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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

#### 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 1 Explanation:
Topic: Understand divisibility rules and why they work (Objective 018).
 Question 2

#### Four children randomly line up, single file.  What is the probability that they are in height order, with the shortest child in front?   All of the children are different heights.

 A $$\large \dfrac{1}{4}$$Hint: Try a simpler question with 3 children -- call them big, medium, and small -- and list all the ways they could line up. Then see how to extend your logic to the problem with 4 children. B $$\large \dfrac{1}{256}$$Hint: Try a simpler question with 3 children -- call them big, medium, and small -- and list all the ways they could line up. Then see how to extend your logic to the problem with 4 children. C $$\large \dfrac{1}{16}$$Hint: Try a simpler question with 3 children -- call them big, medium, and small -- and list all the ways they could line up. Then see how to extend your logic to the problem with 4 children. D $$\large \dfrac{1}{24}$$Hint: The number of ways for the children to line up is $$4!=4 \times 3 \times 2 \times 1 =24$$ -- there are 4 choices for who is first in line, then 3 for who is second, etc. Only one of these lines has the children in the order specified.
Question 2 Explanation:
Topic: Apply knowledge of combinations and permutations to the computation of probabilities (Objective 0026).
 Question 3

#### 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 3 Explanation:
Topic: Integers (Objective 0016)
 Question 4

#### 7.5 meters

Hint:
Here is a picture, note that the large and small right triangles are similar:

One way to do the problem is to note that there is a dilation (scale) factor of 5 on the shadows, so there must be that factor on the heights too. Another way is to note that the shadows are twice as long as the heights.

Hint:
Draw a picture.

Hint:
Draw a picture.

#### 45 meters

Hint:
Draw a picture.
Question 4 Explanation:
Topic: Apply geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to similarity, ; and use these concepts to solve problems (Objective 0024) . Fits in other places too.
 Question 5

#### There are six gumballs in a bag — two red and four green.  Six children take turns picking a gumball out of the bag without looking.   They do not return any gumballs to the bag.  What is the probability that the first two children to pick from the bag pick the red gumballs?

 A $$\large \dfrac{1}{3}$$Hint: This is the probability that the first child picks a red gumball, but not that the first two children pick red gumballs. B $$\large \dfrac{1}{8}$$Hint: Are you adding things that you should be multiplying? C $$\large \dfrac{1}{9}$$Hint: This would be the probability if the gumballs were returned to the bag. D $$\large \dfrac{1}{15}$$Hint: The probability that the first child picks red is 2/6 = 1/3. Then there are 5 gumballs in the bag, one red, so the probability that the second child picks red is 1/5. Thus 1/5 of the time, after the first child picks red, the second does too, so the probability is 1/5 x 1/3 = 1/15.
Question 5 Explanation:
Topic: Calculate the probabilities of simple and compound events and of independent and dependent events (Objective 0026).
 Question 6

#### The graph above represents the equation $$\large 3x+Ay=B$$, where A and B are integers.  What are the values of A and B?

 A $$\large A = -2, B= 6$$Hint: Plug in (2,0) to get B=6, then plug in (0,-3) to get A=-2. B $$\large A = 2, B = 6$$Hint: Try plugging (0,-3) into this equation. C $$\large A = -1.5, B=-3$$Hint: The problem said that A and B were integers and -1.5 is not an integer. Don't try to use slope-intercept form. D $$\large A = 2, B = -3$$Hint: Try plugging (2,0) into this equation.
Question 6 Explanation:
Topic: Find a linear equation that represents a graph (Objective 0022).
 Question 7

#### 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 7 Explanation:
Topic: Compare different data sets (Objective 0025).
 Question 8

#### Intercept = 0.4 cm, Slope = 125 cm/page

Hint:
This would mean that each page of the book was 125 cm thick.

