Regents Integrated Algebra
Twenty Day Countdown
The problem sets are chosen from the Integrated Algebra’s POWER PI’s – problems that
often show up on the regents exam.
As you go through these problems with your class, you should know how to solve/answer the questions
algebraically as well as with the graphing calculator. Remember….our goal is to get all of our students to pass
the Regents Exam. Use these pages to practice, study, and read over. The more work you put into preparing for
the exam, the more you will get out of it!
DAY Power PI
Rational Expressions & Equations
1
2
3
4
5
6
AA.12 , AA.13 & AA.23
Operations with Polynomials and Transforming Formulas
AA.19 & AA.20
Factoring
AA.15 & AA.16
Rational Expressions ( undefined & simplify)
AA.17 & AA.18
Rational Expressions (add/subtract)
AA.17 & AA.18
Rational Expressions (multiply/divide)
AA. 25 & AA.26
Rational Equations
Right Triangle Trigonometry & Pythagorean Theorem
7
8
AA.42, AA.43, & AA.44
Right Triangle Trigonometry
AA.42, AA.43, AA.44, & AA.45
Right Triangle Trigonometry mixed with Pythagorean Theorem
Perimeter, Area, & Volume of Polygons & Relative Error
9
10
11
A.G.1
Area and Perimeter of figures composed of Polygons
A.G.2
Use formulas to calculate volume and surface area of the rectangular solids and cylinders
A.M.1 & A.M.3Calculate rates using appropriate units
Calculate the relative error in measuring square and cubic units, when there is an error in the linear
measure
Equations, Inequalities, & Systems
12
13
14
15
16
AA.6, AA.7, & AA.10
Solve verbal inequalities, linear equations, and systems
AA.6, AA.7, & AA.10
Solve verbal inequalities, linear equations, and systems
AA.9
Exponential growth/decay
AG.7 & AG.9: Graphic Systems
AG.7 & AG.9: Graphic Systems
Probability
17
18
AS. 19, AS.22, & AS.23 Probability: sample spaces, tree diagrams, independent and dependent events
AS. 19, AS.22, & AS.23 Probability: sample spaces, tree diagrams, independent and dependent events
Sets
19
20
AA.29, AA.30, & AA.31 Sets
AA.29, AA.30, & AA.31 Sets
1
Regents Integrated Algebra
Day 1:
AA.12:
AA.13:
AA.23:
Twenty Day Countdown
AA.12, AA.13 & AA.23
Multiply and divide monomial expressions with a common base, using properties of exponents
Add, subtract, and multiply monomials and polynomials
Solve literal equations for a given variable
Solving Equations with Variables in the Answer
Get the variable you are solving for by itself by undoing all the operations that are being
done to it. **Remember it’s like solving regular equations, variables represent numbers.
1. If 3ax  b  c, then x equals
(1) c  b  3a
(2) c  b  3a
(3)
cb
3a
(4)
bc
3a
2. Drew answered the question below on the IA Regents. Is he correct or incorrect?
If he is incorrect please explain what mistake you think he may have made.
The expression, is equivalent to
1)
2)
3)
4)
3. Libby solved the following equation for 𝑦 on her IA Regents Exam. Did she solve it correctly? If not,
please correct and explain what mistake you think she made.
Solve 3𝑥 + 4𝑦 = 7, for 𝑦
4. If
1)
2)
3)
4)
5.
is subtracted from
The expression
1)
2)
3)
4)
3𝑥 + 4𝑦 = 7
4𝑦 = −3𝑥 + 7
−3
𝑦=
𝑥+7
4
, the result is
“From comes First”
is equivalent to
2
Regents Integrated Algebra
Twenty Day Countdown
6.
The expression
7.
is equivalent to
What is the product of
1)
and
?
2)
3)
4)
Practice Questions:
8.
Which expression represents
1)
in simplest form?
2)
3)
4)
9.
What is half of
?
1)
2)
3)
4)
10.
Which expression represents
11.
1)
2)
3)
4)
Which expression is equivalent to
1)
in simplest form?
?
2)
3)
4)
3
Regents Integrated Algebra
12.
13.
Twenty Day Countdown
The formula for potential energy is
, where P is potential energy, m is mass, g is
gravity, and h is height. Which expression can be used to represent g?
1)
g  P  mh
2)
g
3)
g  Pmh
4)
g
Pm
h
P
mh
A formula used for calculating velocity is
. What is a expressed in terms of v and t?
1)
2)
3)
4)
14.
The expression
1)
is equivalent to
2)
3)
4)
15.
The expression
1)
is equivalent to
2)
3)
4)
4
Regents Integrated Algebra
Twenty Day Countdown
16.
The product of
17.
When (6x2 + 3x + 5) is subtracted from (2x2 + 6x + 5), the result is _______________.
18.
The product of
19.
If
and
and
is ______________.
is _______________.
, then a equals ______________________.
5
Regents Integrated Algebra
Twenty Day Countdown
Name: ________________________________________________
Date: ______________
Day 2: AA.19 and AA.20 (Factoring)
AA.19: Identify and factor the difference of two perfect squares
AA.20: Factor algebraic expressions completely, including trinomials with lead coefficient of one (after factoring
a GCF)
GRAPHING CALCULATOR
Method 1: Put the expression in y1 and the answer in choices in y2 and see if the graphs lie
on top of one another.
Method 2: Put the expression on the home screen. Hit 2nd test = than type equivalent
expression. If the logical expression (Boolean Value) 1 = true and 0 = false
1. When
1)
2)
3)
4)
is factored completely, the result is
2. Xavier answered the question below on the IA Regents. Is he correct or incorrect?
If he is incorrect please explain what mistake you think he may have made.
Which expression is equivalent to
?
1)
2)
3)
4)
3. What are the factors of
1)
2)
3)
4)
4.
?
Factored completely, the expression
is equivalent to
1)
2)
3)
4)
Hint: Factor completely means to take out a GCF first
6
Regents Integrated Algebra
Twenty Day Countdown
PRACTICE QUESTIONS:
5.
If Ann correctly factors an expression that is the difference of two perfect squares, her factors
could be
1)
2)
3)
4)
6.
Which expression represents
1)
factored completely?
2)
3)
4)
7.
Factored completely, the expression
1)
is equivalent to
2)
3)
4)
8.
Written in simplest factored form, the binomial 2x2 - 50 can be expressed as
1)
2)
3)
4)
7
Regents Integrated Algebra
9.
Twenty Day Countdown
One of the factors of 4x2 - 9 is
1)
2)
3)
4)
10.
Which expression is equivalent to
1)
?
2)
3)
4)
11.
Which expression is a factor of
1)
?
2)
3)
4)
12.
What are the factors of the expression
1)
?
2)
3)
4)
13.
Factored completely, the expression
1)
is equivalent to
2)
3)
8
Regents Integrated Algebra
Twenty Day Countdown
4)
14.
