## 20 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

#### 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 1 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 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

#### 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 3 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 4

#### 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 4 Explanation:
Topic: Identifying rational and irrational numbers (Objective 0016).
 Question 5

#### Use the samples of a student’s work below to answer the question that follows:

$$\large \dfrac{2}{3}\times \dfrac{3}{4}=\dfrac{4\times 2}{3\times 3}=\dfrac{8}{9}$$

$$\large \dfrac{2}{5}\times \dfrac{7}{7}=\dfrac{7\times 2}{5\times 7}=\dfrac{2}{5}$$

$$\large \dfrac{7}{6}\times \dfrac{3}{4}=\dfrac{4\times 7}{6\times 3}=\dfrac{28}{18}=\dfrac{14}{9}$$

#### It is not valid. It never produces the correct answer.

Hint:
In the middle example,the answer is correct.

#### It is not valid. It produces the correct answer in a few special cases, but it’s still not a valid algorithm.

Hint:
Note that this algorithm gives a/b divided by c/d, not a/b x c/d, but some students confuse multiplication and cross-multiplication. If a=0 or if c/d =1, division and multiplication give the same answer.

#### It is valid if the rational numbers in the multiplication problem are in lowest terms.

Hint:
Lowest terms is irrelevant.

#### It is valid for all rational numbers.

Hint:
Can't be correct as the first and last examples have the wrong answers.
Question 5 Explanation:
Topic: Analyze Non-Standard Computational Algorithms (Objective 0019).
 Question 6

#### The function d(x) gives the result when 12 is divided by x.  Which of the following is a graph of d(x)?

 A Hint: d(x) is 12 divided by x, not x divided by 12. B Hint: When x=2, what should d(x) be? C Hint: When x=2, what should d(x) be? D
Question 6 Explanation:
Topic: Identify and analyze direct and inverse relationships in tables, graphs, algebraic expressions and real-world situations (Objective 0021)
 Question 7

#### 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 7 Explanation:
Topic: Recognize and apply the concept of conditional probability (Objective 0026).
 Question 8

#### 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 8 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 9

#### 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 9 Explanation:
Topic: Recognize and analyze pictorial representations of number operations. (Objective 0019).
 Question 10

#### 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 10 Explanation:
Topic: Use algebra to solve word problems involving fractions, ratios, proportions, and percents (Objective 0020).
 Question 11

#### 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 11 Explanation:
Topics: Laws of exponents (Objective 0019).
 Question 12

#### The picture below represents a board with pegs on it, where the closest distance between two pegs is 1 cm.  What is the area of the pentagon shown?

 A $$\large 8\text{ c}{{\text{m}}^{2}}$$Hint: Don't just count the dots inside, that doesn't give the area. Try adding segments so that the slanted lines become the diagonals of rectangles. B $$\large 11\text{ c}{{\text{m}}^{2}}$$Hint: Try adding segments so that the slanted lines become the diagonals of rectangles. C $$\large 11.5\text{ c}{{\text{m}}^{2}}$$Hint: An easy way to do this problem is to use Pick's Theorem (of course, it's better if you understand why Pick's theorem works): area = # pegs inside + half # pegs on the border - 1. In this case 8+9/2-1=11.5. A more appropriate strategy for elementary classrooms is to add segments; here's one way. There are 20 1x1 squares enclosed, and the total area of the triangles that need to be subtracted is 8.5 D $$\large 12.5\text{ c}{{\text{m}}^{2}}$$Hint: Try adding segments so that the slanted lines become the diagonals of rectangles.
Question 12 Explanation:
Topics: Calculate measurements and derive and use formulas for calculating the areas of geometric shapes and figures (Objective 0023).
 Question 13

#### 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 13 Explanation:
Topics: Understand various representations of functions, and translate among different representations of functional relationships (Objective 0021).
 Question 14

#### A car is traveling at 60 miles per hour.  Which of the expressions below could be used to compute how many feet the car travels in 1 second?  Note that 1 mile = 5,280 feet.