#### Intercept =0.4 cm, Slope = $$\dfrac{1}{125}$$cm/page

Hint:
The intercept is how thick the book would be with no pages in it. The slope is how much 1 extra page adds to the thickness of the book.

#### Intercept = 125 cm, Slope = 0.4 cm

Hint:
This would mean that with no pages in the book, it would be 125 cm thick.

#### Intercept = $$\dfrac{1}{125}$$cm, Slope = 0.4 pages/cm

Hint:
This would mean that each new page of the book made it 0.4 cm thicker.
Question 8 Explanation:
Topic: Interpret the meaning of the slope and the intercepts of a linear equation that models a real-world situation (Objective 0022).
 Question 9

#### 2

Hint:
$$10^3 \times 10^4=10^7$$, and note that if you're guessing when the answers are so closely related, you're generally better off guessing one of the middle numbers.

#### 20

Hint:
$$\dfrac{\left( 4\times {{10}^{3}} \right)\times \left( 3\times {{10}^{4}} \right)}{6\times {{10}^{6}}}=\dfrac {12 \times {{10}^{7}}}{6\times {{10}^{6}}}=$$$$2 \times {{10}^{1}}=20$$

#### 200

Hint:
$$10^3 \times 10^4=10^7$$

#### 2000

Hint:
$$10^3 \times 10^4=10^7$$, and note that if you're guessing when the answers are so closely related, you're generally better off guessing one of the middle numbers.
Question 9 Explanation:
Topics: Scientific notation, exponents, simplifying fractions (Objective 0016, although overlaps with other objectives too).
 Question 10

#### 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 10 Explanation:
Topic: Laws of Exponents (Objective 0019).
 Question 11

#### 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 11 Explanation:
Topic: Using number lines (Objective 0017)
 Question 12

#### 314

Hint:
Try your procedure on a smaller number that you can count to see where you made a mistake.

#### 317

Hint:
Are there ever an odd number of white squares?

#### 320

Hint:
One way to see this is that there are 6 tiles on the left and right ends, and the rest of the white tiles are twice the number of black tiles (there are many other ways to look at it too).

#### 322

Hint:
Try your procedure on a smaller number that you can count to see where you made a mistake.
Question 12 Explanation:
Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic) (Objective 0021).
 Question 13

#### 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 13 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 14

#### 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 14 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 15

#### 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 15 Explanation:
Topic: Distinguish between linear and nonlinear functions (Objective 0022).
 Question 16

#### 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 16 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 17

#### 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.

#### \$70

Hint:
Not a factor of 126 or 266, so couldn't be correct.
Question 17 Explanation:
Topic: Use GCF in real-world context (Objective 0018)
 Question 18

#### Solve for x: $$\large 4-\dfrac{2}{3}x=2x$$

 A $$\large x=3$$Hint: Try plugging x=3 into the equation. B $$\large x=-3$$Hint: Left side is positive, right side is negative when you plug this in for x. C $$\large x=\dfrac{3}{2}$$Hint: One way to solve: $$4=\dfrac{2}{3}x+2x$$ $$=\dfrac{8}{3}x$$.$$x=\dfrac{3 \times 4}{8}=\dfrac{3}{2}$$. Another way is to just plug x=3/2 into the equation and see that each side equals 3 -- on a multiple choice test, you almost never have to actually solve for x. D $$\large x=-\dfrac{3}{2}$$Hint: Left side is positive, right side is negative when you plug this in for x.
Question 18 Explanation:
Topic: Solve linear equations (Objective 0020).
 Question 19

#### 2 pentagons and 5 rectangles.

Hint:
These can be assembled to form a pentagonal prism, not a pentagonal pyramid.

#### 1 square and 5 equilateral triangles.

Hint:
You need a pentagon for a pentagonal pyramid.

#### 1 pentagon and 10 isosceles triangles.

Question 19 Explanation:
Topic:Classify and analyze three-dimensional figures using attributes of faces, edges, and vertices (Objective 0024).
 Question 20

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