If
is a factor of
15.
Factor:
16.
Factor: 4x3 – 2x
17.
Factor completely:
, what is the value of b?
9
Regents Integrated Algebra
Twenty Day Countdown
Day 3: AA.15 & AA.16 (Rational Expressions)
AA.15: Find values of a variable for which an algebraic fraction is undefined
AA.16: Simplify fractions with polynomials in the numerator and denominator by factoring both and renaming
them to lowest terms
GRAPHING CALCULATOR
Rational Expression is Undefined when the denominator equals zero.
To find the value that makes the expression undefined, set the denominator equal to zero.
***Use your calculator to substitute in your choices into the denominator to find which
one gives zero
To Simplify
1. Factor numerator and denominator
(Use GCF, Difference of 2 Perfect Squares or Trinomial Factoring)
2. Cancel like terms
1.
For all values of x for which the expression is defined,
1)
is equivalent to
2)
3)
4)
2. Maria answered the question below on the IA Regents. Is she correct or incorrect?
If she is incorrect please explain what mistake you think she may have made.
x
The function y = x  25 is undefined when the value of x is
2
(1) 0 or 5
(2) 5 or -5
(3) 5, only
(4) -5, only
3.
Which expression represents
1)
in simplest form?
2)
3)
4) -1
10
Regents Integrated Algebra
4.
x2  9
undefined?
x  7 x  10
Which value of x makes the expression
(1) -5
(2) 2
Twenty Day Countdown
2
(3) 3
(4) -3
5. The area of a rectangle is represented by 𝑥 2 − 5𝑥 − 14, if the length is represented by (𝑥 − 7),
Express the width as a binomial?
PRACTICE QUESTIONS:
6.
Which value of x makes the expression
x4
undefined?
x3
1) -4
2) -3
3) 3
4) 0
7.
Which expression represents
2 x 2  12 x
in simplest form?
x6
1) 0
2) 2x
3) 4x
4) 2x + 2
8.
For which value of x will the fraction
3
be undefined?
2x  4
1) -2
2) 2
3) 0
4) -4
9.
For what values of x is the fraction
4 x
undefined?
x2  4
11
Regents Integrated Algebra
10.
Twenty Day Countdown
3 y 2  12 y
The expression
is equivalent to
4y2  y3
3
y
3
2) y
9
3) 4
3 12
4)  2
4 y
1)
11.
Simplify:
x 2  6x  5
x 2  25
12
Regents Integrated Algebra
Twenty Day Countdown
Name: ___________________________
Date: __________
Regents Review – Quick Check – Day 1, Day 2, Day 3
1.
Drew answered the question below on the IA Regents. Is he correct or incorrect?
If he is incorrect please explain what mistake you think he may have made.
The expression, is equivalent to
1)
2)
3)
4)
_______________________________________________________________
_______________________________________________________________
_______________________________________________________________
_______________________________________________________________
2. If
1)
2)
3)
4)
is subtracted from
3. What are the factors of
1)
2)
3)
4)
4.
Factored completely, the expression
1)
2)
3)
4)
, the result is
?
is equivalent to
13
Regents Integrated Algebra
Twenty Day Countdown
5.
Which expression represents
1)
in simplest form?
2)
3)
4) -1
6. Maria answered the question below on the IA Regents. Is she correct or incorrect?
If she is incorrect please explain what mistake you think she may have made.
x
The function y = x  25 is undefined when the value of x is
2
(1) 0 or 5
(2) 5 or -5
(3) 5, only
(4) -5, only
_______________________________________________________________
_______________________________________________________________
_______________________________________________________________
_______________________________________________________________
14
Regents Integrated Algebra
Twenty Day Countdown
Day 4: AA.17 & AA.18 (Rational Expressions: Add/Subtract)
AA.17: Add or Subtract fractional expressions with monomial or binomial denominators
AA.18: Multiply and divide fractional expressions and express answers in simplest form
To Add or Subtract Rational Expressions
1. Find the LCD
2. Multiply each fraction to make the LCD
3. Combine Numerator over the LCD
**Remember once you have a common denominator the denominator doesn’t change
Graphing Calculator: All answers may be checked using the home screen with Boolean
Values. (The 2nd TEST key. 1 = true and 0 = false)
1.
What is the sum of
and
1)
4)
2)
3)
expressed in simplest form?
Lakeisha answered the question above on the IA Regents. Is she correct or incorrect?
If she is incorrect please explain what mistake you think she may have made.
3
2𝑥
+
4
3𝑥
=
7
5𝑥
Did she answer the problem correctly?
2.
Expressed in simplest form,
3.
is equivalent to
CJ worked out the following problem below. He claimed the answer was
the correct answer was
−1𝑥+19
.
2𝑥+6
The expression
−1𝑥+7
2𝑥+6
.
His girlfriend, Jenny said
Who is correct, CJ or Jenny?
is equivalent to
4.
What is the sum of
1)
2)
and
3)
?
4)
15
Regents Integrated Algebra
Twenty Day Countdown
PRACTICE QUESTIONS:
5.
Which expression is equivalent to
a b
?

x 2x
2a  b
2x
2a  b
2)
x
ab
3)
3x
ab
4)
2x
1)
6.
What is the sum of
3
7
and
?
7n
3n
1
n
10
2)
21n
42
3)
21n
58
4)
21n
1)
7.
What is
2 x x2

expressed in simplest form?
5x
5x
1) 0
2
5
4
3)
5x
2x  4
4)
5x
2)
16
Regents Integrated Algebra
8.
Expressed in simplest form,
Twenty Day Countdown
2x  3 2x  3
is equivalent to

3
6
2x  5
6
2x  9
2)
6
 x  12
3)
12
 x  16
4)
12
1)
9.
What is
2 x x2

expressed in simplest form?
5x
5x
1) 0
2
5
4
3)
5x
2x  4
4)
5x
2)
17
Regents Integrated Algebra
Twenty Day Countdown
Day 5: AA.17 & AA.18 (Rational Expressions: Multiply/Divide)
AA.17: Add or Subtract fractional expressions with monomial or binomial denominators
AA.18: Multiply and divide fractional expressions and express answers in simplest form
To Multiply ( or Divide) Rational Expressions
(**** IF division flip the rational after the  symbol )
1. Factor all numerators and denominators
2. Cancel ANY LIKE terms from a numerator and a denominator
3. Multiply Across Numerators then Denominators
4. Simplify if possible
Graphing Calculator: All answers may be checked using the home screen with Boolean Values. (The 2nd TEST
key. 1 = true and 0 = false)
1.
Express in simplest form:
James solved this problem with his partner Miguel. Did they solve it correctly?
(𝑥 + 7)(𝑥 + 2)
3(𝑥 + 2)
÷
(𝑥 + 7)(𝑥 − 7) (𝑥 + 7)(𝑥 − 6)
3(𝑥 + 2)(𝑥 + 2)
(𝑥 − 7)(𝑥 − 6)
2.
