Linear Relationships and Systems Mastery

Grade 8 Mathematics Proportional Relationships, Functions, and Systems 45-Minute Interactive Session

What Are Proportional Relationships?

A relationship where two quantities maintain a constant ratio Can be written as y = kx where k is the constant of proportionality Passes through the origin (0,0) Examples: distance vs time at constant speed, cost vs quantity

Fill-in-the-Blank: Identifying Proportional Relationships

Look at the equation y = 3x. This _____ a proportional relationship because _____. The constant of proportionality is _____. If x = 4, then y = _____. ANSWERS: IS, it passes through origin; 3; 12

Unit Rate as Slope

Slope = rise/run = change in y/change in x In proportional relationships, slope equals the unit rate Unit rate tells us how much y changes for each unit increase in x Example: If y = 5x, the slope is 5 (5 units up for every 1 unit right)

Proportional vs Non-Proportional Linear Relationships

{"left":"Proportional: y = kx\nPasses through origin (0,0)\nConstant ratio between variables\nExamples: y = 2x, y = 0.5x","right":"Non-Proportional: y = mx + b (b ≠ 0)\nDoes NOT pass through origin\nConstant rate of change but not constant ratio\nExamples: y = 2x + 3, y = -x + 5"}

Quick Check: Which is Proportional?

A) y = 4x + 2 B) y = -3x C) y = x + 1 D) Cost = $5 + $2 per item Think about it: Which equation represents a proportional relationship?

What Makes a Function?

A function assigns exactly ONE output (y) to each input (x) Each x-value can only have one y-value Can be represented as ordered pairs, tables, graphs, or mappings Vertical Line Test: If any vertical line crosses the graph more than once, it's NOT a function

Function Detective: Fill-in-the-Blanks

Set A: {(1,2), (2,4), (3,6), (4,8)} _____ a function because _____. Set B: {(1,3), (2,5), (1,7), (3,9)} _____ a function because _____. The vertical line test works because _____. ANSWERS: IS, each x has one y; IS NOT, x=1 has two y-values; it checks if any x has multiple y-values

Direct Variation

Special type of proportional relationship Form: y = kx where k ≠ 0 k is called the constant of variation As one variable increases, the other increases proportionally Graph is always a straight line through the origin

Systems of Linear Equations: Solving Methods

Solve by Graphing: Fill-in-the-Solution

System: y = 2x + 1 and y = -x + 4 Step 1: Graph both lines on the same coordinate plane Step 2: Find where the lines _____ Step 3: The solution is the point (_____, _____) ANSWERS: INTERSECT; (1, 3)

Mathematical Wisdom

"Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding." - William Paul Thurston Linear relationships help us understand how quantities change together in our world