Component One: Patterns in Linear Algebra
1. In this component, you will work with patterns of change
whose graphs make lines. This is known as linear algebra.
The fields of economics, sociology, political science,
medicine and other sciences use data that show trends to
make decisions. There are different ways that data can be
shown or represented. You may have heard teachers talking
about “multiple representations”. This is what you will
address in this first component.
2. Let’s look at our first pattern of change represented as a
geometric pattern. Maria used colored counters to build the
first four stages of the pattern shown on the screen. Use
counters from the pile to build the next stage of the pattern.
3. Does your 5th stage of the pattern match Maria’s? If not,
use words to describe the change you see from the 1st stage
to the 2nd stage, then to the 3rd stage and so on.
Mathematicians “talk their numbers”. It is important to talk
your thinking aloud. Do you see the pattern? As Maria
continues to build stages in the pattern, what trend do you
see in the number of counters she will need?
4. A second way to represent data is in a table. Use the data
from the pattern you and Maria made to complete the table.
5. Does your table look like Maria’s? If not, re-count the
counters she used to make each stage. What trend do you
see in this representation?
6. While you are looking at the data shown in table, think
about the information listed as “Stage #” and that listed as
“Number of Counters Used”. In a minute, we will label one
column as “independent” and the other as “dependent”.
Think about where you will place each label as I define the
terms.
7. The value identified as dependent, depends on its
independent partner. For example, the number of counters
used to build a stage of the pattern, depends on what stage
of the pattern you want to build. Ready? Move the terms to
the correct column in the table.
8. The column labeled “Stage #” lists the independent data
while the column labeled “Number of Counters Used” lists
dependent data. Again, the number of counters Maria needs
depends on which stage of the pattern she is building. This
is especially important because now, we are going to
represent the data in a 3rd way. We will now draw a graph
of the data.
9. First, label the independent column of your table (Stage #)
with the variable x and the dependent column (# of
Counters Used) with the variable y.
10. To complete this 3rd representation of the data, write the
data from the table as ordered pairs (x,y) and graph the
ordered pairs on the coordinate grid.
11. First, write the data from the table as ordered pairs.
12. Did you write the ordered pairs as (0,1), (1, 3), (2,5), (3,
7), (4,9), and (5,11)? The first number in the pair is the
stage # or the x value and the second number in the pair is
the # of tiles or the y value.
13. Remember that an ordered pair is an address on the
coordinate grid. The x value tells you the point’s location
relative to the horizontal or x-axis and the y value tells you
the point’s location relative to the y or vertical axis. Now,
plot the ordered pairs.
14. Did you plot the points as shown?
15. Connect the points to show the graphic representation of
the pattern. What trend do you see from this
representation? Look back at the pattern you saw as you
added colored counters to Maria’s design, to the table you
created that listed the Stage # as x, the independent data,
and the Number of Counters Used as y, the dependent data,
and finally to the line you created from the ordered pairs on
the table. Use words to explain the pattern and how it grew.
16. Did you notice that the pattern began with one counter
and grew two with each new stage? Look back over the
three representations you have done so far (geometric
pattern, table, graph on the coordinate grid) and use words
to describe the trend or pattern you see.
17. Your verbal description is the 4th of the multiple
representations of the data in Maria’s problem. How does
your description compare with these?
18. Use the trend you described to predict how many
counters it would take Maria to build the 7th stage in the
pattern.
19. Did you use the pattern or trend that you described to
predict that she would need 22 counters? Recognizing a
trend in the data enables us to predict and plan for future
events.
20. Finally, look back at the pattern, the table, the graph and
the verbal description to add the 5th and final representation
of the data: the linear equation that describes the trend or
pattern. Complete the mathematical sentence: y =________.
21. Do you need a hint? Think: y (Number of Counters
Used) = _________. Do you need another hint? Think: y
(Number of Counters Used) = 1 + _____________. Or y (Number
of Counters Used) = 1 + the Stage # times 2.
22. Did you write an equation equivalent to y = 1 + 2x ?
23. In the third component, you will learn to write the
equation of the line in slope intercept form. The slope will
describe the rate of change and the intercept will describe
the data when x = 0. The equation that describes the work
we have done in this component, written in slope intercept
form is y = 2x + 1.
23B. Throughout our discussion of the different ways
Maria’s pattern can be represented, we have described the
relation between the stage number and the number of
counters in equation form. Another way to describe this
relation is by the use of function notation. A function, like an
equation, describes a relation between sets of data. A
function is, however, a relation that assigns each value of the
independent variable (x) only one value of the dependent
variable (y). The function that describes Maria’s work
would be f(x) = 2x + 1. If the value of x is 3, the function tells
us that f(3) = 2  3 + 1 = 7. Seven counters are used in stage
3.
24. Now, you know the multiple representations of the
problem Maria addressed with her colored counters. She
could have represented the pattern with the geometric
pattern, with the table showing independent (x) and
dependent (y) data, with the line through points generated
from the table, by a verbal description or, finally, an
algebraic equation.
25. Now, let’s practice using multiple representations to
show a trend or a pattern in another set of data. You will
work to complete the five ways to represent the data in the
following situation. If you have difficulty answering any of
the questions, push the “hint” button for help.
26. Felicity is saving for a summer trip that will cost $560.
She had money in savings to make a $120 deposit. If she
baby-sits 5 hours a week for $8 an hour, how long will it
take her to earn enough money to pay for the trip?
27. Complete the drawing to illustrate the problem.
Consider her deposit to be week 0, then show how much she
adds to her account each week for the first 3 weeks.
28. Hint: Each week that Felicity works, she adds $40 to her
account.
29. Your drawing should look something like this.
30. Complete the table to show the weeks she works and
the total money in her account.
31. Hint: Before she begins to work (week 0), Felicity has
$120 in her account. At the end of week 1, she adds $40 for
a total of $160.
32. Your table should look like this.
33. Write the data from the table as ordered pairs and graph
them on the coordinate grid.
34. Hint: Before she begins to work (x = 0), Felicity paid a
$120 deposit (y = 120). The next week, x = 1 and y = 160 so
plot the ordered pairs (0,120) and (1, 160) on the
coordinate grid.
35. Your graph should look like this. If it does not, check
that your ordered pairs are: (0, 120), (1, 160), (2,200) and
(3, 240).
36. Write a verbal description of the pattern.
37. Hint: Tell how Felicity began with her deposit and then
added to her account each week,
38. Did you write something like, “Felicity began with a
$120 deposit and added $40 each week,” or “Felicity added
$40 each week to her $120 deposit.”
39. Finally, write an equation that describes the pattern.
40. Hint: Think about the columns of data represented by
the variables x and y. The total amount of money in Felicia’s
account (the dependent variable, y) is equal to the deposit
plus $40 times the number of weeks she has worked (the
independent variable, x).
41. Did you write: y = $120 + $40 x or y = 40x + 120? If you
will replace y with the total she needs, $560, can you
determine how many weeks she will work to pay for her
trip? In the next component of this course, we will review
and practice the “rules of algebra” that will help you solve
your equation and communicate your algebraic thinking
with others.