Grade 8 mathematics
The distance a phone has to compute
A straight-line distance is the hypotenuse of a right triangle. GPS needs that kind of distance, measured from satellites instead of paced off on a sidewalk.
- Course
- Grade 8 math, Pythagorean theorem
- Grade
- Grade 8. A high school geometry class can use it on the day the distance formula comes back.
- Length
- 45 minutes. The race is 10.
- When
- Illustrative Mathematics Grade 8, Unit 8, Lesson 11, “Finding Distances in the Coordinate Plane.” Lesson 10, “Applications of the Pythagorean Theorem,” also works if students can already find a missing side.
Host this class opens a private 10-minute lobby, Pythagorean theorem to Global Positioning System, fewest clicks. Students see the start and the target. The answer key shows the chain, so keep it on your screen.
Do not project this page
It contains the route. Put the lobby PIN on the board. Leave this lesson on your own laptop, or print it and hold it.
One device per student, or one per pair. Phones work. Students can join as guests. No accounts.
These articles do not already link. The shortest routes are 2 clicks.
Routes students can take
Pythagorean theorem → Trigonometry → Global Positioning System. Pythagorean theorem → Euclidean geometry → Global Positioning System. A longer wander still counts if they reach the target. These are the short ones to expect.
Individual routes
2 shortest routes. Hover a route to highlight it on the map.
- 12 clicks
Pythagorean theoremIn mathematics, the Pythagorean theorem or Pythagoras's theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle.
TrigonometryTrigonometry (from Ancient Greek τρίγωνον (trígōnon) 'triangle' and μέτρον (métron) 'measure') is a branch of mathematics concerned with relationships between angles and side lengths of triangles.
Global Positioning SystemThe Global Positioning System (GPS) is a satellite-based hyperbolic navigation system owned by the United States Space Force and operated by Mission Delta 31.
- 22 clicks
- Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras's theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle.
Euclidean geometryEuclidean geometry is a mathematical system attributed to Euclid, an ancient Greek mathematician, which he described in his textbook on geometry, Elements.
- Global Positioning System
The Global Positioning System (GPS) is a satellite-based hyperbolic navigation system owned by the United States Space Force and operated by Mission Delta 31.
Who it is for
The grade 8 math teacher who has just taught the theorem and is about to hear “when will I use this?” The honest answer is a distance you cannot tape-measure. A phone finding its place is that job.
In Illustrative Mathematics Grade 8, this is Unit 8. Students already have two proofs of the theorem and can find an unknown side (Lessons 7 and 8). Lesson 11 asks them to treat the gaps in the coordinates as the legs and the straight-line distance as the hypotenuse. Run this lesson that day, before they start the coordinate practice, so the grid problem has a reason.
A second chain on the live Wikipedia articles is Pythagorean theorem, Euclidean geometry, Global Positioning System. Either bridge is a win. Trigonometry is the tool that turns angles and distances into a position. Euclidean geometry is the family the theorem belongs to.
Say this plainly in the debrief, because it is the part students will get wrong: a phone does not draw one triangle on a paper map and stop. It measures how far it is from several satellites, which is called trilateration. On a flat map, each of those distances is a Pythagorean calculation. The theorem is the distance step inside the job, which is exactly 8.G.B.8.
Standards
- CCSS.Math.Content.8.G.B.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
- CCSS.Math.Content.8.G.B.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Before they walk in
They already need to know
- Students can find the third side of a right triangle when they know the other two.
- They have heard the words leg and hypotenuse.
- They do not need trigonometry yet. The race is how they meet that word.
You do this
- Open the answer key once, on your own screen, so you know both bridges. Then leave that page.
- Host the class from the button on this page. It opens a private lobby, Pythagorean theorem to Global Positioning System, 10 minutes, fewest clicks. Private keeps the round off the public list.
- Put one sentence on the board, and nothing else: “Name a distance nobody can pace off.”
The period
The question
Read the board sentence. Take three answers: a river, a flight, the height of a building. Do not say GPS yet. Tell them they are about to race from the Pythagorean theorem to a page that needs a distance of that kind.
Open the room
Show the PIN and the QR code from the lobby screen. One device per student, or one per pair. Guests can join. No accounts. The only moves are the blue links on the page. The article search will not find the target. Fewest clicks wins. Ten minutes.
The race
Walk the room. If someone is stuck on a page of formulas, ask “Which of these words is about measuring a distance or an angle?” Do not name Trigonometry or Euclidean geometry. If someone opens a new tab, send them back. The route only counts inside the game.
Name the bridge
When the round ends, the routes appear in the lobby. Put the chains on the board: Pythagorean theorem to Trigonometry to Global Positioning System, and the geometry twin. Ask why a page about triangles had to pass through trigonometry or Euclidean geometry before it could reach a navigation system. Collect the idea “those are how you compute a distance.”
One triangle, on a grid
School is at (0, 0) on a city grid. The park is 6 blocks east and 8 blocks north. The legs are 6 and 8. The straight-line distance is √(36 + 64) = √100 = 10 blocks. That is 8.G.B.8. Then say the sentence you prepared: the phone measures distances from satellites and computes a place. Each flat distance is this kind of triangle. The theorem does not steer the car. It supplies the length.
Exit
Students write one sentence on a slip: what the theorem computes that a GPS needs. Collect them at the door.
Answers you will need
Why is Trigonometry a reasonable bridge?
Trigonometry relates angles and side lengths in triangles. Navigation uses angles and distances to fix a position. The Pythagorean theorem is the right-triangle case of that same family.
Why is Euclidean geometry also a reasonable bridge?
The theorem is a theorem of Euclidean geometry. GPS calculations of distance live in that geometry. A student who went that way found the subject, not a trivia page.
Does a phone just do a² + b² = c²?
No. Say so. The receiver times signals from several satellites and turns those times into distances, then finds the place that matches all of them. On a flat map, a single one of those distances is a Pythagorean length. That is the piece this unit owns.
What if the fastest student used 4 clicks through Ancient Greece?
Read the route aloud. Ask which click was the one that started being about distance. Praise the arrival, then show the 2-click chain as the one that stays on the idea.
Exit
In one sentence, what does the Pythagorean theorem compute that a GPS needs?
A finished answer has
- The sentence names a distance or a length, or it names the hypotenuse.
- A strong sentence mentions satellites or a distance you cannot walk, and then the triangle that computes it.
Steer these
- “GPS uses triangles, so it knows the streets.” Streets are a map drawn afterward. The measurement comes first.
- “It squares the coordinates of the satellite.” They do not need the algebra of orbit. The 6-8-10 grid is the whole example.
If the period is short
Skip the opening brainstorm. Race for 10 minutes, name the bridge for 8, and do the 6-8-10 triangle. The exit sentence can be said aloud.
If the network fails
Ask them to guess the bridge before you tell it. Then draw the 6-8-10 triangle and say the measured chain is Pythagorean theorem, Trigonometry, Global Positioning System. The triangle is the lesson. The race is the way they find the bridge themselves.