## 45 Random Questions

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

#### What fraction of the area of the picture below is shaded?

 A $$\large \dfrac{17}{24}$$Hint: You might try adding segments so each quadrant is divided into 6 pieces with equal area -- there will be 24 regions, not all the same shape, but all the same area, with 17 of them shaded (for the top left quarter, you could also first change the diagonal line to a horizontal or vertical line that divides the square in two equal pieces and shade one) . B $$\large \dfrac{3}{4}$$Hint: Be sure you're taking into account the different sizes of the pieces. C $$\large \dfrac{2}{3}$$Hint: The bottom half of the picture is 2/3 shaded, and the top half is more than 2/3 shaded, so this answer is too small. D $$\large \dfrac{17}{6}$$Hint: This answer is bigger than 1, so doesn't make any sense. Be sure you are using the whole picture, not one quadrant, as the unit.
Question 1 Explanation:
Topic: Models of Fractions (Objective 0017)
 Question 2

#### In which table below is y a function of x?

 A Hint: If x=3, y can have two different values, so it's not a function. B Hint: If x=3, y can have two different values, so it's not a function. C Hint: If x=1, y can have different values, so it's not a function. D Hint: Each value of x always corresponds to the same value of y.
Question 2 Explanation:
Topic: Understand the definition of function and various representations of functions (e.g., input/output machines, tables, graphs, mapping diagrams, formulas) (Objective 0021).
 Question 3

#### 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 25 Explanation:
Topic: Understand models of operations on numbers (Objective 0019).
 Question 26

#### 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 26 Explanation:
Topic: Demonstrate knowledge of divisibility rules (Objective 0018).
 Question 27

#### 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 27 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 28

#### 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 28 Explanation:
Topic: Find the greatest common factor of a set of numbers (Objective 0018).
 Question 29

#### In March of 2012, 1 dollar was worth the same as 0.761 Euros, and 1 dollar was also worth the same as 83.03 Japanese Yen.  Which of the expressions below gives the number of Yen that are worth 1 Euro?

 A $$\large {83}.0{3}\cdot 0.{761}$$Hint: This equation gives less than the number of yen per dollar, but 1 Euro is worth more than 1 dollar. B $$\large \dfrac{0.{761}}{{83}.0{3}}$$Hint: Number is way too small. C $$\large \dfrac{{83}.0{3}}{0.{761}}$$Hint: One strategy here is to use easier numbers, say 1 dollar = .5 Euros and 100 yen, then 1 Euro would be 200 Yen (change the numbers in the equations and see what works). Another is to use dimensional analysis: we want # yen per Euro, or yen/Euro = yen/dollar $$\times$$ dollar/Euro = $$83.03 \times \dfrac {1}{0.761}$$ D $$\large \dfrac{1}{0.{761}}\cdot \dfrac{1}{{83}.0{3}}$$Hint: Number is way too small.
Question 29 Explanation:
Topic: Analyze the relationships among proportions, constant rates, and linear functions (Objective 0022).
 Question 30

#### 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 30 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 31

#### 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 31 Explanation:
Topic: Identifying rational and irrational numbers (Objective 0016).
 Question 32

#### 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 32 Explanation:
Topic: Solve a variety of measurement problems (e.g., time, temperature, rates, average rates of change) in real-world situations (Objective 0023).
 Question 33

#### Below are several expressions

 I. $$\large \dfrac{1}{2}+\dfrac{1}{3}$$ II. $$\large .400000$$ III. $$\large\dfrac{1}{5}+\dfrac{1}{5}$$ IV. $$\large 40\%$$ V. $$\large 0.25$$ VI. $$\large\dfrac{14}{35}$$

#### I, III, V, VI

Hint:
I and V are not at all how fractions and decimals work.

#### III, VI

Hint:
These are right, but there are more.

#### II, III, VI

Hint:
These are right, but there are more.

