Hints will display for most wrong answers; explanations for most right answers.   You can attempt a question multiple times; it will only be scored correct if you get it right the first time.

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

#### 1.5°

Hint:
Celsius and Fahrenheit don't increase at the same rate.

#### 1.8°

Hint:
That's how much the Fahrenheit temp increases when the Celsius temp goes up by 1 degree.

#### 2.7°

Hint:
Each degree increase in Celsius corresponds to a $$\dfrac{9}{5}=1.8$$ degree increase in Fahrenheit. Thus the increase is 1.8+0.9=2.7.

#### Not enough information.

Hint:
A linear equation has constant slope, which means that every increase of the same amount in one variable, gives a constant increase in the other variable. It doesn't matter what temperature the patient started out at.
Question 1 Explanation:
Topic: Interpret the meaning of the slope and the intercepts of a linear equation that models a real-world situation (Objective 0022).
 Question 2

#### 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 2 Explanation:
Topic: Identifying rational and irrational numbers (Objective 0016).
 Question 3

#### Given that 10 cm is approximately equal to 4 inches, which of the following expressions models a way to find out approximately how many inches are equivalent to 350 cm?

 A $$\large 350\times \left( \dfrac{10}{4} \right)$$Hint: The final result should be smaller than 350, and this answer is bigger. B $$\large 350\times \left( \dfrac{4}{10} \right)$$Hint: Dimensional analysis can help here: $$350 \text{cm} \times \dfrac{4 \text{in}}{10 \text{cm}}$$. The cm's cancel and the answer is in inches. C $$\large (10-4) \times 350$$Hint: This answer doesn't make much sense. Try with a simpler example (e.g. 20 cm not 350 cm) to make sure that your logic makes sense. D $$\large (350-10) \times 4$$Hint: This answer doesn't make much sense. Try with a simpler example (e.g. 20 cm not 350 cm) to make sure that your logic makes sense.
Question 3 Explanation:
Topic: Applying fractions to word problems (Objective 0017) This problem is similar to one on the official sample test for that objective, but it might fit better into unit conversion and dimensional analysis (Objective 0023: Measurement)
 Question 4

#### Which of the lists below is in order from least to greatest value?

 A $$\large -0.044,\quad -0.04,\quad 0.04,\quad 0.044$$Hint: These are easier to compare if you add trailing zeroes (this is finding a common denominator) -- all in thousandths, -0.044, -0.040,0 .040, 0.044. The middle two numbers, -0.040 and 0.040 can be modeled as owing 4 cents and having 4 cents. The outer two numbers are owing or having a bit more. B $$\large -0.04,\quad -0.044,\quad 0.044,\quad 0.04$$Hint: 0.04=0.040, which is less than 0.044. C $$\large -0.04,\quad -0.044,\quad 0.04,\quad 0.044$$Hint: -0.04=-0.040, which is greater than $$-0.044$$. D $$\large -0.044,\quad -0.04,\quad 0.044,\quad 0.04$$Hint: 0.04=0.040, which is less than 0.044.
Question 4 Explanation:
Topic: Ordering decimals and integers (Objective 0017).
 Question 5

#### 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 5 Explanation:
Topic: Recognize and apply the concept of conditional probability (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

#### 212

Hint:
Can the number of toothpicks be even?

#### 213

Hint:
One way to see this is that every new "house" adds 4 toothpicks to the leftmost vertical toothpick -- so the total number is 1 plus 4 times the number of "houses." There are many other ways to look at the problem too.

#### 217

Hint:
Try your strategy with a smaller number of "houses" so you can count and find your mistake.

#### 265

Hint:
Remember that the "houses" overlap some walls.
Question 7 Explanation:
Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic). (Objective 0021).
 Question 8

#### 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 8 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 9

#### A cylindrical soup can has diameter 7 cm and height 11 cm. The can holds g grams of soup.   How many grams of the same soup could a cylindrical can with diameter 14 cm and height 33 cm hold?

 A $$\large 6g$$Hint: You must scale in all three dimensions. B $$\large 12g$$Hint: Height is multiplied by 3, and diameter and radius are multiplied by 2. Since the radius is squared, final result is multiplied by $$2^2\times 3=12$$. C $$\large 18g$$Hint: Don't square the height scale factor. D $$\large 36g$$Hint: Don't square the height scale factor.
Question 9 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 10

#### 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 10 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 11

#### A solution requires 4 ml of saline for every 7 ml of medicine. How much saline would be required for 50 ml of medicine?

