Download Mathematics Curriculum Math III

Bozeman Public Schools
Mathematics Curriculum
Math III
Process Standards: Throughout all content standards described below, students use appropriate
technology and engage in the mathematical processes of problem solving, reasoning,
communication, connections and representations.
Content Standards: Symbolic reasoning and calculations with symbols are central in algebra.
Through the study of algebra, a student develops an understanding of symbolic language of
mathematics and the sciences. In addition, algebraic skills and concepts are developed and used
in a wide variety of problem-solving situations.
Note: The following concepts should be reviewed at the beginning of the school year: computing
using the four basic arithmetic operations; comparing, ordering and computing fractions,
decimals, percents and integers; primes, factors, and multiples; and applying order of operations.
(N): Numbers and Operations: Students demonstrate understanding of and an ability to use
numbers and operations.
N.1.0 Students will be able to simplify radical terms. Students will be able to perform the
four arithmetic operations using like and unlike radical terms and express the result
in simplest form.
N.2.0 Students will be able to understand and use scientific notation to compute products
and quotients of numbers.
Example 1:
The average U.S. dollar bill has a thickness of 4.3 x 10-3 inch. What decimal is
equivalent to 4.3 x 10-3?
A.
0.43
B.
0.043
C.
0.0043
D.
0.00043
Example 2:
Scientists consider water unsafe if it contains more than 1.5 x 10-5 grams of lead per liter.
How is this number written in standard form?
A.
0.0000015
B.
0.000015
C.
0.00015
D.
0.0015
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(A): Algebraic Concepts: Students use algebraic concepts, processes, and language to model
and solve a variety of real-world and mathematical problems. Students demonstrate
understanding of and an ability to use patterns, relations and functions.
A.1.0 Students will be able to analyze and solve problems that involve quadratic
equations.
Example:
For which value or values of n is the following statement true?
1
n2 = n
2
1
A.
only
2
1
B.
and 0 only
2 !
C.
The statement is never true.
D.
The statement is true of all real numbers.
A.2.0 Students will be able to multiply and divide monomial expressions with a common
base, using the properties of exponents
Example:
Which expression is equivalent to 3x2 • 2x3?
A.
5x5
B.
5x6
C.
6x5
D.
6x6
A.3.0 Students will be able to add, subtract, multiply, and divide monomials and
polynomials.
A.4.0 Students will be able to simplify fractions with polynomials in the numerator and
denominator by factoring both and remaining both and renaming them to lowest
terms.
A.5.0 Students will be able to add or subtract factional expressions with monomial or like
binomial denominators.
Example:
Evaluate:
2 1 3
+ "
3 6 4
A.6.0 Students will be able to multiply and divide algebraic fractions and express the
product or quotient in simplest form.
!
A.7.0 Students will be able to identify and factor the difference of two perfect squares.
A.8.0 Students will be able to factor second- and simple third degrees polynomials. These
techniques include finding a common factor for all terms in a polynomial,
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recognizing the difference of two squares, and recognizing perfect squares of
binomials.
A.9.0 Students will be able to graph, and solve quadratic equations using the quadratic
formula.
A.10.0 Students will be able to solve equations involving fractional expressions.
A.11.0 Students will be able to solve algebraic proportions in one variable which result in
linear or quadratic equations.
A.12.0 Students will understand and apply the multiplication property of zero to solve
quadratic equations with integral coefficients and integral roots.
A.13.0 Students will be able to determine the vertex and axis of symmetry of a parabola,
given its equation.
(G): Geometry: Students demonstrate understanding of shape and an ability to use geometry.
G.1.0 Students will be able to identify and graph linear, quadratic (parabolic), and
absolute value.
G.2.0 Students will be able to find the roots of a parabolic function graphically.
G.3.0 Students will be able to solve systems of linear and quadratic equations graphically.
Example:
A system of two linear equations is graphed on the coordinate plane below.
Which is the best estimate of the x-value of the solution to this system of equations?
A.
2.5
B.
-1.5
C.
-2
D.
-3
G.4.0 Students will be able to determine a vertex and axis of symmetry of a parabola, given
its graph.