What is the product of
1)
and
expressed in simplest form?
and
in simplest form.
2)
3)
4)
3.
Express in simplest form:
4.
Express the product of
18
Regents Integrated Algebra
Twenty Day Countdown
PRACTICE QUESTIONS:
5. Perform the indicated operation and express in simplest form:
x2  x
6
 2
3
x 1
6. Perform the indicated operation and express in simplest form:
x 2  16
x4

2
x  x  20 x  4
7. Perform the indicated operation and simplify:
3x  6 x 2  4

4 x  12 x  3
8. Express in simplest form:
x2 9 3 x

2x  8 x  4
19
Regents Integrated Algebra
Twenty Day Countdown
Day 6: AA.25 & AA.26 (Rational Equations) Many students have difficulty solving rational equations
AA.25: Solve equations involving fractional expressions.Note: Expressions which result in linear equations in one
variable.
AA.26: Solve algebraic proportions in one variable which result in linear or quadratic equations
To Solve a Rational Equation
Method 1 (works for all)
1. Find the LCD
2. Multiply ALL parts of the equation by the LCD
3. Solve the remaining equation
(**If there is a squared term get one side to = 0 and factor, then set each factor =0
and solve)
Method 2 (works for proportions---fraction = fraction)
1. Cross Multiply
2. Solve remaining equation
(**If there is a squared term get one side to = 0 and factor, then set each factor =0
and solve)
Graphing Calculator: Method 1
1. Enter left side into y = , then enter right side into y = and graph
***remember to use ( ) for each fraction
2. To find where they intersect, hit 2nd CALC then 5 (intersect)
3. When finding the intersect point you must use the left/right arrow keys to get close to the point of intersection on
the 1st equation, then hit enter. Repeat this process for the second equation. Finally hit enter when it asks for guess.
4. Your answer is the value for x = (no matter the variable you are solving for)
**sometimes the graph looks crazy but it will still give you a correct x value
Method 2: Plug in your choices for x and see which one makes the left side = the right side.
1.
Which value of x is the solution of
2x 1 7x  2
 
?
5 3
15
(1)
3
5
(3) 3
(2)
31
26
(4) 7
2.
What is the solution set of
(1) {– 2, 3}
(3) {– 1, 6}
(2) {– 3, – 2}
(4) {– 6, 1}
?
3.
What is the value of x in the equation
1
(1) – 8
(3) 8
1
(2) − 8
?
(4) 8
20
Regents Integrated Algebra
Twenty Day Countdown
4.
What is the value of w in the equation:
?
(Is there a way I can use my calculator to see if my answer is correct?
5. Jose answered the question below on the IA Regents. Is he correct or incorrect?
If he is incorrect please explain what mistake you think he may have made.
6 ∙ 6∙
Which value of x is the solution of
(1) 1
(3) 3
(2) – 1
(4) – 3
6∙
?
2x + 3x + 1 = 6x
5x + 1 =6x
1=x
Practice Questions:
6.
What is the value of x in the equation
?
1) 12
2) 8
3) 3
4)
7.
What is the value of x in the equation
?
1)
2) 16
3)
4) 4
21
Regents Integrated Algebra
8.
What is the solution of
Twenty Day Countdown
?
1) 1
2) 5
3) 6
4) 14
9.
Which value of x is the solution of
?
1)
2)
3)
4) 4
10.
What is the value of x in the equation
?
1)
2)
3)
4) 8
11.
Solve for x:
22
Regents Integrated Algebra
12.
Twenty Day Countdown
Solve for y:
Name: _______________________________________
Ticket out the Door:
1. Which value of x is the solution of the equation
Date: _____________
?
1) 6
2) 10
3) 15
4) 30
2. Solve for y:
23
Regents Integrated Algebra
Twenty Day Countdown
Day 7: AA.42, AA.43, AA.44 (Right Triangle Trigonometry)
AA.42: Find the sine, cosine, and tangent ratios of an angle of a right triangle, given the lengths of the sides
AA.43: Using Trigonometry to Find an Angle: Determine the measure of an angle of a right triangle, given the
length of any two sides of the triangle
AA.44: Using Trigonometry to Find a Side: Find the measure of a side of a right triangle, given an acute angle and
the length of another side
Students will be given the Trigonometric Ratios on the IA Reference Sheet
Graphing Calculator:
 Make sure calculator is in degree mode
 To find the measure of an angle: 2nd sin-1( ratio here) ENTER
 To find an a length of a side: sin (angle) = ratio
1. Which ratio represents
1)
2)
3)
in the accompanying diagram of
4)
2. In triangle MCT, the measure of
represents the sine of
?
1)
2)
3)
?
,
,
, and
. Which ratio
4)
3.
24
Regents Integrated Algebra
4. In right triangle ABC,
of
.
,
,
Twenty Day Countdown
, and
. Find, to the nearest degree, the measure
5. A 100 foot wire is extended from the ground to the top of a 60 foot pole, which is perpendicular to the level
ground. To the nearest degree, what is the measure of the angle that the wire makes with the ground?
HINT: Draw a picture
6. A tree casts a 25-foot shadow on a sunny day, as shown in the diagram below.
If the angle of elevation from the tip of the shadow to the top of the tree is 32°, what is the height of the
tree to the nearest tenth of a foot?
1) 13.2
2) 15.6
3) 21.2
4) 40.0
Practice questions:
7.
In the accompanying diagram of right triangle ABC, AB = 8, BC = 15, AC = 17, and m  ABC = 90.
What is tan  C?
8
15
17
(2)
15
(1)
8
17
15
(4)
17
(3)
25
Regents Integrated Algebra
8.
Which ratio represents sin A in the accompanying diagram of ABC ?
5
13
12
(2)
13
(1)
9.
Twenty Day Countdown
12
5
13
(4)
5
(3)
The accompanying diagram shows a ramp 30 feet long leaning against a wall at a construction site.
If the ramp forms an angle of 32° with the ground, how high above the ground, to the nearest tenth, is
the top of the ramp?
(1) 15.9ft
(3) 25.4ft
(2) 18.7ft
(4) 56.6 ft
10.
Find, to the nearest tenth of a foot, the height of the
tree represented in the accompanying diagram.
26
Regents Integrated Algebra
Twenty Day Countdown
11.
A person standing on level ground is 2,000 feet away from the foot of a 420-foot-tall building, as
shown in the accompanying diagram. To the nearest degree, what is the value of x?
12.
Joe is holding his kite string 3 feet above the ground, as shown in
the accompanying diagram. The distance between his hand and a
point directly under the kite is 95 feet. If the angle of elevation to
the kite is 50°, find the height, h, of his kite, to the nearest foot.
