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Course 1, Unit 1 - Patterns of Change

Overview
The intent of this unit, which begins Core-Plus Mathematics Course 1, is to focus student attention on the variety of types of change inherent in problem situations. This unit will provide students with a broad picture of patterns of change. Students will explore linear, quadratic, inverse variation, and exponential patterns of change throughout the unit. Within this unit there is an effort to make a distinction between cause-and-effect change relationships and change-over-time relationships. In the third unit of this course, linear functions will be analyzed as a class of functions with a specific pattern of change. The unit should be completed in under 4 weeks of classes that meet approximately 50 minutes each day.

Key Ideas from Course 1, Unit 1

Linear: Linear functions have graphs that are straight lines, rules that can be written in the form y = a + bx, and tables of (x, y) values in which the ratio of change in y to change in x is constant. These ideas are formally developed in Unit 3. (See student book pages 150-167.)
Exponential: Exponential functions have curved graphs showing the dependent variable increasing at an increasing rate (for exponential growth) and decreasing at a decreasing rate (for exponential decay) and rules that can be written in the form y = a(b)^x, where b is the constant growth or decay factor. In tables of (x, y) values for exponential functions, if successive x values differ by 1, then the ratio of corresponding y values is b. Ideas about exponential growth and decay will be developed more formally in Course 1, Unit 5. (See student book pages 289-303, 322-331.)
Quadratic: Quadratic functions have graphs that are parabolas, rules that can be written in the form y = ax² + bx + c, and tables of (x, y) values in which y values change in a symmetric pattern centered at a maximum or minimum value. For example, y = x² - 4.

x	y
-3	5
-2	0
-1	-3
0	-4
1	-3
2	0
3	5

NOW-NEXT rules (pages 26-33): In many problem situations it is important to study the pattern of change in a single variable that changes with passage of time. Observing values of that variable at regular time intervals, it is natural to look for a pattern relating each value of the variable to the next value. The NOW-NEXT language is an informal way of capturing this perspective on patterns of change. Writing linear and exponential patterns of change in NOW-NEXT form highlights the constant additive and constant multiplicative patterns of change that characterize those two fundamental quantitative relationships. These ideas are developed further in Course 1, Units 3 and 5. (See student book pages 157-161.) Examples:

x	y
0	2
1	5
2	8
3	11
4	14

Linear Relationship

To get NEXT y, add 3 to the current y-value.
Two symbolic ways to represent this pattern are
NEXT = NOW + 3, starting at 2, and y = 3x + 2.

x	y
0	2
1	6
2	18
3	54
4	162

Exponential Relationship

To get NEXT y, multiply the current y-value by 3.
Two symbolic ways to represent this pattern are
NEXT = 3NOW, starting at 2, and y = 2(3^x).

Examples of other patterns introduced:

y = 3/x
y = x³

This work with NOW-NEXT patterns of change is also a precursor to work with sequences and series in future units (see Course 3, Unit 7, Recursion and Iteration).