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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

#### In each expression below  N represents a negative integer. Which expression could have a negative value?

 A $$\large {{N}^{2}}$$Hint: Squaring always gives a non-negative value. B $$\large 6-N$$Hint: A story problem for this expression is, if it was 6 degrees out at noon and N degrees out at sunrise, by how many degrees did the temperature rise by noon? Since N is negative, the answer to this question has to be positive, and more than 6. C $$\large -N$$Hint: If N is negative, then -N is positive D $$\large 6+N$$Hint: For example, if $$N=-10$$, then $$6+N = -4$$
Question 1 Explanation:
If you are stuck on a question like this, try a few examples to eliminate some choices and to help you understand what the question means. Topic: Characteristics of integers (Objective 0016).
 Question 2

#### 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 2 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 3

#### 21 cm

Hint:
How many miles would correspond to 24 cm on the map? Try adjusting from there.

#### 22 cm

Hint:
How many miles would correspond to 24 cm on the map? Try adjusting from there.

#### 23 cm

Hint:
One way to solve this without a calculator is to note that 4 groups of 6 cm is 2808 miles, which is 100 miles too much. Then 100 miles would be about 1/7 th of 6 cm, or about 1 cm less than 24 cm.

#### 24 cm

Hint:
4 groups of 6 cm is over 2800 miles on the map, which is too much.
Question 3 Explanation:
Topic: Apply proportional thinking to estimate quantities in real world situations (Objective 0019).
 Question 4

#### 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 4 Explanation:
Topic: Converting between fractions and decimals (Objective 0017)
 Question 5

#### 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 5 Explanation:
Topic: Analyze and justify standard and non-standard computational techniques (Objective 0019).
 Question 6

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

#### 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 7 Explanation:
Topic: Calculate the probabilities of simple and compound events and of independent and dependent events (Objective 0026).
 Question 8

#### The least common multiple of 60 and N is 1260. Which of the following could be the prime factorization of N?

 A $$\large2\cdot 5\cdot 7$$Hint: 1260 is divisible by 9 and 60 is not, so N must be divisible by 9 for 1260 to be the LCM. B $$\large{{2}^{3}}\cdot {{3}^{2}}\cdot 5 \cdot 7$$Hint: 1260 is not divisible by 8, so it isn't a multiple of this N. C $$\large3 \cdot 5 \cdot 7$$Hint: 1260 is divisible by 9 and 60 is not, so N must be divisible by 9 for 1260 to be the LCM. D $$\large{{3}^{2}}\cdot 5\cdot 7$$Hint: $$1260=2^2 \cdot 3^2 \cdot 5 \cdot 7$$ and $$60=2^2 \cdot 3 \cdot 5$$. In order for 1260 to be the LCM, N has to be a multiple of $$3^2$$ and of 7 (because 60 is not a multiple of either of these). N also cannot introduce a factor that would require the LCM to be larger (as in choice b).
Question 8 Explanation:
Topic: Least Common Multiple (Objective 0018)
 Question 9

#### 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 9 Explanation:
Topic: Recognize and apply the concept of conditional probability (Objective 0026).
 Question 10

#### What is the probability that two randomly selected people were born on the same day of the week?  Assume that all days are equally probable.

 A $$\large \dfrac{1}{7}$$Hint: It doesn't matter what day the first person was born on. The probability that the second person will match is 1/7 (just designate one person the first and the other the second). Another way to look at it is that if you list the sample space of all possible pairs, e.g. (Wed, Sun), there are 49 such pairs, and 7 of them are repeats of the same day, and 7/49=1/7. B $$\large \dfrac{1}{14}$$Hint: What would be the sample space here? Ie, how would you list 14 things that you pick one from? C $$\large \dfrac{1}{42}$$Hint: If you wrote the seven days of the week on pieces of paper and put the papers in a jar, this would be the probability that the first person picked Sunday and the second picked Monday from the jar -- not the same situation. D $$\large \dfrac{1}{49}$$Hint: This is the probability that they are both born on a particular day, e.g. Sunday.
Question 10 Explanation:
Topic: Calculate the probabilities of simple and compound events and of independent and dependent events (Objective 0026).
 Question 11

#### 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 11 Explanation:
Topics: Understand various representations of functions, and translate among different representations of functional relationships (Objective 0021).
 Question 12

