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## MTEL General Curriculum Mathematics Practice

 Question 1

#### Which of the following is equivalent to $$\dfrac{3}{4}-\dfrac{1}{8}+\dfrac{2}{8}\times \dfrac{1}{2}?$$

 A $$\large \dfrac{7}{16}$$Hint: Multiplication comes before addition and subtraction in the order of operations. B $$\large \dfrac{1}{2}$$Hint: Addition and subtraction are of equal priority in the order of operations -- do them left to right. C $$\large \dfrac{3}{4}$$Hint: $$\dfrac{3}{4}-\dfrac{1}{8}+\dfrac{2}{8}\times \dfrac{1}{2}$$=$$\dfrac{3}{4}-\dfrac{1}{8}+\dfrac{1}{8}$$=$$\dfrac{3}{4}+-\dfrac{1}{8}+\dfrac{1}{8}$$=$$\dfrac{3}{4}$$ D $$\large \dfrac{3}{16}$$Hint: Multiplication comes before addition and subtraction in the order of operations.
Question 1 Explanation:
Topic: Operations on Fractions, Order of Operations (Objective 0019).
 Question 2

#### 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 2 Explanation:
Topic: Identify and analyze direct and inverse relationships in tables, graphs, algebraic expressions and real-world situations (Objective 0021)
 Question 3

#### 4 congruent sides

Hint:
The most common definition of a rhombus is a quadrilateral with 4 congruent sides.

#### A center of rotational symmetry

Hint:
The diagonal of a rhombus separates it into two congruent isosceles triangles. The center of this line is a center of 180 degree rotational symmetry that switches the triangles.

#### 4 congruent angles

Hint:
Unless the rhombus is a square, it does not have 4 congruent angles.

#### 2 sets of parallel sides

Hint:
All rhombi are parallelograms.
Question 3 Explanation:
Topic: Classify and analyze polygons using attributes of sides and angles, and symmetry (Objective 0024).
 Question 4

#### II and III

Hint:
Problem I is partitive (or partitioning or sharing) -- we put 12 objects into 3 groups. Problems II and III are quotative (or measurement) -- we put 12 objects in groups of 3.

#### All three problems model the same meaning of division

Question 4 Explanation:
Topic: Understand models of operations on numbers (Objective 0019).
 Question 5

#### 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 5 Explanation:
Topic: Translate among different representations (e.g., tables, graphs, algebraic expressions, verbal descriptions) of functional relationships (Objective 0021).
 Question 6