 A $$\large 60\dfrac{\text{miles}}{\text{hour}}\cdot 5280\dfrac{\text{feet}}{\text{mile}}\cdot 60\dfrac{\text{minutes}}{\text{hour}}\cdot 60\dfrac{\text{seconds}}{\text{minute}}$$Hint: This answer is not in feet/second. B $$\large 60\dfrac{\text{miles}}{\text{hour}}\cdot 5280\dfrac{\text{feet}}{\text{mile}}\cdot \dfrac{1}{60}\dfrac{\text{hour}}{\text{minutes}}\cdot \dfrac{1}{60}\dfrac{\text{minute}}{\text{seconds}}$$Hint: This is the only choice where the answer is in feet per second and the unit conversions are correct. C $$\large 60\dfrac{\text{miles}}{\text{hour}}\cdot \dfrac{1}{5280}\dfrac{\text{foot}}{\text{miles}}\cdot 60\dfrac{\text{hours}}{\text{minute}}\cdot \dfrac{1}{60}\dfrac{\text{minute}}{\text{seconds}}$$Hint: Are there really 60 hours in a minute? D $$\large 60\dfrac{\text{miles}}{\text{hour}}\cdot \dfrac{1}{5280}\dfrac{\text{mile}}{\text{feet}}\cdot 60\dfrac{\text{minutes}}{\text{hour}}\cdot \dfrac{1}{60}\dfrac{\text{minute}}{\text{seconds}}$$Hint: This answer is not in feet/second.
Question 14 Explanation:
Topic: Use unit conversions and dimensional analysis to solve measurement problems (Objective 0023).
 Question 15

#### There are 15 students for every teacher.  Let t represent the number of teachers and let s represent the number of students.  Which of the following equations is correct?

 A $$\large t=s+15$$Hint: When there are 2 teachers, how many students should there be? Do those values satisfy this equation? B $$\large s=t+15$$Hint: When there are 2 teachers, how many students should there be? Do those values satisfy this equation? C $$\large t=15s$$Hint: This is a really easy mistake to make, which comes from transcribing directly from English, "1 teachers equals 15 students." To see that it's wrong, plug in s=2; do you really need 30 teachers for 2 students? To avoid this mistake, insert the word "number," "Number of teachers equals 15 times number of students" is more clearly problematic. D $$\large s=15t$$
Question 15 Explanation:
Topic: Select the linear equation that best models a real-world situation (Objective 0022).
 Question 16

#### The prime factorization of  n can be written as n=pqr, where p, q, and r are distinct prime numbers.  How many factors does n have, including 1 and itself?

 A $$\large3$$Hint: 1, p, q, r, and pqr are already 5, so this isn't enough. You might try plugging in p=2, q=3, and r=5 to help with this problem. B $$\large5$$Hint: Don't forget pq, etc. You might try plugging in p=2, q=3, and r=5 to help with this problem. C $$\large6$$Hint: You might try plugging in p=2, q=3, and r=5 to help with this problem. D $$\large8$$Hint: 1, p, q, r, pq, pr, qr, pqr.
Question 16 Explanation:
Topic: Recognize uses of prime factorization of a number (Objective 0018).
 Question 17

#### 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 17 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 18

#### 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 18 Explanation:
Topic: Understand models of operations on numbers (Objective 0019).
 Question 19

#### Commutative Property.

Hint:
For addition, the commutative property is $$a+b=b+a$$ and for multiplication it's $$a \times b = b \times a$$.

#### Associative Property.

Hint:
For addition, the associative property is $$(a+b)+c=a+(b+c)$$ and for multiplication it's $$(a \times b) \times c=a \times (b \times c)$$

#### Identity Property.

Hint:
0 is the additive identity, because $$a+0=a$$ and 1 is the multiplicative identity because $$a \times 1=a$$. The phrase "identity property" is not standard.

#### Distributive Property.

Hint:
$$(25+1) \times 16 = 25 \times 16 + 1 \times 16$$. This is an example of the distributive property of multiplication over addition.
Question 19 Explanation:
Topic: Analyze and justify mental math techniques, by applying arithmetic properties such as commutative, distributive, and associative (Objective 0019). Note that it's hard to write a question like this as a multiple choice question -- worthwhile to understand why the other steps work too.
 Question 20

#### 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 20 Explanation:
Topic: Understand the definition of function and various representations of functions (e.g., input/output machines, tables, graphs, mapping diagrams, formulas) (Objective 0021).
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