20 Random Questions

Hints will display for most wrong answers; explanations for most right answers. You can attempt a question multiple times; it will only be scored correct if you get it right the first time.

I used the official objectives and sample test to construct these questions, but cannot promise that they accurately reflect what’s on the real test. Some of the sample questions were more convoluted than I could bear to write. See terms of use. See the MTEL Practice Test main page to view questions on a particular topic or to download paper practice tests.

MTEL General Curriculum Mathematics Practice

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Question 1

A sales companies pays its representatives $2 for each item sold, plus 40% of the price of the item. The rest of the money that the representatives collect goes to the company. All transactions are in cash, and all items cost $4 or more. If the price of an item in dollars is p, which expression represents the amount of money the company collects when the item is sold?

A	$ \large \dfrac{3}{5}p-2$ Hint: The company gets 3/5=60% of the price, minus the $2 per item.
B	$ \large \dfrac{3}{5}\left( p-2 \right)$ Hint: This is sensible, but not what the problem states.
C	$ \large \dfrac{2}{5}p+2$ Hint: The company pays the extra $2; it doesn't collect it.
D	$ \large \dfrac{2}{5}p-2$ Hint: This has the company getting 2/5 = 40% of the price of each item, but that's what the representative gets.

Question 1 Explanation:

Topic: Use algebra to solve word problems involving fractions, ratios, proportions, and percents (Objective 0020).

Question 2

Which of the lines depicted below is a graph of $ \large y=2x-5$?

A	a Hint: The slope of line a is negative.
B	b Hint: Wrong slope and wrong intercept.
C	c Hint: The intercept of line c is positive.
D	d Hint: Slope is 2 -- for every increase of 1 in x, y increases by 2. Intercept is -5 -- the point (0,-5) is on the line.

Question 2 Explanation:

Topic: Find a linear equation that represents a graph (Objective 0022).

Question 3

What set of transformations will transform the leftmost image into the rightmost image?

A	A 90 degree clockwise rotation about (2,1) followed by a translation of two units to the right. Hint: Part of the figure would move below the x-axis with these transformations.
B	A translation 3 units up, followed by a reflection about the line y=x. Hint: See what happens to the point (5,1) under this set of transformations.
C	A 90 degree clockwise rotation about (5,1), followed by a translation of 2 units up.
D	A 90 degree clockwise rotation about (2,1) followed by a translation of 2 units to the right. Hint: See what happens to the point (3,3) under this set of transformations.

Question 3 Explanation:

Topic:Analyze and apply geometric transformations (e.g., translations, rotations, reflections, dilations) (Objective 0024).

Question 4

Use the expression below to answer the question that follows.

$ \large 3\times {{10}^{4}}+2.2\times {{10}^{2}}$

Which of the following is closest to the expression above?

A	Five million Hint: Pay attention to the exponents. Adding 3 and 2 doesn't work because they have different place values.
B	Fifty thousand Hint: Pay attention to the exponents. Adding 3 and 2 doesn't work because they have different place values.
C	Three million Hint: Don't add the exponents.
D	Thirty thousand Hint: $ 3\times {{10}^{4}} = 30,000;$ the other term is much smaller and doesn't change the estimate.

Question 4 Explanation:

Topics: Place value, scientific notation, estimation (Objective 0016)

Question 5

Exactly one of the numbers below is a prime number. Which one is it?

A	$ \large511 $ Hint: Divisible by 7.
B	$ \large517$ Hint: Divisible by 11.
C	$ \large519$ Hint: Divisible by 3.
D	$ \large521$

Question 5 Explanation:

Topics: Identify prime and composite numbers and demonstrate knowledge of divisibility rules (Objective 0018).

Question 6

What is the probability that two randomly selected people were born on the same day of the week? Assume that all days are equally probable.

A	$ \large \dfrac{1}{7}$ Hint: It doesn't matter what day the first person was born on. The probability that the second person will match is 1/7 (just designate one person the first and the other the second). Another way to look at it is that if you list the sample space of all possible pairs, e.g. (Wed, Sun), there are 49 such pairs, and 7 of them are repeats of the same day, and 7/49=1/7.
B	$ \large \dfrac{1}{14}$ Hint: What would be the sample space here? Ie, how would you list 14 things that you pick one from?
C	$ \large \dfrac{1}{42}$ Hint: If you wrote the seven days of the week on pieces of paper and put the papers in a jar, this would be the probability that the first person picked Sunday and the second picked Monday from the jar -- not the same situation.
D	$ \large \dfrac{1}{49}$ Hint: This is the probability that they are both born on a particular day, e.g. Sunday.

