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Frogs, Fleas and Painted Cubes Practice Final
1. A rectangle has a perimeter of 100 centimeters and a side of length l centimeters.
a. Draw a rectangle to represent this situation. Label each side of the rectangle in terms of l.
b. Write an equation for the area A of the rectangle in terms of l.
c. Make a table showing how the area changes as the length of a side changes.
d. Sketch a graph of the relationship between the length of a side and the area.
e. Estimate the greatest area possible for a rectangle with this perimeter. What are the
dimensions of the rectangle with this maximum area?
f. Study your graph, table, and equation for the area of rectangles with a fixed perimeter of
100 centimeters. Which representation is the most useful for predicting the maximum area?
Explain your choice.
2. a. When an equation is in factored form, explain how you know whether it represents a quadratic
relationship.
b. When an equation is in expanded form, explain how you know whether it represents a
quadratic relationship.
c. Explain how you can tell whether a graph represents a quadratic relationship.
3. The patterns below represent the first four numbers in a sequence.
a. What are the next two numbers in the sequence?
b. Describe the arrangement of squares representing the nth number in the sequence.
c. Write an equation for calculating the nth number in the sequence.
d. Make a table and a graph of the first ten numbers in this sequence. Describe the pattern of
change from one number to the next in the sequence.
4. A square has sides of length x centimeters. A new rectangle is created by increasing one dimension
by 2 centimeters and decreasing the other dimension by 3 centimeters.
a. Write two expressions, one in factored form and one in expanded form, for the area of the
new rectangle.
b. Write an equation for the area, A, of the rectangle. Graph the equation, and describe the
graph.
5. Find an equivalent expression in factored form for
.
Multiple Choice
Identify the choice that best completes the statement or answers the question.
Simplify the expression.
____
____
6.
a.
b.
a.
b.
7.
____
8. At what points does the graph of f(x) = 2x2 – 4x – 16 intersect the x-axis?
____
a. (–4, 0) and (2, 0)
b. (–2, 0), and (4, 0)
9. Which of the following equations best represents the data in the table?
a. y = x2 + 3
x
y
-2
-1
0
1
2
-1
2
3
2
-1
b. y = -x2 + 3
Simplify the product.
____ 10.
a.
b.
____ 11. What does 4x2 – 64 look like when it is in its completely factored form?
a. (4x + 4)(x – 16)
b. 4(x – 4)(x + 4)
Frogs, Fleas and Painted Cubes Practice Final
Answer Section
SHORT ANSWER
1. ANS:
a.
b.
c. Possible table:
d.
e. The greatest area is 625 cm2 for a square with a side length of 25 cm.
f.
Answers will vary. Some students may claim the table is the most useful because it is easy to find where
the data “turns around.” Others may say the graph is the most useful because it is easy to locate the
maximum point, which is where the maximum area is represented. Less likely is the claim that the
equation is the most helpful, unless a student reasons that half of half the perimeter is equal to the side
length of the rectangle with the greatest area and that value of half the perimeter, 50, can be found in the
equation.
PTS: 1
DIF: L2
REF: Frogs Fleas Painted Cubes | Check-Up 1
OBJ: Investigation 1: Introduction to Quadratic Relationships
NAT: NAEP A1a| NAEP A1b| NAEP A1e| NAEP A2g
STA: 8CT MA.8.8b| 8CT MA.8.8d| 8CT MA.8.8e| 8CT MA.8.9b| 8CT MA.8.10c
TOP: Problem 1.1 Maximizing Area
KEY: perimeter | writing equations | rectangle | table | area |
graph
2. ANS:
a. A factored equation is quadratic if it consists of exactly two linear factors. No terms in either factor can
contain an
or an x raised to a power greater than 2. Examples:
,
,
and
.
b. An expanded equation is quadratic if it contains an
greater than 2. Examples:
,
term and no other term with x raised to a power
,
, and
.
c. A graph that represents a quadratic relationship will have the shape of a parabola, with exactly one
maximum or minimum point.
PTS: 1
DIF: L2
REF: Frogs Fleas Painted Cubes | Question Bank
OBJ: Investigation 2: Quadratic Expressions
STA: 8CT MA.8.8b| 8CT MA.8.9b
TOP: Problem 2.4 The Distributive Property and Quadratic Expressions
KEY: factor | foil | expanded form | graph | quadratic relationship
3. ANS:
a. 27; 38
b. A square of size n plus 2 more 1 by 1 squares, one located on the side of the square in the top left corner
and the other located on the bottom of the square in the bottom right corner.
c. n2 + 2
d.
n
y
1
3
2
6
3
11
4
18
5
27
6
38
7
51
8
66
9
83
10
102
To get from one number to the next in the sequence, just keep adding consecutive odd whole
numbers, starting with 3.
