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# Teaching Math in the Intermediate Grades

Provides strategies and resources for educators to effectively teach mathematical concepts to students in the intermediate grade, and critical thinking skills.

## creativity

Is cos (α + β) = cos α cos β - sin α sin β?

• True
• False

Which of the following is equal to tan x?

• cot x
• sin⁡ x cos ⁡x
• 1/cot⁡x
• 1/tan⁡x

The value of x in the function k(x) = -x + 3 is -3.

• True
• False

What is the equivalent of -3.5π in degrees?

• 6800
• 5800
• 6300
• 6500

cos θ is equal to 1/sec⁡θ .

• True
• False

Which of the following table of values shows an example of exponential function?

The initial side is found in either x or y - axis.

• True
• False

Identify ¾ as irrational or rational.

• Identify ¾ as irrational or rational.
• Select one:
• Question text

Evaluate the polynomial function f(3) = 2x3 + 3x2 - 7x - 51.

• 9
• 54
• -36
• -27

What is the next step if you’re going to derived the identity 1 – cos2 x = sin2 x and to equated to 1?

• Subtract 1 to both sides
• Subtract sin2 x to both sides
• Add 1 to both sides
• Add cos2 x to both sides

Coterminal angles are the same degrees.

• True
• False

If the length of the hypotenuse and one leg of a right triangle were known, the right triangle can be solve.

• True
• False

It refers to the position of the terminal sides of different degrees and yet located at the same position.

• Reference Angles
• Standard Position
• Coterminal Angles

In what quadrant does -2100, can be found?

Base is where the exponent affixes

• True
• False

What is the cofunction of cotangent?

• tangent
• sine
• cosine
• secant

An exponential function is represented as g(x) = x3.

• True
• False

1.7 is a rational number

• True
• False

The value of sin 300 is

• 1/2
• -1/2
• √3/2
• -√3/2

What is the value of cos – 300?

• 1/2
• -1/2
• √3/2
• (-√3)/2

Mr. Roberts stimulates students' curiosity and encourages them to investigate further by asking them questions that begin with, "What would happen if..?"

• More Effective Instructional Decisions

Given the rational function (x+10)/2x = 5, what is the first step to simplify it?

• Multiply both sides to 2x
• Subtract 5 to both sides
• Divide the right side to 2x.
• Switch 5 and 2x.

The simplified form of sec2 x – 1 + tan2 x is

• 1
• 2 tan2 x
• tan2 x
• sec2 x

What is the leading coefficient of the polynomial function h(x) = x3 + 2x4 - x7 + 12 -13x?

• 7
• -13
• -1
• 12

What is the value of the exponential function h(-3) = 3x – 2?

• 3/5
• 1/243
• 3-5
• 3

K is the capital letter for kappa.

• True
• False

Which of the following is considered as an exponential function?

• f(x) =(5/2)x
• f(x) = 2/-x
• f(x) = x/3 + 3x2
• f(x) = x3

Find y in the exponential function 7y - 8 = 49?

• 41
• 15
• 10
• 57

The ratio for cosecant is

• hypotenuse/opposite

The value of the unknown side is ____.

• 15.9
• 13.4
• 14.1
• 12.7

cosine – 600 is equal to - √3/2.

• True
• False

The sum of the trigonometric functions 2 sin x + 5 and 4 - sin x is

• sin x + 9
• 3 sin x + 9
• 6 sin x + 4 sin x
• 2 sin x + 9 - sin x

Using the exponential function g(x) = 22x + 3. Complete the table below.

• -2, 1, 2, 8
• 1/2 , 2, 8, 32
• 1/2 , 2, 8, 16
• 2, 8, 16, 32

In what quadrant does 5π/9 belong?

If you convert -43 = -64 to logarithmic form, the answer: is log -4 -64 = 3

• True
• False

Ms. Lopez ensures that her students do not get lost by requiring them to stop when they finish an assignment and wait for others to finish

• Less Effective Instructional Decisions

Evaluate sin(900 + 300), using sum and difference formula.

• 1
• √3/2
• √3/4
• 1/2

Using the exponential function, e(x) = 4x, continue the value of f(x) from the table below.

• 16, 256, 1024
• 16, 256, 4096
• 16, 64, 256
• 4, 16, 64

What is the value of g in the logarithmic form log 4 1024 = g?