27
Regents Integrated Algebra
Twenty Day Countdown
Name: ___________________________________
Date: ___________
Day 8: AA.42, AA.43, AA.44, & AA.45 (Right Triangle Trigonometry Plus Pythagorean Theorem)
AA.45: Pythagorean Theorem: Determine the measure of a third side of a right triangle using the Pythagorean
Theorem, given the lengths of any two sides
1.
Nancy’s rectangular garden is represented in the diagram below.
If a diagonal walkway crosses her garden, what is its length, in feet?
2.
28
Regents Integrated Algebra
Twenty Day Countdown
3.
4.
A stake is to be driven into the ground away from the base of a 50-foot pole, as shown in the diagram
below. A wire from the stake on the ground to the top of the pole is to be installed at an angle of
elevation of 52°.
How far away from the base of the pole should the stake be driven in, to the nearest foot?
What will be the length of the wire from the stake to the top of the pole, to the nearest foot?
Practice questions:
5.
Ron and Francine are building a ramp for
performing skateboard stunts, as shown in the
accompanying diagram. The ramp is 7 feet
long and 3 feet high. What is the measure of
the angle, x, that the ramp makes with the
ground, to the nearest tenth of a degree?
29
Regents Integrated Algebra
6.
Twenty Day Countdown
As seen in the accompanying diagram, a
person can travel from New York City to
Buffalo by going north 170 miles to Albany
and then west 280 miles to Buffalo.
a If an engineer wants to design a highway
to connect New York City directly to Buffalo,
at what angle, x, would she need to build the
highway? Find the angle to the nearest
degree.
b To the nearest mile, how many miles
would be saved by traveling directly from
New York City to Buffalo rather than by
traveling first to Albany and then to Buffalo?
7.
A surveyor needs to determine the distance across the pond shown in the accompanying diagram.
She determines that the distance from her position to point P on the south shore of the pond is 175
meters and the angle from her position to point X on the north shore is 32°. Determine the distance,
PX, across the pond, rounded to the nearest meter.
30
Regents Integrated Algebra
Twenty Day Countdown
8.
A tree casts a shadow that is 20 feet long. The angle of elevation from the end of the shadow to the
top of the tree is 66°. Determine the height of the tree, to the nearest foot.
9.
A 10-foot ladder is to be placed against the side of a building. The base of the ladder must be
placed at an angle of 72° with the level ground for a secure footing. Find, to the nearest inch, how
far the base of the ladder should be from the side of the building and how far up the side of the
building the ladder will reach.
10.
In the accompanying diagram, the base of a
15-foot ladder rests on the ground 4 feet from
a 6-foot fence.
a If the ladder touches the top of the fence
and the side of a building, what angle, to the
nearest degree, does the ladder make with the
ground?
b Using the angle found in part a, determine
how far the top of the ladder reaches up the
side of the building, to the nearest foot.
31
Regents Integrated Algebra
Twenty Day Countdown
Name: ____________________________________
Date: __________
Day 9: AG.1 (Perimeter and Area of Polygons)
AG.1: Compositions of Polygons and Circles: Find the area and/or perimeter of figures composed of polygons
and circles or sectors of a circle Note: Figures may include triangles, rectangles, squares, parallelograms,
rhombuses, trapezoids, circles, semi-circles, quarter-circles, and regular polygons (perimeter only) NOTE: The
problems we have chosen to review include Pythagorean Theorem, probability, and percent. The problems are
from Regents Course I and IA.
Formulas To Remember (NOT on Reference Sheet)
Area Formulas (Cover)


Area of
or
-- multiply length times width
Area of
-- multiply radius squared (r2) times
**if you only need part of the circle then divide

Area of
-- multiply the base times the height then divide by 2
Perimeter/Circumference (around the outside) For
for
A= lw
 A =  r2
multiply the radius by 2 and then times by
or
A=
bh
2
add up the sides on the outside and
 . C = 2r 
1.
32
Regents Integrated Algebra
Twenty Day Countdown
2. A figure is made up of a rectangle and a semicircle as shown in the diagram below.
What is the area of the figure, to the nearest tenth of a square centimeter?
What is the perimeter?
3.
4.
33
Regents Integrated Algebra
Twenty Day Countdown
Practice Problems:
5.
Serena’s garden is a rectangle joined with a semicircle, as shown in the diagram below. Line
segment AB is the diameter of semicircle P. Serena wants to put a fence around her garden.
Calculate the length of fence Serena needs to the nearest tenth of a foot.
6.
A window is made up of a single piece of glass in the shape of a semicircle and a rectangle, as
shown in the diagram below. Tess is decorating for a party and wants to put a string of lights all
the way around the outside edge of the window.
To the nearest foot, what is the length of the string of lights that Tess will need to decorate the
window?
34
Regents Integrated Algebra
7.
Twenty Day Countdown
A playground in a local community consists of a rectangle and two semicircles, as shown in the
diagram below.
Which expression represents the amount of fencing, in yards, that would be needed to completely
enclose the playground?
1)
2)
3)
4)
8.
Luis is going to paint a basketball court on his driveway, as shown in the diagram below. This
basketball court consists of a rectangle and a semicircle.
Which expression represents the area of this basketball court, in square feet?
1) 80
2) 80 + 8π
3) 80 + 16π
4) 80 + 64π
35
Regents Integrated Algebra
9.
Twenty Day Countdown
In the diagram below, circle O is inscribed in square ABCD. The square has an area of 36.
What is the area of the circle?
1) 9π
2) 6π
3) 3π
4) 36π
10. In the accompanying diagram, a circle with radius 4 is inscribed in a square.
What is the area of the shaded region?
1)
2)
3)
4)
36
Regents Integrated Algebra
Twenty Day Countdown
11. Virginia has a circular rug on her square living room floor, as represented in the accompanying
diagram. If her entire living room floor measures 100 square feet, what is the area of the part of
the floor covered by the rug?
37
Regents Integrated Algebra
Twenty Day Countdown
Name: ____________________________________
Date: __________
Day 10: A.G.2 (Using the IA Regents formula sheet to calculate surface area
and volume)
A.G.2: Use formulas to calculate volume and surface area of the rectangular solids and cylinders
Note: Several students cannot complete these problems because they do not know there is a formula sheet and
are not familiar with it. Stress the format and formulas on the reference sheet.
Formulas To Remember (NOT on Formula Sheet)
Volume (fill) rectangular prism 𝒍 ∙ 𝒘 ∙ 𝒉 (in cubic units)
1. Jasmine answered the question below on the IA Regents. Is she correct or incorrect?
If she is incorrect please explain what mistake you think she may have made.
Mrs. Ayer is painting the outside of her son’s toy box, including the top and bottom.
The toy box measures 3 feet long, 1.5 feet wide, and 2 feet high. What is the total surface area she will
paint?
(1) 9.0 ft 2
(2) 135
(3) 22.5 ft 2
. ft 2
Jasmine said, “The formula for a toy box is length times width times height”
(4) 27.0 ft 2
2. The diagram to the right represents Joe's two fish tanks.
Joe's larger tank is completely filled with water.
