Course
1, Unit 2 - Patterns of Change
Summary
In the "Patterns of Change" unit, students extend their understanding
and skill in algebra in three ways. They learn how to recognize relationships
between independent and dependent variables in problems and experiments,
and describe patterns in quantitative variables that change over time.
They learn how to read and construct data tables and graphs that display
relationships between variables. They begin developing symbol sense -
the ability to express and reason about linear, exponential,
and quadratic functions using symbolic formulas and equations. All of
this is an essential precursor to more formal work with algebraic symbols.
Key
Ideas from Course 1, Unit 2
- 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 98-101, 121-132.)
- 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
6. (See
student book pages 110-115.)
- 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 equations: 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
6. (See
student book pages 141-144.) 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:
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).
|
|
