Week 11 · Two Equations at Once

Wednesday

Puzzle Lab: strings on the floor

// Where the lines cross

⏱ about 20 min

Wednesday: Puzzle Lab, Strings on the Floor

Raven tapes the big floor grid flat. Rocket brings two strings, one red and one blue. "Lab day."

"Red string: y = 1/2x + 2," Raven reads from a card. "Blue string: y = 2x - 1."

Rocket pins the red string through (0, 2) and (4, 4). Then the blue through (0, -1) and (2, 3).

"They cross at (2, 3)," he says. "What do you notice, Raven?"

"Half of 2 is 1, plus 2 is 3. And 2 × 2 - 1 is 3. The strings agree with the algebra."

Nova hovers low over the grid, her light following the red string. "What if a string were pinned a little off?"

"Then the crossing would be a little off," Rocket says. "Graphing is a check, not a proof."

"Substitution is the proof," Raven says. "Four systems. Strings first, then algebra."

What you need

Graph paper taped together into a big grid, or a grid drawn on a large sheet, taped flat to the floor.
Two strings of different colors, about 2 meters each, and masking tape to pin their ends.
Four puzzle cards, each with two equations in y = mx + b form.
Your Puzzle Notebook and a pencil.

Safety first

Tape every edge of the grid down so no one trips. Keep shoes on in the garage.
Strings lie flat and are taped at both ends. Never wrap string around fingers or necks.
Scissors for cutting string are used by a grown-up or with one watching.

Run the lab

Draw and label the axes on your floor grid from 0 to 8 across and up.
For the first equation, find two points, such as the intercept and one more. Tape the string through them.
Do the same with the second string for the second equation.
Read the square where the strings cross and write it as (x, y).
Now solve the system by substitution. Compare the answer with the strings.
Record the system, the string reading and the algebra answer in your lab table.

The crew's lab log

Here are the four systems the crew laid out. The numbers are the crew's own, chosen so each crossing lands on a grid point.

System	Equation 1	Equation 2	Strings crossed at	Substitution gives
1	y = x + 1	y = -x + 5	(2, 3)	(2, 3)
2	y = 2x	y = x + 3	(3, 6)	(3, 6)
3	y = 1/2x + 2	y = 2x - 1	?	?
4	y = 3x - 2	y = x + 4	?	?

$A grid from 0 to 6 with a red line and a blue line crossing at a marked point, (2, 3).$

READ THE CREW'S STRINGS

Read the question.
Tap your answer.

System 3: y = 1/2x + 2 and y = 2x - 1. Where do the strings cross?

System 4: y = 3x - 2 and y = x + 4. Where do the strings cross?

Which point lies on both strings of system 4, y = 3x - 2 and y = x + 4?

System 3 is y = 1/2x + 2 and y = 2x - 1. What is the x of the crossing point? Type the number.

WHY THIS EXERCISEThe strings estimate the crossing. Substitution confirms it exactly.

Statement	True or false?
A string laid through two correct points shows the whole line.	?
(2, 3) is a solution of y = 1/2x + 2 and y = 2x - 1.	?
(2, 4) is a solution of y = 3x - 2 and y = x + 4.	?
If a string is pinned a little off, the crossing point moves a little too.	?
2 × 2 - 1 = 1/2 × 2 + 2	?

WHY THIS EXERCISEThe lab shows graphing as a check and substitution as the exact answer.

Try it

Lay strings for y = x and y = -x + 6. Predict the crossing before you look. Then check with substitution.
Now move the blue string to y = x + 2. What happens to the crossing? Say why.

Draw your floor grid with the two strings of system 3. Mark the crossing and write the point.

Puzzle Notebook: record your four systems with the string reading and the substitution answer for each.Did any string reading disagree with the algebra? Write what you think caused it.Write one system whose strings would never cross, and lay it out to see.

Great lab work. Tomorrow the crew writes systems from puzzle stories with two unknowns.

Grown-up checked my answers

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