• 4
• 3
• 7
• 5

The name of the Greek letter ξ is _______.

• Zeta
• Xi
• Psi
• Chi

The sin value of 1800 is

• -0
• -1
• 1
• 0

The leading coefficient of the polynomial function b(x) = 8 + 5x3 - 7x8 + 12x5 - x is 8.

• True
• False

What is the first step in solving the exponential function -3x + 5 = 81?

• Disregard all the bases and solve for x.
• Transform 81 to exponential form
• Divide 81 to -3
• Subtract x - 5 to the right side

A rational function is the quotient of two polynomials

• True
• False

The exponential function of the graph below is:

• f(x) = 10x+ 1
• f(x) = -10x
• f(x) = 10x
• f(x) = 10-x

If the smallest leg of a right triangle measure 5cm and the other leg measure 12 cm, how long is the longest side?

• 15cm
• 13 cm
• 14cm
• 17cm

Is any number which can name a location on a number line Every location on a number line can be called in some way by a ___________.

• Real Numbers

Is sin (α + β) = sin α cos β - cos α sin β?

• True
• False

36250 can be found at quadrant?

Students read about the history of the Pythagorean Theorem. After reading, they solve problems using the theorem. Students then write about what they did compared to original uses of the Pythagorean Theorem.

• More Effective Instructional Decisions

The following measurements can be found in Quadrant III, except

• -2400
• 120
• (-2π)/3
• (-5π)/6

Hypotenuse is the longest side of a right triangle.

• True
• False

The given polynomial function is h(x) = (x - 3)(x + 5) (x - 1)(2x - 1)(x + 7), what is the degree of the function?

• 5
• 4
• 6
• 7

Using the polynomial function f(x) = 5x2 + 21, what is the value of the unfilled box in the table of values below.

• 41
• 26
• 21
• 16

The Law of Cosines may apply if the given are two angles and the included side.

• True
• False

If the hypotenuse and one acute angle were given, all unknowns of the oblique triangle may be solve.

• True
• False

n degrees, what is the equivalent value of 7π/8 ?

• 112.50
• 157.50
• 52.50
• 1670

Evaluate cos(900 + 900)

• 1
• 1/2
• -1
• 2

Convert 53 = y to logarithmic form.

• log 3 y = 5
• log y 5 = 3
• log 5 y = 3
• log y 3 = 5

The value of x is ____.

• 9.68
• 6.98
• 8.69
• 9.86

What is the value of x?

• 12.35
• 11.67
• 13.78
• 10.2

What is the value of x in (x+8)/(2x-5) = -3?

• -6
• 2
• -3
• 1

if you’re going to sketch the graph of f(x) = 7sin Θ. 7 stands for

• intercept
• minimum value
• maximum value
• amplitude

In simplifying the exponential function 32x + 7 = 27, the value of x is

• 2x = -4
• x = - 2
• x = 2
• x = -4

A polynomial function is defined as the sum of monomials.

• True
• False

What is the trigonometric function with a ratio adjacent/opposite

• cosecant
• cosine
• tangent
• cotangent

π is equivalent to 1800.

• True
• False

Which of the following is coterminal of 900?

• 2700
• 4500
• -2700
• 2250

What is the value of b in 6b = 1/36?

• -2
• -1/2
• 2
• 1/2

The symbol i is called ________.

• iota
• tau
• ita
• chi

The following are all exponential functions, except

• 3ax - 1
• (a + b) a - b
• (2x + 1)3 + (-2)
• -2by

Solve: 2 csc x(3 – 2csc x + csc2 x)

• 6 – 4 csc x + 2 csc2 x
• csc3 x – 4 csc2 x + 6 csc x
• 6 csc x – 4 csc x + 2 csc2 x
• 6 csc x – 4 csc2 x + 2 csc3 x

What is the unknown measurement of the side of the right triangle below?

• √55
• 3
• 6
• √33

Simplify: f(-5) = 4b2 + 8b - 35

• -115
• 25
• 45
• -175

Any real number that cannot be expressed in fractional form is an __________.

• Irrational Numbers

What is the measurement of the other leg if the hypotenuse is 12 and the shortest leg is 3?