#### Which of the numbers below is not equivalent to 4%?

 A $$\large \dfrac{1}{25}$$Hint: 1/25=4/100, so this is equal to 4% (be sure you read the question correctly). B $$\large \dfrac{4}{100}$$Hint: 4/100=4% (be sure you read the question correctly). C $$\large 0.4$$Hint: 0.4=40% so this is not equal to 4% D $$\large 0.04$$Hint: 0.04=4/100, so this is equal to 4% (be sure you read the question correctly).
Question 12 Explanation:
Converting between fractions, decimals, and percents (Objective 0017).
 Question 13

#### 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 13 Explanation:
Topic: Compare different data sets (Objective 0025).
 Question 14

#### 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 14 Explanation:
Topic: Use algebra to solve word problems involving fractions, ratios, proportions, and percents (Objective 0020).
 Question 15

#### Which of the following is a correct equation for the graph of the line depicted above?

 A $$\large y=-\dfrac{1}{2}x+2$$Hint: The slope is -1/2 and the y-intercept is 2. You can also try just plugging in points. For example, this is the only choice that gives y=1 when x=2. B $$\large 4x=2y$$Hint: This line goes through (0,0); the graph above does not. C $$\large y=x+2$$Hint: The line pictured has negative slope. D $$\large y=-x+2$$Hint: Try plugging x=4 into this equation and see if that point is on the graph above.
Question 15 Explanation:
Topic: Find a linear equation that represents a graph (Objective 0022).
 Question 16

#### 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 16 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).
 Question 17

#### Tetrahedron

Hint:
All the faces of a tetrahedron are triangles.

#### Triangular Prism

Hint:
A prism has two congruent, parallel bases, connected by parallelograms (since this is a right prism, the parallelograms are rectangles).

#### Triangular Pyramid

Hint:
A pyramid has one base, not two.

#### Trigon

Hint:
A trigon is a triangle (this is not a common term).
Question 17 Explanation:
Topic: Classify and analyze three-dimensional figures using attributes of faces, edges, and vertices (Objective 0024).
 Question 18

#### A

Hint:
$$\frac{34}{135} \approx \frac{1}{4}$$ and $$\frac{53}{86} \approx \frac {2}{3}$$. $$\frac {1}{4}$$ of $$\frac {2}{3}$$ is small and closest to A.

#### B

Hint:
Estimate with simpler fractions.

#### C

Hint:
Estimate with simpler fractions.

#### D

Hint:
Estimate with simpler fractions.
Question 18 Explanation:
Topic: Understand meaning and models of operations on fractions (Objective 0019).
 Question 19

#### II and III

Hint:
Problem I is partitive (or partitioning or sharing) -- we put 12 objects into 3 groups. Problems II and III are quotative (or measurement) -- we put 12 objects in groups of 3.

#### All three problems model the same meaning of division

Question 36 Explanation:
Topic: Understand models of operations on numbers (Objective 0019).
 Question 37

#### Which of the following values of x satisfies the inequality $$\large \left| {{(x+2)}^{3}} \right|<3?$$

 A $$\large x=-3$$Hint: $$\left| {{(-3+2)}^{3}} \right|$$=$$\left | {(-1)}^3 \right |$$=$$\left | -1 \right |=1$$ . B $$\large x=0$$Hint: $$\left| {{(0+2)}^{3}} \right|$$=$$\left | {2}^3 \right |$$=$$\left | 8 \right |$$ =$$8$$ C $$\large x=-4$$Hint: $$\left| {{(-4+2)}^{3}} \right|$$=$$\left | {(-2)}^3 \right |$$=$$\left | -8 \right |$$ =$$8$$ D $$\large x=1$$Hint: $$\left| {{(1+2)}^{3}} \right|$$=$$\left | {3}^3 \right |$$=$$\left | 27 \right |$$ = $$27$$
Question 37 Explanation:
Topics: Laws of exponents, order of operations, interpret absolute value (Objective 0019).
 Question 38

#### The teacher can be sure that the mean and median will be the same without doing any computation.

Hint:
Does this make sense? How likely is it that the mean and median of any large data set will be the same?

#### The teacher can be sure that the mean is bigger than the median without doing any computation.

Hint:
This is a skewed distribution, and very large countries like China and India contribute huge numbers to the mean, but are counted the same as small countries like Luxembourg in the median (the same thing happens w/data on salaries, where a few very high income people tilt the mean -- that's why such data is usually reported as medians).

#### The teacher can be sure that the median is bigger than the mean without doing any computation.

Hint:
Think about a set of numbers like 1, 2, 3, 4, 10,000 -- how do the mean/median compare? How might that relate to countries of the world?

