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

 Question 1

#### P divides 30

Hint:
2, 3, and 5 are the prime factors of 240, and all divide 30.

#### P divides 48

Hint:
P=5 doesn't work.

#### P divides 75

Hint:
P=2 doesn't work.

#### P divides 80

Hint:
P=3 doesn't work.
Question 1 Explanation:
Topic: Find the prime factorization of a number and recognize its uses (Objective 0018).
 Question 2

#### 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 2 Explanation:
Topic: Use unit conversions and dimensional analysis to solve measurement problems (Objective 0023).
 Question 3

#### 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 3 Explanation:
Topic: Apply LCM in "real-world" situations (according to standardized tests....) (Objective 0018).
 Question 4

#### 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.

#### 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.

#### 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

#### The speed of sound in dry air at 68 degrees F is 343.2 meters per second.  Which of the expressions below could be used to compute the number of kilometers that a sound wave travels in 10 minutes (in dry air at 68 degrees F)?

 A $$\large 343.2\times 60\times 10$$Hint: In kilometers, not meters. B $$\large 343.2\times 60\times 10\times \dfrac{1}{1000}$$Hint: Units are meters/sec $$\times$$ seconds/minute $$\times$$ minutes $$\times$$ kilometers/meter, and the answer is in kilometers. C $$\large 343.2\times \dfrac{1}{60}\times 10$$Hint: Include units and make sure answer is in kilometers. D $$\large 343.2\times \dfrac{1}{60}\times 10\times \dfrac{1}{1000}$$Hint: Include units and make sure answer is in kilometers.
Question 5 Explanation:
Topic: Use unit conversions and dimensional analysis to solve measurement problems (Objective 0023).
 Question 6

#### 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 6 Explanation:
Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic) (Objective 0021)
 Question 7

#### What is the least common multiple of 540 and 216?

 A $$\large{{2}^{5}}\cdot {{3}^{6}}\cdot 5$$Hint: This is the product of the numbers, not the LCM. B $$\large{{2}^{3}}\cdot {{3}^{3}}\cdot 5$$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 for each prime that's a factor of either number, use the largest exponent that appears in one of the factorizations. You can also take the product of the two numbers divided by their GCD. C $$\large{{2}^{2}}\cdot {{3}^{3}}\cdot 5$$Hint: 216 is a multiple of 8. D $$\large{{2}^{2}}\cdot {{3}^{2}}\cdot {{5}^{2}}$$Hint: Not a multiple of 216 and not a multiple of 540.
Question 7 Explanation:
Topic: Find the least common multiple of a set of numbers (Objective 0018).
 Question 8

#### 2,000

Hint:
The answer is bigger than 7,000.

#### 20,000

Hint:
Estimate 896/216 first.

#### 3,000

Hint:
The answer is bigger than 7,000.

#### 30,000

Hint:
$$\dfrac{896}{216} \approx 4$$ and $$7154 \times 4$$ is over 28,000, so this answer is closest.
Question 8 Explanation:
Topics: Estimation, simplifying fractions (Objective 0016, overlaps with other objectives).
 Question 9

#### Which of the following does not represent the number of gumdrops in a row of h houses?

 A $$\large 2+3h$$Hint: Think of this as start with 2 gumdrops on the left wall, and then add 3 gumdrops for each house. B $$\large 5+3(h-1)$$Hint: Think of this as start with one house, and then add 3 gumdrops for each of the other h-1 houses. C $$\large h+(h+1)+(h+1)$$Hint: Look at the gumdrops in 3 rows: h gumdrops for the "rooftops," h+1 for the tops of the vertical walls, and h+1 for the floors. D $$\large 5+3h$$Hint: This one is not a correct equation (which makes it the correct answer!). Compare to choice A. One of them has to be wrong, as they differ by 3.
Question 9 Explanation:
Topic: Translate among different representations (e.g., tables, graphs, algebraic expressions, verbal descriptions) of functional relationships (Objective 0021).
 Question 10

#### 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.

#### 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).

#### An equilateral quadrilateral that is not equiangular.

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

#### An equiangular hexagon that is not equilateral.

Hint:
This hexagon has equal angles, but it doesn't have equal sides:
Question 10 Explanation:
Topic: Classify and analyze polygons using attributes of sides and angles (Objective 0024).
 Question 11

#### Point B is halfway between two tick marks.  What number is represented by Point B?

 A $$\large 0.645$$Hint: That point is marked on the line, to the right. B $$\large 0.6421$$Hint: That point is to the left of point B. C $$\large 0.6422$$Hint: That point is to the left of point B. D $$\large 0.6425$$
Question 11 Explanation:
Topic: Using Number Lines (Objective 0017)
 Question 12

#### 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.

#### 120

Hint:
The floor is 120 sq feet, and the tiles are smaller than 1 sq foot.

Hint:

#### 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 12 Explanation:
Topic: Estimate and calculate measurements, use unit conversions to solve measurement problems, solve measurement problems in real-world situations (Objective 0023).
 Question 13

#### The quotient is $$3\dfrac{1}{2}$$. There are 3 whole blocks each representing $$\dfrac{2}{3}$$ and a partial block composed of 3 small rectangles. The 3 small rectangles represent $$\dfrac{3}{6}$$ of a whole, or $$\dfrac{1}{2}$$.

Hint:
We are counting how many 2/3's are in
2 1/2: the unit becomes 2/3, not 1.

