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

#### 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 1 Explanation:
Topics: Estimation, simplifying fractions (Objective 0016, overlaps with other objectives).
 Question 2

#### 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 2 Explanation:
Topic: Characteristics of composite numbers (Objective 0018).
 Question 3

#### 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 3 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 4

#### 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 4 Explanation:
Topic: Analyze and justify standard and non-standard computational techniques (Objective 0019).
 Question 5

#### 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 5 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 6

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

#### 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 7 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 8

#### A sales companies pays its representatives $2 for each item sold, plus 40% of the price of the item. The rest of the money that the representatives collect goes to the company. All transactions are in cash, and all items cost$4 or more.   If the price of an item in dollars is p, which expression represents the amount of money the company collects when the item is sold?

 A $$\large \dfrac{3}{5}p-2$$Hint: The company gets 3/5=60% of the price, minus the $2 per item. B $$\large \dfrac{3}{5}\left( p-2 \right)$$Hint: This is sensible, but not what the problem states. C $$\large \dfrac{2}{5}p+2$$Hint: The company pays the extra$2; it doesn't collect it. D $$\large \dfrac{2}{5}p-2$$Hint: This has the company getting 2/5 = 40% of the price of each item, but that's what the representative gets.
Question 8 Explanation:
Topic: Use algebra to solve word problems involving fractions, ratios, proportions, and percents (Objective 0020).
 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

#### 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 10 Explanation:
Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic) (Objective 0021).
 Question 11

#### P divides 30

Hint:
2, 3, and 5 are the prime factors of 240, and all divide 30.

#### P divides 48

Hint:
P=5 doesn't work.

#### P divides 75

Hint:
P=2 doesn't work.

#### P divides 80

Hint:
P=3 doesn't work.
Question 11 Explanation:
Topic: Find the prime factorization of a number and recognize its uses (Objective 0018).
 Question 12

#### 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 12 Explanation:
Topic: Apply knowledge of combinations and permutations to the computation of probabilities (Objective 0026).
 Question 13

#### Based on the data given above, what was the probability that a randomly chosen girl in 1990 drank milk?

 A $$\large \dfrac{502}{1222}$$Hint: This is the probability that a randomly chosen girl who drinks milk was in the 1989-1991 food survey. B $$\large \dfrac{502}{2149}$$Hint: This is the probability that a randomly chosen girl from the whole survey drank milk and was also surveyed in 1989-1991. C $$\large \dfrac{502}{837}$$ D $$\large \dfrac{1222}{2149}$$Hint: This is the probability that a randomly chosen girl from any year of the survey drank milk.
Question 13 Explanation:
Topic: Recognize and apply the concept of conditional probability (Objective 0026).
 Question 14

#### 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 14 Explanation:
Topic: Estimate and calculate measurements using customary, metric, and nonstandard units of measurement (Objective 0023).
 Question 15

#### 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 15 Explanation:
Topic: Recognize and apply the concepts of variable, function, equality, and equation to express relationships algebraically (Objective 0020).
 Question 16

#### 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 16 Explanation:
Topic: Derive and use formulas for calculating the lengths, perimeters, areas, volumes, and surface areas of geometric shapes and figures (Objective 0023).
 Question 17

#### This student divides fractions by first finding a common denominator, then dividing the numerators.

$$\large \dfrac{2}{3} \div \dfrac{3}{4} \longrightarrow \dfrac{8}{12} \div \dfrac{9}{12} \longrightarrow 8 \div 9 = \dfrac {8}{9}$$ $$\large \dfrac{2}{5} \div \dfrac{7}{20} \longrightarrow \dfrac{8}{20} \div \dfrac{7}{20} \longrightarrow 8 \div 7 = \dfrac {8}{7}$$ $$\large \dfrac{7}{6} \div \dfrac{3}{4} \longrightarrow \dfrac{14}{12} \div \dfrac{9}{12} \longrightarrow 14 \div 9 = \dfrac {14}{9}$$

#### It is not valid. Common denominators are for adding and subtracting fractions, not for dividing them.

Hint:
Don't be so rigid! Usually there's more than one way to do something in math.

#### It got the right answer in these three cases, but it isn‘t valid for all rational numbers.

Hint:
Did you try some other examples? What makes you say it's not valid?

#### It is valid if the rational numbers in the division problem are in lowest terms and the divisor is not zero.

Hint:
Lowest terms doesn't affect this problem at all.

#### It is valid for all rational numbers, as long as the divisor is not zero.

Hint:
When we have common denominators, the problem is in the form a/b divided by c/b, and the answer is a/c, as the student's algorithm predicts.
Question 17 Explanation:
Topic: Analyze Non-Standard Computational Algorithms (Objective 0019).
 Question 18

#### 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 18 Explanation:
Topic: Place Value (Objective 0016)
 Question 19

#### What was the mean score on the quiz?

 A $$\large 2.75$$Hint: There were 20 students who took the quiz. Total points earned: $$2 \times 1+6 \times 2+ 7\times 3+5 \times 4=55$$, and 55/20 = 2.75. B $$\large 2$$Hint: How many students are there total? Did you count them all? C $$\large 3$$Hint: How many students are there total? Did you count them all? Be sure you're finding the mean, not the median or the mode. D $$\large 2.5$$Hint: How many students are there total? Did you count them all? Don't just take the mean of 1, 2, 3, 4 -- you have to weight them properly.
Question 19 Explanation:
Topics: Analyze and interpret various graphic representations, and use measures of central tendency (e.g., mean, median, mode) and spread to describe and interpret real-world data (Objective 0025).
 Question 20

#### 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 20 Explanation:
Topic: Distinguish between linear and nonlinear functions (Objective 0022).
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