 A $$\large 28 \dfrac{4}{7}$$ mlHint: 49 ml of medicine requires 28 ml of saline. The extra ml of saline requires 4 ml saline/ 7 ml medicine = 4/7 ml saline per 1 ml medicine. B $$\large 28 \dfrac{1}{4}$$ mlHint: 49 ml of medicine requires 28 ml of saline. How much saline does the extra ml require? C $$\large 28 \dfrac{1}{7}$$ mlHint: 49 ml of medicine requires 28 ml of saline. How much saline does the extra ml require? D $$\large 87.5$$ mlHint: 49 ml of medicine requires 28 ml of saline. How much saline does the extra ml require?
Question 11 Explanation:
Topic: Apply proportional thinking to estimate quantities in real world situations (Objective 0019).
 Question 12

#### 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 12 Explanation:
Topic: Analyze geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to concepts of symmetry (Objective 0024).
 Question 13

#### 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 13 Explanation:
Topic: Using number lines (Objective 0017)
 Question 14

#### If two fair coins are flipped, what is the probability that one will come up heads and the other tails?

 A $$\large \dfrac{1}{4}$$Hint: Think of the coins as a penny and a dime, and list all possibilities. B $$\large \dfrac{1}{3}$$Hint: This is a very common misconception. There are three possible outcomes -- both heads, both tails, and one of each -- but they are not equally likely. Think of the coins as a penny and a dime, and list all possibilities. C $$\large \dfrac{1}{2}$$Hint: The possibilities are HH, HT, TH, TT, and all are equally likely. Two of the four have one of each coin, so the probability is 2/4=1/2. D $$\large \dfrac{3}{4}$$Hint: Think of the coins as a penny and a dime, and list all possibilities.
Question 14 Explanation:
Topic: Calculate the probabilities of simple and compound events and of independent and dependent events (Objective 0026).
 Question 15

#### 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 15 Explanation:
Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic) (Objective 0021).
 Question 16

#### 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 16 Explanation:
Topic:Classify and analyze three-dimensional figures using attributes of faces, edges, and vertices (Objective 0024).
 Question 17

#### A sphere

Hint:
All views would be circles.

#### A cone

Hint:
Two views would be triangles, not rectangles.

#### A pyramid

Hint:
How would one view be a circle?
Question 17 Explanation:
Topic: Match three-dimensional figures and their two-dimensional representations (e.g., nets, projections, perspective drawings) (Objective 0024).
 Question 18

#### Which of the following equations could also represent A  for the values shown?

 A $$\large A(n)=n+4$$Hint: For a question like this, you don't have to find the equation yourself, you can just try plugging the function machine inputs into the equation, and see if any values come out wrong. With this equation n= -1 would output 3, not 0 as the machine does. B $$\large A(n)=n+2$$Hint: For a question like this, you don't have to find the equation yourself, you can just try plugging the function machine inputs into the equation, and see if any values come out wrong. With this equation n= 2 would output 4, not 6 as the machine does. C $$\large A(n)=2n+2$$Hint: Simply plug in each of the four function machine input values, and see that the equation produces the correct output, e.g. A(2)=6, A(-1)=0, etc. D $$\large A(n)=2\left( n+2 \right)$$Hint: For a question like this, you don't have to find the equation yourself, you can just try plugging the function machine inputs into the equation, and see if any values come out wrong. With this equation n= 2 would output 8, not 6 as the machine does.
Question 18 Explanation:
Topics: Understand various representations of functions, and translate among different representations of functional relationships (Objective 0021).
 Question 19

#### 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 19 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 20

#### The student used a method that worked for this problem and can be generalized to any subtraction problem.

Hint:
Note that this algorithm is taught as the "standard" algorithm in much of Europe (it's where the term "borrowing" came from -- you borrow on top and "pay back" on the bottom).

#### The student used a method that worked for this problem and that will work for any subtraction problem that only requires one regrouping; it will not work if more regrouping is required.

Hint:
Try some more examples.

#### The student used a method that worked for this problem and will work for all three-digit subtraction problems, but will not work for larger problems.

Hint:
Try some more examples.

#### The student used a method that does not work. The student made two mistakes that cancelled each other out and was lucky to get the right answer for this problem.

Hint:
Remember, there are many ways to do subtraction; there is no one "right" algorithm.
Question 20 Explanation:
Topic: Analyze and justify standard and non-standard computational techniques (Objective 0019).
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