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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

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 1 Explanation: 
Topic: Understand meaning and models of operations on fractions (Objective 0019).
Question 2

Here are some statements:

I) 5 is an integer    II)\( -5 \)  is an integer    III) \(0\) is an integer

Which of the statements are true?

A

I only

B

I and II only

C

I and III only

D

I, II, and III

Hint:
The integers are ...-3, -2, -1, 0, 1, 2, 3, ....
Question 2 Explanation: 
Topic: Characteristics of Integers (Objective 0016)
Question 3

If two fair coins are flipped, what is the probability that one will come up heads and the other tails?

A
\( \large \dfrac{1}{4}\)
Hint:
Think of the coins as a penny and a dime, and list all possibilities.
B
\( \large \dfrac{1}{3} \)
Hint:
This is a very common misconception. There are three possible outcomes -- both heads, both tails, and one of each -- but they are not equally likely. Think of the coins as a penny and a dime, and list all possibilities.
C
\( \large \dfrac{1}{2}\)
Hint:
The possibilities are HH, HT, TH, TT, and all are equally likely. Two of the four have one of each coin, so the probability is 2/4=1/2.
D
\( \large \dfrac{3}{4}\)
Hint:
Think of the coins as a penny and a dime, and list all possibilities.
Question 3 Explanation: 
Topic: Calculate the probabilities of simple and compound events and of independent and dependent events (Objective 0026).
Question 4

Which of the following is closest to the height of a college student in centimeters?

A

1.6 cm

Hint:
This is more the height of a Lego toy college student -- less than an inch!
B

16 cm

Hint:
Less than knee high on most college students.
C

160 cm

Hint:
Remember, a meter stick (a little bigger than a yard stick) is 100 cm. Also good to know is that 4 inches is approximately 10 cm.
D

1600 cm

Hint:
This college student might be taller than some campus buildings!
Question 4 Explanation: 
Topic: Estimate and calculate measurements using customary, metric, and nonstandard units of measurement (Objective 0023).
Question 5

The pattern below consists of a row of black squares surrounded by white squares.

 How many white squares would surround a row of 157 black squares?

A

314

Hint:
Try your procedure on a smaller number that you can count to see where you made a mistake.
B

317

Hint:
Are there ever an odd number of white squares?
C

320

Hint:
One way to see this is that there are 6 tiles on the left and right ends, and the rest of the white tiles are twice the number of black tiles (there are many other ways to look at it too).
D

322

Hint:
Try your procedure on a smaller number that you can count to see where you made a mistake.
Question 5 Explanation: 
Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic) (Objective 0021).
Question 6

Aya and Kendra want to estimate the height of a tree. On a sunny day, Aya measures Kendra's shadow as 3 meters long, and Kendra measures the tree's shadow as 15 meters long. Kendra is 1.5 meters tall. How tall is the tree?

A

7.5 meters

Hint:
Here is a picture, note that the large and small right triangles are similar:

One way to do the problem is to note that there is a dilation (scale) factor of 5 on the shadows, so there must be that factor on the heights too. Another way is to note that the shadows are twice as long as the heights.
B

22.5 meters

Hint:
Draw a picture.
C

30 meters

Hint:
Draw a picture.
D

45 meters

Hint:
Draw a picture.
Question 6 Explanation: 
Topic: Apply geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to similarity, ; and use these concepts to solve problems (Objective 0024) . Fits in other places too.
Question 7

Use the table below to answer the question that follows:

Each number in the table above represents a value W that is determined by the values of x and y.  For example, when x=3 and y=1, W=5.  What is the value of W when x=9 and y=14?  Assume that the patterns in the table continue as shown.

A
\( \large W=-5\)
Hint:
When y is even, W is even.
B
\( \large W=4\)
Hint:
Note that when x increases by 1, W increases by 2, and when y increases by 1, W decreases by 1. At x=y=0, W=0, so at x=9, y=14, W has increased by \(9 \times 2\) and decreased by 14, or W=18-14=4.
C
\( \large W=6\)
Hint:
Try fixing x or y at 0, and start by finding W for x=0 y=14 or x=9, y=0.
D
\( \large W=32\)
Hint:
Try fixing x or y at 0, and start by finding W for x=0 y=14 or x=9, y=0.
Question 7 Explanation: 
Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic) (Objective 0021)
Question 8

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 8 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 9

Below is a portion of a number line.