#### The quotient is $$\dfrac{4}{15}$$. There are four whole blocks separated into a total of 15 small rectangles.

Hint:
This explanation doesn't make much sense. Probably you are doing "invert and multiply," but inverting the wrong thing.

#### This picture cannot be used to find the quotient because it does not show how to separate $$2\dfrac{1}{2}$$ into equal sized groups.

Hint:
Study the measurement/quotative model of division. It's often very useful with fractions.
Question 13 Explanation:
Topic: Recognize and analyze pictorial representations of number operations. (Objective 0019).
 Question 14

#### 0 years

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

#### 12 years

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

#### 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.

#### 42 years

Hint:
Question 14 Explanation:
Topic: Compare different data sets (Objective 0025).
 Question 15

#### 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 15 Explanation:
Topic: Find the greatest common factor of a set of numbers (Objective 0018).
 Question 16

#### a

Hint:
The slope of line a is negative.

#### b

Hint:
Wrong slope and wrong intercept.

#### c

Hint:
The intercept of line c is positive.

#### 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 16 Explanation:
Topic: Find a linear equation that represents a graph (Objective 0022).
 Question 17

#### 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 17 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 18

#### An above-ground swimming pool is in the shape of a regular hexagonal prism, is one meter high, and holds 65 cubic meters of water.  A second pool has a base that is also a regular hexagon, but with sides twice as long as the sides in the first pool.  This second pool is also one meter high.  How much water will the second pool hold?

 A $$\large 65\text{ }{{\text{m}}^{3}}$$Hint: A bigger pool would hold more water. B $$\large 65\cdot 2\text{ }{{\text{m}}^{3}}$$Hint: Try a simpler example, say doubling the sides of the base of a 1 x 1 x 1 cube. C $$\large 65\cdot 4\text{ }{{\text{m}}^{3}}$$Hint: If we think of the pool as filled with 1 x 1 x 1 cubes (and some fractions of cubes), then scaling to the larger pool changes each 1 x 1 x 1 cube to a 2 x 2 x 1 prism, or multiplies volume by 4. D $$\large 65\cdot 8\text{ }{{\text{m}}^{3}}$$Hint: Try a simpler example, say doubling the sides of the base of a 1 x 1 x 1 cube.
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

#### This student divides fractions by first finding a common denominator, then dividing the numerators.

$$\large \dfrac{2}{3} \div \dfrac{3}{4} \longrightarrow \dfrac{8}{12} \div \dfrac{9}{12} \longrightarrow 8 \div 9 = \dfrac {8}{9}$$ $$\large \dfrac{2}{5} \div \dfrac{7}{20} \longrightarrow \dfrac{8}{20} \div \dfrac{7}{20} \longrightarrow 8 \div 7 = \dfrac {8}{7}$$ $$\large \dfrac{7}{6} \div \dfrac{3}{4} \longrightarrow \dfrac{14}{12} \div \dfrac{9}{12} \longrightarrow 14 \div 9 = \dfrac {14}{9}$$

#### It is not valid. Common denominators are for adding and subtracting fractions, not for dividing them.

Hint:
Don't be so rigid! Usually there's more than one way to do something in math.

#### It got the right answer in these three cases, but it isn‘t valid for all rational numbers.

Hint:
Did you try some other examples? What makes you say it's not valid?

#### It is valid if the rational numbers in the division problem are in lowest terms and the divisor is not zero.

Hint:
Lowest terms doesn't affect this problem at all.

#### It is valid for all rational numbers, as long as the divisor is not zero.

Hint:
When we have common denominators, the problem is in the form a/b divided by c/b, and the answer is a/c, as the student's algorithm predicts.
Question 19 Explanation:
Topic: Analyze Non-Standard Computational Algorithms (Objective 0019).
 Question 20

#### 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 20 Explanation:
Topic: Recognize and apply the concept of conditional probability (Objective 0026).
 Question 21

#### 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 21 Explanation:
Topic: Understand and apply principles of number theory (Objective 0018).
 Question 22

#### 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 22 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 23

#### 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 23 Explanation:
Topic: Using number lines (Objective 0017)
 Question 24

#### 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 24 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 25

#### Which of the graphs below represent functions?

I. II. III. IV.

#### I and IV only.

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

#### 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).

#### II and III only.

Hint:
Learn about the vertical line test.

#### I, II, and IV only.

Hint:
There are vertical lines that go through 2 points in II.
Question 25 Explanation:
Understand the definition of function and various representations of functions (e.g., input/output machines, tables, graphs, mapping diagrams, formulas). (Objective 0021).
 Question 26

#### Commutative Property.

Hint:
For addition, the commutative property is $$a+b=b+a$$ and for multiplication it's $$a \times b = b \times a$$.

#### Associative Property.

Hint:
For addition, the associative property is $$(a+b)+c=a+(b+c)$$ and for multiplication it's $$(a \times b) \times c=a \times (b \times c)$$

#### 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.

#### 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 26 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 27

#### 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? ####$2

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

#### $7 Hint: A possible fee, but not the largest possible fee. Check the other choices to see which are factors of all three numbers. ####$14

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

#### 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.

#### It is too high by a factor of 100

Question 42 Explanation:
Topics: Estimation, Scientific Notation in the real world (Objective 0016).
 Question 43

#### 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 43 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 44

#### 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 44 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 45

#### 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?

#### 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?

#### 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.

#### 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 45 Explanation:
Topic: Analyze geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to concepts of symmetry (Objective 0024).
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