#### II, III, IV, VI

Question 33 Explanation:
Topic: Converting between fractions, decimals, and percents (Objective 0017)
 Question 34

#### George left home at 10:00 and drove to work on a crooked path. He was stopped in traffic at 10:30 and 10:45. He drove 30 miles total.

Hint:
Just because he ended up 30 miles from home doesn't mean he drove 30 miles total.

#### George drove to work. On the way to work there is a little hill and a big hill. He slowed down for them. He made it to work at 11:15.

Hint:
The graph is not a picture of the roads.

#### George left home at 10:15. He drove 10 miles, then realized he’d forgotten something at home. He turned back and got what he’d forgotten. Then he drove in a straight line, at many different speeds, until he got to work around 11:15.

Hint:
A straight line on a distance versus time graph means constant speed.

#### George left home at 10:15. He drove 10 miles, then realized he’d forgotten something at home. He turned back and got what he’d forgotten. Then he drove at a constant speed until he got to work around 11:15.

Question 34 Explanation:
Topic: Use qualitative graphs to represent functional relationships in the real world (Objective 0021).
 Question 35

#### What is the perimeter of the window glass?

 A $$\large 3x+\dfrac{\pi x}{2}$$Hint: By definition, $$\pi$$ is the ratio of the circumference of a circle to its diameter; thus the circumference is $$\pi d$$. Since we have a semi-circle, its perimeter is $$\dfrac{1}{2} \pi x$$. Only 3 sides of the square contribute to the perimeter. B $$\large 3x+2\pi x$$Hint: Make sure you know how to find the circumference of a circle. C $$\large 3x+\pi x$$Hint: Remember it's a semi-circle, not a circle. D $$\large 4x+2\pi x$$Hint: Only 3 sides of the square contribute to the perimeter.
Question 35 Explanation:
Topic: Derive and use formulas for calculating the lengths, perimeters, areas, volumes, and surface areas of geometric shapes and figures (Objective 0023).
 Question 36

#### 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 36 Explanation:
Converting between fractions, decimals, and percents (Objective 0017).
 Question 37

#### 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 37 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 38

#### 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 38 Explanation:
Topic: Analyze and justify standard and non-standard computational techniques (Objective 0019).
 Question 39

#### 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 39 Explanation:
Topic: Calculate the probabilities of simple and compound events and of independent and dependent events (Objective 0026).
 Question 40

#### 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 40 Explanation:
Topic: Characteristics of composite numbers (Objective 0018).
 Question 41

#### 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 41 Explanation:
Topic: Calculate the probabilities of simple and compound events and of independent and dependent events (Objective 0026).
 Question 42

#### If the polygon shown above is reflected about the y axis and then rotated 90 degrees clockwise about the origin, which of the following graphs is the result?

 A Hint: Try following the point (1,4) to see where it goes after each transformation. B C Hint: Make sure you're reflecting in the correct axis. D Hint: Make sure you're rotating the correct direction.
Question 42 Explanation:
Topic: Analyze and apply geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to concepts of symmetry, similarity, and congruence; and use these concepts to solve problems (Objective 0024).
 Question 43

#### $$\large A-B+C\div D\times E$$?

 A $$\large A-B-\dfrac{C}{DE}$$Hint: In the order of operations, multiplication and division have the same priority, so do them left to right; same with addition and subtraction. B $$\large A-B+\dfrac{CE}{D}$$Hint: In practice, you're better off using parentheses than writing an expression like the one in the question. The PEMDAS acronym that many people memorize is misleading. Multiplication and division have equal priority and are done left to right. They have higher priority than addition and subtraction. Addition and subtraction also have equal priority and are done left to right. C $$\large \dfrac{AE-BE+CE}{D}$$Hint: Use order of operations, don't just compute left to right. D $$\large A-B+\dfrac{C}{DE}$$Hint: In the order of operations, multiplication and division have the same priority, so do them left to right
Question 43 Explanation:
Topic: Justify algebraic manipulations by application of the properties of order of operations (Objective 0020).
 Question 44

#### 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 44 Explanation:
Topic: Using number lines (Objective 0017)
 Question 45

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