Question 6 Explanation:

Topic: Calculate the probabilities of simple and compound events and of independent and dependent events (Objective 0026).

Question 7

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 N2+205-100\div 10=N$ Hint: Use parentheses or else order of operations is off.
B	$ \large \left( \left( 2N+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 7 Explanation:

Topic: Recognize and apply the concepts of variable, function, equality, and equation to express relationships algebraically (Objective 0020).

Question 8

There are 15 students for every teacher. Let t represent the number of teachers and let s represent the number of students. Which of the following equations is correct?

A	$ \large t=s+15$ Hint: When there are 2 teachers, how many students should there be? Do those values satisfy this equation?
B	$ \large s=t+15$ Hint: When there are 2 teachers, how many students should there be? Do those values satisfy this equation?
C	$ \large t=15s$ Hint: This is a really easy mistake to make, which comes from transcribing directly from English, "1 teachers equals 15 students." To see that it's wrong, plug in s=2; do you really need 30 teachers for 2 students? To avoid this mistake, insert the word "number," "Number of teachers equals 15 times number of students" is more clearly problematic.
D	$ \large s=15t$

Question 8 Explanation:

Topic: Select the linear equation that best models a real-world situation (Objective 0022).

Question 9

The "houses" below are made of toothpicks and gum drops.

How many toothpicks are there in a row of 53 houses?

A	212 Hint: Can the number of toothpicks be even?
B	213 Hint: One way to see this is that every new "house" adds 4 toothpicks to the leftmost vertical toothpick -- so the total number is 1 plus 4 times the number of "houses." There are many other ways to look at the problem too.
C	217 Hint: Try your strategy with a smaller number of "houses" so you can count and find your mistake.
D	265 Hint: Remember that the "houses" overlap some walls.

Question 9 Explanation:

Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic). (Objective 0021).

Question 10

What is the greatest common factor of 540 and 216?

A	$ \large{{2}^{2}}\cdot {{3}^{3}}$ Hint: One way to solve this is to factor both numbers: $540=2^2 \cdot 3^3 \cdot 5$ and $216=2^3 \cdot 3^3$. Then take the smaller power for each prime that is a factor of both numbers.
B	$ \large2\cdot 3$ Hint: This is a common factor of both numbers, but it's not the greatest common factor.
C	$ \large{{2}^{3}}\cdot {{3}^{3}}$ Hint: $2^3 = 8$ is not a factor of 540.
D	$ \large{{2}^{2}}\cdot {{3}^{2}}$ Hint: This is a common factor of both numbers, but it's not the greatest common factor.

Question 10 Explanation:

Topic: Find the greatest common factor of a set of numbers (Objective 0018).

Question 11

A car is traveling at 60 miles per hour. Which of the expressions below could be used to compute how many feet the car travels in 1 second? Note that 1 mile = 5,280 feet.

A	$ \large 60\dfrac{\text{miles}}{\text{hour}}\cdot 5280\dfrac{\text{feet}}{\text{mile}}\cdot 60\dfrac{\text{minutes}}{\text{hour}}\cdot 60\dfrac{\text{seconds}}{\text{minute}} $ Hint: This answer is not in feet/second.
B	$ \large 60\dfrac{\text{miles}}{\text{hour}}\cdot 5280\dfrac{\text{feet}}{\text{mile}}\cdot \dfrac{1}{60}\dfrac{\text{hour}}{\text{minutes}}\cdot \dfrac{1}{60}\dfrac{\text{minute}}{\text{seconds}} $ Hint: This is the only choice where the answer is in feet per second and the unit conversions are correct.
C	$ \large 60\dfrac{\text{miles}}{\text{hour}}\cdot \dfrac{1}{5280}\dfrac{\text{foot}}{\text{miles}}\cdot 60\dfrac{\text{hours}}{\text{minute}}\cdot \dfrac{1}{60}\dfrac{\text{minute}}{\text{seconds}}$ Hint: Are there really 60 hours in a minute?
D	$ \large 60\dfrac{\text{miles}}{\text{hour}}\cdot \dfrac{1}{5280}\dfrac{\text{mile}}{\text{feet}}\cdot 60\dfrac{\text{minutes}}{\text{hour}}\cdot \dfrac{1}{60}\dfrac{\text{minute}}{\text{seconds}}$ Hint: This answer is not in feet/second.

