Course 1, Unit 1 - Patterns of Change
Overview
The intent of this unit, which begins Core-Plus Mathematics Course1, 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 50minutes 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 Unit3. (See student book pages150-167.)
- 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 xvalues differ by 1, then the ratio of corresponding yvalues is b. Ideas about exponential growth and decay will be developed more formally in Course1, Unit5. (See student book pages289-303, 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 yvalues 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 (pages26-33): 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 Course1, Units3 and 5. (See student book pages157-161.) 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(3x).
- 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 Course3, Unit7, Recursion and Iteration).