He takes water from it to completely fill the small tank.
Determine how many cubic inches of water
will remain in the larger tank.
38
Regents Integrated Algebra
Twenty Day Countdown
3. A soup can is in the shape of a cylinder. The can has a volume of 342 cm3 and adiameter of 6 cm.
Express the height of the can in terms of π.
Determine the maximum number of soup cans that can be stacked on their base between two shelves if the
distance between the shelves is exactly 36 cm. Explain your answer.
4. A plastic storage box in the shape of a rectangular prism has a length of 𝑥 + 3 and a width of 𝑥 − 4, and a
height of 5. Represent the surface area as a trinomial in terms of 𝑥.
What is a trinomial???
5. Find the volume, in cubic centimeters, and the surface area, in square centimeters,
of the rectangular prism shown to the right.
39
Regents Integrated Algebra
Twenty Day Countdown
Practice Questions:
6.
Lenny made a cube in technology class. Each edge measured 1.5 cm. What is the volume of the
cube in cubic centimeters?
1) 2.25
2) 3.375
3) 9.0
4) 13.5
7.
A cylindrical container has a diameter of 12 inches and a height of 15 inches, as illustrated in the
diagram below.
What is the volume of this container to the nearest tenth of a cubic inch?
1) 6,785.8
2) 4,241.2
3) 2,160.0
4) 1,696.5
8.
A cylinder has a diameter of 10 inches and a height of 2.3 inches. What is the volume of this
cylinder, to the nearest tenth of a cubic inch?
1) 72.3
2) 83.1
3) 180.6
4) 722.6
40
Regents Integrated Algebra
9.
The volume of a cylindrical can in
radius, in inches?
1) 8
Twenty Day Countdown
cubic inches. If the height of the can is 2 inches, what is its
2) 2
3) 16
4) 4
10. If the length of a side of a cube is 7x, which expression represents the cube's volume?
1)
2)
3)
4)
11. How many square inches of wrapping paper are needed to entirely cover a box that is 2 inches by 3
inches by 4 inches?
1) 18
2) 24
3) 26
4) 52
12. The length and width of the base of a rectangular prism are 5.5 cm and 3 cm. The height of the
prism is 6.75 cm. Find the exact value of the surface area of the prism, in square centimeters.
41
Regents Integrated Algebra
Twenty Day Countdown
13. Find the volume, in cubic centimeters, and the surface area, in square centimeters, of the
rectangular prism shown below.
14. As shown in the accompanying diagram, the length, width, and height of Richard’s fish tank are 24
inches, 16 inches, and 18 inches, respectively. Richard is filling his fish tank with water from a hose
at the rate of 500 cubic inches per minute. How long will it take, to the nearest minute, to fill the
tank to a depth of 15 inches?
42
Regents Integrated Algebra
Twenty Day Countdown
15. A cardboard box has length
, width
, and height .
a Write an expression, in terms of x, to represent the volume of the box.
b If
16.
centimeters, what is the number of cubic centimeters in the volume of the box?
Mike buys his ice cream packed in a rectangular prism-shaped carton, while Carol buys hers in a
cylindrical-shaped carton. The dimensions of the prism are 5 inches by 3.5 inches by 7 inches. The
cylinder has a diameter of 5 inches and a height of 7 inches. Which container holds more ice
cream? Justify your answer. Determine, to the nearest tenth of a cubic inch, how much more ice
cream the larger container holds.
43
Regents Integrated Algebra
Twenty Day Countdown
Day 11: A.M.1 & A.M.3 (Rates and Relative Error)
A.M.1: Calculate rates using appropriate units
A.M.3: Calculate the relative error in measuring square and cubic units, when there is an error in the linear
measure
A rate is a ratio that compares two different kinds of numbers, such as miles per hour, or inches per minute. A
unit rate compares a quantity to its unit of measure.
REMEMEMBERDISTANCE = RATE x TIME
1. Tom drove 290 miles from his college to home and used 23.2 gallons of gasoline. His sister, Ann, drove 225
miles from her college to home and used 15 gallons of gasoline. Whose vehicle had better gas mileage?
Justify your answer
Lydia answered the question on the IA Regents exam. Check her work for mistakes.
290
Tom = 23.2 = 12.5 𝑚𝑝𝑔
smaller amount of gas
Ann =
225
15
= 15 𝑚𝑝ℎ Tom gets the better gas mileage because that is the
2. A three toed sloth can travel at an average rate of 0.15 mile per hour. What is the rate of speed of the sloth
in feet per minute?
3. Jessica measured a piece of paper to be 22.8 cm by 29.6 cm. The piece of paper is actually
22.7 cm by 29.5 cm.
Determine the number of square centimeters in the area of the piece of paper using Jessica’s
measurements.
Determine the number of square centimeters in the actual area of the piece of paper.
Determine the relative error in calculating the area. Express your answer as a decimal to the
nearest thousandth.
Jessica does not think there is a significant amount of error. Do you agree or disagree? Justify
your answer.
44
Regents Integrated Algebra
Twenty Day Countdown
4. Ashley measured the dimensions of a rectangular prism to be 6 cm by 10 cm by 1.5 cm. The actual
dimensions are 5.9 cm by 10.3 cm by 1.7 cm. Determine the relative error, to the nearest thousandth, in
calculating the volume of the prism.
5. An oil company distributes oil in a metal can shaped like a cylinder that has an actual radius of 5.1 cm and a
height of 15.1 cm. A worker incorrectly measured the radius as 5 cm and the height as 15 cm. Determine
the relative error in calculating the surface area, to the nearest thousandth.
Practice Questions:
6.
A cell phone can receive 120 messages per minute. At this rate, how many messages can the
phone receive in 150 seconds?
1) 48
2) 75
3) 300
4) 18,000
7.
Nicole’s aerobics class exercises to fast-paced music. If the rate of the music is 120 beats per
minute, how many beats would there be in a class that is 0.75 hour long?
1) 90
2) 160
3) 5,400
4) 7,200
45
Regents Integrated Algebra
Twenty Day Countdown
8.
Joseph typed a 1,200-word essay in 25 minutes. At this rate, determine how many words he can
type in 45 minutes.
9.
Carrie bought new carpet for her living room. She calculated the area of the living room to be
174.2 square feet. The actual area was 149.6 square feet. What is the relative error of the area to
the nearest ten-thousandth?
1) 0.1412
2) 0.1644
3) 1.8588
4) 2.1644
10. Students calculated the area of a playing field to be 8,100 square feet. The actual area of the field
is 7,678.5 square feet. Find the relative error in the area, to the nearest thousandth.
11. Ryan estimates the measurement of the volume of a popcorn container to be 282 cubic inches.
The actual volume of the popcorn container is 289 cubic inches. What is the relative error of
Ryan's measurement to the nearest thousandth?
