Lesson Description
This lesson introduces simple linear patterns and how they appear in tables, graphs, and equations. Students will identify constant rate of change and connect patterns to slope and linear equations.
Learning Objectives
- Identify linear patterns in tables of values.
- Determine the constant rate of change.
- Represent simple linear patterns using equations and graphs.
Topics Covered
- Constant rate of change
- Writing linear equations from patterns
- Graphing linear relationships
Key Terms
- Linear Pattern: A pattern that increases or decreases at a constant rate.
- Rate of Change: The amount a quantity changes compared to another quantity.
- y-intercept: The value of y when x equals zero.
Lesson Examples
Example 1: Identifying a Linear Pattern in a Table
- Given table: x = 1,2,3,4 and y = 3,5,7,9.
- Find differences in y-values: +2 each time.
- The pattern is linear with rate of change 2.
Example 2: Writing an Equation
- From Example 1, slope (m) = 2.
- When x = 1, y = 3. Substitute into y = mx + b.
- 3 = 2(1) + b → b = 1. Equation: y = 2x + 1.
Practice Problems
- Determine if the pattern 4, 7, 10, 13 is linear.
- Find the rate of change for y-values: 2, 6, 10, 14.
- Write an equation for a pattern with slope 3 and y-intercept 5.
- Graph the equation y = x + 4.
Tips & Common Mistakes
- Check that differences are consistent to confirm linearity.
- Do not confuse slope with the y-intercept.
- Always verify your equation with at least one point.
Summary
Simple linear patterns change at a constant rate. They can be identified in tables, represented on graphs, and written as equations in the form y = mx + b. Recognizing these connections strengthens understanding of linear relationships.
Challenge Problems
- A taxi charges a $4 base fee plus $2 per mile. Write an equation.
- Determine if the points (1,4), (2,8), (3,12) form a linear pattern.
- Create your own linear pattern and write its equation.