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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

The histogram below shows the number of pairs of footware owned by a group of college students.

Which of the following statements can be inferred from the graph above?

A

The median number of pairs of footware owned is between 50 and 60 pairs.

Hint:
The same number of data points are less than the median as are greater than the median -- but on this histogram, clearly more than half the students own less than 50 pairs of shoes, so the median is less than 50.
B

The mode of the number of pairs of footware owned is 20.

Hint:
The mode is the most common number of pairs of footwear owned. We can't tell it from this histogram because each bar represents 10 different numbers-- perhaps 8 students each own each number from 10 to 19, but 40 students own exactly 6 pairs of shoes.... or perhaps not....
C

The mean number of pairs of footware owned is less than the median number of pairs of footware owned.

Hint:
This is a right skewed distribution, and so the mean is bigger than the median -- the few large values on the right pull up the mean, but have little effect on the median.
D

The median number of pairs of footware owned is between 10 and 20.

Hint:
There are approximately 230 students represented in this survey, and the 41st through 120th lowest values are between 10 and 20 -- thus the middle value is in that range.
Question 1 Explanation: 
Topics: Analyze and interpret various graphic and data representations, and use measures of central tendency (e.g., mean, median, mode) and spread to describe and interpret real-world data (Objective 0025).
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

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 3 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 4

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 4 Explanation: 
Topic:Analyze and apply geometric transformations (e.g., translations, rotations, reflections, dilations) (Objective 0024).
Question 5

How many factors does 80 have?

A
\( \large8\)
Hint:
Don't forget 1 and 80.
B
\( \large9\)
Hint:
Only perfect squares have an odd number of factors -- otherwise factors come in pairs.
C
\( \large10\)
Hint:
1,2,4,5,8,10,16,20,40,80
D
\( \large12\)
Hint:
Did you count a number twice? Include a number that isn't a factor?
Question 5 Explanation: 
Topic: Understand and apply principles of number theory (Objective 0018).
Question 6

Which of the following is not possible?

A

An equiangular triangle that is not equilateral.

Hint:
The AAA property of triangles states that all triangles with corresponding angles congruent are similar. Thus all triangles with three equal angles are similar, and are equilateral.
B

An equiangular quadrilateral that is not equilateral.

Hint:
A rectangle is equiangular (all angles the same measure), but if it's not a square, it's not equilateral (all sides the same length).
C

An equilateral quadrilateral that is not equiangular.

Hint:
This rhombus has equal sides, but it doesn't have equal angles:
D

An equiangular hexagon that is not equilateral.

Hint:
This hexagon has equal angles, but it doesn't have equal sides:
Question 6 Explanation: 
Topic: Classify and analyze polygons using attributes of sides and angles (Objective 0024).
Question 7

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 7 Explanation: 
Topic: Select the linear equation that best models a real-world situation (Objective 0022).
Question 8

What is the length of side \(\overline{BD}\) in the triangle below, where \(\angle DBA\) is a right angle?

A
\( \large 1\)
Hint:
Use the Pythagorean Theorem.
B
\( \large \sqrt{5}\)
Hint:
\(2^2+e^2=3^2\) or \(4+e^2=9;e^2=5; e=\sqrt{5}\).
C
\( \large \sqrt{13}\)
Hint:
e is not the hypotenuse.
D
\( \large 5\)
Hint:
Use the Pythagorean Theorem.
Question 8 Explanation: 
Topic: Derive and use formulas for calculating the lengths, perimeters, areas, volumes, and surface areas of geometric shapes and figures (Objective 0023), and recognize and apply connections between algebra and geometry (e.g., the use of coordinate systems, the Pythagorean theorem) (Objective 0024).
Question 9

Use the graph below to answer the question that follows.

 

Which of the following is a correct equation for the graph of the line depicted above?

 
A
\( \large y=-\dfrac{1}{2}x+2\)
Hint:
The slope is -1/2 and the y-intercept is 2. You can also try just plugging in points. For example, this is the only choice that gives y=1 when x=2.
B
\( \large 4x=2y\)
Hint:
This line goes through (0,0); the graph above does not.
C
\( \large y=x+2\)
Hint:
The line pictured has negative slope.
D
\( \large y=-x+2\)
Hint:
Try plugging x=4 into this equation and see if that point is on the graph above.
Question 9 Explanation: 
Topic: Find a linear equation that represents a graph (Objective 0022).
Question 10

The table below gives data from various years on how many young girls drank milk.

Based on the data given above, what was the probability that a randomly chosen girl in 1990 drank milk?

A
\( \large \dfrac{502}{1222}\)
Hint:
This is the probability that a randomly chosen girl who drinks milk was in the 1989-1991 food survey.
B
\( \large \dfrac{502}{2149}\)
Hint:
This is the probability that a randomly chosen girl from the whole survey drank milk and was also surveyed in 1989-1991.
C
\( \large \dfrac{502}{837}\)
D
\( \large \dfrac{1222}{2149}\)
Hint:
This is the probability that a randomly chosen girl from any year of the survey drank milk.
Question 10 Explanation: 
Topic: Recognize and apply the concept of conditional probability (Objective 0026).
Question 11

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 11 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 12

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 12 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 13

A class is using base-ten block to represent numbers.  A large cube represents 1000, a flat represents 100, a rod represents 10, and a little cube represents 1.  Which of these is not a correct representation for 2,347?