1) 0.024
2) 0.025
3) 0.096
4) 1.025
12. Using his ruler, Howell measured the sides of a rectangular prism to be 5 cm by 8 cm by 4 cm. The
actual measurements are 5.3 cm by 8.2 cm by 4.1 cm. Find Howell’s relative error in calculating
the volume of the prism, to the nearest thousandth.
46
Regents Integrated Algebra
Twenty Day Countdown
13.
47
Regents Integrated Algebra
Twenty Day Countdown
Day 12: AA.6, AA.7, & AA.10 (Solve verbal inequalities, linear equations, and systems)
AA.6: Modeling Equations: Analyze and solve verbal problems whose solution requires solving a linear equation
in one variable or linear inequality in one variable
AA.7& AA.10:Writing Linear Systems: Analyze and solve verbal problems whose solution requires solving
systems of linear equations in two variables IA Regents is due for a verbal system
1. Michael works as a painter. He earns $35 per hour. He pays his helper $55 per day from his earnings.
Michael would like to earn more than $225 today. How many hours will he need to work?
2. Jasmine wants to buy a television that costs $796, including tax. She makes a down payment of $250. If she
pays the rest in 6 monthly payments, how much will each payment be?
3. Shawna compares the prices to download music from two different web sites.
Write an equation to determine the number of songs, s, that need to be purchased so the two plans will
cost the same.
Equation___________________________________
Use your equation to find the number of songs must be downloaded for both music plans to cost the same?
Show your work.
4. Jeff bought 5 oranges and 5 apples for a total cost of $7.70. Gene bought 5 oranges and 3 apples for a total
cost of $6.40. What is the cost of each orange and each apple?
48
Regents Integrated Algebra
Twenty Day Countdown
Practice Questions:
5.
Mr. Cash bought d dollars worth of stock. During the first year, the value of the stock tripled.
Thenext year, the value of the stock decreased by$1200.
(a) Write an expression in terms of d to representthe value of the stock after two years.
(b) If an initial investment is $1,000, determine itsvalue at the end of 2 years.
6.
An online music club has a one-time registration fee of $13.95 and charges $0.49 to buy each song.
If Emma has $50.00 to join the club and buy songs, what is the maximum number of songs she can
buy?
1) 73
2) 74
3) 131
4) 130
7.
Sara's telephone service costs $21 per monthplus $0.25 for each local call, and longdistancecalls are
extra. Last month, Sara'sbill was $36.64, and it included $6.14 in longdistancecharges. How many
local calls didshe make?
8.
A prom ticket at Smith High School is $120.Tom is going to save money for the ticket bywalking his
neighbor’s dog for $15 per week. If Tom already has saved $22, what is theminimum number of
weeks Tom must walkthe dog to earn enough to pay for the promticket?
49
Regents Integrated Algebra
9.
Twenty Day Countdown
Julia went to the movies and bought one jumbopopcorn and two chocolate chip cookies for
$5.00.Marvin went to the same movie and bought onejumbo popcorn and four chocolate chip
cookies for$6.00. How much does one chocolate chip cookiecost?
1) $0.50
2) $0.75
3) $1.00
4) $2.00
10. The cost of 3 markers and 2 pencils is $1.80. Thecost of 4 markers and 6 pencils is $2.90. What isthe
cost of each item? Include appropriate units inyour answer.
11. Juan has a cellular phone that costs $12.95 permonth plus 25¢ per minute for each call. Tiffany has a
cellular phone that costs $14.95 per monthplus 15¢ per minute for each call. For what numberof
minutes do the two plans cost the same?
50
Regents Integrated Algebra
Twenty Day Countdown
Day 13: AA.6, AA.7, & AA.10 (Solve verbal inequalities, linear equations, and systems)
AA.6: Modeling Equations: Analyze and solve verbal problems whose solution requires solving a linear equation
in one variable or linear inequality in one variable
AA.7& 10:Writing Linear Systems: Analyze and solve verbal problems whose solution requires solving systems of
linear equations in two variables DUE to CCSSM: IA Regents is due for a verbal system
1. Linda’s Video Store sold three times as many Titanic CDs as Godzilla CDs. The price of a Titanic CD is $20.
And the price of a Godzilla CD is $15. If her total sales for these CDs was $2250, what was the total number
of each CD she sold?
2. Rachel answered the question below on the state test. Is she correct or incorrect?
If she is incorrect please explain what mistake you think she may have made.
Shana wants to buy a new bicycle that has a retail price of $259.99. She knows that it will be on sale
next week for 30% off the retail price. If the tax rate is 7%, find the total amount, to the nearest cent,
that she will save by waiting until next week
259(.30) = 77.7
259 – 77.7= 181.3
181.3(.07) = 12.69
181.3 + 12.69 = 195.99
Shanna paid
$195.99
3. OLD Course IR Problem
51
Regents Integrated Algebra
Twenty Day Countdown
4.
Note: This is an inequality system It is not in the Integrated Algebra curriculum; however it is good
practice for setting up an inequality
Practice Questions:
5.
The ninth grade class at a local high school needs to purchase a park permit for $350.00 for
their upcoming class picnic. Each ninth grader attending the picnic pays $0.50. Each guest
pays $1.75. If 250 ninth graders attend the picnic, which inequality can be used to determine
the number of guests, x, needed to cover the cost of the permit?
(1) (0.5)(250) + 1.75x ≥ 350.00
(2) 0.5x + (1.75)(250) ≥ 350.00
(3) (0.5)(250) − 1.75x ≥ 350.00
(4) 0.5x − (1.75)(250) ≥ 350.00
6. In a hockey league, 57 players play on five different teams. Each team has at least 11 players. What
is the largest possible number of players on any one team?
1) 15
2) 21
3) 14
4) 13
52
Regents Integrated Algebra
Twenty Day Countdown
7. Thelma and Laura start a lawn-mowing business and buy a lawnmower for $180. They plan to
charge $10 to mow one lawn. What is the minimum number of lawns they need to mow if they wish
to earn a profit of at least $600?
8. A prom ticket at Smith High School is $105.Tom is going to save money for the ticket bywalking his
neighbor’s dog for $12 per week. If Tom already has saved $17, what is theminimum number of
weeks Tom must walkthe dog to earn enough to pay for the promticket?
9. Mr. Braun has $80.00 to spend on pizzas andsoda pop for a picnic. Pizzas cost $8.75 each and the
drinks cost $0.50 each. Four times as many drinks as pizzas are needed. What isthe maximum
number of pizzas that Mr.Braun can buy?
53
Regents Integrated Algebra
Twenty Day Countdown
Day 14: AA.9 (Solve exponential growth/decay)
A.A.9: Exponential Functions: Analyze and solve verbal problems that involve exponential growth and decay
Formula for exponential growth or decay: 𝐴(𝑡) = 𝐴𝑜 (1 ± 𝑟)𝑡 where A(t)= ending
value, Ao = initial value, r = rate (written as a decimal), and t = time
1. Michelle is making a rectangular canvas for a painting. Her original plan was to make the canvas 10 inches
long and 7 inches wide. Then she decided to increase the length and width by the same number of inches
so that the area would be 130 square inches. How many inches will she add to the length and width?