Point A is one-quarter of the distance from 0.26 to 0.28.  What number is represented by point A?

A
\( \large0.26\)
Hint:
Please reread the question.
B
\( \large0.2625\)
Hint:
This is one-quarter of the distance between 0.26 and 0.27, which is not what the question asked.
C
\( \large0.265\)
D
\( \large0.27\)
Hint:
Please read the question more carefully. This answer would be correct if Point A were halfway between the tick marks, but it's not.
Question 9 Explanation: 
Topic: Using number lines (Objective 0017)
Question 10

Which of the following is an irrational number?

A
\( \large \sqrt[3]{8}\)
Hint:
This answer is the cube root of 8. Since 2 x 2 x 2 =8, this is equal to 2, which is rational because 2 = 2/1.
B
\( \large \sqrt{8}\)
Hint:
It is not trivial to prove that this is irrational, but you can get this answer by eliminating the other choices.
C
\( \large \dfrac{1}{8}\)
Hint:
1/8 is the RATIO of two integers, so it is rational.
D
\( \large -8\)
Hint:
Negative integers are also rational, -8 = -8/1, a ratio of integers.
Question 10 Explanation: 
Topic: Identifying rational and irrational numbers (Objective 0016).
Question 11

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 11 Explanation: 
Topic: Apply LCM in "real-world" situations (according to standardized tests....) (Objective 0018).
Question 12

Below are four inputs and outputs for a function machine representing the function A:

Which of the following equations could also represent A  for the values shown?

A
\( \large A(n)=n+4\)
Hint:
For a question like this, you don't have to find the equation yourself, you can just try plugging the function machine inputs into the equation, and see if any values come out wrong. With this equation n= -1 would output 3, not 0 as the machine does.
B
\( \large A(n)=n+2\)
Hint:
For a question like this, you don't have to find the equation yourself, you can just try plugging the function machine inputs into the equation, and see if any values come out wrong. With this equation n= 2 would output 4, not 6 as the machine does.
C
\( \large A(n)=2n+2\)
Hint:
Simply plug in each of the four function machine input values, and see that the equation produces the correct output, e.g. A(2)=6, A(-1)=0, etc.
D
\( \large A(n)=2\left( n+2 \right)\)
Hint:
For a question like this, you don't have to find the equation yourself, you can just try plugging the function machine inputs into the equation, and see if any values come out wrong. With this equation n= 2 would output 8, not 6 as the machine does.
Question 12 Explanation: 
Topics: Understand various representations of functions, and translate among different representations of functional relationships (Objective 0021).
Question 13

At a school fundraising event, people can buy a ticket to spin a spinner like the one below.  The region that the spinner lands in tells which, if any, prize the person wins.

If 240 people buy tickets to spin the spinner, what is the best estimate of the number of keychains that will be given away?

A

40

Hint:
"Keychain" appears on the spinner twice.
B

80

Hint:
The probability of getting a keychain is 1/3, and so about 1/3 of the time the spinner will win.
C

100

Hint:
What is the probability of winning a keychain?
D

120

Hint:
That would be the answer for getting any prize, not a keychain specifically.
Question 13 Explanation: 
Topic: I would call this topic expected value, which is not listed on the objectives. This question is very similar to one on the sample test. It's not a good question in that it's oversimplified (a more difficult and interesting question would be something like, "The school bought 100 keychains for prizes, what is the probability that they will run out before 240 people play?"). In any case, I believe the objective this is meant for is, "Recognize the difference between experimentally and theoretically determined probabilities in real-world situations. (Objective 0026)." This is not something easily assessed with multiple choice .
Question 14

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 14 Explanation: 
Topics: Identify prime and composite numbers and demonstrate knowledge of divisibility rules (Objective 0018).
Question 15

Use the solution procedure below to answer the question that follows:

\( \large {\left( x+3 \right)}^{2}=10\)

\( \large \left( x+3 \right)\left( x+3 \right)=10\)

\( \large {x}^{2}+9=10\)

\( \large {x}^{2}+9-9=10-9\)

\( \large {x}^{2}=1\)

\( \large x=1\text{ or }x=-1\)

Which of the following is incorrect in the procedure shown above?

