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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


Your answers are highlighted below.
Question 1

How many lines of reflective symmetry and how many centers of rotational symmetry does the parallelogram depicted below have?

 
A

4 lines of reflective symmetry, 1 center of rotational symmetry.

Hint:
Try cutting out a shape like this one from paper, and fold where you think the lines of reflective symmetry are (or put a mirror there). Do things line up as you thought they would?
B

2 lines of reflective symmetry, 1 center of rotational symmetry.

Hint:
Try cutting out a shape like this one from paper, and fold where you think the lines of reflective symmetry are (or put a mirror there). Do things line up as you thought they would?
C

0 lines of reflective symmetry, 1 center of rotational symmetry.

Hint:
The intersection of the diagonals is a center of rotational symmetry. There are no lines of reflective symmetry, although many people get confused about this fact (best to play with hands on examples to get a feel). Just fyi, the letter S also has rotational, but not reflective symmetry, and it's one that kids often write backwards.
D

2 lines of reflective symmetry, 0 centers of rotational symmetry.

Hint:
Try cutting out a shape like this one from paper. Trace onto another sheet of paper. See if there's a way to rotate the cut out shape (less than a complete turn) so that it fits within the outlines again.
Question 1 Explanation: 
Topic: Analyze geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to concepts of symmetry (Objective 0024).
Question 2

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 2 Explanation: 
Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic) (Objective 0021)
Question 3

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 3 Explanation: 
Topic: Understand meaning and models of operations on fractions (Objective 0019).
Question 4

Which of the following is equal to one million three hundred thousand?

A
\(\large1.3\times {{10}^{6}}\)
B
\(\large1.3\times {{10}^{9}}\)
Hint:
That's one billion three hundred million.
C
\(\large1.03\times {{10}^{6}}\)
Hint:
That's one million thirty thousand.
D
\(\large1.03\times {{10}^{9}}\)
Hint:
That's one billion thirty million
Question 4 Explanation: 
Topic: Scientific Notation (Objective 0016)
Question 5

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 5 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 6

M is a multiple of 26.  Which of the following cannot be true?

A

M is odd.

Hint:
All multiples of 26 are also multiples of 2, so they must be even.
B

M is a multiple of 3.

Hint:
3 x 26 is a multiple of both 3 and 26.
C

M is 26.

Hint:
1 x 26 is a multiple of 26.
D

M is 0.

Hint:
0 x 26 is a multiple of 26.
Question 6 Explanation: 
Topic: Characteristics of composite numbers (Objective 0018).
Question 7

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 7 Explanation: 
Topic: Place Value (Objective 0016)
Question 8

The first histogram shows the average life expectancies for women in different countries in Africa in 1998; the second histogram gives similar data for Europe:

  

How much bigger is the range of the data for Africa than the range of the data for Europe?

A

0 years

Hint:
Range is the maximum life expectancy minus the minimum life expectancy.
B

12 years

Hint:
Are you subtracting frequencies? Range is about values of the data, not frequency.
C

18 years

Hint:
It's a little hard to read the graph, but it doesn't matter if you're consistent. It looks like the range for Africa is 80-38= 42 years and for Europe is 88-64 = 24; 42-24=18.
D

42 years

Hint:
Read the question more carefully.
Question 8 Explanation: 
Topic: Compare different data sets (Objective 0025).
Question 9

A map has a scale of 3 inches = 100 miles.  Cities A and B are 753 miles apart.  Let d be the distance between the two cities on the map.  Which of the following is not correct?

A
\( \large \dfrac{3}{100}=\dfrac{d}{753}\)
Hint:
Units on both side are inches/mile, and both numerators and denominators correspond -- this one is correct.
B
\( \large \dfrac{3}{100}=\dfrac{753}{d}\)
Hint:
Unit on the left is inches per mile, and on the right is miles per inch. The proportion is set up incorrectly (which is what we wanted). Another strategy is to notice that one of A or B has to be the answer because they cannot both be correct proportions. Then check that cross multiplying on A gives part D, so B is the one that is different from the other 3.
C
\( \large \dfrac{3}{d}=\dfrac{100}{753}\)
Hint:
Unitless on each side, as inches cancel on the left and miles on the right. Numerators correspond to the map, and denominators to the real life distances -- this one is correct.
D
\( \large 100d=3\cdot 753\)
Hint:
This is equivalent to part A.
Question 9 Explanation: 
Topic: Analyze the relationships among proportions, constant rates, and linear functions (Objective 0022).
Question 10

The expression \( \large {{7}^{-4}}\cdot {{8}^{-6}}\) is equal to which of the following?