A

23 flats, 4 rods, 7 little cubes

Hint:
Be sure you read the question carefully: 2300+40+7=2347
B

2 large cubes, 3 flats, 47 rods

Hint:
2000+300+470 \( \neq\) 2347
C

2 large cubes, 34 rods, 7 little cubes

Hint:
Be sure you read the question carefully: 2000+340+7=2347
D

2 large cubes, 3 flats, 4 rods, 7 little cubes

Hint:
Be sure you read the question carefully: 2000+300+40+7=2347
Question 13 Explanation: 
Topic: Place Value (Objective 0016)
Question 14

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 14 Explanation: 
Topic: Recognize and apply the concepts of variable, function, equality, and equation to express relationships algebraically (Objective 0020).
Question 15

Which of the following inequalities describes all values of x  with \(\large  \dfrac{x}{2}\le \dfrac{x}{3}\)?

A
\( \large x < 0\)
Hint:
If x =0, then x/2 = x/3, so this answer can't be correct.
B
\( \large x \le 0\)
C
\( \large x > 0\)
Hint:
If x =0, then x/2 = x/3, so this answer can't be correct.
D
\( \large x \ge 0\)
Hint:
Try plugging in x = 6.
Question 15 Explanation: 
Topics: Inequalities, operations (Objective 0019) (not exactly sure how to classify, but this is like one of the problems on the official sample test).
Question 16

Taxicab fares in Boston (Spring 2012) are $2.60 for the first \(\dfrac{1}{7}\) of a mile or less and $0.40 for each \(\dfrac{1}{7}\) of a mile after that.

Let d represent the distance a passenger travels in miles (with \(d>\dfrac{1}{7}\)). Which of the following expressions represents the total fare?

A
\( \large \$2.60+\$0.40d\)
Hint:
It's 40 cents for 1/7 of a mile, not per mile.
B
\( \large \$2.60+\$0.40\dfrac{d}{7}\)
Hint:
According to this equation, going 7 miles would cost $3; does that make sense?
C
\( \large \$2.20+\$2.80d\)
Hint:
You can think of the fare as $2.20 to enter the cab, and then $0.40 for each 1/7 of a mile, including the first 1/7 of a mile (or $2.80 per mile).

Alternatively, you pay $2.60 for the first 1/7 of a mile, and then $2.80 per mile for d-1/7 miles. The total is 2.60+2.80(d-1/7) = 2.60+ 2.80d -.40 = 2.20+2.80d.
D
\( \large \$2.60+\$2.80d\)
Hint:
Don't count the first 1/7 of a mile twice.
Question 16 Explanation: 
Topic: Identify variables and derive algebraic expressions that represent real-world situations (Objective 0020), and select the linear equation that best models a real-world situation (Objective 0022).
Question 17

Which of the lists below is in order from least to greatest value?

A
\( \large \dfrac{1}{2},\quad \dfrac{1}{3},\quad \dfrac{1}{4},\quad \dfrac{1}{5}\)
Hint:
This is ordered from greatest to least.
B
\( \large \dfrac{1}{3},\quad \dfrac{2}{7},\quad \dfrac{3}{8},\quad \dfrac{4}{11}\)
Hint:
1/3 = 2/6 is bigger than 2/7.
C
\( \large \dfrac{1}{4},\quad \dfrac{2}{5},\quad \dfrac{2}{3},\quad \dfrac{4}{5}\)
Hint:
One way to look at this: 1/4 and 2/5 are both less than 1/2, and 2/3 and 4/5 are both greater than 1/2. 1/4 is 25% and 2/5 is 40%, so 2/5 is greater. The distance from 2/3 to 1 is 1/3 and from 4/5 to 1 is 1/5, and 1/5 is less than 1/3, so 4/5 is bigger.
D
\( \large \dfrac{7}{8},\quad \dfrac{6}{7},\quad \dfrac{5}{6},\quad \dfrac{4}{5}\)
Hint:
This is in order from greatest to least.
Question 17 Explanation: 
Topic: Ordering Fractions (Objective 0017)
Question 18

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 18 Explanation: 
Topic: Apply LCM in "real-world" situations (according to standardized tests....) (Objective 0018).
Question 19

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 19 Explanation: 
Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic) (Objective 0021)
Question 20

The histogram below shows the frequency of a class's scores on a 4 question quiz.

What was the mean score on the quiz?

A
\( \large 2.75\)
Hint:
There were 20 students who took the quiz. Total points earned: \(2 \times 1+6 \times 2+ 7\times 3+5 \times 4=55\), and 55/20 = 2.75.
B
\( \large 2\)
Hint:
How many students are there total? Did you count them all?
C
\( \large 3\)
Hint:
How many students are there total? Did you count them all? Be sure you're finding the mean, not the median or the mode.
D
\( \large 2.5\)
Hint:
How many students are there total? Did you count them all? Don't just take the mean of 1, 2, 3, 4 -- you have to weight them properly.
Question 20 Explanation: 
Topics: Analyze and interpret various graphic representations, and use measures of central tendency (e.g., mean, median, mode) and spread to describe and interpret real-world data (Objective 0025).
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