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MTEL General Curriculum Mathematics Practice


Your answers are highlighted below.
Question 1

Which of the lists below contains only irrational numbers?

A
\( \large\pi , \quad \sqrt{6},\quad \sqrt{\dfrac{1}{2}}\)
B
\( \large\pi , \quad \sqrt{9}, \quad \pi +1\)
Hint:
\( \sqrt{9}=3\)
C
\( \large\dfrac{1}{3},\quad \dfrac{5}{4},\quad \dfrac{2}{9}\)
Hint:
These are all rational.
D
\( \large-3,\quad 14,\quad 0\)
Hint:
These are all rational.
Question 1 Explanation: 
Topic: Identifying rational and irrational numbers (Objective 0016).
Question 2

The table below gives the result of a survey at a college, asking students whether they were residents or commuters:

Based on the above data, what is the probability that a randomly chosen commuter student is a junior or a senior?

 
A
\( \large \dfrac{34}{43}\)
B
\( \large \dfrac{34}{71}\)
Hint:
This is the probability that a randomly chosen junior or senior is a commuter student.
C
\( \large \dfrac{34}{147}\)
Hint:
This is the probability that a randomly chosen student is a junior or senior who is a commuter.
D
\( \large \dfrac{71}{147}\)
Hint:
This is the probability that a randomly chosen student is a junior or a senior.
Question 2 Explanation: 
Topic: Recognize and apply the concept of conditional probability (Objective 0026).
Question 3

Which of the following values of x satisfies the inequality \( \large \left| {{(x+2)}^{3}} \right|<3?\)

A
\( \large x=-3\)
Hint:
\( \left| {{(-3+2)}^{3}} \right|\)=\( \left | {(-1)}^3 \right | \)=\( \left | -1 \right |=1 \) .
B
\( \large x=0\)
Hint:
\( \left| {{(0+2)}^{3}} \right|\)=\( \left | {2}^3 \right | \)=\( \left | 8 \right | \) =\( 8\)
C
\( \large x=-4\)
Hint:
\( \left| {{(-4+2)}^{3}} \right|\)=\( \left | {(-2)}^3 \right | \)=\( \left | -8 \right | \) =\( 8\)
D
\( \large x=1\)
Hint:
\( \left| {{(1+2)}^{3}} \right|\)=\( \left | {3}^3 \right | \)=\( \left | 27 \right | \) = \(27\)
Question 3 Explanation: 
Topics: Laws of exponents, order of operations, interpret absolute value (Objective 0019).
Question 4

Below is a portion of a number line:

 Point B is halfway between two tick marks.  What number is represented by Point B?

 
A
\( \large 0.645\)
Hint:
That point is marked on the line, to the right.
B
\( \large 0.6421\)
Hint:
That point is to the left of point B.
C
\( \large 0.6422\)
Hint:
That point is to the left of point B.
D
\( \large 0.6425\)
Question 4 Explanation: 
Topic: Using Number Lines (Objective 0017)
Question 5

Given that 10 cm is approximately equal to 4 inches, which of the following expressions models a way to find out approximately how many inches are equivalent to 350 cm?

A
\( \large 350\times \left( \dfrac{10}{4} \right)\)
Hint:
The final result should be smaller than 350, and this answer is bigger.
B
\( \large 350\times \left( \dfrac{4}{10} \right)\)
Hint:
Dimensional analysis can help here: \(350 \text{cm} \times \dfrac{4 \text{in}}{10 \text{cm}}\). The cm's cancel and the answer is in inches.
C
\( \large (10-4) \times 350 \)
Hint:
This answer doesn't make much sense. Try with a simpler example (e.g. 20 cm not 350 cm) to make sure that your logic makes sense.
D
\( \large (350-10) \times 4\)
Hint:
This answer doesn't make much sense. Try with a simpler example (e.g. 20 cm not 350 cm) to make sure that your logic makes sense.
Question 5 Explanation: 
Topic: Applying fractions to word problems (Objective 0017) This problem is similar to one on the official sample test for that objective, but it might fit better into unit conversion and dimensional analysis (Objective 0023: Measurement)
Question 6

The Americans with Disabilties Act (ADA) regulations state that the maximum slope for a wheelchair ramp in new construction is 1:12, although slopes between 1:16 and 1:20 are preferred.  The maximum rise for any run is 30 inches.   The graph below shows the rise and runs of four different wheelchair ramps.  Which ramp is in compliance with the ADA regulations for new construction?

