solution in the real world will make sense, so it’s important to critically think about the results.

​As with any project, I do use a rubric, which is 100% editable for teachers. This project is evaluated on the following criteria: neatness/organization, accurate research, writing a systems of equations and finding the correct solution, the graph, analyzing the results, and the visuals. I also include an answer key for the questions and all of the possible scenarios/solutions depending on the growth rate of each tree in comparison with the growth rate of the teacher’s tree. This is a HUGE time saver for teachers!

​If you were to visit my classroom, you would see a lot of different ways students learn: guided notes, games, stations, activities, projects and more! Project Based Learning (PBL) is a great way for students to critically think, problem solve, and, in general, see math differently. Therefore, I try to integrate a project into every unit and make them as “real” as possible.
 
One of my favorites, and my students, is the Parabola Selfie Project. In this project, students take math outside of the classroom and explore the real world to find a parabola.

​Let’s take a quick look at how this project is broken down…

Students find a parabola in the real world and then take a selfie with it. Why take a selfie, you say? Well, first of all, it makes it fun for them since selfies are something they do often and share on Facebook, Snapchat, Instagram, or some other form of social media. Second, I want to make sure they don’t just Google search a picture online. That would take all of the fun out of this project. There is so much in the world to investigate, so I want them to go out there and see math as much as possible.

​Next we pop that picture right into Desmos, which is an online graphing calculator. Students adjust the scale of the graph to match the dimensions of the real life parabola. I’ve even created a video showing students how to do this, in case they are not familiar with Desmos. Then they write an equation for their parabola and analyze the parabola by finding characteristics such as the axis of symmetry, vertex, domain, range, etc. 

In almost every project I try to create a fun little twist that involves students observing or reviewing each other’s project. In this one, I have students exchange their graphs with each other and use the Parabola Swap table to record their information. This will give students an opportunity to identify characteristics of another parabola and also receive feedback on the accuracy of their data.

I rarely have students complete a project and then individually present to the class. It takes up too much class time and sometimes it can be difficult to see and understand the data when in a slideshow. I’d rather students take their information and put it on a poster or in a report format. Then we do some kind of walk around to view all the projects up close.
 
For this project, since students already swapped parabola graphs with another group and filled out the characteristics table for that graph, I don’t have them fill out another form when they do the walk around. Instead, I have them view each project and then vote on who found the most unique parabola in the real world. I give out a prize to the first and second place winners. You could give out a homework pass, food, or anything that your students enjoy. This gives each student a little more incentive to really find a fun and unique parabola.

​As with any project, I do use a rubric. I evaluate each project on the following criteria: neatness/organization, the parabola selfie, the graph, the characteristics of the graph, and the quadratic equation.

 Here are some of my student’s Parabola Selfies: 

Doesn’t the Parabola Selfie Project look like fun?!....and educational! Click on the project below that you'd like to try in your classroom:

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I use to despise teaching piecewise functions. You know when it’s raining out and you forgot your umbrella? What do you do? Well, you run as fast as you can to your car, and as soon as you get in you take a deep breath and think, I made it!...but I’m pretty wet. That’s how I use to teach piecewise functions. I wasn’t prepared, I did it fast, and the results were not great.

​A couple of years ago I decided to seriously take a look at how I could teach piecewise functions best. I did some research and decided to make sure I covered how to evaluate, graph, and write the functions. In the past, I had just focused on graphing them and assumed that if students could do that, then they could do anything with piecewise functions. Yeah, I know, wishful thinking. Therefore, I created a lesson that clearly covered those three areas. Then I had my kids dive right into piecewise functions with a project. Let’s be honest, after that one lesson they weren't loving piecewise functions yet. This project was just as important as the lesson because it made piecewise functions come alive.

PART A
Students create a graph of a roller coaster in regards to time and height using linear, absolute value, and quadratic functions. I tell them they can pretend that their graph is what the roller coaster looks like as well, because it makes it a little more fun for them. However, we know that physics would not necessarily allow this to be true. (Downhill will not be as fast as uphill as you would see in an absolute value or quadratic function).

PART B
Students write a function that represents their graph. Some of my students use Desmos to check and see if their function and graph match. This really helps make sure all equations in the piecewise function are correct, and if they don’t, they can adjust their graph accordingly. Another way is to have them check on their graphing calculator. Of course, there will always be some students who don’t do either. I use Desmos when grading the project and this is sometimes what I see...

YIKES! Teacher => face plant into desk.

Hmmmm....something is not quite right.

