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HSPA – PREP ACTIVITY #4: Patterns, Functions and Algebra
What do HSPA Patterns, Functions and Algebra questions look like?
Multiple-choice questions
Example 1: Patterns
What is the next number in the
sequence?
4 -8 16 -32
___
Solution to example 1: Look at the sequence; notice
if the ‘numbers’ are getting larger or smaller. If they
are getting larger you probably need to add or
multiply to get from one number to the next.
If the sign changes every other number it is probably
A. -49
B. 46
C. 64
because each number is multiplied by a negative
number. Remember, when multiplying
(-)(+) = - and
(+)(-) = -
(-)(-) = + and (+)(+) = +
(-32) (-2) = 64
D. -64
Solution: (C) 64
Solution to example 2:
Example 2: Patterns
Here you see the numbers begin as positive and then
Is this an arithmetic or geometric
sequence?
6
2
-2
-6
-10 -14
all become negative. By trial and error you see that
the pattern is to add a -4 to each number to get the
next number. When you add to continue a sequence
this is called an arithmetic sequence.
A. Arithmetic
B. Geometric
C. Both
D. Neither
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PART-I
(Part-I contains mostly Pre-Algebra and Basic-Algebra type examples.)
PRACTICE with Patterns and Numbers
Which value is the missing number in the sequence?
1.
61
57
53
A. 52
2.
___
45
41
B. 51
C. 49
D. 47
Use the pattern to find the units digit of 412
41 = 4
42 = 16
A. 2
B. 4
43 = 64
44 = 256
C. 6
45 = 1,024
D. 18
3.
If x + 3 is the rule, what are the next three numbers?
4.
If a – 2 is the rule, what are the missing numbers?
5.
What is the next number in the sequence
6.
Continue the sequence; fill in the next three numbers:
… 7, 10, 13, ____, ____, ____,
100, 98, 96, ___, 92 , ___, 88
3, 15, 75, 325, ____,
1000, 500, 250, ____, ____, ____ …
7.
What is the rule to continue the pattern?
… 3, - 6, 12, -24, …
8.
What is the rule to continue the pattern?
… 25, -12.5, 6.25, …
9.
What is the rule to continue the pattern?
… 2, -8, 32, -128,…
10.
3x – 2 is the rule; continue the pattern.
… 10, 28, ____, ____, ____, ____
11.
x2 is the rule; continue the pattern.
… 2, 4, 16, ____, ____
12.
x2 +1 is the rule; continue the pattern.
…2, 5, 26,
_____
PRACTICE Solving linear equations
1.
3x = -21
A. x = 7
B. x = -7
C. x = 18
D. x = -18
2.
12 = 2a – 4
A. a = 16
B. a = -8
C. a = 8
D. a = -16
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3.
2b – 9 = 5b
A. b = -3
B. b = 3
C. b = -6
D. b = -9/7
4.
3(4+x) = 24
A. x = -12
B. x = 12
C. x = 9
D. x = 4
5.
3w – (5)2 = 20
A. w = -15
B. w = 15
C. w = 10
D. w = -10
6.
3x + 4 = 12x – 8
A. x =
7.
2(a-3) = 3(2a + 10)
A. a = -1
B. a = -4
C. –9 = a
D. –8 = x
8.
¼ z = 20
A. z = 5
B. z = 16
C. z = 80
D. z = -16
9.
x(5-3)2 = -64
A. x = 4
B. x = -4
C. x = 16
D. x = -16
10.
16 + ½ y = -24
A. y = -80
B. y = 80
C. y = 20
D. y = -20
3
2
B. x =
4
3
C. x =
3
4
D. x =
3
2
PRACTICE recognizing Lines and Slope
1. Sketch two lines as described below:

