Two Trains

Hook 5 min

1. Say: "Two trains leave at noon from opposite ends of a 300-mile line. One goes 60 miles per hour, the other 40. Where do they pass?" Write a few guesses on the board.
2. Ask: "Is there any way two trains on one straight track could pass each other twice?" Take a hands-up vote and leave it up.

Explore 15 min

1. Pairs open Yard. The default, Head on, is the hook's setup. Before anyone presses Play: "Where will each train be at 1 pm?" (A at mile 60, B at mile 260.)
2. Press Play. Point at the route ladder beside the graph: "Why does each train sit at the height of its line?" (The height is how far it is from Ashford.) The trains pass at Pine Hollow, 3:00 pm, mile 180. Check the hook guesses.
3. Pairs work through the Presets in order and fill in the table for each: Catch up (meet 3:00 pm at Millbrook, mile 120), Same speed, Same schedule, Too slow to catch.
4. On Same speed, turn on Show gap and drag the clock. "What happens to the gap?" (It stays 45 miles at every time.)

Make sense 10 min

1. On the projector, load Head on and tick Show the algebra. "Why do we set the two equations equal?" (We want the time when both trains are at the same mile.) Walk through 60t = −40t + 300, so 100t = 300 and t = 3.
2. Load Same speed: the algebra ends "45 = 0. That is never true: no solution." "What do the two lines look like?" (Parallel. Same slope, different starting point.)
3. Load Same schedule: "0 = 0", infinitely many solutions. "Why is every point a solution?" (It is the same line, so the trains are side by side from 2:00 pm to 5:00 pm.)
4. Load Too slow to catch. "The algebra gives t = −4.5. Did the trains meet?" (No. That is 4.5 hours before noon at mile −180, before either train left and off the track. Right for the equations, wrong for the story.)
5. Back to the hook vote. Write the three cases on the board: lines cross once (one solution), parallel (no solution), same line (infinitely many).

Apply 10 min

1. Pairs switch to Dispatch and do ticket 1, Meet at Millbrook: set B's speed, then press Check. If they are stuck, Show a hint.
2. Everyone does ticket 6, Call it first, on paper before the screen. A leaves Ashford at 1:00 pm at 75; B leaves Brookport at noon at 50. Pairs type their answers in Your meeting time and Your mile, then Play. "What equation did you solve?" (75(t − 1) = 300 − 50t, so t = 3: 3:00 pm at mile 150.)
3. Then ticket 7, Meet twice. Pairs try, then decide whether to press It can't be done. "Why can't two straight lines cross twice?" (Two different lines cross at most once. The same line meets everywhere, not twice.) Settle the hook vote.
4. Fast pairs: ticket 8, Junction at four (both speeds; A 15, B 60).

Exit check 5 min

On a card: "Train A: d = 50t. Train B: d = −30t + 240. When and where do they meet? Then change one number in B's equation so the trains never meet, and say why." (50t = −30t + 240, so 80t = 240, t = 3: 3:00 pm at mile 150. Never meet: any B with rate 50 and a different starting number, such as d = 50t + 240, because the lines are parallel.)