Week 11 · Matrices
Dock Day
Dock Day: the family hull
// The lane card grid
⏱ about 20 min
Dock Day: The Family Hull
Saturday the cardboard hull and a sheet of graph paper come to the kitchen table.
"Four corners," Comet says, tracing it. "(0, 0), (4, 0), (3, 1), (0, 1)."
Wren hands out matrix cards. "Each person multiplies the corners by a card and plots the new hull."
Someone draws [2 0; 0 1]. "Twice as long, same height. Determinant 2, so the area doubles."
Someone else draws [1 0; 0 -1]. "Upside down. Determinant -1. Same area, turned over."
Nova dims her light to a hint. "Would you like a hint? Predict the area from ad - bc before you count."
"And one card has determinant zero," Wren says. "Whoever draws it gets a hull squashed flat."
"Then we know that card has no inverse," Comet says. "No way to stretch the hull back."
- A matrix is a table of numbers named as one object. Its size is rows by columns.
- Same-size matrices add entry by entry. A scalar multiplies every entry.
- A product is row times column, pair by pair, added. AB and BA can differ. I acts like 1 and the zero matrix like 0.
- A 2 by 2 matrix moves every point of the plane. The absolute value of its determinant is the area factor.
- The inverse exists exactly when the determinant is not zero, and [x; y] = A⁻¹b solves the matrix equation.
◇ FAMILY HULL CARDS
Trace a cardboard hull on graph paper with whole-number corners. Count its area.
Write five matrix cards: [2 0; 0 1], [1 0; 0 -1] and [1 1; 0 1]. Add [0 1; 1 0] and [2 4; 1 2].
Each person draws a card, says the determinant and predicts the new area, then multiplies the corners and plots the hull.
Count the new area and compare. Trade cards until everyone has plotted a stretch, a flip and the flat one.
Write five matrix cards: [2 0; 0 1], [1 0; 0 -1] and [1 1; 0 1]. Add [0 1; 1 0] and [2 4; 1 2].
Each person draws a card, says the determinant and predicts the new area, then multiplies the corners and plots the hull.
Count the new area and compare. Trade cards until everyone has plotted a stretch, a flip and the flat one.
For a grown-up
This is a drawing and counting activity. Cardboard is cut with scissors only, by someone sitting down.
A counted area that misses the prediction is a good moment: check one corner's multiplication together.
The hull and the cards are the family's own game, not facts about any real boat.
A counted area that misses the prediction is a good moment: check one corner's multiplication together.
The hull and the cards are the family's own game, not facts about any real boat.
DOCK DAY CHECK
- Read the question.
- Tap your answer.
![A blue shape with vertices (0, 0), (3, 1) and its red image under [1 0; 0 -1], vertices (0, 0), (3, -1)](/assets/math/transform-1_0-0_n1-0_0-3_1.svg)
Apply the flip card [1 0; 0 -1] to the corner ⟨3, 1⟩. Where does it land?![A blue shape with vertices (0, 0), (1, 0), (1, 1), (0, 1) and its red image under [2 0; 0 1], vertices (0, 0), (2, 0), (2, 1), (0, 1)](/assets/math/transform-2_0-0_1-0_0-1_0-1_1-0_1.svg)
The stretch card [2 0; 0 1] is drawn. By what factor does it multiply the hull's area?Someone draws [2 4; 1 2]. What is its determinant?
The swap card [0 1; 1 0] is drawn. What is its determinant? Type the number.
ad - bc with a = 0, b = 1, c = 1, d = 0.
WHY THIS EXERCISEA determinant of -1 keeps the area and turns the hull over, which is what swapping x and y does.
Draw the family hull and two of its images, with each card's matrix and determinant written beside it.