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## MTEL General Curriculum Mathematics Practice

 Question 1

#### 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 1 Explanation:
Understand the definition of function and various representations of functions (e.g., input/output machines, tables, graphs, mapping diagrams, formulas). (Objective 0021).
 Question 2

#### 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 2 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 3

#### M is odd.

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

#### M is a multiple of 3.

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

#### M is 26.

Hint:
1 x 26 is a multiple of 26.

#### M is 0.

Hint:
0 x 26 is a multiple of 26.
Question 3 Explanation:
Topic: Characteristics of composite numbers (Objective 0018).
 Question 4

#### A family has four children.  What is the probability that two children are girls and two are boys?  Assume the the probability of having a boy (or a girl) is 50%.

 A $$\large \dfrac{1}{2}$$Hint: How many different configurations are there from oldest to youngest, e.g. BGGG? How many of them have 2 boys and 2 girls? B $$\large \dfrac{1}{4}$$Hint: How many different configurations are there from oldest to youngest, e.g. BGGG? How many of them have 2 boys and 2 girls? C $$\large \dfrac{1}{5}$$Hint: Some configurations are more probable than others -- i.e. it's more likely to have two boys and two girls than all boys. Be sure you are weighting properly. D $$\large \dfrac{3}{8}$$Hint: There are two possibilities for each child, so there are $$2 \times 2 \times 2 \times 2 =16$$ different configurations, e.g. from oldest to youngest BBBG, BGGB, GBBB, etc. Of these configurations, there are 6 with two boys and two girls (this is the combination $$_{4}C_{2}$$ or "4 choose 2"): BBGG, BGBG, BGGB, GGBB, GBGB, and GBBG. Thus the probability is 6/16=3/8.
Question 4 Explanation:
Topic: Apply knowledge of combinations and permutations to the computation of probabilities (Objective 0026).
 Question 5

#### A

Hint:
Rise is more than 30 inches.

#### B

Hint:
Run is almost 24 feet, so rise can be almost 2 feet.

#### C

Hint:
Run is 12 feet, so rise can be at most 1 foot.

#### D

Hint:
Slope is 1:10 -- too steep.
Question 5 Explanation:
Topic: Interpret meaning of slope in a real world situation (Objective 0022).
 Question 6

#### If the polygon shown above is reflected about the y axis and then rotated 90 degrees clockwise about the origin, which of the following graphs is the result?

 A Hint: Try following the point (1,4) to see where it goes after each transformation. B C Hint: Make sure you're reflecting in the correct axis. D Hint: Make sure you're rotating the correct direction.
Question 6 Explanation:
Topic: Analyze and apply geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to concepts of symmetry, similarity, and congruence; and use these concepts to solve problems (Objective 0024).
 Question 7

#### 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

Hint:
Estimate with simpler fractions.

#### C

Hint:
Estimate with simpler fractions.

#### D

Hint:
Estimate with simpler fractions.
Question 7 Explanation:
Topic: Understand meaning and models of operations on fractions (Objective 0019).
 Question 8

#### The polygon depicted below is drawn on dot paper, with the dots spaced 1 unit apart.  What is the perimeter of the polygon?

 A $$\large 18+\sqrt{2} \text{ units}$$Hint: Be careful with the Pythagorean Theorem. B $$\large 18+2\sqrt{2}\text{ units}$$Hint: There are 13 horizontal or vertical 1 unit segments. The longer diagonal is the hypotenuse of a 3-4-5 right triangle, so its length is 5 units. The shorter diagonal is the hypotenuse of a 45-45-90 right triangle with side 2, so its hypotenuse has length $$2 \sqrt{2}$$. C $$\large 18 \text{ units}$$Hint: Use the Pythagorean Theorem to find the lengths of the diagonal segments. D $$\large 20 \text{ units}$$Hint: Use the Pythagorean Theorem to find the lengths of the diagonal segments.
Question 8 Explanation:
Topic: Recognize and apply connections between algebra and geometry (e.g., the use of coordinate systems, the Pythagorean theorem) (Objective 0024).
 Question 9

#### 1.6 cm

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

#### 16 cm

Hint:
Less than knee high on most college students.

#### 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.

#### 1600 cm

Hint:
This college student might be taller than some campus buildings!
Question 9 Explanation:
Topic: Estimate and calculate measurements using customary, metric, and nonstandard units of measurement (Objective 0023).
 Question 10

#### The function d(x) gives the result when 12 is divided by x.  Which of the following is a graph of d(x)?

 A Hint: d(x) is 12 divided by x, not x divided by 12. B Hint: When x=2, what should d(x) be? C Hint: When x=2, what should d(x) be? D
Question 10 Explanation:
Topic: Identify and analyze direct and inverse relationships in tables, graphs, algebraic expressions and real-world situations (Objective 0021)
 Question 11

#### 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 11 Explanation:
Topic: Understand divisibility rules and why they work (Objective 018).
 Question 12

#### 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 12 Explanation:
Topics: Laws of exponents (Objective 0019).
 Question 13

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

Question 13 Explanation:
Topics: Estimation, Scientific Notation in the real world (Objective 0016).
 Question 14

#### 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 14 Explanation:
Topic: Apply LCM in "real-world" situations (according to standardized tests....) (Objective 0018).
 Question 15

#### The prime factorization of  n can be written as n=pqr, where p, q, and r are distinct prime numbers.  How many factors does n have, including 1 and itself?

 A $$\large3$$Hint: 1, p, q, r, and pqr are already 5, so this isn't enough. You might try plugging in p=2, q=3, and r=5 to help with this problem. B $$\large5$$Hint: Don't forget pq, etc. You might try plugging in p=2, q=3, and r=5 to help with this problem. C $$\large6$$Hint: You might try plugging in p=2, q=3, and r=5 to help with this problem. D $$\large8$$Hint: 1, p, q, r, pq, pr, qr, pqr.
Question 15 Explanation:
Topic: Recognize uses of prime factorization of a number (Objective 0018).
 Question 16

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

#### Which of the numbers below is not equivalent to 4%?

 A $$\large \dfrac{1}{25}$$Hint: 1/25=4/100, so this is equal to 4% (be sure you read the question correctly). B $$\large \dfrac{4}{100}$$Hint: 4/100=4% (be sure you read the question correctly). C $$\large 0.4$$Hint: 0.4=40% so this is not equal to 4% D $$\large 0.04$$Hint: 0.04=4/100, so this is equal to 4% (be sure you read the question correctly).
Question 18 Explanation:
Converting between fractions, decimals, and percents (Objective 0017).
 Question 19

#### 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 19 Explanation:
Topic: Place Value (Objective 0016)
 Question 20

#### Store B

Hint:
This saves 15% and C saves 25%.

#### Store D

Hint:
This is about 20% off, which is less of a discount than C.
Question 20 Explanation:
Topic: Understand the meanings and models of integers, fractions, decimals,percents, and mixed numbers and apply them to the solution of word problems (Objective 0017).
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