• 9
• 3
• 3√15
• √10

Subtract 4 cot2 x – 3 cot x + 12 from -5 cot2 x – 3 cot x – 11.

• –9cot2 x - 13
• 1
• 0
• -1

From the figure below, the trigonometric function to be use if the given ratio 37/35 is

• sec π
• sin ϕ
• csc π
• cos π

What is the ratio of tan λ, using the figure below

• f/g
• g/e
• g/f
• f/e

Given the rational function (x+3)/5 = (x-7)/(x+1), what is the first step to solve it?

• Subtract x + 7 on both sides
• Multiply both sides by (5)(x + 1)
• Add x + 1 on both sides
• Divide both sides by x + 3

When do graph of exponential function goes down?

• If the exponent is positive
• If the base is negative
• If the base is positive
• If the exponent is negative

What is the value of x in the function g(x) = 2x - 1?

• 1/2
• -1
• 2
• (-1)/2

Subtract:

• -18tan4 x + 6 tan2 x
• - 6 tan2 x + 2 tan x - 14
• – 6 tan2 x 6 tan x
• 18 tan2 x + 6 tan x + 14

sin 2100 is equal to -1/2

• True
• False

The unknown value of q in the exponential function 3q = 243 is 5.

• True
• False

The unknown measurement of the side of the triangle below is

• 23
• 20
• 24
• 18

Two students are working problems on the board while the rest of the class watches.

• Less Effective Instructional Decisions

Which of the following polynomials has 3 terms?

• c(x) = 3x3 + 3x2 – 3x - 3
• a(x) = 3 + 2x – 7x2
• b(x) = -12 + 7x2 – 3x3 -x
• d(x) = 3x3

The answer: for -73 = ___ is?

• 343
• -343
• -21
• 21

What is the value of x in the function h(x) = -2x + 12?

• 3
• 6
• -4
• 12

Which of the following is the graph of a polynomial function if the leading coefficient is positive and the degree is even?

When you substitute -3 in the polynomial function a(x) = 7x3 + 8x2 + 20, the value of x is 135.

• True
• False

Convert 34 = d in logarithmic form.

• log d 4 = 3
• log 4 3 = d
• log 3 4 = d
• log 3 d = 4

If we solve f(x) = x - 7, the value of x is 7.

• True
• False

What is the trigonometric function to be use to solve for y?

• tangent
• Pythagorean Theorem
• sine
• cosine

What is the quadrant of -1800?

• Along y - axis
• Along x - axis

The unknown value of m in m7 = 2187 is 3.

• True
• False

What is the remainder when you divide x3 - 1 to x - 1?

• None
• 2
• 1
• -1

-3000 is equal to _____ radians.

• (-5π)/3
• (-4π)/3
• π/4
• (-3π)/2

The unit circle is similar with coordinate plane.

• True
• False

The reference angle of the given figure below is:

• 1180
• 620
• 280
• 1520

The reciprocal of cosine is sine.

• True
• False

The following are the steps in solving the x of the function 32x + 7 = 27. What is the correct step?

• i, ii, iii, iv
• i, iv, iii, ii
• ii, iii, iv, i
• iii, i, ii, iv

-900 is equivalent to - π/2 radians.

• True
• False

A triangular parcel of land with points E, F, G was to be fenced. ∠E = 47° and ∠F = 78°. EG = 180m. What is the measurement of ∠G?

• 50°
• 55°
• 45°

All angles has a reference angle of 300, except

• 1500
• 3900
• 2200
• 2100

Students have done their work on chart paper and are holding the chart paper while explaining to the class how their reached their conclusions.

• More Effective Instructional Decisions

The graph of f(x) = 7x and f(x) = 7x – 3 swift to the right and going down.

• True
• False

What is the coterminal of -700 ?

• -2900
• 1600
• 700
• 2900

From the exponential function g(x) = ax, either a or x cannot be negative integer.

• True
• False

is any number which can be written in a fractional form.

• Rational Numbers

The symbol of omega in lower case and upper case are ω and Ω

• True
• False

In arranging the polynomial function b(x) = 8 + 5x3 - 7x8 + 12x5 - x in descending order, the last term is - x.

• True
• False

The Pythagorean Theorem was formulated by Pythagoras of Italy.

• True
• False

Which of the following angle is in standard position?

