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

#### 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 1 Explanation:
Topic: Compare different data sets (Objective 0025).
 Question 2

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

 A $$\large \dfrac{1}{2},\quad \dfrac{1}{3},\quad \dfrac{1}{4},\quad \dfrac{1}{5}$$Hint: This is ordered from greatest to least. B $$\large \dfrac{1}{3},\quad \dfrac{2}{7},\quad \dfrac{3}{8},\quad \dfrac{4}{11}$$Hint: 1/3 = 2/6 is bigger than 2/7. C $$\large \dfrac{1}{4},\quad \dfrac{2}{5},\quad \dfrac{2}{3},\quad \dfrac{4}{5}$$Hint: One way to look at this: 1/4 and 2/5 are both less than 1/2, and 2/3 and 4/5 are both greater than 1/2. 1/4 is 25% and 2/5 is 40%, so 2/5 is greater. The distance from 2/3 to 1 is 1/3 and from 4/5 to 1 is 1/5, and 1/5 is less than 1/3, so 4/5 is bigger. D $$\large \dfrac{7}{8},\quad \dfrac{6}{7},\quad \dfrac{5}{6},\quad \dfrac{4}{5}$$Hint: This is in order from greatest to least.
Question 2 Explanation:
Topic: Ordering Fractions (Objective 0017)
 Question 3

#### Let p represent the regular price of these t-shirt. Which of the following equations is correct?

 A $$\large 0.8p=\8.73$$Hint: 80% of the regular price = $8.73. B $$\large \8.73+0.2*\8.73=p$$Hint: The 20% off was off of the ORIGINAL price, not off the$8.73 (a lot of people make this mistake). Plus this is the same equation as in choice c. C $$\large 1.2*\8.73=p$$Hint: The 20% off was off of the ORIGINAL price, not off the \$8.73 (a lot of people make this mistake). Plus this is the same equation as in choice b. D $$\large p-0.2*\8.73=p$$Hint: Subtract p from both sides of this equation, and you have -.2 x 8.73 =0.
Question 3 Explanation:
Topics: Use algebra to solve word problems involving percents and identify variables, and derive algebraic expressions that represent real-world situations (Objective 0020).
 Question 4

#### 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 4 Explanation:
Topic: Converting between fraction and decimal representations (Objective 0017)
 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

#### 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 6 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 7

#### 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 7 Explanation:
Topic: Analyze geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to concepts of symmetry (Objective 0024).
 Question 8

#### 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 8 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 9

#### 0.38

Hint:
If you are just writing the numerator next to the denominator then your technique is way off, but by coincidence your answer is close; try with 2/3 and 0.23 is nowhere near correct.

#### 0.125

Hint:
This is 1/8, not 3/8.

#### 0.83

Hint:
3/8 is less than a half, and 0.83 is more than a half, so they can't be equal.
Question 9 Explanation:
Topic: Converting between fractions and decimals (Objective 0017)
 Question 10

#### 23 flats, 4 rods, 7 little cubes

Hint:
Be sure you read the question carefully: 2300+40+7=2347

#### 2 large cubes, 3 flats, 47 rods

Hint:
2000+300+470 $$\neq$$ 2347

#### 2 large cubes, 34 rods, 7 little cubes

Hint:
Be sure you read the question carefully: 2000+340+7=2347

#### 2 large cubes, 3 flats, 4 rods, 7 little cubes

Hint:
Be sure you read the question carefully: 2000+300+40+7=2347
Question 10 Explanation:
Topic: Place Value (Objective 0016)
 Question 11

#### The median number of pairs of footware owned is between 50 and 60 pairs.

Hint:
The same number of data points are less than the median as are greater than the median -- but on this histogram, clearly more than half the students own less than 50 pairs of shoes, so the median is less than 50.

#### The mode of the number of pairs of footware owned is 20.

Hint:
The mode is the most common number of pairs of footwear owned. We can't tell it from this histogram because each bar represents 10 different numbers-- perhaps 8 students each own each number from 10 to 19, but 40 students own exactly 6 pairs of shoes.... or perhaps not....

#### The mean number of pairs of footware owned is less than the median number of pairs of footware owned.

Hint:
This is a right skewed distribution, and so the mean is bigger than the median -- the few large values on the right pull up the mean, but have little effect on the median.