A

The commutative property is used incorrectly.

Hint:
The commutative property is \(a+b=b+a\) or \(ab=ba\).
B

The associative property is used incorrectly.

Hint:
The associative property is \(a+(b+c)=(a+b)+c\) or \(a \times (b \times c)=(a \times b) \times c\).
C

Order of operations is done incorrectly.

D

The distributive property is used incorrectly.

Hint:
\((x+3)(x+3)=x(x+3)+3(x+3)\)=\(x^2+3x+3x+9.\)
Question 15 Explanation: 
Topic: Justify algebraic manipulations by application of the properties of equality, the order of operations, the number properties, and the order properties (Objective 0020).
Question 16

A biology class requires a lab fee, which is a whole number of dollars, and the same amount for all students. On Monday the instructor collected $70 in fees, on Tuesday she collected $126, and on Wednesday she collected $266. What is the largest possible amount the fee could be?

A

$2

Hint:
A possible fee, but not the largest possible fee. Check the other choices to see which are factors of all three numbers.
B

$7

Hint:
A possible fee, but not the largest possible fee. Check the other choices to see which are factors of all three numbers.
C

$14

Hint:
This is the greatest common factor of 70, 126, and 266.
D

$70

Hint:
Not a factor of 126 or 266, so couldn't be correct.
Question 16 Explanation: 
Topic: Use GCF in real-world context (Objective 0018)
Question 17

Which of the following is equal to eleven billion four hundred thousand?

A
\( \large 11,400,000\)
Hint:
That's eleven million four hundred thousand.
B
\(\large11,000,400,000\)
C
\( \large11,000,000,400,000\)
Hint:
That's eleven trillion four hundred thousand (although with British conventions; this answer is correct, but in the US, it isn't).
D
\( \large 11,400,000,000\)
Hint:
That's eleven billion four hundred million
Question 17 Explanation: 
Topic: Place Value (Objective 0016)
Question 18

A cylindrical soup can has diameter 7 cm and height 11 cm. The can holds g grams of soup.   How many grams of the same soup could a cylindrical can with diameter 14 cm and height 33 cm hold?

A
\( \large 6g\)
Hint:
You must scale in all three dimensions.
B
\( \large 12g\)
Hint:
Height is multiplied by 3, and diameter and radius are multiplied by 2. Since the radius is squared, final result is multiplied by \(2^2\times 3=12\).
C
\( \large 18g\)
Hint:
Don't square the height scale factor.
D
\( \large 36g\)
Hint:
Don't square the height scale factor.
Question 18 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 19

What fraction of the area of the picture below is shaded?

A
\( \large \dfrac{17}{24}\)
Hint:
You might try adding segments so each quadrant is divided into 6 pieces with equal area -- there will be 24 regions, not all the same shape, but all the same area, with 17 of them shaded (for the top left quarter, you could also first change the diagonal line to a horizontal or vertical line that divides the square in two equal pieces and shade one) .
B
\( \large \dfrac{3}{4}\)
Hint:
Be sure you're taking into account the different sizes of the pieces.
C
\( \large \dfrac{2}{3}\)
Hint:
The bottom half of the picture is 2/3 shaded, and the top half is more than 2/3 shaded, so this answer is too small.
D
\( \large \dfrac{17}{6} \)
Hint:
This answer is bigger than 1, so doesn't make any sense. Be sure you are using the whole picture, not one quadrant, as the unit.
Question 19 Explanation: 
Topic: Models of Fractions (Objective 0017)
Question 20

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