Question 11 Explanation:

Topic: Use unit conversions and dimensional analysis to solve measurement problems (Objective 0023).

Question 12

The letters A, and B represent digits (possibly equal) in the ten digit number x=1,438,152,A3B. For which values of A and B is x divisible by 12, but not by 9?

A	$ \large A = 0, B = 4$ Hint: Digits add to 31, so not divisible by 3, so not divisible by 12.
B	$ \large A = 7, B = 2$ Hint: Digits add to 36, so divisible by 9.
C	$ \large A = 0, B = 6$ Hint: Digits add to 33, divisible by 3, not 9. Last digits are 36, so divisible by 4, and hence by 12.
D	$ \large A = 4, B = 8$ Hint: Digits add to 39, divisible by 3, not 9. Last digits are 38, so not divisible by 4, so not divisible by 12.

Question 12 Explanation:

Topic: Demonstrate knowledge of divisibility rules (Objective 0018).

Question 13

The chairs in a large room can be arranged in rows of 18, 25, or 60 with no chairs left over. If C is the smallest possible number of chairs in the room, which of the following inequalities does C satisfy?

A	$ \large C\le 300$ Hint: Find the LCM.
B	$ \large 300 < C \le 500 $ Hint: Find the LCM.
C	$ \large 500 < C \le 700 $ Hint: Find the LCM.
D	$ \large C>700$ Hint: The LCM is 900, which is the smallest number of chairs.

Question 13 Explanation:

Topic: Apply LCM in "real-world" situations (according to standardized tests....) (Objective 0018).

Question 14

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 14 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 15

The equation $ \large F=\frac{9}{5}C+32$ is used to convert a temperature measured in Celsius to the equivalent Farentheit temperature.

A patient's temperature increased by 1.5° Celcius. By how many degrees Fahrenheit did her temperature increase?

A	1.5° Hint: Celsius and Fahrenheit don't increase at the same rate.
B	1.8° Hint: That's how much the Fahrenheit temp increases when the Celsius temp goes up by 1 degree.
C	2.7° Hint: Each degree increase in Celsius corresponds to a $\dfrac{9}{5}=1.8$ degree increase in Fahrenheit. Thus the increase is 1.8+0.9=2.7.
D	Not enough information. Hint: A linear equation has constant slope, which means that every increase of the same amount in one variable, gives a constant increase in the other variable. It doesn't matter what temperature the patient started out at.

Question 15 Explanation:

Topic: Interpret the meaning of the slope and the intercepts of a linear equation that models a real-world situation (Objective 0022).

Question 16

In January 2011, the national debt was about 14 trillion dollars and the US population was about 300 million people. Someone reading these figures estimated that the national debt was about $5,000 per person. Which of these statements best describes the reasonableness of this estimate?

A	It is too low by a factor of 10 Hint: 14 trillion $ \approx 15 \times {{10}^{12}} $ and 300 million $ \approx 3 \times {{10}^{8}}$, so the true answer is about $ 5 \times {{10}^{4}} $ or $50,000.
B	It is too low by a factor of 100
C	It is too high by a factor of 10
D	It is too high by a factor of 100

Question 16 Explanation:

Topics: Estimation, Scientific Notation in the real world (Objective 0016).

Question 17

A family went on a long car trip. Below is a graph of how far they had driven at each hour.

Which of the following is closest to their average speed driving on the trip?

A	$ \large d=20t$ Hint: Try plugging t=7 into the equation, and see how it matches the graph.
B	$ \large d=30t$ Hint: Try plugging t=7 into the equation, and see how it matches the graph.
C	$ \large d=40t$
D	$ \large d=50t$ Hint: Try plugging t=7 into the equation, and see how it matches the graph.

Question 17 Explanation:

Topic: Select the linear equation that best models a real-world situation (Objective 0022).

Question 18

Here is a student's work on several multiplication problems:

For which of the following problems is this student most likely to get the correct solution, even though he is using an incorrect algorithm?

A	58 x 22 Hint: This problem involves regrouping, which the student does not do correctly.
B	16 x 24 Hint: This problem involves regrouping, which the student does not do correctly.
C	31 x 23 Hint: There is no regrouping with this problem.
D	141 x 32 Hint: This problem involves regrouping, which the student does not do correctly.