#### The median number of pairs of footware owned is between 10 and 20.

Hint:
There are approximately 230 students represented in this survey, and the 41st through 120th lowest values are between 10 and 20 -- thus the middle value is in that range.
Question 11 Explanation:
Topics: Analyze and interpret various graphic and data representations, and use measures of central tendency (e.g., mean, median, mode) and spread to describe and interpret real-world data (Objective 0025).
 Question 12

#### 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 12 Explanation:
Topic: Demonstrate knowledge of divisibility rules (Objective 0018).
 Question 13

#### A family on vacation drove the first 200 miles in 4 hours and the second 200 miles in 5 hours.  Which expression below gives their average speed for the entire trip?

 A $$\large \dfrac{200+200}{4+5}$$Hint: Average speed is total distance divided by total time. B $$\large \left( \dfrac{200}{4}+\dfrac{200}{5} \right)\div 2$$Hint: This seems logical, but the problem is that it weights the first 4 hours and the second 5 hours equally, when each hour should get the same weight in computing the average speed. C $$\large \dfrac{200}{4}+\dfrac{200}{5}$$Hint: This would be an average of 90 miles per hour! D $$\large \dfrac{400}{4}+\dfrac{400}{5}$$Hint: This would be an average of 180 miles per hour! Even a family of race car drivers probably doesn't have that average speed on a vacation!
Question 13 Explanation:
Topic: Solve a variety of measurement problems (e.g., time, temperature, rates, average rates of change) in real-world situations (Objective 0023).
 Question 14

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

#### 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 16 Explanation:
Topic: Recognize and apply the concept of conditional probability (Objective 0026).
 Question 17

#### 1.6 cm

Hint:
This is more the height of a Lego toy college student -- less than an inch!

#### 16 cm

Hint:
Less than knee high on most college students.

#### 160 cm

Hint:
Remember, a meter stick (a little bigger than a yard stick) is 100 cm. Also good to know is that 4 inches is approximately 10 cm.

#### 1600 cm

Hint:
This college student might be taller than some campus buildings!
Question 17 Explanation:
Topic: Estimate and calculate measurements using customary, metric, and nonstandard units of measurement (Objective 0023).
 Question 18

#### The column below consists of two cubes and a cylinder.  The cylinder has diameter y, which is also the length of the sides of each cube.   The total height of the column is 5y.  Which of the formulas below gives the volume of the column?

 A $$\large 2{{y}^{3}}+\dfrac{3\pi {{y}^{3}}}{4}$$Hint: The cubes each have volume $$y^3$$. The cylinder has radius $$\dfrac{y}{2}$$ and height $$3y$$. The volume of a cylinder is $$\pi r^2 h=\pi ({\dfrac{y}{2}})^2(3y)=\dfrac{3\pi {{y}^{3}}}{4}$$. Note that the volume of a cylinder is analogous to that of a prism -- area of the base times height. B $$\large 2{{y}^{3}}+3\pi {{y}^{3}}$$Hint: y is the diameter of the circle, not the radius. C $$\large {{y}^{3}}+5\pi {{y}^{3}}$$Hint: Don't forget to count both cubes. D $$\large 2{{y}^{3}}+\dfrac{3\pi {{y}^{3}}}{8}$$Hint: Make sure you know how to find the volume of a cylinder.
Question 18 Explanation:
Topic: Derive and use formulas for calculating the lengths, perimeters, areas, volumes, and surface areas of geometric shapes and figures (Objective 0023).
 Question 19

#### 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 19 Explanation:
Topic: Integers (Objective 0016)
 Question 20

#### The student‘s solution is correct.

Hint:
Try plugging into the original solution.

#### The student did not correctly use properties of equality.

Hint:
After $$x=-2x+10$$, the student subtracted 2x on the left and added 2x on the right.

#### The student did not correctly use the distributive property.

Hint:
Distributive property is $$a(b+c)=ab+ac$$.

#### The student did not correctly use the commutative property.

Hint:
Commutative property is $$a+b=b+a$$ or $$ab=ba$$.
Question 20 Explanation:
Topic: Justify algebraic manipulations by application of the properties of equality, the order of operations, the number properties, and the order properties (Objective 0020).
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