Applications of the Pythagorean Theorem

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I enjoy spending time with the Pythagorean Theorem.  Proving it and showing how the squares of each leg add up to the square of the hypotenuse can provide a clear picture to your students.  I have used square crackers, square napkins or even the desks in my room to show how the Pythagorean Theorem works, but also to help students figure out how to apply the equations and figure out a missing side.

Once students understand the equation and see how it works, I like to show them how much we use the right triangles in our lives.  What are some real life applications of the Pythagorean Theorem?  I remember walking on some paths at the park.  My brother was ahead of me on his bike and had already turned the corner.  I wanted to catch up to him, but I needed to cut the corner and find the shortest distance.  I remember my father telling me to run across the grass.  He even said, “Take the hypotenuse, that’s the diagonal cut across the grass and you can catch up faster!”  My father was a high school math teacher, so I guess hypotenuse was part of his everyday vocabulary.  But do you know what?  It’s a great word…and his strategy worked!

When I started to explore real world applications of the Pythagorean Theorem, I discovered that maps can be a useful way to show the hypotenuse as well as the right angles.  But right triangles are used in many other designs as well, including buildings and bridges.  When I thought about the importance of knowing the pythagorean theorem for certain jobs, it hit me.  Did you know that firefighters need to know the Pythagorean Theorem?  They may have several different lengths of ladders on the truck, and even ladders that can extend to different heights, but firefighters need to know which ladder will reach which height.  And they don’t want to be playing a trial and error game, they want to make sure they pick the correct ladder the first time so they have the best chance of saving people as quickly as possible.

What better way to practice applications of the Pythagorean Theorem than with ladders, heights and real world jobs.  Kindergarten classes get to go on field trips to the fire station or have visits from a firetruck.  I think that middle schoolers should too.  Maybe we can learn some tricks of the trade about how they use the Pythagorean Theorem?

   

Set up for the Activity: Application of the Pythagorean Theorem

The set up will take some time in order to make stations around the room.  The scenarios are located on pages 3,4, and 5 of the worksheet and can be cut out and provided at each station for students to pick up.  (This would prevent students from trying to work ahead before arriving at the stations).

The stations will represent calls to the fire station that the students have to respond to/visit and determine which size ladder they have to use.  You can label the height of the object, or rely on the students reading each scenario.  You also want to set up your props at certain heights that the students can make measurements too.

The ladders and stations will be in inches. Tell the students that the scale is 1 inch = 1 foot.  (The scenarios are written in feet to represent a real fire call).

  • For each group: Cut strips of paper or thin dowels with the following lengths:  These will be the “ladders”.  Students can begin by cutting their own ladders, but it’s a good idea to equip them with blank strips of different lengths, then they can measure them when doing the calculations.

5 inches

13 inches

15 inches

26 inches

10 inches

39 inches

50 inches

Create the following stations around the room: (Scenarios are located on the worksheet with images as well)

Station 1: Place a “cat” or picture of a cat, 12 inches off the ground.  Put a piece of tape 5 inches away to mark on the ground where the bottom of the ladder can be placed.

A cat is stuck in a tree.  She is on a branch that is 12 feet off the ground.  If you can set your ladder 5 feet from the base of the tree, what length ladder do you need? (Answer: Students will need the 13ft ladder or 13 inch strip of paper “ladder”)

Station 2: Draw a picture of a building or a burning window.  Tape it 36 inches off the ground.

There is a fire on the third story of a building.  You can see flames coming out of a window, so you decide to put a ladder to the window next to it.  The window is 36ft high.  What size ladder do you need if you place the ladder 15 ft from the bottom of the building? (Answer: Students will need the 39 ft ladder or the 39 inch strip of paper “ladder”)

Station 3: Put a piece of tape on the wall 48 inches off the ground.

An air conditioning unit on the roof of the school building needs to be inspected.  If you use your 50 ft ladder, and place it on the ground 14 feet from the edge of the building, how tall is the school roof? (Answer: The school roof is 48 ft.)

Station 4: On the board, draw a picture of a firetruck, with a ladder extended and a flag flying from the top.

At a town celebration, a ladder truck is parked and has an American flag flying from one of the ladders that is sticking diagonally from the truck into the air.  If the flag is flying from a 60ft ladder, and the truck is 36 ft long, how high is the flag from the truck? (Answer: The flag is 48 ft.)

Station 5: Place “people” on a desk, box, or stack of books that is 24 inches high.  If you have action figures or toys, this could be fun, or you can put objects there or drawings.  Place a piece of tape 10 inches from the bottom of the box.

The fire has been extinguished, but the stairway is blocked.  Firefighters must get the people out of the building.  There’s a balcony 24 feet up, what size ladder do you need? (Answer: Students will need the 26 ft ladder or 26 inch strip of paper ladder)

Station 6: Place a cow or object on the wall or on a desk 8 inches high. Do not place tape on the floor.

Did you know that a cow can walk up stairs but not down them if they are too steep?  There’s a cow 8 feet up in the barn and she needs help!  If you can get up there to lower down a ramp and lead her down, then she will be safe. What size ladder do you need?  Be sure to use a ladder from your choices and also show how far the ladder needs to be placed. (Answer: Students will need a 10 ft ladder placed 6 feet from the barn, or a 10 in strip of paper ladder placed 6 in from the wall/desk.)