A

A

Hint:
Rise is more than 30 inches.
B

B

Hint:
Run is almost 24 feet, so rise can be almost 2 feet.
C

C

Hint:
Run is 12 feet, so rise can be at most 1 foot.
D

D

Hint:
Slope is 1:10 -- too steep.
Question 6 Explanation: 
Topic: Interpret meaning of slope in a real world situation (Objective 0022).
Question 7

An above-ground swimming pool is in the shape of a regular hexagonal prism, is one meter high, and holds 65 cubic meters of water.  A second pool has a base that is also a regular hexagon, but with sides twice as long as the sides in the first pool.  This second pool is also one meter high.  How much water will the second pool hold?

A
\( \large 65\text{ }{{\text{m}}^{3}}\)
Hint:
A bigger pool would hold more water.
B
\( \large 65\cdot 2\text{ }{{\text{m}}^{3}}\)
Hint:
Try a simpler example, say doubling the sides of the base of a 1 x 1 x 1 cube.
C
\( \large 65\cdot 4\text{ }{{\text{m}}^{3}}\)
Hint:
If we think of the pool as filled with 1 x 1 x 1 cubes (and some fractions of cubes), then scaling to the larger pool changes each 1 x 1 x 1 cube to a 2 x 2 x 1 prism, or multiplies volume by 4.
D
\( \large 65\cdot 8\text{ }{{\text{m}}^{3}}\)
Hint:
Try a simpler example, say doubling the sides of the base of a 1 x 1 x 1 cube.
Question 7 Explanation: 
Topic: Determine how the characteristics (e.g., area, volume) of geometric figures and shapes are affected by changes in their dimensions (Objective 0023).
Question 8

A homeowner is planning to tile the kitchen floor with tiles that measure 6 inches by 8 inches.  The kitchen floor is a rectangle that measures 10 ft by 12 ft, and there are no gaps between the tiles.  How many tiles does the homeowner need?

A

30

Hint:
The floor is 120 sq feet, and the tiles are smaller than 1 sq foot. Also, remember that 1 sq foot is 12 \(\times\) 12=144 sq inches.
B

120

Hint:
The floor is 120 sq feet, and the tiles are smaller than 1 sq foot.
C

300

Hint:
Recheck your calculations.
D

360

Hint:
One way to do this is to note that 6 inches = 1/2 foot and 8 inches = 2/3 foot, so the area of each tile is 1/2 \(\times\) 2/3=1/3 sq foot, or each square foot of floor requires 3 tiles. The area of the floor is 120 square feet. Note that the tiles would fit evenly oriented in either direction, parallel to the walls.
Question 8 Explanation: 
Topic: Estimate and calculate measurements, use unit conversions to solve measurement problems, solve measurement problems in real-world situations (Objective 0023).
Question 9

The following story situations model \( 12\div 3\):

I)  Jack has 12 cookies, which he wants to share equally between himself and two friends.  How many cookies does each person get?

II) Trent has 12 cookies, which he wants to put into bags of 3 cookies each.  How many bags can he make?

III) Cicely has $12.  Cookies cost $3 each.  How many cookies can she buy?

Which of these questions illustrate the same model of division, either partitive (partioning) or measurement (quotative)?

A

I and II

B

I and III

C

II and III

Hint:
Problem I is partitive (or partitioning or sharing) -- we put 12 objects into 3 groups. Problems II and III are quotative (or measurement) -- we put 12 objects in groups of 3.
D

All three problems model the same meaning of division

Question 9 Explanation: 
Topic: Understand models of operations on numbers (Objective 0019).
Question 10

The letters A, B, and C represent digits (possibly equal) in the twelve digit number x=111,111,111,ABC.  For which values of A, B, and C is x divisible by 40?