#### There is no way for the teacher to know the relative size of the mean and median without computing them.

Hint:
Knowing the shape of the distribution of populations does give us enough info to know the relative size of the mean and median, even without computing them.
Question 38 Explanation:
Topic: Use measures of central tendency (e.g., mean, median, mode) and spread to describe and interpret real-world data (Objective 0025).
 Question 39

#### What is the greatest common factor of 540 and 216?

 A $$\large{{2}^{2}}\cdot {{3}^{3}}$$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 take the smaller power for each prime that is a factor of both numbers. B $$\large2\cdot 3$$Hint: This is a common factor of both numbers, but it's not the greatest common factor. C $$\large{{2}^{3}}\cdot {{3}^{3}}$$Hint: $$2^3 = 8$$ is not a factor of 540. D $$\large{{2}^{2}}\cdot {{3}^{2}}$$Hint: This is a common factor of both numbers, but it's not the greatest common factor.
Question 39 Explanation:
Topic: Find the greatest common factor of a set of numbers (Objective 0018).
 Question 40

#### 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 40 Explanation:
Topic: Integers (Objective 0016)
 Question 41

#### The chairs in a large room can be arranged in rows of 18, 25, or 60 with no chairs left over. If C is the smallest possible number of chairs in the room, which of the following inequalities does C satisfy?

 A $$\large C\le 300$$Hint: Find the LCM. B $$\large 300 < C \le 500$$Hint: Find the LCM. C $$\large 500 < C \le 700$$Hint: Find the LCM. D $$\large C>700$$Hint: The LCM is 900, which is the smallest number of chairs.
Question 41 Explanation:
Topic: Apply LCM in "real-world" situations (according to standardized tests....) (Objective 0018).
 Question 42

#### The expression $$\large {{7}^{-4}}\cdot {{8}^{-6}}$$ is equal to which of the following?

 A $$\large \dfrac{8}{{{\left( 56 \right)}^{4}}}$$Hint: The bases are whole numbers, and the exponents are negative. How can the numerator be 8? B $$\large \dfrac{64}{{{\left( 56 \right)}^{4}}}$$Hint: The bases are whole numbers, and the exponents are negative. How can the numerator be 64? C $$\large \dfrac{1}{8\cdot {{\left( 56 \right)}^{4}}}$$Hint: $$8^{-6}=8^{-4} \times 8^{-2}$$ D $$\large \dfrac{1}{64\cdot {{\left( 56 \right)}^{4}}}$$
Question 42 Explanation:
Topics: Laws of exponents (Objective 0019).
 Question 43

#### The quotient is $$3\dfrac{1}{2}$$. There are 3 whole blocks each representing $$\dfrac{2}{3}$$ and a partial block composed of 3 small rectangles. The 3 small rectangles represent $$\dfrac{3}{6}$$ of a whole, or $$\dfrac{1}{2}$$.

Hint:
We are counting how many 2/3's are in
2 1/2: the unit becomes 2/3, not 1.

#### The quotient is $$\dfrac{4}{15}$$. There are four whole blocks separated into a total of 15 small rectangles.

Hint:
This explanation doesn't make much sense. Probably you are doing "invert and multiply," but inverting the wrong thing.

#### This picture cannot be used to find the quotient because it does not show how to separate $$2\dfrac{1}{2}$$ into equal sized groups.

Hint:
Study the measurement/quotative model of division. It's often very useful with fractions.
Question 43 Explanation:
Topic: Recognize and analyze pictorial representations of number operations. (Objective 0019).
 Question 44

#### The commutative property is used incorrectly.

Hint:
The commutative property is $$a+b=b+a$$ or $$ab=ba$$.

#### The associative property is used incorrectly.

Hint:
The associative property is $$a+(b+c)=(a+b)+c$$ or $$a \times (b \times c)=(a \times b) \times c$$.

#### The distributive property is used incorrectly.

Hint:
$$(x+3)(x+3)=x(x+3)+3(x+3)$$=$$x^2+3x+3x+9.$$
Question 44 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).
 Question 45

#### Which of the following is an irrational number?

 A $$\large \sqrt[3]{8}$$Hint: This answer is the cube root of 8. Since 2 x 2 x 2 =8, this is equal to 2, which is rational because 2 = 2/1. B $$\large \sqrt{8}$$Hint: It is not trivial to prove that this is irrational, but you can get this answer by eliminating the other choices. C $$\large \dfrac{1}{8}$$Hint: 1/8 is the RATIO of two integers, so it is rational. D $$\large -8$$Hint: Negative integers are also rational, -8 = -8/1, a ratio of integers.
Question 45 Explanation:
Topic: Identifying rational and irrational numbers (Objective 0016).
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