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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

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 1 Explanation: 
Topic: Characteristics of composite numbers (Objective 0018).
Question 2

The column below consists of two cubes and a cylinder.  The cylinder has diameter y, which is also the length of the sides of each cube.   The total height of the column is 5y.  Which of the formulas below gives the volume of the column?

 
A
\( \large 2{{y}^{3}}+\dfrac{3\pi {{y}^{3}}}{4}\)
Hint:
The cubes each have volume \(y^3\). The cylinder has radius \(\dfrac{y}{2}\) and height \(3y\). The volume of a cylinder is \(\pi r^2 h=\pi ({\dfrac{y}{2}})^2(3y)=\dfrac{3\pi {{y}^{3}}}{4}\). Note that the volume of a cylinder is analogous to that of a prism -- area of the base times height.
B
\( \large 2{{y}^{3}}+3\pi {{y}^{3}}\)
Hint:
y is the diameter of the circle, not the radius.
C
\( \large {{y}^{3}}+5\pi {{y}^{3}}\)
Hint:
Don't forget to count both cubes.
D
\( \large 2{{y}^{3}}+\dfrac{3\pi {{y}^{3}}}{8}\)
Hint:
Make sure you know how to find the volume of a cylinder.
Question 2 Explanation: 
Topic: Derive and use formulas for calculating the lengths, perimeters, areas, volumes, and surface areas of geometric shapes and figures (Objective 0023).
Question 3

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 3 Explanation: 
Topic: Estimate and calculate measurements, use unit conversions to solve measurement problems, solve measurement problems in real-world situations (Objective 0023).
Question 4

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 4 Explanation: 
Topic: Identifying rational and irrational numbers (Objective 0016).
Question 5

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 5 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 6

The picture below shows identical circles drawn on a piece of paper.  The rectangle represents an index card that is blocking your view of \( \dfrac{3}{5}\) of the circles on the paper.  How many circles are covered by the rectangle?

A

4

Hint:
The card blocks more than half of the circles, so this number is too small.
B

5

Hint:
The card blocks more than half of the circles, so this number is too small.
C

8

Hint:
The card blocks more than half of the circles, so this number is too small.
D

12

Hint:
2/5 of the circles or 8 circles are showing. Thus 4 circles represent 1/5 of the circles, and \(4 \times 5=20\) circles represent 5/5 or all the circles. Thus 12 circles are hidden.
Question 6 Explanation: 
Topic: Models of Fractions (Objective 0017)
Question 7

The chart below gives percentiles for the number of sit-ups that boys of various ages can do in 60 seconds (source , June 24, 2011)

 

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

A

95% of 12 year old boys can do 56 sit-ups in 60 seconds.

Hint:
The 95th percentile means that 95% of scores are less than or equal to 56, and 5% are greater than or equal to 56.
B

At most 25% of 7 year old boys can do 19 or more sit-ups in 60 seconds.

Hint:
The 25th percentile means that 25% of scores are less than or equal to 19, and 75% are greater than or equal to 19.
C

Half of all 13 year old boys can do less than 41 sit-ups in 60 seconds and half can do more than 41 sit-ups in 60 seconds.

Hint:
Close, but not quite. There's no accounting for boys who can do exactly 41 sit ups. Look at these data: 10, 20, 41, 41, 41, 41, 50, 60, 90. The median is 41, but more than half can do 41 or more.
D

At least 75% of 16 year old boys can only do 51 or fewer sit-ups in 60 seconds.

Hint:
The "at least" is necessary due to duplicates. Suppose the data were 10, 20, 51, 51. The 75th percentile is 51, but 100% of the boys can only do 51 or fewer situps.
Question 7 Explanation: 
Topic: Analyze and interpret various graphic and nongraphic data representations (e.g., frequency distributions, percentiles) (Objective 0025).
Question 8

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 8 Explanation: 
Topic: Use GCF in real-world context (Objective 0018)
Question 9

Each individual cube that makes up the rectangular solid depicted below has 6 inch sides.  What is the surface area of the solid in square feet?