A
\( \large A = 3, B = 2, C=0\)
Hint:
Note that it doesn't matter what the first 9 digits are, since 1000 is divisible by 40, so DEF,GHI,JKL,000 is divisible by 40 - we need to check the last 3.
B
\( \large A = 0, B = 0, C=4\)
Hint:
Not divisible by 10, since it doesn't end in 0.
C
\( \large A = 4, B = 2, C=0\)
Hint:
Divisible by 10 and by 4, but not by 40, as it's not divisible by 8. Look at 40 as the product of powers of primes -- 8 x 5, and check each. To check 8, either check whether 420 is divisible by 8, or take ones place + twice tens place + 4 * hundreds place = 18, which is not divisible by 8.
D
\( \large A =1, B=0, C=0\)
Hint:
Divisible by 10 and by 4, but not by 40, as it's not divisible by 8. Look at 40 as the product of powers of primes -- 8 x 5, and check each. To check 8, either check whether 100 is divisible by 8, or take ones place + twice tens place + 4 * hundreds place = 4, which is not divisible by 8.
Question 10 Explanation: 
Topic: Understand divisibility rules and why they work (Objective 018).
Question 11

Which of the following is equivalent to

\( \large A-B+C\div D\times E\)?

A
\( \large A-B-\dfrac{C}{DE} \)
Hint:
In the order of operations, multiplication and division have the same priority, so do them left to right; same with addition and subtraction.
B
\( \large A-B+\dfrac{CE}{D}\)
Hint:
In practice, you're better off using parentheses than writing an expression like the one in the question. The PEMDAS acronym that many people memorize is misleading. Multiplication and division have equal priority and are done left to right. They have higher priority than addition and subtraction. Addition and subtraction also have equal priority and are done left to right.
C
\( \large \dfrac{AE-BE+CE}{D}\)
Hint:
Use order of operations, don't just compute left to right.
D
\( \large A-B+\dfrac{C}{DE}\)
Hint:
In the order of operations, multiplication and division have the same priority, so do them left to right
Question 11 Explanation: 
Topic: Justify algebraic manipulations by application of the properties of order of operations (Objective 0020).
Question 12

The function d(x) gives the result when 12 is divided by x.  Which of the following is a graph of d(x)?

 
A
Hint:
d(x) is 12 divided by x, not x divided by 12.
B
Hint:
When x=2, what should d(x) be?
C
Hint:
When x=2, what should d(x) be?
D
Question 12 Explanation: 
Topic: Identify and analyze direct and inverse relationships in tables, graphs, algebraic expressions and real-world situations (Objective 0021)
Question 13

Which of the following points is closest to \( \dfrac{34}{135} \times \dfrac{53}{86}\)?

A

A

Hint:
\(\frac{34}{135} \approx \frac{1}{4}\) and \( \frac{53}{86} \approx \frac {2}{3}\). \(\frac {1}{4}\) of \(\frac {2}{3}\) is small and closest to A.
B

B

Hint:
Estimate with simpler fractions.
C

C

Hint:
Estimate with simpler fractions.
D

D

Hint:
Estimate with simpler fractions.
Question 13 Explanation: 
Topic: Understand meaning and models of operations on fractions (Objective 0019).
Question 14

The Venn Diagram below gives data on the number of seniors, athletes, and vegetarians in the student body at a college:

How many students at the college are seniors who are not vegetarians?

A
\( \large 137\)
Hint:
Doesn't include the senior athletes who are not vegetarians.
B
\( \large 167\)
C
\( \large 197\)
Hint:
That's all seniors, including vegetarians.
D
\( \large 279\)
Hint:
Includes all athletes who are not vegetarians, some of whom are not seniors.
Question 14 Explanation: 
Topic: Venn Diagrams (Objective 0025)
Question 15

Here is a mental math strategy for computing 26 x 16:

Step 1: 100 x 16 = 1600

Step 2: 25 x 16 = 1600 ÷· 4 = 400

Step 3: 26 x 16 = 400 + 16 = 416

Which property best justifies Step 3 in this strategy?

A

Commutative Property.

Hint:
For addition, the commutative property is \(a+b=b+a\) and for multiplication it's \( a \times b = b \times a\).
B

Associative Property.

Hint:
For addition, the associative property is \((a+b)+c=a+(b+c)\) and for multiplication it's \((a \times b) \times c=a \times (b \times c)\)
C

Identity Property.

Hint:
0 is the additive identity, because \( a+0=a\) and 1 is the multiplicative identity because \(a \times 1=a\). The phrase "identity property" is not standard.
D

Distributive Property.