Line AB with slope =
1
3

Line CD with slope =
3
1

What is the relationship of these lines? Explain your answer.
2. If the slope of one line equals the slope of another line then
A. the two lines are intersecting lines
B. the two lines are perpendicular lines
C. the two lines are parallel lines
D. the two lines always have very steep slopes
3. Given two lines:
line a: 2y -4x = 12
and
line b: y = 2x + 5
A. They are parallel
B. Line-a has a negative slope and Line-b has a positive slope
C. They are perpendicular
D. They are intersecting lines but are not perpendicular
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REMEMBER: The slope-intercept form of an equation is y = m x + b.
The m is the slope of the line, and the b is the point where the line intersects the y-axis.
PRACTICE with SLOPE
4.
Look at the following linear equation. It is not yet in the correct form. Put it in the
correct form, then, determine the slope of the line represented by this equation?
2y = 6x + 10
A. 6
B. 3
C. 10
D. 5
5.
Here is another linear equation that is not yet in correct form. Put it in standard form,
then determine the slope of the line it represents. Y - 6 = 4x
2
A. –6
B. 6
C.
D. 4
3
6.
This linear equation is not yet in correct form.
What is the slope of this line? y – 2x = 14
A. –2
B. 2
C. 14
D. 7
7.
The following two points are on the same line: (3, 0) and (2, 4). What is the slope of this
line?
4
4
1
1
1
4
A.
or
B.
or
C.
D.
1
4
2
1
1
4
8.
You are told that the slope of a line is
1
. Which two of the following pairs of
3
coordinate points are on this line? (There are two correct answers.)
A. (2, 4) (-5, -3)
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B.(2, 4) (5, 3)
C. (-2, -4) (5, 3)
D. (-4, 4) (5, 1)
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9. Open-Ended Question:
Table-A
Table-B
Shows
linear growth
Shows
exponential growth
Input
Output
x
2x + 1
0
1
1
3
2
5
3
7
4
?
5
11
6
13
7
?
100
201
Input
x
0
1
2
3
4
5
6
7
100
Output
x2
0
2
4
In both tables there is a rule that determines the
output in each situation.

In Table-A the rule is 2x+1 (take the input
number, multiply it by 2 and add 1).

In Table-B the rule is x2 (take the input
number and square it).
?
16
?
36
?
10,000
9. Let’s say these tables represent the way two different banks determine the interest you would
receive on money deposited at their bank.

Use the rule for each table and fill in the missing numbers (?) in the shaded cells. (Replace ?)

In Table-A: How much money would you receive if you deposited $50?

In Table-B: How much money would you receive if you deposited $50?

If you deposit $50 in each bank explain why do you receive so much more interest in one
bank than in the other?
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PRACTICE combining algebra and geometry
(Be sure to answer each bulleted item. Draw and label a diagram first.)
10.
The perimeter of a rectangle is 22 inches; width is X-2; length is X+5.

Write an equation to represent this perimeter.

Find the value of x.

Find the value of each side.

Check your work.

Find the area of the rectangle.
11. The perimeter of an isosceles triangle is 100 ft. The base = 2X, one side is given as 3X+2.




Write an equation to represent the perimeter.
Find the value of x.
Find the value of each side.
Check your work
12. The perimeter an equilateral triangle is 108 meters. You are given that one side is X + 6.




Write an equation to represent the perimeter of this equilateral triangle.
Find the value of x.
Find the length of each side
Check your work.
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13. Draw a triangle with  A = xo,  B=2xo,  C=3xo.

Write an equation to represent the total of these three angles.

Find the value of x.