Question 18 Explanation:

Topic: Analyze computational algorithms (Objective 0019).

Question 19

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 19 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 20

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 20 Explanation:

Topic: Identify and analyze direct and inverse relationships in tables, graphs, algebraic expressions and real-world situations (Objective 0021)

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A	\( \large \dfrac{3}{5}p-2\) Hint: The company gets 3/5=60% of the price, minus the $2 per item.
B	\( \large \dfrac{3}{5}\left( p-2 \right)\) Hint: This is sensible, but not what the problem states.
C	\( \large \dfrac{2}{5}p+2\) Hint: The company pays the extra $2; it doesn't collect it.
D	\( \large \dfrac{2}{5}p-2\) Hint: This has the company getting 2/5 = 40% of the price of each item, but that's what the representative gets.

A	\( \large511 \) Hint: Divisible by 7.
B	\( \large517\) Hint: Divisible by 11.
C	\( \large519\) Hint: Divisible by 3.
D	\( \large521\)

A	\( \large \dfrac{1}{7}\) Hint: It doesn't matter what day the first person was born on. The probability that the second person will match is 1/7 (just designate one person the first and the other the second). Another way to look at it is that if you list the sample space of all possible pairs, e.g. (Wed, Sun), there are 49 such pairs, and 7 of them are repeats of the same day, and 7/49=1/7.
B	\( \large \dfrac{1}{14}\) Hint: What would be the sample space here? Ie, how would you list 14 things that you pick one from?
C	\( \large \dfrac{1}{42}\) Hint: If you wrote the seven days of the week on pieces of paper and put the papers in a jar, this would be the probability that the first person picked Sunday and the second picked Monday from the jar -- not the same situation.
D	\( \large \dfrac{1}{49}\) Hint: This is the probability that they are both born on a particular day, e.g. Sunday.

A	\( \large N2+205-100\div 10=N\) Hint: Use parentheses or else order of operations is off.
B	\( \large \left( \left( 2N+20 \right)5-100 \right)\div 10=N\)
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.

A	\( \large t=s+15\) Hint: When there are 2 teachers, how many students should there be? Do those values satisfy this equation?
B	\( \large s=t+15\) Hint: When there are 2 teachers, how many students should there be? Do those values satisfy this equation?
C	\( \large t=15s\) Hint: This is a really easy mistake to make, which comes from transcribing directly from English, "1 teachers equals 15 students." To see that it's wrong, plug in s=2; do you really need 30 teachers for 2 students? To avoid this mistake, insert the word "number," "Number of teachers equals 15 times number of students" is more clearly problematic.
D	\( \large s=15t\)

A	\( \large{{2}^{2}}\cdot {{3}^{3}}\) Hint: One way to solve this is to factor both numbers: \(540=2^2 \cdot 3^3 \cdot 5\) and \(216=2^3 \cdot 3^3\). Then take the smaller power for each prime that is a factor of both numbers.
B	\( \large2\cdot 3\) Hint: This is a common factor of both numbers, but it's not the greatest common factor.
C	\( \large{{2}^{3}}\cdot {{3}^{3}}\) Hint: \(2^3 = 8\) is not a factor of 540.
D	\( \large{{2}^{2}}\cdot {{3}^{2}}\) Hint: This is a common factor of both numbers, but it's not the greatest common factor.

A	\( \large 60\dfrac{\text{miles}}{\text{hour}}\cdot 5280\dfrac{\text{feet}}{\text{mile}}\cdot 60\dfrac{\text{minutes}}{\text{hour}}\cdot 60\dfrac{\text{seconds}}{\text{minute}} \) Hint: This answer is not in feet/second.
B	\( \large 60\dfrac{\text{miles}}{\text{hour}}\cdot 5280\dfrac{\text{feet}}{\text{mile}}\cdot \dfrac{1}{60}\dfrac{\text{hour}}{\text{minutes}}\cdot \dfrac{1}{60}\dfrac{\text{minute}}{\text{seconds}} \) Hint: This is the only choice where the answer is in feet per second and the unit conversions are correct.
C	\( \large 60\dfrac{\text{miles}}{\text{hour}}\cdot \dfrac{1}{5280}\dfrac{\text{foot}}{\text{miles}}\cdot 60\dfrac{\text{hours}}{\text{minute}}\cdot \dfrac{1}{60}\dfrac{\text{minute}}{\text{seconds}}\) Hint: Are there really 60 hours in a minute?
D	\( \large 60\dfrac{\text{miles}}{\text{hour}}\cdot \dfrac{1}{5280}\dfrac{\text{mile}}{\text{feet}}\cdot 60\dfrac{\text{minutes}}{\text{hour}}\cdot \dfrac{1}{60}\dfrac{\text{minute}}{\text{seconds}}\) Hint: This answer is not in feet/second.