2. The current student population at the Edison Campus is 2,500. The enrollment at the school increases at a
rate of 4% each year. To the nearest whole number, what will the student population be closest to in 3
years'?
3. A bank is advertising that new customers can open a savings account with a2 1 % interest rate compounded
2
annually. Roberto invests $5,000 in an account at this rate. If he makes no additional deposits or
withdrawals on his account, find the amount of money he will have, to the nearest cent, after five years.
4. Justin’s Copy Stop purchased a new copier for $45,000. Each year it depreciates (loses value) at a rate of 6%.
What will its approximate value be at the end of the fifth year?
(1) $60,220.15
(2) $33,079.14
(3) $33,025.68
(4) $460.80
54
Regents Integrated Algebra
Twenty Day Countdown
Practice Questions:
5.
Mr. Smith invested $2,500 in a savings account that earns 3% interest compounded annually. He
made no additional deposits or withdrawals. Which expression can be used to determine the
number of dollars in this account at the end of 4 years?
1)
2)
3)
4)
6.
Cassandra bought an antique dresser for $500. If the value of her dresser increases 6% annually,
what will be the value of Cassandra's dresser at the end of 3 years to the nearest dollar?
1) $415
2) $590
3) $596
4) $770
7.
Kathy plans to purchase a car that depreciates (loses value) at a rate of 14% per year. The initial
cost of the car is $21,000. Which equation represents the value, v, of the car after 3 years?
1)
2)
3)
4)
55
Regents Integrated Algebra
8.
Twenty Day Countdown
A car depreciates (loses value) at a rate of 4.5% annually. Greg purchased a car for $12,500. Which
equation can be used to determine the value of the car, V, after 5 years?
1)
2)
3)
4)
9.
The value of a car purchased for $20,000 decreases at a rate of 12% per year. What will be the
value of the car after 3 years?
1) $12,800.00
2) $13,629.44
3) $17,600.00
4) $28,098.56
10. The Booster Club raised $30,000 for a sports fund. No more money will be placed into the fund.
Each year the fund will decrease by 5%. Determine the amount of money, to the nearest cent, that
will be left in the sports fund after 4 years.
56
Regents Integrated Algebra
Twenty Day Countdown
Day 15: AG.7 & AG.9: Systems
AG.7: Graph and solve systems of linear equations and inequalities with rational coefficients in two variables
AG.9: Solve systems of linear and quadratic equations graphically Note: Only use systems of linear and
quadratic equations that lead to solutions whose coordinates are integers
To Solve a System of Equations using your Calculator:
1. Must have y by itself for both equations. Enter the first equation into y = , then enter second equation into y =
and graph
2. To find where they intersect, hit 2nd trace then 5 (intersect)
3. When finding the intersect point you must use the left/right arrow keys to get close to the point of intersection
on the 1st equation, then hit enter. Repeat this process for the second equation. Finally hit enter when it asks
for guess.
4. The x and y value is you’re your answer or point (x,y).
To Solve a System of Inequalities using your Calculator:
**Use the left arrow to get on the left side of y = and hit enter to get to the up or down shading triangle
1. On the set of axes below, solve the following system of equations graphically. State the coordinates of the
solution.
2. Which ordered pair is in the solution set of the system of equations y   x 1 and
y  x 2  5x  6?
(1) (5, 2)
(2) (5, -4)
(3) (-5, 6)
(4) (-5, -1)
57
Regents Integrated Algebra
Twenty Day Countdown
3.
4. What is the value of the y-coordinate of the solution to the system of equations
x  2 y  9 and
x  y  3?
(1) 5
(2) 3
(3) 2
(4) 6
58
Regents Integrated Algebra
Twenty Day Countdown
Practice Questions:
5. What is the value of y in the following system of equations?
2x + 3y = 6
2x + y = -2
1) 1
2) 2
3) -3
4) 4
6. What point is the intersection of the graphs of the lines 2x - y = 3 and x + y = 3?
1) (2,1)
2) (1,2)
3) (3,0)
4) (3,3)
59
Regents Integrated Algebra
Twenty Day Countdown
Day 16: AG.7 & AG.9: Systems
AG.7: Graph and solve systems of linear equations and inequalities with rational coefficients in two variables
AG.9: Solve systems of linear and quadratic equations graphically Note: Only use systems of linear and
quadratic equations that lead to solutions whose coordinates are integers
1.
2.
60
Regents Integrated Algebra
Twenty Day Countdown
3.
4. On the set of axes below, solve the following system of equations graphically for all
values of x and y.
61
Regents Integrated Algebra
Twenty Day Countdown
Practice Questions:
5. What is the value of x in the following system of equations?
2x + 3y = 6
2x + y = -2
6. What is the solution of the system of equations c + 3d = 8 and c = 4d - 6?
7. On the set of axes below, graph the following system of inequalities and state the coordinates of
apoint in the solution set.
2x − y ≥ 6
x >2
62
Regents Integrated Algebra
Twenty Day Countdown
Day 17: AS. 19, AS.22, & AS.23 Probability: sample spaces, tree diagrams, independent and dependent
events
AS.19: Determine the number of elements in a sample space and the number of favorable events
AS.22: Determine, based on calculated probability of a set of events, if:
*some or all are equally likely to occur
* one is more likely to occur than another
* whether or not an event is certain to happen or not to happen
AS.23: Calculate the probability of: a series of independent events; a series of dependent events; two mutually
exclusive events; two events that are not mutually exclusive
Note: Course IR problems will be mixed in for review. Most problem cover more than one PI.
1.
2. This problem mixes verbal linear equations with probability
63
Regents Integrated Algebra
Twenty Day Countdown
3. Bob and Laquisha have volunteered to serve on the Junior Prom Committee. The
names of twenty volunteers, including Bob and Laquisha, are put into a bowl. If
two names are randomly drawn from the bowl without replacement, what is the
probability that Bob’s name will be drawn first and Laquisha’s name will be
drawn second?
1)
2)
3)
4)
4. A sandwich consists of one type of bread, one type of meat, and one type of
cheese. The possible choices are listed below.
Bread: white, rye
Meat: ham, turkey, beef
Cheese: American, Swiss
Draw a tree diagram or list a sample space of all the possible different
sandwiches consisting of one type of bread, one type of meat, and one type of
cheese. Determine the number of sandwiches that will not include turkey.
Determine the number of sandwiches that will include rye bread and Swiss
cheese.
64
Regents Integrated Algebra
Twenty Day Countdown
Practice questions:
5.
The probability that the Cubs win their first game is
second game is
1
. The probability that the Cubs win their
3
3
. What is the probability that theCubs win both games?
7
16
21
1
2)
7
6
3)
7
2
4)
5
1)
6.