 
A
\( \large 11\text{ f}{{\text{t}}^{2}}\)
Hint:
Check your units and make sure you're using feet and inches consistently.
B
\( \large 16.5\text{ f}{{\text{t}}^{2}}\)
Hint:
Each square has surface area \(\dfrac{1}{2} \times \dfrac {1}{2}=\dfrac {1}{4}\) sq feet. There are 9 squares on the top and bottom, and 12 on each of 4 sides, for a total of 66 squares. 66 squares \(\times \dfrac {1}{4}\) sq feet/square =16.5 sq feet.
C
\( \large 66\text{ f}{{\text{t}}^{2}}\)
Hint:
The area of each square is not 1.
D
\( \large 2376\text{ f}{{\text{t}}^{2}}\)
Hint:
Read the question more carefully -- the answer is supposed to be in sq feet, not sq inches.
Question 9 Explanation: 
Topics: Use unit conversions to solve measurement problems, and derive and use formulas for calculating surface areas of geometric shapes and figures (Objective 0023).
Question 10

Use the samples of a student's work below to answer the question that follows:

\( \large \dfrac{2}{3}\times \dfrac{3}{4}=\dfrac{4\times 2}{3\times 3}=\dfrac{8}{9}\) \( \large \dfrac{2}{5}\times \dfrac{7}{7}=\dfrac{7\times 2}{5\times 7}=\dfrac{2}{5}\) \( \large \dfrac{7}{6}\times \dfrac{3}{4}=\dfrac{4\times 7}{6\times 3}=\dfrac{28}{18}=\dfrac{14}{9}\)

Which of the following best describes the mathematical validity of the algorithm the student is using?

A

It is not valid. It never produces the correct answer.

Hint:
In the middle example,the answer is correct.
B

It is not valid. It produces the correct answer in a few special cases, but it‘s still not a valid algorithm.

Hint:
Note that this algorithm gives a/b divided by c/d, not a/b x c/d, but some students confuse multiplication and cross-multiplication. If a=0 or if c/d =1, division and multiplication give the same answer.
C

It is valid if the rational numbers in the multiplication problem are in lowest terms.

Hint:
Lowest terms is irrelevant.
D

It is valid for all rational numbers.

Hint:
Can't be correct as the first and last examples have the wrong answers.
Question 10 Explanation: 
Topic: Analyze Non-Standard Computational Algorithms (Objective 0019).
Question 11

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 11 Explanation: 
Topics: Laws of exponents (Objective 0019).
Question 12

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

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

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 16 Explanation: 
Topic: Recognize and apply the concept of conditional probability (Objective 0026).
Question 17

Use the expression below to answer the question that follows.

                 \( \large \dfrac{\left( 4\times {{10}^{3}} \right)\times \left( 3\times {{10}^{4}} \right)}{6\times {{10}^{6}}}\)

Which of the following is equivalent to the expression above?

A

2

Hint:
\(10^3 \times 10^4=10^7\), and note that if you're guessing when the answers are so closely related, you're generally better off guessing one of the middle numbers.
B

20

Hint:
\( \dfrac{\left( 4\times {{10}^{3}} \right)\times \left( 3\times {{10}^{4}} \right)}{6\times {{10}^{6}}}=\dfrac {12 \times {{10}^{7}}}{6\times {{10}^{6}}}=\)\(2 \times {{10}^{1}}=20 \)
C

200

Hint:
\(10^3 \times 10^4=10^7\)
D

2000

Hint:
\(10^3 \times 10^4=10^7\), and note that if you're guessing when the answers are so closely related, you're generally better off guessing one of the middle numbers.
Question 17 Explanation: 
Topics: Scientific notation, exponents, simplifying fractions (Objective 0016, although overlaps with other objectives too).
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

Which of the graphs below represent functions?

I. II. III. IV.   
A

I and IV only.

Hint:
There are vertical lines that go through 2 points in IV .
B

I and III only.

Hint:
Even though III is not continuous, it's still a function (assuming that vertical lines between the "steps" do not go through 2 points).
C

II and III only.

Hint:
Learn about the vertical line test.
D

I, II, and IV only.

Hint:
There are vertical lines that go through 2 points in II.
Question 19 Explanation: 
Understand the definition of function and various representations of functions (e.g., input/output machines, tables, graphs, mapping diagrams, formulas). (Objective 0021).
Question 20

There are six gumballs in a bag — two red and four green.  Six children take turns picking a gumball out of the bag without looking.   They do not return any gumballs to the bag.  What is the probability that the first two children to pick from the bag pick the red gumballs?

A
\( \large \dfrac{1}{3}\)
Hint:
This is the probability that the first child picks a red gumball, but not that the first two children pick red gumballs.
B
\( \large \dfrac{1}{8}\)
Hint:
Are you adding things that you should be multiplying?
C
\( \large \dfrac{1}{9}\)
Hint:
This would be the probability if the gumballs were returned to the bag.
D
\( \large \dfrac{1}{15}\)
Hint:
The probability that the first child picks red is 2/6 = 1/3. Then there are 5 gumballs in the bag, one red, so the probability that the second child picks red is 1/5. Thus 1/5 of the time, after the first child picks red, the second does too, so the probability is 1/5 x 1/3 = 1/15.
Question 20 Explanation: 
Topic: Calculate the probabilities of simple and compound events and of independent and dependent events (Objective 0026).
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