Launch the Applications of the Pythagorean Theorem Activity

When your students come to class greet them with a big “Welcome to the Pythagorean Fire Company!”

Then explain that today they will be using the Pythagorean Theorem to help them find the best length ladder for each job.

Depending on whether the students need review or not, you can review the Pythagorean Theorem and give the students an example to practice finding a missing side:

Here’s an example:  Draw 2 legs of a triangle: 3 and 4.  Then connect them with a diagonal and ask the students if they remember the name of the side opposite the right angle.  (Hypotenuse).  Then review the proof of the pythagorean theorem by drawing a 3×3 square, and a 4×4 square off the legs. Then ask:

  • What is the area of a 3×3 square?  9
  • What is the area of the 4×4 square?  16
  • So what is the area of the hypotenuse square?  25
  • This means that the length of that side will be?  5

Show the 3,4,5 triangle.  Then review how to solve it using the equation:

a2 + b2 = c2

Review which is the hypotenuse (you can use the previous example to help).  Then say that if they know any 2 sides, they can figure out the missing side.  In this case we knew the 2 legs were length 3 and 4.

So the equation would be:  (3)2 + (4)2 = c2

Solve:  9 + 16 = c2

25 = c2   so c= 5

If they need more practice, you can also provide an example with other numbers or other missing sides:

Here’s an example:  A triangle with lengths 5, b, 13

a2 + b2 = c2  would be (5)2 + b2 = (13)2

25 + b2 = 169

b2  = 144  so b = 12

Tell your students that today they are going to see the Pythagorean Theorem in action.  You can ask them if they remember visiting a firehouse or meeting a firefighter when they were younger, or you could ask if anyone has ever thought about becoming a firefighter or if a member of their family is a firefighter.  Then ask “Do you know how firefighters use the Pythagorean Theorem?”

Today, they get to become the firefighters and use the Pythagorean Theorem to help figure out which ladder to use!  Will their ladder choice help save the day?

Pythagorean Firefighters to the Rescue: Application of Pythagorean Theorem Activity

The students will set up at each station and move to the next station after testing out their “ladder”.  You may choose to set a timer, 5-6 minutes for each station.  This will allow for rotations, as well as put the “five-alarm fire” urgency into the activity.

Make sure that the students are recording their data and sketching right triangles to show which side they are solving for as well as the equation for the Pythagorean Theorem.

Students will:

  1. Read the scenario.
  2. Take measurements.
  3. Record their data, showing a sketch or model of the scenario.
  4. Solve for the missing side.
  5. Test their solution with one of the ladders.

The Pythagorean Firefighters to the Rescue: Application of Pythagorean Theorem Worksheet has space for the students to show their work.

Here are the scenarios:

Station 1:

A cat is stuck in a tree.  She is on a branch that is 12 feet off the ground.  If you can set your ladder anywhere from 5 feet from the base of the tree, what length ladder do you need? (Answer: Students will need the 13ft ladder of 13 inch strip of paper “ladder”)

Station 2:

There is a fire on the third story of a building.  You can see flames coming out of a window, so you decide to put a ladder to the window next to it.  The window is 36ft high.  What size ladder do you need if you place the ladder 15 ft from the bottom of the building? (Answer: 39 ft)

Station 3:

An air conditioning unit on the roof of the school building needs to be inspected.  If you use your 50 ft ladder, and place it on the ground 14 feet from the edge of the building, how tall is the school roof? (Answer: 48 ft height of school roof)           

Station 4:

At a town celebration, a ladder truck is parked and has an American flag flying from one of the ladders that is sticking into the air.  If the flag is flying from a 60ft ladder, and the truck is 36 ft long, how high is the flag from the truck? (Answer: 48 ft above the truck) 

Station 5:

The fire has been extinguished, but the stairway is blocked.  Firefighters must get the people out of the building.  There’s a balcony 24 feet up, what size ladder do you need? (Answer: 26 ft ladder)

Station 6:

Did you know that a cow can walk up stairs but not down them if they are too steep?  There’s a cow 8 feet up in the barn and she needs help!  If you can get up there to lower down a ramp and lead her down, then she will be safe. What size ladder do you need?  Be sure to use a ladder from your choices and also show how far the ladder needs to be placed.  (10 ft ladder placed 6 feet from the barn)

Reflecting on the Application of the Pythagorean Theorem Activity

After the students have chosen their ladders, talk about the activity.  You can have each group share one of the scenarios and how they solved it.

Also ask questions like:

  • Which side was the easiest to find?
  • Did you have to find the length of the ladder every time?
  • How is this similar to real life use of ladders?
  • Did you notice any patterns?
  • Is there a station that was difficult for your group? Were still able to save the day? How did you work through the challenging part?
  • How would this be different for firefighters on a real call?

Extensions:

  • Take this to the next level by finding real ladders and using them outside (the building custodian could probably help with this!). Measure the length of the ladder, then measure set distances from the edge of the building to different spots on the ground.  Predict what height the ladder will reach depending on how far the ladder is from the building.  Don’t have students actually climb the ladders for safety reasons.
  • Students can research safe distances and placement of ladders. They can also research fire ladders and their lengths to help determine how high some ladders can reach, and what length they are.  Here is a video of a firefighter student, explaining the Pythagorean Theorem.
  • Students can use a shoe box and measure the sides to determine which length “ladder” or staircase could fit diagonally inside the box.

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