A
\( \large \dfrac{8}{{{\left( 56 \right)}^{4}}}\)
Hint:
The bases are whole numbers, and the exponents are negative. How can the numerator be 8?
B
\( \large \dfrac{64}{{{\left( 56 \right)}^{4}}}\)
Hint:
The bases are whole numbers, and the exponents are negative. How can the numerator be 64?
C
\( \large \dfrac{1}{8\cdot {{\left( 56 \right)}^{4}}}\)
Hint:
\(8^{-6}=8^{-4} \times 8^{-2}\)
D
\( \large \dfrac{1}{64\cdot {{\left( 56 \right)}^{4}}}\)
Question 10 Explanation: 
Topics: Laws of exponents (Objective 0019).
Question 11

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 11 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).
Question 12

Here is a student's work solving an equation:

\( x-4=-2x+6\)

\( x-4+4=-2x+6+4\)

\( x=-2x+10\)

\( x-2x=10\)

\( x=10\)

Which of the following statements is true?

A

The student‘s solution is correct.

Hint:
Try plugging into the original solution.
B

The student did not correctly use properties of equality.

Hint:
After \( x=-2x+10\), the student subtracted 2x on the left and added 2x on the right.
C

The student did not correctly use the distributive property.

Hint:
Distributive property is \(a(b+c)=ab+ac\).
D

The student did not correctly use the commutative property.

Hint:
Commutative property is \(a+b=b+a\) or \(ab=ba\).
Question 12 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 13

A teacher has a list of all the countries in the world and their populations in March 2012.  She is going to have her students use technology to compute the mean and median of the numbers on the list.   Which of the following statements is true?

A

The teacher can be sure that the mean and median will be the same without doing any computation.

Hint:
Does this make sense? How likely is it that the mean and median of any large data set will be the same?
B

The teacher can be sure that the mean is bigger than the median without doing any computation.

Hint:
This is a skewed distribution, and very large countries like China and India contribute huge numbers to the mean, but are counted the same as small countries like Luxembourg in the median (the same thing happens w/data on salaries, where a few very high income people tilt the mean -- that's why such data is usually reported as medians).
C

The teacher can be sure that the median is bigger than the mean without doing any computation.

Hint:
Think about a set of numbers like 1, 2, 3, 4, 10,000 -- how do the mean/median compare? How might that relate to countries of the world?
D

There is no way for the teacher to know the relative size of the mean and median without computing them.

Hint:
Knowing the shape of the distribution of populations does give us enough info to know the relative size of the mean and median, even without computing them.
Question 13 Explanation: 
Topic: Use measures of central tendency (e.g., mean, median, mode) and spread to describe and interpret real-world data (Objective 0025).
Question 14

Use the expression below to answer the question that follows.

                 \(\large \dfrac{\left( 155 \right)\times \left( 6,124 \right)}{977}\)

Which of the following is the best estimate of the expression above?

A

100

Hint:
6124/977 is approximately 6.
B

200

Hint:
6124/977 is approximately 6.
C

1,000

Hint:
6124/977 is approximately 6. 155 is approximately 150, and \( 6 \times 150 = 3 \times 300 = 900\), so this answer is closest.
D

2,000

Hint:
6124/977 is approximately 6.
Question 14 Explanation: 
Topics: Estimation, simplifying fractions (Objective 0016).
Question 15

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 15 Explanation: 
Topic: Classify and analyze polygons using attributes of sides and angles (Objective 0024).
Question 16

Use the graph below to answer the question that follows:

The graph above represents the equation \( \large 3x+Ay=B\), where A and B are integers.  What are the values of A and B?

A
\( \large A = -2, B= 6\)
Hint:
Plug in (2,0) to get B=6, then plug in (0,-3) to get A=-2.
B
\( \large A = 2, B = 6\)
Hint:
Try plugging (0,-3) into this equation.
C
\( \large A = -1.5, B=-3\)
Hint:
The problem said that A and B were integers and -1.5 is not an integer. Don't try to use slope-intercept form.
D
\( \large A = 2, B = -3\)
Hint:
Try plugging (2,0) into this equation.
Question 16 Explanation: 
Topic: Find a linear equation that represents a graph (Objective 0022).
Question 17

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 17 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 18

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 18 Explanation: 
Topic: Understand divisibility rules and why they work (Objective 018).
Question 19

A publisher prints a series of books with covers made of identical material and using the same thickness of paper for each page.  The covers of the book together are 0.4 cm thick, and 125 pieces of the paper used together are 1 cm thick.

The publisher uses a linear function to determine the total thickness, T(n) of a book made with n sheets of paper.   What are the slope and intercept of T(n)?

A

Intercept = 0.4 cm, Slope = 125 cm/page

Hint:
This would mean that each page of the book was 125 cm thick.
B

Intercept =0.4 cm, Slope = \(\dfrac{1}{125}\)cm/page

Hint:
The intercept is how thick the book would be with no pages in it. The slope is how much 1 extra page adds to the thickness of the book.
C

Intercept = 125 cm, Slope = 0.4 cm

Hint:
This would mean that with no pages in the book, it would be 125 cm thick.
D

Intercept = \(\dfrac{1}{125}\)cm, Slope = 0.4 pages/cm

Hint:
This would mean that each new page of the book made it 0.4 cm thicker.
Question 19 Explanation: 
Topic: Interpret the meaning of the slope and the intercepts of a linear equation that models a real-world situation (Objective 0022).
Question 20

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 20 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)
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