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G.5.0 Students demonstrate understanding by identifying and giving examples of
undefined terms, axioms, theorems, and inductive and deductive reasoning.
Example:
Use the table below to answer the following:
Term number (n) Value (p)
1
3
2
5
3
7
4
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Which rule describes the pattern shown in the table?
A. p = n + 2
B. p = 2n + 1
C. p = 3n – 1
D. p = 5n -2
G.6.0 Students write geometric proofs.
G.7.0 Students construct and judge the validity of a logical argument and give
counterexamples to disprove a statement.
Example 1:
Steve claims that if two angles are supplementary, then they are adjacent. Which pair of
angles is a counterexample to Steve’s claim, proving that is it FALSE?
Example 2:
Which figures could be constructed to prove that this statement is FALSE?
A. two squares
B. two equilateral triangles
C. two rectangles that are not squares
D. two regular pentagons
Example 3:
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James hears a classmate say that all trapezoids have congruent base angles. Which
figure could James use to show that his classmate’s statement is incorrect?
G.8.0 Students know, derive, and solve problems involving the perimeter, circumference,
arc length.
Example 1:
Mina is designing a kite for a competition. She begins with the scale drawing shown
below.
Mina plans to decorate the outer edge of the kite with tape. If her actual kite is 96 cm
wide, how much tape will she need?
A. 324 cm
B. 162 cm
C. 96 cm
D. 54 cm
Example 2:
The door of the cabinet shown below can open to a maximum angle of 90°.
5
What is the length of the arc through which the door swings?
A.
23.6 inches
B.
47.1 inches
C.
62.8 inches
D.
78.5 inches
G.9.0 Students compute areas of circles.
Example 1:
The congruent circles in the square below are tangent to one another and to the square:
What is the approximate area of the shaded region in this figure?
A.
243 square inches
B.
239 square inches
C.
154 square inches
D.
70 square inches
Example 2:
A window has a rectangular bottom and a semicircular top. The dimensions of the
window are shown in the diagram below.
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What is the approximate area of the window?
A.
540 square inches
B.
629 square inches
C.
667 square inches
D.
794 square inches
Example 3:
Many movies use computer-generated animation. To convey the sense of movement on
the screen, an image will grow or shrink over time. The figures below show the initial
image of a circular window, together with its radius and area, as well as the growth of
the initial image over various numbers of seconds.
a.
b.
c.
d.
e.
According to this pattern, what would be the radius and area at 4 seconds?
Write an equation showing the relationship between the radius, r, of the circle
and the number of seconds, t.
Use the equation you wrote in part b to find how many centimeters long the
radius of the circle will be after 20 seconds. Show or explain how you found your
answer.
Use the equation you wrote in part b to find the number of seconds it will take for
the image to have a radius of 65 centimeters. Show or explain how you found
your answer.
Explain why you would describe the relationship between time and area as linear
or nonlinear.
G.10.0 Students compute the volumes and surface areas of prisms, pyramids, cylinders,
cones, and spheres.
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Example 1:
John needed to order sand for a playground sandbox. The sandbox is 3 feet wide, 4 feet
long, and 18 inches deep. To find how much sand to order, John performed the following
steps.
• He multiplied 3 feet by 4 feet to get 12 square feet.
• He then multiplied 12 square feet by 1.5 feet to get 18 cubic feet.
• He divided 18 cube feet by 3 feet in a yard to get 6 cubic yards of sand.
When the sand was delivered, he knew he had made a mistake. What should ge have
done differently?
A.
He should have divided by 27 instead of 3 to get cubic yards.
B.
He should have multiplied by 18 instead of by 1.5 to find the volume.
C.
He should have multiplied by 3 rather than dividing by 3 to get cubic years.
D.
He should be added 3 + 4 and multiplied by 2 instead of multiplying 3 x 4.
Example 2:
The length of each side of a cube is the cube root of its volume. If a cube has a volume of
64 cubic feet, what must be the length of each side of the cube?
A.
4 feet
B.
8 feet
C.
21 feet
D.
192 feet
Example 3:
To serve as the base for an asphalt driveway, KM Construction Company uses crushed
1
rock at a depth of foot, as shown below.