YAY! You got it!

PART C
Let’s go for a ride….

This is a fun way for them to analyze the characteristics of their function.

PART D
The students create an answer key for their function that identifies the function attributes, which will be used in the stations activity. 

STATIONS ACTIVITY
I do not have students present their roller coaster to the class because it’s not only a bore, but also very time-consuming. Instead, students walk around to a few roller coasters and identify the domain, range, intervals of increase/decrease, minima, maxima, and evaluate for x-values. They get a chance to see other roller coasters and get some review.

EXTENSION - FIELD TRIP
And finally, the best part! You can take your students on a field trip to an amusement park. The students video record a roller coaster and sketch the graph relating time and height. They love this aspect of the project….a day away from school! Here is one of the videos from our field trip:

I know what you are thinking…I turned 1-2 days of instruction into 5 days. Right, you may not have that kind of time. No worries! You can use the parts of the project that work best for you. Use the video above instead of going on a field trip. Each year I find a way to get as much in as possible because the results are amazing! The new precalculus teacher this year came to me the other day and said, “Wow, the students from Algebra 2 last year really know piecewise functions.” And I said, “Yes, yes they do,” with a big grin on my face. Some things are just worth it.

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It’s that time of year when seniors are receiving their college acceptance letters. Woohoo, so exciting! The school where I work is very small, so we acknowledge each acceptance for every senior at monthly assemblies. It’s so neat to hear all the different schools that our students have been accepted! Often I get a little teary-eyed because I am so proud of them.
 
This makes it a perfect time for me to whip out the College Trend Project involving linear and exponential lines of best fit. Students research different universities to see if the tuition is trending as a linear or exponential function and predict future tuition costs.

So, if you are looking for a "real world" project that is engaging and meaningful, look no further!

Students will be able to demonstrate their knowledge and understanding of the following skills:

• Gather accurate data and create a table
• Graph a scatter plot
• Find the linear and exponential line of best fit and graph
• Determine which equation (linear or exponential) best models the data
• Predict future tuition costs using the line of best fit function
• Research background information about each university/college  

​Of all the projects we do, this is definitely my students’ favorite. They LOVE to research universities they are interested in attending and are shocked by how much the cost has changed over the years. I also incorporate a stations activity, so students can review each other’s projects. The students have fun looking at other universities they may not have thought of or even heard of before. Plus, these student sheets help you grade the projects too, which is a nice perk!
 
Most students do this project in 9th or 10th grade, so by the time they are seniors, we love to look back and talk about the schools they thought they would attend and how their plans may have changed.
 
If you are interested in learning more about this project and using it in your class, click on the example below. The project and rubric are 100% editable, so you can adjust the years. Plus, I’ve included two bonus calculator reference sheets (linear regression and exponential regression) to help your students with the line of best fit portion of the project. Oh, and I forgot to mention there is a ton of technology integrated into this project!

​​I am always trying to find fun and easy ways to make math more applicable for my students. The idea of visiting other places is a great way to get their attention. I created a frequent flyer mile template that can be used in any math course that touches on slope, linear equations, or direct variation. The activity can be used with a whole-group, small-group, or even a take home assignment. One of the best features is, it’s set up to give the students lots of choices. They get to choose the airline they use, the dates they fly, and where they are going to travel. Their goal is to find out if they can earn a free flight. Of course, there is lots of math they need to do along the way!

​Here is how I organize the activity for collaborative, small groups:

FIRST: Create groups of 2 or 3, and give a list of airlines they can choose from. Be sure to check beforehand that the airline has a points program. Some groups can use the same airline, it really doesn’t matter.

SECOND: Have them research their airline to see how the frequent flyer mile program works. This will help them determine what x- and y- should represent and create their equation. I usually pre-determine the goal points they need to earn for the free flight, as that may be more difficult for them to find, but it’s up to you.

THIRD: Now, they get to decide which places they will travel! They will calculate their points along the way. You will observe students discussing places to travel and strategies on how to make sure they travel far enough to earn a free flight.

LAST: Once they are finished traveling, they need to add the total miles (x) they flew and the total points (y) they earned. They will use these numbers as their right endpoint on their graph. It will also help them determine if they earned a free flight.
 
I use a similar model as above for a whole-group activity, except we vote on which places to visit. We have fun googling each place to see how cool it would be to go there! This would even be a good activity to tie into a history course…hmmmm, something to think about for next time.
 
Sometimes taking a break from the standard lesson is a good thing!

​Grab this FREE activity sheet here:

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