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## MTEL General Curriculum Mathematics Practice

 Question 1

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

#### Exactly one of the numbers below is a prime number.  Which one is it?

 A $$\large511$$Hint: Divisible by 7. B $$\large517$$Hint: Divisible by 11. C $$\large519$$Hint: Divisible by 3. D $$\large521$$
Question 3 Explanation:
Topics: Identify prime and composite numbers and demonstrate knowledge of divisibility rules (Objective 0018).
 Question 4

#### 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 4 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 5

#### 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 5 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 6

#### In the triangle below, $$\overline{AC}\cong \overline{AD}\cong \overline{DE}$$ and $$m\angle CAD=100{}^\circ$$.  What is $$m\angle DAE$$?

 A $$\large 20{}^\circ$$Hint: Angles ACD and ADC are congruent since they are base angles of an isosceles triangle. Since the angles of a triangle sum to 180, they sum to 80, and they are 40 deg each. Thus angle ADE is 140 deg, since it makes a straight line with angle ADC. Angles DAE and DEA are base angles of an isosceles triangle and thus congruent-- they sum to 40 deg, so are 20 deg each. B $$\large 25{}^\circ$$Hint: If two sides of a triangle are congruent, then it's isosceles, and the base angles of an isosceles triangle are equal. C $$\large 30{}^\circ$$Hint: If two sides of a triangle are congruent, then it's isosceles, and the base angles of an isosceles triangle are equal. D $$\large 40{}^\circ$$Hint: Make sure you're calculating the correct angle.
Question 6 Explanation:
Topic: Classify and analyze polygons using attributes of sides and angles, including real-world applications. (Objective 0024).
 Question 7

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

#### 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 8 Explanation:
Topic: Interpret meaning of slope in a real world situation (Objective 0022).
 Question 9

#### 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 9 Explanation:
Topic: Compare different data sets (Objective 0025).
 Question 10

#### George left home at 10:00 and drove to work on a crooked path. He was stopped in traffic at 10:30 and 10:45. He drove 30 miles total.

Hint:
Just because he ended up 30 miles from home doesn't mean he drove 30 miles total.

#### George drove to work. On the way to work there is a little hill and a big hill. He slowed down for them. He made it to work at 11:15.

Hint:
The graph is not a picture of the roads.

#### George left home at 10:15. He drove 10 miles, then realized he‘d forgotten something at home. He turned back and got what he‘d forgotten. Then he drove in a straight line, at many different speeds, until he got to work around 11:15.

Hint:
A straight line on a distance versus time graph means constant speed.

#### George left home at 10:15. He drove 10 miles, then realized he‘d forgotten something at home. He turned back and got what he‘d forgotten. Then he drove at a constant speed until he got to work around 11:15.

Question 10 Explanation:
Topic: Use qualitative graphs to represent functional relationships in the real world (Objective 0021).
 Question 11

#### 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 11 Explanation:
Topic: Select the linear equation that best models a real-world situation (Objective 0022).
 Question 12

#### 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 12 Explanation:
Topic: Understand meaning and models of operations on fractions (Objective 0019).
 Question 13

#### 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 13 Explanation:
Topics: Estimation, simplifying fractions (Objective 0016).
 Question 14

#### 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.

#### Store A

Hint:
This would save about \$2.50. You can quickly see that D saves more.

#### 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 28 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).
 Question 29

#### 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 29 Explanation:
Converting between fractions, decimals, and percents (Objective 0017).
 Question 30

#### The result is always the number that you started with! Suppose you start by picking N. Which of the equations below best demonstrates that the result after Step 6 is also N?

 A $$\large N*2+20*5-100\div 10=N$$Hint: Use parentheses or else order of operations is off. B $$\large \left( \left( 2*N+20 \right)*5-100 \right)\div 10=N$$ C $$\large \left( N+N+20 \right)*5-100\div 10=N$$Hint: With this answer you would subtract 10, instead of subtracting 100 and then dividing by 10. D $$\large \left( \left( \left( N\div 10 \right)-100 \right)*5+20 \right)*2=N$$Hint: This answer is quite backwards.
Question 30 Explanation:
Topic: Recognize and apply the concepts of variable, function, equality, and equation to express relationships algebraically (Objective 0020).
 Question 31

#### 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 31 Explanation:
Topic: Classify and analyze polygons using attributes of sides and angles, and symmetry (Objective 0024).
 Question 32

#### 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 32 Explanation:
Topic: Use unit conversions and dimensional analysis to solve measurement problems (Objective 0023).
 Question 33

#### What is the probability that two randomly selected people were born on the same day of the week?  Assume that all days are equally probable.