 A $$\large 3.10m=400+0.2m$$Hint: These are the numbers in the problem, but this equation doesn't make sense. If you don't know how to make an equation, try plugging in an easy number like m=500 minutes to see if each side equals what it should. B $$\large 3+0.1m=29.99+.20m$$Hint: Doesn't account for the 400 free minutes. C $$\large 3+0.1m=400+29.99+.20(m-400)$$Hint: Why would you add 400 minutes and $29.99? If you don't know how to make an equation, try plugging in an easy number like m=500 minutes to see if each side equals what it should. D $$\large 3+0.1m=29.99+.20(m-400)$$Hint: The left side is$3 plus $0.10 times the number of minutes. The right is$29.99 plus $0.20 times the number of minutes over 400. Question 6 Explanation: Identify variables and derive algebraic expressions that represent real-world situations (Objective 0020).  Question 7 #### 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? #### 40 Hint: "Keychain" appears on the spinner twice. #### 80 Hint: The probability of getting a keychain is 1/3, and so about 1/3 of the time the spinner will win. #### 100 Hint: What is the probability of winning a keychain? #### 120 Hint: That would be the answer for getting any prize, not a keychain specifically. Question 7 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 8 #### 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? #### I only #### I and II only #### I and III only #### I, II, and III Hint: The integers are ...-3, -2, -1, 0, 1, 2, 3, .... Question 8 Explanation: Topic: Characteristics of Integers (Objective 0016)  Question 9 #### Which of the following is an irrational number?  A $$\large \sqrt[3]{8}$$Hint: This answer is the cube root of 8. Since 2 x 2 x 2 =8, this is equal to 2, which is rational because 2 = 2/1. B $$\large \sqrt{8}$$Hint: It is not trivial to prove that this is irrational, but you can get this answer by eliminating the other choices. C $$\large \dfrac{1}{8}$$Hint: 1/8 is the RATIO of two integers, so it is rational. D $$\large -8$$Hint: Negative integers are also rational, -8 = -8/1, a ratio of integers. Question 9 Explanation: Topic: Identifying rational and irrational numbers (Objective 0016).  Question 10 #### 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 10 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 11 #### 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 11 Explanation: Topic: Understand and apply principles of number theory (Objective 0018).  Question 12 #### Kendra is trying to decide which fraction is greater, $$\dfrac{4}{7}$$ or $$\dfrac{5}{8}$$. Which of the following answers shows the best reasoning? #### $$\dfrac{4}{7}$$ is $$\dfrac{3}{7}$$away from 1, and $$\dfrac{5}{8}$$ is $$\dfrac{3}{8}$$away from 1. Since eighth‘s are smaller than seventh‘s, $$\dfrac{5}{8}$$ is closer to 1, and is the greater of the two fractions. #### $$7-4=3$$ and $$8-5=3$$, so the fractions are equal. Hint: Not how to compare fractions. By this logic, 1/2 and 3/4 are equal, but 1/2 and 2/4 are not. #### $$4\times 8=32$$ and $$7\times 5=35$$. Since $$32<35$$ , $$\dfrac{5}{8}<\dfrac{4}{7}$$ Hint: Starts out as something that works, but the conclusion is wrong. 4/7 = 32/56 and 5/8 = 35/56. The cross multiplication gives the numerators, and 35/56 is bigger. #### $$4<5$$ and $$7<8$$, so $$\dfrac{4}{7}<\dfrac{5}{8}$$ Hint: Conclusion is correct, logic is wrong. With this reasoning, 1/2 would be less than 2/100,000. Question 12 Explanation: Topics: Comparing fractions, and understanding the meaning of fractions (Objective 0017).  Question 13 #### Which of the following is not possible? #### 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 13 Explanation: Topic: Classify and analyze polygons using attributes of sides and angles (Objective 0024).  Question 14 #### Use the samples of a student€™s work below to answer the question that follows: #### This student divides fractions by first finding a common denominator, then dividing the numerators. $$\large \dfrac{2}{3} \div \dfrac{3}{4} \longrightarrow \dfrac{8}{12} \div \dfrac{9}{12} \longrightarrow 8 \div 9 = \dfrac {8}{9}$$ $$\large \dfrac{2}{5} \div \dfrac{7}{20} \longrightarrow \dfrac{8}{20} \div \dfrac{7}{20} \longrightarrow 8 \div 7 = \dfrac {8}{7}$$ $$\large \dfrac{7}{6} \div \dfrac{3}{4} \longrightarrow \dfrac{14}{12} \div \dfrac{9}{12} \longrightarrow 14 \div 9 = \dfrac {14}{9}$$ #### Which of the following best describes the mathematical validity of the algorithm the student is using? #### 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 14 Explanation: Topic: Analyze Non-Standard Computational Algorithms (Objective 0019).  Question 15 #### 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? #### 100 Hint: 6124/977 is approximately 6. #### 200 Hint: 6124/977 is approximately 6. #### 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. #### 2,000 Hint: 6124/977 is approximately 6. Question 15 Explanation: Topics: Estimation, simplifying fractions (Objective 0016).  Question 16 #### 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 16 Explanation: Converting between fractions, decimals, and percents (Objective 0017).  Question 17 #### 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 17 Explanation: Topic: Analyze the relationships among proportions, constant rates, and linear functions (Objective 0022).  Question 18 #### 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 18 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 19

#### 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 19 Explanation:
Topic: Find a linear equation that represents a graph (Objective 0022).
 Question 20

#### How many students at the college are seniors who are not vegetarians?

 A $$\large 137$$Hint: Doesn't include the senior athletes who are not vegetarians. B $$\large 167$$ C $$\large 197$$Hint: That's all seniors, including vegetarians. D $$\large 279$$Hint: Includes all athletes who are not vegetarians, some of whom are not seniors.
Question 20 Explanation:
Topic: Venn Diagrams (Objective 0025)
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