PTS: 1
DIF: L2
REF: Frogs Fleas Painted Cubes | Additional Practice Investigation 3
OBJ: Investigation 3: Quadratic Patterns of Change
NAT: NAEP N5e
STA: 8CT MA.8.8b| 8CT MA.8.8d| 8CT MA.8.8e| 8CT MA.8.9b
TOP: Problem 3.1 Exploring Triangular Numbers
KEY: sequence | finding patterns | finite sequence | writing equations | graph | table
4. ANS:
a. The area can be represented as
or
The graph is a parabola with x-intercepts at –2 and 3.
b.
PTS: 1
DIF: L2
REF: Frogs Fleas Painted Cubes | Question Bank
OBJ: Investigation 2: Quadratic Expressions
NAT: NAEP A1a| NAEP A1b| NAEP A1e| NAEP A2g
STA: 8CT MA.8.2e| 8CT MA.8.8b| 8CT MA.8.9b
TOP: Problem 2.3 Changing Both Dimensions
KEY: area | rectangle | writing equations | factor | standard form | expanded form
5. ANS:
(x + 5)(x + 3)
PTS:
OBJ:
NAT:
STA:
TOP:
1
DIF: L2
REF: Frogs Fleas Painted Cubes | Check-Up 2
Investigation 2: Quadratic Expressions
NAEP A1a| NAEP A1b| NAEP A1e| NAEP A2g
8CT MA.8.2e| 8CT MA.8.8b| 8CT MA.8.9b
Problem 2.3 Changing Both Dimensions
KEY: equivalent expression | factor
MULTIPLE CHOICE
6. ANS: A
PTS: 1
DIF: L1
REF: Frogs Fleas Painted Cubes | Skills Practice Investigation 1
OBJ: Investigation 1: Introduction to Quadratic Relationships
NAT: NAEP A1a| NAEP A1b| NAEP A1e| NAEP A2g
STA: 8CT MA.8.8b| 8CT MA.8.8d| 8CT MA.8.8e| 8CT MA.8.9b
TOP: Problem 1.2 Reading Tables and Graphs
KEY: combining like terms | algebraic expression
MSC: NAEP A3b | CAT5.LV18.54 | CTBS.LV18.54 | ITBS.LV14.A | S9.Adv1.PRA | S10.Adv1.PRA |
TV.LV18.12 | TV.LV18.16 | TV.LV18.52
7. ANS: B
PTS: 1
DIF: L1
REF: Frogs Fleas Painted Cubes | Skills Practice Investigation 2
OBJ: Investigation 2: Quadratic Expressions
NAT: NAEP A1a| NAEP A1b| NAEP A1e| NAEP A2g
STA: 8CT MA.8.8b| 8CT MA.8.9b
TOP: Problem 2.1 Representing Areas of Rectangles | Problem 2.2 Changing One DImension | Problem 2.3
Changing Both Dimensions
KEY: monomial | binomial
MSC: CAT5.LV18.54 | CTBS.LV18.54 | ITBS.LV14.A | S9.Adv1.PRA | S10.Adv1.PRA | TV.LV18.16 |
TV.LV18.52
8. ANS: B
PTS: 1
DIF: L2
REF: Frogs Fleas Painted Cubes | Multiple Choice
OBJ: Investigation 2: Quadratic Expressions
NAT: NAEP N5e
STA: 8CT MA.8.8b| 8CT MA.8.9b
TOP: Problem 2.5 A Closer Look at Parabolas
KEY: x-intercept | quadratic equation | solve
9. ANS: B
PTS: 1
DIF: L2
REF: Frogs Fleas Painted Cubes | Multiple Choice
OBJ: Investigation 1: Introduction to Quadratic Relationships
STA: 8CT MA.8.8b| 8CT MA.8.8d| 8CT MA.8.8e| 8CT MA.8.9b
TOP: Problem 1.3 Writing and Equation KEY: using equations
10. ANS: B
PTS: 1
DIF: L1
REF: Frogs Fleas Painted Cubes | Skills Practice Investigation 2
OBJ: Investigation 2: Quadratic Expressions
NAT: NAEP A1a| NAEP A1b| NAEP A1e| NAEP A2g
STA: 8CT MA.8.2e| 8CT MA.8.8b| 8CT MA.8.9b
TOP: Problem 2.3 Changing Both Dimensions
KEY: binomial | multiplication of polynomials
MSC: NAEP A3b | CAT5.LV18.54 | CTBS.LV18.54 | ITBS.LV14.A | S9.Adv1.PRA | S10.Adv1.PRA |
TV.LV18.16
11. ANS: B
PTS: 1
DIF: L2
REF: Frogs Fleas Painted Cubes | Multiple Choice
OBJ: Investigation 2: Quadratic Expressions
STA: 8CT MA.8.8b| 8CT MA.8.9b
TOP: Problem 2.4 The Distributive Property and Quadratic Expressions
KEY: factor