3
!
If the width in feet of a rectangular driveway is represented by w and its length in feet by
l, which expression can be used to find the number of cubic feet of crushed rock required
for the driveway?
1
A.
wl
3
1
B.
(w + l)
3
1
W+l
! C.
3
1
w+ l
! D.
3
! Example 4:
The surface area, S, of a sphere with radius r is given by the formula below.
!
S = 4πr2
The radius of a basketball is 6 times the radius of a Ping-Pong ball. How many times
greater is the surface area of the basketball than the surface area of the Ping-Pong ball?
A) 3 B) 6 C) 12 D) 36
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G.11.0 Students determine how changes in dimensions affect the perimeter, area, and
volume of common geometric figures and solids.
Example 1:
A food manufacturer must decide which of two cylindrical cans to use for a new product.
The larger can is the same height as the smaller can but has a radius that is 1.5 times the
radius of the smaller can. How many times greater is the volume of the larger can than
the volume of the smaller can?
A.
1.5
B.
2.25
C.
3.0
D.
3.375
Example 2:
A school ordered a new fish tank shaped like a rectangular prism. It has the same depth
as the old fish tank, which was also shaped like a rectangular prism. The new fish tank is
twice as long and twice as wide as the old fish tank. About how much more water will be
needed to fill the new tank than was needed to fill the old tank?
A.
twice as much
B.
four times as much
C.
six times as much
D.
eight times as much
G.12.0 Students visualize/draw 3-D figures including prisms, pyramids, spheres and cones.
Example 1:
A paper, drinking cup is pictured below.
Which of the following could be a “net” for the cup?
Example 2:
A structure is created using identical cubes. The top view and right side views are
shown below.
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If the model contains exactly 6 cubes, which figure could represent the front view?
Example 3:
A plane cuts vertically through the center of a drinking cup, as shown in the figure
below.
Which figure shows the cross section created by the cup and the plane?
G.13.0 Students know the definitions of the basic trigonometric functions defined by the
angles of a right triangle.
G.14.0 Students use trigonometric functions to solve for an unknown length of a side of a
right triangle, given an angle and a length of a side.
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Example 1:
A mountain has a vertical drop of 900 feet. From the base of the mountain the angle of
elevation is 48°, as shown in the figure below.
Which expression can be evaluated to find the length l?
Example 2:
A highway sign indicates that a stretch of road has a grade of 6 percent, which indicates
that the road rises 6 feet for every 100 feet of run as shown in the diagram below.
Which expression can be evaluated to find the degree measure of the angle x?
Example 3:
Engineers are attaching a guy wire to the top of an 80-foot radio tower. The guy wire
should make a 20° angle with a vertical line through the top of the tower, as shown
below.
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Which equation could be solved to find the final length, x, in feet, of the guy wire?
80
A.
sin(20°) =
x
80
B.
cos(20°) =
x
x
C.
sin(20°)
=
!
80
x
D.
cos(20°)
=
!
80
Example!4:
Use the picture below to answer the question.
!
The instructions for a painter’s ladder indicate the following: For the ladder to be at the
proper angle with the ground, the distance from the bottom of the wall to the bottom of
the ladder should be one-fourth the length of the ladder. Which equation could be solved
to find the measure of the angle that the ladder should make with the ground?
A.
cos(A) = 4
1
B.
cos(A) =
4
C.
sin(A) = 4
1
D.
cos(A) =
4
!
G.15.0 Students know and are able to use angle and side relationships in problems with
special right triangles, such as 30°, 60°, and 90° triangles and 45°, 45°, and 90°
!
triangles.
G.16.0 Students prove and solve problems regarding relationships among chords, secants,
tangents, inscribed angles, and inscribed and circumscribed polygons of circles.
Example 1:
The circles in the figure below are concentric with center ).
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!
If "AOC # "BOD , which congruence theorem justifies that "AOC # "BOD ?
A.
Angle-Angle-Side (AAS)
B.
Angle-Side-Angle (ASA)
C.
Side-Angle-Side (SAS)
!
D.
Side-Side-Side (SSS)
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