Hint:
\( (25+1) \times 16 = 25 \times 16 + 1 \times 16 \). This is an example of the distributive property of multiplication over addition.
Question 15 Explanation: 
Topic: Analyze and justify mental math techniques, by applying arithmetic properties such as commutative, distributive, and associative (Objective 0019). Note that it's hard to write a question like this as a multiple choice question -- worthwhile to understand why the other steps work too.
Question 16

Which of the following is the equation of a linear function?

A
\( \large y={{x}^{2}}+2x+7\)
Hint:
This is a quadratic function.
B
\( \large y={{2}^{x}}\)
Hint:
This is an exponential function.
C
\( \large y=\dfrac{15}{x}\)
Hint:
This is an inverse function.
D
\( \large y=x+(x+4)\)
Hint:
This is a linear function, y=2x+4, it's graph is a straight line with slope 2 and y-intercept 4.
Question 16 Explanation: 
Topic: Distinguish between linear and nonlinear functions (Objective 0022).
Question 17

A family on vacation drove the first 200 miles in 4 hours and the second 200 miles in 5 hours.  Which expression below gives their average speed for the entire trip?

A
\( \large \dfrac{200+200}{4+5}\)
Hint:
Average speed is total distance divided by total time.
B
\( \large \left( \dfrac{200}{4}+\dfrac{200}{5} \right)\div 2\)
Hint:
This seems logical, but the problem is that it weights the first 4 hours and the second 5 hours equally, when each hour should get the same weight in computing the average speed.
C
\( \large \dfrac{200}{4}+\dfrac{200}{5} \)
Hint:
This would be an average of 90 miles per hour!
D
\( \large \dfrac{400}{4}+\dfrac{400}{5} \)
Hint:
This would be an average of 180 miles per hour! Even a family of race car drivers probably doesn't have that average speed on a vacation!
Question 17 Explanation: 
Topic: Solve a variety of measurement problems (e.g., time, temperature, rates, average rates of change) in real-world situations (Objective 0023).
Question 18

In each expression below  N represents a negative integer. Which expression could have a negative value?

A
\( \large {{N}^{2}}\)
Hint:
Squaring always gives a non-negative value.
B
\( \large 6-N\)
Hint:
A story problem for this expression is, if it was 6 degrees out at noon and N degrees out at sunrise, by how many degrees did the temperature rise by noon? Since N is negative, the answer to this question has to be positive, and more than 6.
C
\( \large -N\)
Hint:
If N is negative, then -N is positive
D
\( \large 6+N\)
Hint:
For example, if \(N=-10\), then \(6+N = -4\)
Question 18 Explanation: 
If you are stuck on a question like this, try a few examples to eliminate some choices and to help you understand what the question means. Topic: Characteristics of integers (Objective 0016).
Question 19

In which table below is y a function of x?

A
Hint:
If x=3, y can have two different values, so it's not a function.
B
Hint:
If x=3, y can have two different values, so it's not a function.
C
Hint:
If x=1, y can have different values, so it's not a function.
D
Hint:
Each value of x always corresponds to the same value of y.
Question 19 Explanation: 
Topic: Understand the definition of function and various representations of functions (e.g., input/output machines, tables, graphs, mapping diagrams, formulas) (Objective 0021).
Question 20

Here is a number trick:

 1) Pick a whole number

 2) Double your number.

 3) Add 20 to the above result.

 4) Multiply the above by 5

 5) Subtract 100

 6) Divide by 10

The result is always the number that you started with! Suppose you start by picking N. Which of the equations below best demonstrates that the result after Step 6 is also N?

A
\( \large N*2+20*5-100\div 10=N\)
Hint:
Use parentheses or else order of operations is off.
B
\( \large \left( \left( 2*N+20 \right)*5-100 \right)\div 10=N\)
C
\( \large \left( N+N+20 \right)*5-100\div 10=N\)
Hint:
With this answer you would subtract 10, instead of subtracting 100 and then dividing by 10.
D
\( \large \left( \left( \left( N\div 10 \right)-100 \right)*5+20 \right)*2=N\)
Hint:
This answer is quite backwards.
Question 20 Explanation: 
Topic: Recognize and apply the concepts of variable, function, equality, and equation to express relationships algebraically (Objective 0020).
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