Throughout history, many people have contributed to the development of mathematics. These
mathematicians include Pythagoras, Euclid, Hypatia, Euler, Einstein, Agnesi, Fibonacci, and Pascal.
What is the probability that a mathematician’s name selected at random fromthose listed will
start with either the letter E or theletter A?
2
8
3
2)
8
4
3)
8
6
4)
8
1)
7.
A spinner that is equally divided into eight numbered sectors is spun 20 times. The tablebelow
shows the number of times the arrow landed in each numbered sector.
Based on the table, what is the empirical probability that the
spinner will land on a prime number on the next spin?
9
20
11
2)
20
12
3)
20
14
4)
20
1)
65
Regents Integrated Algebra
8.
Twenty Day Countdown
The Grimaldis have three children born in different years.
a Draw a tree diagram or list a sample space toshow all the possible arrangements of boy and girl
children in the Grimaldi family.
b Using your information from part a, what is the probability that the Grimaldis have three boys?
66
Regents Integrated Algebra
Name: _______________________________________
Twenty Day Countdown
Date: ___________
Day 18: AS. 19, AS.22, & AS.23 Probability: sample spaces, tree diagrams, independent and dependent
events
AS.19: Determine the number of elements in a sample space and the number of favorable events
AS.22: Determine, based on calculated probability of a set of events, if:
*some or all are equally likely to occur
* one is more likely to occur than another
* whether or not an event is certain to happen or not to happen
AS.23: Calculate the probability of: a series of independent events; a series of dependent events; two mutually
exclusive events; two events that are not mutually exclusive
Note: Course IR problems will be mixed in for review. Most problem cover more than one PI.
1. Three storage bins contain colored blocks. Bin 1 contains 15 red and 14 blue blocks. Bin 2 contains 16
white and 15 blue blocks. Bin 3 contains 15 red and 15 white blocks. All of the blocks from the three bins
are placed into one box. If one block is randomly selected from the box, which color block would most
likely be picked? Justify your answer.
2.
67
Regents Integrated Algebra
Twenty Day Countdown
3.
68
Regents Integrated Algebra
Twenty Day Countdown
4.
5. Brianna is using the two spinners shown below to play her new board game. She spins the arrow on each
spinner once. Brianna uses the first spinner to determine how many spaces to move. She uses the
second spinner to determine whether her move from the first spinner will be forward or backward.
Find the probability that Brianna will move fewer than four spaces and backward.
69
Regents Integrated Algebra
Twenty Day Countdown
Practice Questions:
6.
7.
8.
70
Regents Integrated Algebra
Twenty Day Countdown
9.
10.
11.
71
Regents Integrated Algebra
Twenty Day Countdown
Name: _______________________________________
Date: ___________
Day 19:AA.29, AA.30, & AA.31 Sets
AA.29: Use set-builder notation and/or interval notation to illustrate the elements of a set, given the elements in
roster form
AA.30: Find the complement of a subset of a given set, within a given universe
AA.31: Find the intersection of sets (no more than three sets) and/or union of sets (no more than three sets
Here is a chart for review:
Methods of Representing Intervals
Words
Number Line
Inequality
Interval
Notation
Set Builder Notation
Numbers
_______ than 4
Numbers
_______ than or
_______ -6
Numbers
______than -8
and _____than
6
Numbers
______than or
______to -6
and _____than 8
Numbers
______than or
______to -2
and _____than
or ______to 6
1.
In roster form, write sets that match the following descriptions:
(a)all odd integers greater than zero and less than 15
(b) all months whose names begin with the letter M
(c) the solution set for x > 4
72
Regents Integrated Algebra
Twenty Day Countdown
2. Use interval notation to represent the following:
(a) the set of all numbers greater than 1 and less than 9
(b) the set of all numbers less than or equal to – 4
(c) the set of all real numbers from – 5 through 4, inclusive
(d)
(e)
3. Which notation describes {4, 5, 6}?
(1) {x│4 ≤ x< 6, where x is an integer}
(2) {x│3 ≤ x ≤ 6, where x is an integer}
(3) {x│4 <x< 6, where x is an integer}
(4) {x│3 <x ≤ 6, where x is an integer}
4. Which notation describes
1)
2)
3)
4)
?
Practice Questions:
5.
6.
73
Regents Integrated Algebra
7.
Twenty Day Countdown
8.
9.
Write the set of all numbers greater than or equal to 8 and less than 21 in interval
notation.
10.
Write the set of all numbers from 2 to 11 inclusive, in interval notation.
11.
Consider the set of numbers {-5, -4, -3, -2, -1, 0, 1}
a. Write the set of numbers in interval notation.
b. Write the set of numbers in set-builder notation.
12.
Consider the set of numbers {-10, -9, -8, -7, -6}
a. Write the set of numbers in interval notation.
b. Write the set of numbers in set-builder notation.
74
Regents Integrated Algebra
Twenty Day Countdown
Name: _______________________________________
Date: ___________
Day 20:AA.29, AA.30, & AA.31 Sets
AA.29: Use set-builder notation and/or interval notation to illustrate the elements of a set, given the elements in
roster form
AA.30: Find the complement of a subset of a given set, within a given universe
AA.31: Find the intersection of sets (no more than three sets) and/or union of sets (no more than three sets
Below is a chart that you may use for union and intersection review.
1.
a) Consider the set of integers greater than – 3 and less than or equal to 6. A subset of this set is the
positive factors of 6. What is the complement of this subset?
b) Given: Set T: {W, A,T, K, I, N, S}
Set B: { A, I, N }
Set B is a subset of set T. What is the complement of set B?
c) Given: U = { 0, 1, 2, 3, 4, 5, 6, 7}
A = { 1, 2, 4, 5, 6 }
Set A is a subset of set U. What is the complement of set A?
75
Regents Integrated Algebra
Twenty Day Countdown
2. Given :
A = { All odd integers from 1 to 19, inclusive}
B = { 11, 13, 15, 17}
What is the complement of set B within the universe of set A?
3. Alfredo answered the question below on the IA Regents. Is he correct or incorrect?
If he is incorrect please explain what mistake you think he may have made.
In interval notation, the set of all real numbers greater than -6 and less than or equal to 14 is represented
by
1)
2)
3)
4)
4. Twelve players make up a high school basketball team. The team jerseys are numbered 1 through 12. The
players wearing the jerseys numbered 3, 6, 7, 8, and 11 are the only players who start a game. Using set
notation, list the complement of this subset.
Malayshia answered this question on the IA Regents Exam. Is she correct or incorrect?
The complement is {1, 2, 3, 4, 5, 9, 10, 12}
Practice Questions:
5.
6.
76
Regents Integrated Algebra
Twenty Day Countdown
7.
8.
9.
10.
11.
12.
12.
13.
77
Regents Integrated Algebra
14.
16.
Twenty Day Countdown
15.
17.
78