2π is equals to 3600.

• True
• False

- 2π and 2π can be found at two opposite direction

• True
• False

When arrange in ascending order, the last term of the polynomial function m(x) = -6x5 + x4 – 3x7 + 10 must be?

• + x4
• +10
• -3x7
• -6x5

What is the coterminal of 00?

• None
• 1800
• 900
• 7200

3 sin x + 2 sin2x cannot be added because they don’t have similar terms

• True
• False

• -5π

Solve for y

• 8
• 5√5
• √125
• 13

How many terms does the polynomial function g(x) = 7x2 + 3x3 - 12 + 24x6 - x4 has ?

• 6
• 5
• 7
• 1

The graph below shows

• That the amplitude is 4
• Graph of sin function
• Graph of cos function
• That one of the minimum point is (0, 3)

Which of the following is the cofunction of secant?

• cosecant
• cotangent
• cosine
• tangent

A rational function is the product of two polynomials.

• True
• False

If sin α= 2/7, then cos α = 7/2.

• True
• False

A clockwise direction is same as the movements of the hands of a clock.

• True
• False

In △PAN, P = 27.5°, A = 90°, what is the angle measurement of N?

• 52.5°
• 27.5°
• 72.5°
• 62.5°

The exponential form of log b d2 = e is

• be = d2
• d2 = eb
• bd2 = e
• eb = d2

In arranging the polynomial function h(x) = 2x + 3x5 - 7x3 + 22, in ascending order, the first term is 3x5.

• True
• False

All angles belongs to Quadrant III, which is not

• - 4600
• - 2000
• 9050
• 1810

Is sin2θ cos2θ = 1?

• True
• False

The following are the ordered pairs in a unit circle, which is NOT?

• (-1, 0)
• (1, -1)
• (0, 0)
• (0, -1)

The counting numbers (1, 2, 3, ...), and their counterparts (-1, -2, -3, ...), and zero are __________.

• Integers

If the length of two legs were given, then the right triangle may be solve.

• True
• False

Radian uses the symbol π .

• True
• False

Simplify: log 7 49 = x – 4.

• 10
• 2
• 6
• 7

What is the length of side y?

• 19.38
• 15.84
• 20.15
• 18.54

Simplify: log 3 92x = x + 12

• 9
• 18
• 12
• 4

The reciprocal of sine is

• cosine
• secant
• cosecant
• cotangent

Given the polynomial function f(x) = (x + 5)(x - 7)(x + 3), what is the first step to graph it.

• Get the ordered pair for each factor
• Plot the points in the coordinate plane
• Create a table of values and assign values for x.
• Equate all the factors to zero to obtain the values of x.

The expression log g k = w is read as "The logarithm of g base k is w."

• True
• False

The following angles belong to quadrant IV, except

• -10
• -950
• 6400
• 2850

Identify √2 as s irrational or rational.

• Identify √2 as s irrational or rational.
• Select one:
• Question text

What is the function to be use to solve x?

• cosine
• tangent
• secant
• sine

From the unit circle, the ordered pair (0, 1) can be found along x-axis at the left.

• True
• False

The Greek name for σ is rho.

• True
• False

The ratio of cot θ is ____.

• 2/3
• 3/√13
• 3/2
• 2√13/13

The ascending order of a polynomial function is if the exponents are arranged from smallest to highest.

• True
• False

The following degree measurements are found in Quadrant I, except

• 3700
• -150
• -2750
• 280

You may apply the Law of Sines if two sides and an angle opposite one of these sides are given or the SSA.

• True
• False

What is (-3π)/11 in degree measures?

• – 49.090
• 530
• 860
• 490

Which of the following is the symbol for eta?

• ε
• β
• θ
• H

The trigonometric function of the ratio 7/25 is

• csc δ
• sin δ
• sec δ
• cos δ

John Napier is the one who first used the term logarithm.

• True
• False

The log b 25 = 2 is equivalent to 2b = 25.

• True
• False

Given the triangle below, the legs are

• AB & BC
• BC & AC
• AC & CB
• AB & BA

The following are the fundamental identities except

• Quotient identity
• Pythagorean Identity
• Cofunction identity
• Reciprocal identity

The following are coterminal of 900, except

• -2700
• 2700
• -6300
• 4500

Which of the following function is properly arranged in descending order?

