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## MTEL General Curriculum Mathematics Practice

Your answers are highlighted below.
 Question 1

#### 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 1 Explanation:
Topic: Recognize and apply the concepts of variable, function, equality, and equation to express relationships algebraically (Objective 0020).
 Question 2

#### 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 2 Explanation:
Topic: Venn Diagrams (Objective 0025)
 Question 3

#### 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 3 Explanation:
Topic: Identify prime and composite numbers (Objective 0018).
 Question 4

#### If  x  is an integer, which of the following must also be an integer?

 A $$\large \dfrac{x}{2}$$Hint: If x is odd, then $$\dfrac{x}{2}$$ is not an integer, e.g. 3/2 = 1.5. B $$\large \dfrac{2}{x}$$Hint: Only an integer if x = -2, -1, 1, or 2. C $$\large-x$$Hint: -1 times any integer is still an integer. D $$\large\sqrt{x}$$Hint: Usually not an integer, e.g. $$\sqrt{2} \approx 1.414$$.
Question 4 Explanation:
Topic: Integers (Objective 0016)
 Question 5

#### 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 5 Explanation:
Topic: Using Number Lines (Objective 0017)
 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

#### 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 7 Explanation:
Topic: Use unit conversions and dimensional analysis to solve measurement problems (Objective 0023).
 Question 8

#### Which of the following is the equation of a linear function?

 A $$\large y={{x}^{2}}+2x+7$$Hint: This is a quadratic function. B $$\large y={{2}^{x}}$$Hint: This is an exponential function. C $$\large y=\dfrac{15}{x}$$Hint: This is an inverse function. D $$\large y=x+(x+4)$$Hint: This is a linear function, y=2x+4, it's graph is a straight line with slope 2 and y-intercept 4.
Question 8 Explanation:
Topic: Distinguish between linear and nonlinear functions (Objective 0022).
 Question 9

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

#### 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 11 Explanation:
Topic: Find the least common multiple of a set of numbers (Objective 0018).
 Question 12

#### 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 12 Explanation:
Topic: Understand divisibility rules and why they work (Objective 018).
 Question 13

#### 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 13 Explanation:
Understand the definition of function and various representations of functions (e.g., input/output machines, tables, graphs, mapping diagrams, formulas). (Objective 0021).
 Question 14

#### The letters A, and B represent digits (possibly equal) in the ten digit number x=1,438,152,A3B.   For which values of A and B is x divisible by 12, but not by 9?

 A $$\large A = 0, B = 4$$Hint: Digits add to 31, so not divisible by 3, so not divisible by 12. B $$\large A = 7, B = 2$$Hint: Digits add to 36, so divisible by 9. C $$\large A = 0, B = 6$$Hint: Digits add to 33, divisible by 3, not 9. Last digits are 36, so divisible by 4, and hence by 12. D $$\large A = 4, B = 8$$Hint: Digits add to 39, divisible by 3, not 9. Last digits are 38, so not divisible by 4, so not divisible by 12.
Question 14 Explanation:
Topic: Demonstrate knowledge of divisibility rules (Objective 0018).
 Question 15

#### 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 15 Explanation:
Topics: Estimation, simplifying fractions (Objective 0016, overlaps with other objectives).
 Question 16

#### Use the samples of a student€™s work below to answer the question that follows:

$$\large \dfrac{2}{3}\times \dfrac{3}{4}=\dfrac{4\times 2}{3\times 3}=\dfrac{8}{9}$$

$$\large \dfrac{2}{5}\times \dfrac{7}{7}=\dfrac{7\times 2}{5\times 7}=\dfrac{2}{5}$$

$$\large \dfrac{7}{6}\times \dfrac{3}{4}=\dfrac{4\times 7}{6\times 3}=\dfrac{28}{18}=\dfrac{14}{9}$$

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

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

#### 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.

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

Hint:
Lowest terms is irrelevant.

#### It is valid for all rational numbers.

Hint:
Can't be correct as the first and last examples have the wrong answers.
Question 16 Explanation:
Topic: Analyze Non-Standard Computational Algorithms (Objective 0019).
 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

#### Which of the following inequalities describes all values of x  with $$\large \dfrac{x}{2}\le \dfrac{x}{3}$$?

 A $$\large x < 0$$Hint: If x =0, then x/2 = x/3, so this answer can't be correct. B $$\large x \le 0$$ C $$\large x > 0$$Hint: If x =0, then x/2 = x/3, so this answer can't be correct. D $$\large x \ge 0$$Hint: Try plugging in x = 6.
Question 18 Explanation:
Topics: Inequalities, operations (Objective 0019) (not exactly sure how to classify, but this is like one of the problems on the official sample test).
 Question 19

#### 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 19 Explanation:
Topic: Translate among different representations (e.g., tables, graphs, algebraic expressions, verbal descriptions) of functional relationships (Objective 0021).
 Question 20

#### 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 20 Explanation:
Topics: Identify prime and composite numbers and demonstrate knowledge of divisibility rules (Objective 0018).
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