 A $$\large \dfrac{1}{7}$$Hint: It doesn't matter what day the first person was born on. The probability that the second person will match is 1/7 (just designate one person the first and the other the second). Another way to look at it is that if you list the sample space of all possible pairs, e.g. (Wed, Sun), there are 49 such pairs, and 7 of them are repeats of the same day, and 7/49=1/7. B $$\large \dfrac{1}{14}$$Hint: What would be the sample space here? Ie, how would you list 14 things that you pick one from? C $$\large \dfrac{1}{42}$$Hint: If you wrote the seven days of the week on pieces of paper and put the papers in a jar, this would be the probability that the first person picked Sunday and the second picked Monday from the jar -- not the same situation. D $$\large \dfrac{1}{49}$$Hint: This is the probability that they are both born on a particular day, e.g. Sunday.
Question 33 Explanation:
Topic: Calculate the probabilities of simple and compound events and of independent and dependent events (Objective 0026).
 Question 34

#### 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 34 Explanation:
Topic: Classify and analyze polygons using attributes of sides and angles (Objective 0024).
 Question 35

#### 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 35 Explanation:
Topic: Identifying rational and irrational numbers (Objective 0016).
 Question 36

#### There are 15 students for every teacher.  Let t represent the number of teachers and let s represent the number of students.  Which of the following equations is correct?

 A $$\large t=s+15$$Hint: When there are 2 teachers, how many students should there be? Do those values satisfy this equation? B $$\large s=t+15$$Hint: When there are 2 teachers, how many students should there be? Do those values satisfy this equation? C $$\large t=15s$$Hint: This is a really easy mistake to make, which comes from transcribing directly from English, "1 teachers equals 15 students." To see that it's wrong, plug in s=2; do you really need 30 teachers for 2 students? To avoid this mistake, insert the word "number," "Number of teachers equals 15 times number of students" is more clearly problematic. D $$\large s=15t$$
Question 36 Explanation:
Topic: Select the linear equation that best models a real-world situation (Objective 0022).
 Question 37

#### 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 37 Explanation:
Topic: Apply knowledge of combinations and permutations to the computation of probabilities (Objective 0026).
 Question 38

#### All natural numbers from 2 to 266.

Hint:
She only needs to check primes -- checking the prime factors of any composite is enough to look for divisors. As a test taking strategy, the other three choices involve primes, so worth thinking about.

#### All primes from 2 to 266 .

Hint:
Remember, factors come in pairs (except for square root factors), so she would first find the smaller of the pair and wouldn't need to check the larger.

#### All primes from 2 to 133 .

Hint:
She doesn't need to check this high. Factors come in pairs, and something over 100 is going to be paired with something less than 3, so she will find that earlier.

#### All primes from $$\large 2$$ to $$\large \sqrt{267}$$.

Hint:
$$\sqrt{267} \times \sqrt{267}=267$$. Any other pair of factors will have one factor less than $$\sqrt{267}$$ and one greater, so she only needs to check up to $$\sqrt{267}$$.
Question 38 Explanation:
Topic: Identify prime and composite numbers (Objective 0018).
 Question 39

#### The expression $$\large{{8}^{3}}\cdot {{2}^{-10}}$$ is equal to which of the following?

 A $$\large 2$$Hint: Write $$8^3$$ as a power of 2. B $$\large \dfrac{1}{2}$$Hint: $$8^3 \cdot {2}^{-10}={(2^3)}^3 \cdot {2}^{-10}$$ =$$2^9 \cdot {2}^{-10} =2^{-1}$$ C $$\large 16$$Hint: Write $$8^3$$ as a power of 2. D $$\large \dfrac{1}{16}$$Hint: Write $$8^3$$ as a power of 2.
Question 39 Explanation:
Topic: Laws of Exponents (Objective 0019).
 Question 40

#### 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 40 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 41

#### 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 41 Explanation:
Topic: Recognize uses of prime factorization of a number (Objective 0018).
 Question 42

#### Which of the numbers below is a fraction equivalent to $$0.\bar{6}$$?

 A $$\large \dfrac{4}{6}$$Hint: $$0.\bar{6}=\dfrac{2}{3}=\dfrac{4}{6}$$ B $$\large \dfrac{3}{5}$$Hint: This is equal to 0.6, without the repeating decimal. Answer is equivalent to choice c, which is another way to tell that it's wrong. C $$\large \dfrac{6}{10}$$Hint: This is equal to 0.6, without the repeating decimal. Answer is equivalent to choice b, which is another way to tell that it's wrong. D $$\large \dfrac{1}{6}$$Hint: This is less than a half, and $$0.\bar{6}$$ is greater than a half.
Question 42 Explanation:
Topic: Converting between fraction and decimal representations (Objective 0017)
 Question 43

#### $$\large A-B+C\div D\times E$$?

 A $$\large A-B-\dfrac{C}{DE}$$Hint: In the order of operations, multiplication and division have the same priority, so do them left to right; same with addition and subtraction. B $$\large A-B+\dfrac{CE}{D}$$Hint: In practice, you're better off using parentheses than writing an expression like the one in the question. The PEMDAS acronym that many people memorize is misleading. Multiplication and division have equal priority and are done left to right. They have higher priority than addition and subtraction. Addition and subtraction also have equal priority and are done left to right. C $$\large \dfrac{AE-BE+CE}{D}$$Hint: Use order of operations, don't just compute left to right. D $$\large A-B+\dfrac{C}{DE}$$Hint: In the order of operations, multiplication and division have the same priority, so do them left to right
Question 43 Explanation:
Topic: Justify algebraic manipulations by application of the properties of order of operations (Objective 0020).
 Question 44

#### 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 44 Explanation:
Topic: Recognize and analyze pictorial representations of number operations. (Objective 0019).
 Question 45

#### 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 45 Explanation:
Topics: Estimation, simplifying fractions (Objective 0016, overlaps with other objectives).
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