• j(x) = x2 + 2x3 - 3x4 + 4x5
• f(x) = x7 + 5x4 – 4x3 – 3x2 + 2x - 1
• h(x) = x3 + x2 - x4 + 15
• g(x) = 10x3 - 8x4 + 6x5 - 4x6 + 2x

What is the ratio of sin β?

• 2√13/13
• 3√13/13
• 3/√13
• 2/√13

The unknown value of q in 4q = 4096 is 7.

• True
• False

Degree is a unit measure for angles.

• True
• False

The ratio for cosine is

• hypotenuse/opposite
• opposite/hypotenuse

A counterclockwise direction makes a negative degree measurement.

• True
• False

Students are acting out a problem in front of the class. Others in the class participate in a discussion of the problem.

• More Effective Instructional Decisions

What must be the value of the unknown in the box to make it an exponential function?

• -32
• 24
• -28
• 32

The first three trigonometric functions are sine, cosine and tangent.

• True
• False

Which of the following is the graph of a polynomial function?

The degree of the polynomial function in one variable is the largest exponent of its variable.

• True
• False

The inverse of 2x + 5 is 2/x-5

• True
• False

In what quadrant do -1000 lies?

1800 is equivalent to p radians.

• True
• False

Is csc2 θ = 1 – cot2 θ?

• True
• False

cosine 900 is equal to 1.

• True
• False

What is the degree of the polynomial function m(x) = 5x3 + 3x7 - 2x4 + x9 - 18?

• 5
• 9
• 1
• 3

Reference angles are angles that are less than 900 and always positive.

• True
• False

The students in Mr. McCollum's class are talking to each other about math problems.

• More Effective Instructional Decisions

Students are in groups. One student in the group works out the problem while the others closely observe.

• Less Effective Instructional Decisions

The following are equal to 1, except

• cos x sec x
• tan x cot x
• sin x cos x
• sin x csc x

To verify if k(x) = x/2 - 7 and j(x) = 2x + 7 are inverses of each other, the first step is:

• Substitute j(x) to the x of k(x).
• Subtract 7 to j(x)
• Simplify the function

What is the reference angle of 1600?

• 200
• 250
• 600
• 300

Using trigonometric functions, it can find and solve the unknown angles.

• True
• False

If the polynomial functions leading coefficient is negative and the degree is odd, where do the graph starts?

• From right going up
• From left going down
• From left going up
• From right going down

What is the remainder if 2x5 - 2x4 + 12 is divided by x - 2?

• 12
• 68
• 44
• 32

Which of the following polynomial function has a degree of 3?

• f(x) = x⁄3 + 3x2
• f(x) = -3x + 3x3 - x6
• f(x) = 3x2 + 2x3 - 12
• f(x) = -x - 4x4 + 3

The cofunction of cotangent is tangent.

• True
• False

A clockwise direction is a positive measurement.

• True
• False

Mr. Swanson believes that all students should get the same instruction at the same time. To accomplish this he only uses whole group instruction.

• Less Effective Instructional Decisions

Given the polynomial function: p(x) = 2x + 7x4 – x2 + 3x3 -13 + x5, which of the following is the correct arrangement in ascending order?

• p(x) =– x2 + x5 + 2x + 3x3 + 7x4- 13
• p(x) = x5 + 7x4 + 3x3 – x2 + 2x - 13
• p(x) = -13+ 2x - x2 + 3x3 + 7x4 + x5
• p(x) = – 13 + 7x4 + 3x + 2x 3 – x2 + x5

- 3π/4 is equal to how many degrees?

• -1350
• -2250
• -3150
• -450

How many terms does the polynomial function n(z) = 4 + z5 – 5z2 – 0 + 12z7 has?

• 8
• 5
• 7
• 4

The graph of the cosine functions always passes through the origin

• True
• False

The reciprocal of tangent

• secant
• cosecant
• cosine
• cotangent

Which of the following is the inverse of d(x) = 3/x + 4?

• r(x) = 3x - 4
• p(x) = 3/x-4
• q(x) = x/3 - 4
• o(x) = -3/x - 4

Zero and positive integers are the __________.

• Whole Numbers

Identify √5 as irrational or rational.

