Question

name: ____________ date: ____________
notice and wonder
(graph with multiple lines through the origin, axes labeled from -10 to 10)
what do you notice? | what do you wonder?
(blank table for responses)

Explanation:

For "What do you notice?": All lines pass through the origin \((0,0)\), so they are proportional relationships (since \(y = mx\) for some slope \(m\)). The lines have different slopes, meaning different rates of change. Some lines are steeper (larger absolute slope) and some are less steep. For "What do you wonder?": What determines the steepness of each line? Are these lines representing linear functions with different slopes, and how does the slope relate to the angle they make with the \(x\)-axis?

Answer:

What do you notice?

• All lines pass through the origin \((0,0)\), so they represent proportional relationships (\(y = mx\) form).
• Lines have different slopes (steepness): some are steeper (e.g., the green/blue/red lines), others are less steep (e.g., the gray/purple lines).
• The lines are symmetric across the origin (for every line in the first quadrant, there is a corresponding line in the third quadrant with the same slope).

What do you wonder?

• What is the slope (rate of change) of each line?
• How does the slope relate to the angle the line makes with the \(x\)-axis?
• Do these lines represent specific linear functions (e.g., \(y = 2x\), \(y = 0.5x\), etc.)?
• Why are the lines colored differently—does color relate to slope or another property?