Lesson 6 — Matrices as Many Equations at Once

Summary

Learners rewrite a system of equations as an augmented matrix and solve it by row reduction — the same elimination moves from Lesson 5, performed on a grid — then interpret the solution in context. They connect the grid to the ancient Chinese fangcheng array and the Lo Shu square as early array-like objects, and see the matrix as the form a computer actually solves.

Objectives

• Model systems of relationships with matrices and interpret the meaning of a solution in a real context. (D05.S2.12.01)

Connection

When there are many equations and many unknowns, writing them all out becomes a wall of symbols. A matrix squeezes the whole system into one grid — the numbers in rows and columns — so that a few clean moves solve it. The same grid sits under every spreadsheet, every supply plan, and every recommendation system you touch: a table of numbers that represents many relationships at once, and can be reduced to see the answer.

Materials

• Matrix-system sheet
• Math journal
• Low-tech: grid paper and counters

Preparation

• Copy or draw the sheet with a system written three ways.
• Retrieval: from Lesson 5, elimination; from Grade 11 (D05.S1.11.02), matrix rows, columns, and addition. Today a matrix is the system.
• Prepare the row-reduction worked example to model first (S-011).

Facilitator note

This lesson is written to the learner (“you”). The idea to land: an augmented matrix holds the coefficients and the answers side by side; row operations (swap, scale, add a multiple) preserve the solution while clearing one unknown at a time, and the reduced matrix reads out the solution. Teach the moves explicitly with a worked example and guided practice (S-011). The interpretation — what the numbers mean in the original situation — is the goal, not the row operations alone.

The ethics lens: row operations are a public, checkable record — the same answer anyone else can verify, which is what makes a computed result trustworthy rather than opaque. The egalitarianism lens: the matrix puts every unknown in the same grid, so no quantity is privileged; the method is a shared skill, not an arcane one. The global lens: the fangcheng array of the Nine Chapters (China, c. 200 BCE) arranged systems in rows and columns and eliminated unknowns — an ancestor of Gaussian elimination (S-289, S-429); the Lo Shu square is an early Chinese array-like object whose rows, columns, and diagonals sum to 15 (S-496). The technology lens: computers solve enormous systems by exactly these row operations — the matrix is the working form of numerical computation. The environment lens: resource and energy balances (supply meeting demand across many nodes of a grid) are matrices; a solution that needs negative or impossible amounts signals a mismatch between plan and planet (S-005). Preview: Lesson 7 turns from solving to algorithms.

Procedure

1. Recall (5 min). From Lesson 5, the system r + b = 50 and 2r + 3b = 120. Today we pack it into one grid.
2. Write the matrix (10 min). The augmented matrix holds coefficients on the left and the answers on the right:

   [ 1  1 |  50 ]
   [ 2  3 | 120 ]

   Each row is one equation; each column is one unknown (then the answer column). The system is this grid.
3. Worked example — row reduction (15 min). Use row operations that keep the solution the same: scale a row, swap rows, or add a multiple of one row to another. Subtract 2×(row 1) from row 2 to clear the 2:

   [ 1  1 |  50 ]         [ 1  1 | 50 ]
   [ 2  3 | 120 ]  R2-2R1 [ 0  1 | 20 ]

   Now read from the bottom up: b = 20; then r + 20 = 50, so r = 30. Same answer as Lesson 5, done on a grid.
4. Name the goal (6 min). Row reduction aims for a staircase — a leading 1 in each row, zeros below — so each row reveals one unknown. A full row of zeros on the left with a nonzero answer means no solution; a zero row with a zero answer means infinitely many.
5. Guided practice (12 min). With a partner, write a 2×2 system as an augmented matrix, reduce it to the staircase, and read off the solution. Interpret the answer in the original context, and check it by substitution. Swap and verify.
6. Retrieve and connect (5 min). Write one sentence on how the matrix and the equations say the same thing in two forms, and one on why the grid is what a computer prefers.
7. Close (2 min). Say — or write, sign, gesture, or use AAC to express — what the reduced matrix tells you that the original does not make obvious.

Differentiation

• Support: Reduce only to “clear one coefficient” and finish by substitution, using counters to see the move.
• Extension: Reduce a 3×3 system and explain, for each kind of zero row, whether the system is dependent, inconsistent, or uniquely solved.
• Number access (dyscalculia): Provide the step-2 augmented-matrix grid pre-printed with empty cells and the step-3 row-reduction worked example partially completed (the R2 − 2R1 move already applied) to copy; offload the row arithmetic (2×1, 3 − 2, 120 − 100) to a partner or calculator; and take the verbal route — read each row aloud as a sentence (“one rice plus one bean is fifty”) and say “after clearing the 2, the bottom row reads b = 20” to reach the solution without hand-computing.
• Communication access: Every spoken step — saying, naming, reading aloud, discussing, or closing — can be done in writing, sign, gesture, or AAC instead. Learners who are non-speaking, d/Deaf, hard-of-hearing, or who use AAC complete every task in their preferred mode; no step requires producing or hearing sound.

Assessment

• Formative (peer + self): Can the learner write an augmented matrix, reduce it to a staircase, read off the solution, and interpret it in context (in speech, writing, sign, or AAC) with a check?
• Portfolio artifact (unit): The matrix written, reduced, and interpreted, added to the math journal.

Home connection

Find a two-column table at home (prices, distances, a schedule) and write the rows as a matrix. Add or scale a row and write one sentence on what the changed numbers would mean.

Resources

• On worked examples and guided practice: Kirschner, Sweller & Clark (2006) (S-011).
• On the fangcheng array and the history of systems: MacTutor — Nine Chapters (S-289); Boyer & Merzbach, A History of Mathematics (S-429). On the Lo Shu square: Wikipedia — Luoshu Square (S-496).