• Question text

The following may be seen in a right triangle, EXCEPT

• leg
• square
• hypoteneus
• angle

What is the direction of the graph of the exponential function f(x) = 5-x?

• Starts from left going down to the right
• Starts from left going up to the right.
• Starts from right going up to the left
• Starts from right going down to the left

Which of the following is the sum of 3 cos x + 12 and 5cos x - 8?

• 8 cos x + 4
• 2 cos x + 4
• 8 cos x - 4
• 2 cos x - 4

Is tanθ/cotθ = 1?

• True
• False

Which of the following is the ordered pair for 3600?

• (1, 0)
• (0, 1)
• (-1, 0)
• (0, -1)

Simplify sin x sec x/tanx .

• sec x
• tan x
• 1
• sin x

Express the sum and difference of cos 300.

• √3/2
• √6/4
• √3/4
• 1/2

In Pythagorean identity, The following are equal to 1, except

• sec2 x - tan2 x
• sin2 x + cos2 x
• tan2 x + sec2 x
• csc2 x – cot2 x

The following are all exponential function except

• g(x) = x5
• g(x) = bx
• –g(x) = 2xx-1
• g(x) = -3x

Initial side of an angle is the one that rotates.

• True
• False

What is the exponential function of the table below?

• g(x) = 2x
• g(x) = -2x
• g(x) = -3x
• g(x) = -2x- 1

In 900 position of the terminal side in a unit circle, the value of cosine is

• -1
• 1
• -0
• 0

Using the triangle below, which is the hypotenuse?

• BC
• AC
• ABC
• AB

At the end of class Ms. Stark collects everyone's worksheet and grades them.

• At the end of class Ms. Stark collects everyone's worksheet and grades them.
• Less Effective Instructional Decisions

The remainder theorem is used to find the remainder of two polynomials.

• True
• False

The parts of the right triangle are

• Angles and legs
• Two legs and hypotenuse
• Three legs
• Leg and hypotenuse

What is the other identity you can derived from 1 + cot2 x = csc2 x?

• csc2 x = cot2 x = - 1
• - csc2 x = cot2 x + 1
• csc2 x = cot2 x - 1
• csc2 x – cot2 x = 1

- 4π/3 is equivalent

• -2700
• -2100
• -2400
• -1200

Find the value of z in 2/(z+5) = 12

• -26/9
• -58/12
• -29/6
• -15/4

The cofunction of cosecant is cosine.

• True
• False

Cos 1150 is equal to cos( 300 + 600 + 150)

• True
• False

the counting numbers, this set holds all of the whole numbers except zero (1, 2, 3, ....)

• Natural Numbers

The graph of an exponential functions has 3 or more turning points

• True
• False

What is the logarithmic form of dr = ba ?

• log r ba = d
• log d b= ra
• log d ba = r
• log r d= ba

– 3120 can be found at Quadrant IV.

• True
• False

A polynomial function is defined as the product of monomials.

• True
• False

The following are all coterminal of 1350, except

• -8550
• -5850
• -1350
• -2250

Simplify the function k(-3) = 3y4 + 2y5 - 21 11. Simplify the function k(-3) = 3y4 + 2y5 - 21

• 312
• 128
• -264
• -243

Oblique triangles are solved using the Pythagorean Theorem.

• True
• False

Divide 12 tan4 x + 8 tan3 x – 10 tan2 x – 6 tan x by 2 tan x.

• 6 tan3 x + 4 tan2 x – 5 tan x + 3
• 6 tan3 x - 4 tan2 x + 5 tan x – 3 tan x
• 6 tan3 x + 4 tan2 x – 5 tan x - 3
• 6 tan4 x + 4 tan3 x – 5 tan2 x – 3 tan x

What is the degree of the polynomial function m(x) = -6x5 + x4 – 3x7 + 10

• -2
• 7
• -6
• 10

The inverse of the function of f(x) = x -12 is g(x) = x + 12.

• True
• False

The maximum point is the highest point a graph reach.

• True
• False

What is the exponential form of log 12 d = k

• k12 = d
• kd = 12
• dk = 12
• 12k = d

π radian is equivalent to 1800

• True
• False

The trigonometric function to be use in order to solve x is ____.

• cosine
• tangent
• sine
• secant

The following are in standard position, except