Find the value of  B

Find the value of  C

What is the longest side of this triangle? Explain.
PRACTICE writing expressions and equations
1.
Which expression represents six less than the product of 4 and some number?
A. 6 – 4 + n
2.
D. 4 – 6n
B. n < n-4
C. 4 – n >
n
D. 4  n < n
Which of the following is the equation that says that the product of a number and ten is
equal to 20 percent of that number?
A.10n = 2.0 n
4.
C. 6 – 4n
Which of the following says that the square root of a number is greater than 4 less than
the number?
A. n > n – 4
3.
B. 4n – 6
B. 10 + n = .20n
C. n + .20n = 10
D. 10n = 0.20n
Which of the following represents a number squared is equal to twenty more than nine
times that number.
A. x2 = 9x + 20
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B. x2 = (20) (x) + 9
C. x2 > 9 + 20x
D. x2 = 9 + x + 20
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REAL-LIFE APPLICATIONS
Very often it is easiest to solve some basic everyday situations using algebraic equations and
special formulas.
Sample Example:
If Christine put $2,000 into a savings account that received 3% interest compounded annually,
how much money would she have in that account after 5 years?
The formula to determine compound interest
is A = P (1 + r)t
If you do not remember this formula use the HSPA Reference Sheet. A is the amount she
would have after five years, P is the principal amount in the account now, r is the rate (or the
percent) of interest she would receive and t is the time (in this case t = 5 years).
A = 2000 (1 + .03)5
= 2000 (1.03) 5
A = 2000 (1.159274) = 2318.548 or $2,318.55
At the end of five years Christine would have $2,318.55 in her savings account.
PRACTICE (Show your work even when you use a calculator!)
1.
The freshman class just collected money from their major fund-raising for the year. They
have $ 5,300 and plan to put this money into a savings account for 3 years; the bank gives
2.9% interest annually. How much money can they expect to have at the end of the four
years? Use the formula A = P (1 + r)t
A. $ 5,329
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B. $ 5,453.70
C. $ 5,774.60
D. $ 5,940.07
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MIXED PRACTICE (These are some of the easier Pre-Algebra and Algebra-I type questions
that could be on the HSPA).
1. Which of the following is a geometric series?
A.
1
2
1
4
C. 0.5
1.0
1
8
1
16
B. 12
1.5
2.0
D.
1
2
10
8
5
8
6
3
4
7
8
2. In the following equation what is the first step in isolating the variable? -8x -34 = 14
A.
B.
C.
D.
Subtract 34 from both sides
Divide both sides by -8
Add 34 to both sides
Multiply both sides by 8
3. Which equation does not have a solution of 12?
A.
4a + 3 = 51
B.
14 – 2b = -10
C.
3(x-5) = 21
D.
3(6 + y) = 30
4. Jacob’s mom is paid twice her usual hourly wage for each hour she works over 40 hours a
week. This week she worked 50 hours and earned $873.00. What is her hourly wage?
A. $9.96
B. $14.55
C. $ 17.46
D. $ 21.83
5. Judy and Janet took their 3 young children to the movies in nearby Westwood. An adult
ticket costs 3 times the cost of a child’s ticket. If they paid a total of $27.00, what is the
cost of each adult ticket?
A. $ 3.00
B. $ 4.00
C. $ 6.00
D. $ 9.00
6. Which equation matches the following? Six less than some number is five squared.
A. 52 = n – 6
B. n + 6 = 52
C. 6 – n = 52
D. 52 –6 = n
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7. Which statement is true about this equation?
8.
y +
2
x = 10
5
A.
The graph of this line crosses the origin.
B.
The graph of this line has a positive slope.
C.
This line is perpendicular to y =
D.
When graphed this line will be parallel to the line y =
5
x + 5
2
5
x+3
2
Write the following equation in slope-intercept form: 3x – 9y = 12
A.
3x = 9y + 12
B.
y=
C.
x = 9y + 12
D.
x – 3y = 4
1
4
x–
3
3
9. Solve for ‘c’ in the following equation: a c + 4 = b
A.
c=
b4
a
B.
c=
b
+4
a
C.
c=
b4
a
D.
c = 4b - a
10. Solve the following equation. 2x + 3(x – 2) = 24
A. x = 6
B. x = -6
C. x = 25
D. x =
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4
5
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11. How many of the following lines have a slope of 4?
y = 4x + 3
A.
1
B.
2
C.
3
D.
4
y = 4x
4y = x
16x – 4y = 24
y + 4x = 6
PART-II (open-ended questions)
DIRECTIONS FOR QUESTIONS 13 AND 14:
Respond fully to the open-ended question that follows. Show your work and clearly explain
your answer. You will be graded on the correctness of your method as well as the accuracy
of your answer.
13. Given a triangle with perimeter = 100 cm.
side AB=x, side BC=2x+4, and side CA=3x


Draw and label the triangle.
Write an equation to represent the perimeter
of the triangle.