A	\( \large A = 0, B = 4\) Hint: Digits add to 31, so not divisible by 3, so not divisible by 12.
B	\( \large A = 7, B = 2\) Hint: Digits add to 36, so divisible by 9.
C	\( \large A = 0, B = 6\) Hint: Digits add to 33, divisible by 3, not 9. Last digits are 36, so divisible by 4, and hence by 12.
D	\( \large A = 4, B = 8\) Hint: Digits add to 39, divisible by 3, not 9. Last digits are 38, so not divisible by 4, so not divisible by 12.

A	\( \large C\le 300\) Hint: Find the LCM.
B	\( \large 300 < C \le 500 \) Hint: Find the LCM.
C	\( \large 500 < C \le 700 \) Hint: Find the LCM.
D	\( \large C>700\) Hint: The LCM is 900, which is the smallest number of chairs.

A	\( \large d=20t\) Hint: Try plugging t=7 into the equation, and see how it matches the graph.
B	\( \large d=30t\) Hint: Try plugging t=7 into the equation, and see how it matches the graph.
C	\( \large d=40t\)
D	\( \large d=50t\) Hint: Try plugging t=7 into the equation, and see how it matches the graph.

MTEL General Curriculum Mathematics Practice

Which of the lines depicted below is a graph of \( \large y=2x-5\)?

a

b

c

d

What set of transformations will transform the leftmost image into the rightmost image?

A 90 degree clockwise rotation about (2,1) followed by a translation of two units to the right.

A translation 3 units up, followed by a reflection about the line y=x.

A 90 degree clockwise rotation about (5,1), followed by a translation of 2 units up.

A 90 degree clockwise rotation about (2,1) followed by a translation of 2 units to the right.

Use the expression below to answer the question that follows.

\( \large 3\times {{10}^{4}}+2.2\times {{10}^{2}}\)

Which of the following is closest to the expression above?

Five million

Fifty thousand

Three million

Thirty thousand

Exactly one of the numbers below is a prime number. Which one is it?

What is the probability that two randomly selected people were born on the same day of the week? Assume that all days are equally probable.

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?

There are 15 students for every teacher. Let t represent the number of teachers and let s represent the number of students. Which of the following equations is correct?

The "houses" below are made of toothpicks and gum drops.

How many toothpicks are there in a row of 53 houses?

212

213

217

265

What is the greatest common factor of 540 and 216?

A car is traveling at 60 miles per hour. Which of the expressions below could be used to compute how many feet the car travels in 1 second? Note that 1 mile = 5,280 feet.

The letters A, and B represent digits (possibly equal) in the ten digit number x=1,438,152,A3B. For which values of A and B is x divisible by 12, but not by 9?

The chairs in a large room can be arranged in rows of 18, 25, or 60 with no chairs left over. If C is the smallest possible number of chairs in the room, which of the following inequalities does C satisfy?

In which table below is y a function of x?

The equation \( \large F=\frac{9}{5}C+32\) is used to convert a temperature measured in Celsius to the equivalent Farentheit temperature.

A patient's temperature increased by 1.5° Celcius. By how many degrees Fahrenheit did her temperature increase?

1.5°

1.8°

2.7°

Not enough information.

In January 2011, the national debt was about 14 trillion dollars and the US population was about 300 million people. Someone reading these figures estimated that the national debt was about $5,000 per person. Which of these statements best describes the reasonableness of this estimate?

It is too low by a factor of 10

It is too low by a factor of 100

It is too high by a factor of 10

It is too high by a factor of 100

A family went on a long car trip. Below is a graph of how far they had driven at each hour.

Which of the following is closest to their average speed driving on the trip?

Here is a student's work on several multiplication problems:

For which of the following problems is this student most likely to get the correct solution, even though he is using an incorrect algorithm?

58 x 22

16 x 24

31 x 23

141 x 32

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?

Commutative Property.

Associative Property.

Identity Property.

Distributive Property.

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