Solve for x.

Find the length of each side.

14. The following equation was given to three
different students to solve: 2 (x – 4) = 16
Each student made at least one mistake.

Describe each student’s mistake.

Then solve the equation properly.
Student-A
Student-B
Student-C
2 (x - 4) = 16
2 (x - 4) = 16
2 ( x - 4) =16
2x + 8 = 16
-8 -8
2x = 8
2x – 4 = 16
+4 +4
2x = 22
2x – 8 = 16
+8
+8
2x = 24
2x = 8
2
2
2x = 22
2
2
2x = 24
-2
-2
x=4
x = 11
x = -12
If BC is the base of the triangle, and the
height of this triangle is 8 cm. long, what is
the area of the triangle?
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PART-III
The next set of examples is somewhat more challenging than in Part-I; they are similar to
HSPA, PSAT, SAT and ACT examples.
13.
This is a difficulty level-2 problem (on a 1-10 scale with 10 being the most difficult.) On a recent PSAT
about 90% of our 10th graders and 93% of our 11th graders answered this correctly. Let’s make it 100%!
If x = 10, what is the value of
A. 10
14.
B. 20
x( x  1)  x( x  1)
?
x
C. 100
D. 119
E. 200
This is a difficulty level-3 problem (with 10 being the most difficult.) On a recent PSAT about 75% of our
10th graders and 85% of our 11th graders answered this correctly. Let’s do better this time.
If 2n = k, where n is a positive integer, which of the following could be a value of k?
A. 1
15.
C. 32
D. 100
E. 250
This is also a difficulty level-3 problem (with 10 being the most difficult.) On a recent PSAT about 70% of
our 10th graders and 80% of our 11th graders answered this correctly.
If
1
= 2, then r + s =
rs
A. -2
16.
B. 9
B. -1/2
C. ¼
D. ½
E. 4
This is a difficulty level-4 problem (with 10 being the most difficult.) On a recent PSAT 68% of our 10th
graders and 70% of our 11th graders answered this correctly.
If t + u + v = 42 and 2u + 2v = 60, what is the value of t ?
A. 12
B. 14
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C. 18
D. 24
E. 27
Page 12 of 15
17.
This also is a difficulty level-4 problem (with 10 being the most difficult.) On a recent PSAT about 70% or
our 10th graders and 84% of our 11th graders answered this correctly.
If ( (x  2)  2)  2 is an integer, which of the following could be a value of x ?
A. 27
18.
B. 36
C. 42
D. 45
E. 56
This is a difficulty level-5 problem. Yet, it should be easy for us. On last October’s PSAT almost half of
our 10th and 11th graders did not get this correct.
What is the slope of line a in the figure drawn?
A. -2
a
(0, 2)
B. -1
C. -1/2
D. 1
(2, 0)
E. 2
19.
On a recent PSAT about 23% of our 10th graders and 30% of our 11th graders answered this correctly.
This question has a difficulty level of 7 on a scale of 1-10 with 10 being the most difficult. Do it again!
If (x + y)2 – (x – y)2 = 84 and x and y are positive integers, which of the following
could be a value of x + y ?
A. 10
B. 12
C. 14
D. 16
E. 18
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20.
This should be an easy question for us. Yet only 25-33% of our 10th graders, and fewer than 40% of 11th
graders answered this correctly on last October’s PSAT.
In the xy-plane, the line y = ax + 5 is parallel to the line 3x +8y = 10. What is the
value of a?
A. -6
21.
B. 
8
3
C. 
3
8
D. 
3
8
E.
8
3
(85% of our 10th graders, and 92% of our11th graders answered this correctly on last October’s PSAT.
Let’s make it 100% correct this time!)
If 2x + 4 = 10, what is the value of 4x – 20?
A. -8
22.
B. -2
C. 2
D. 7
E. 8
(About 42% of our 10th graders, and 52% of our11th graders answered this correctly on last October’s
PSAT. Let’s do much better this time!)
If yz = 10 and 2xyz = 5, what is the value of 2x?
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2x = _______
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