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


Your answers are highlighted below.
Question 1

Use the four figures below to answer the question that follows:

How many of the figures pictured above have at least one line of reflective symmetry?

A
\( \large 1\)
B
\( \large 2\)
Hint:
The ellipse has 2 lines of reflective symmetry (horizontal and vertical, through the center) and the triangle has 3. The other two figures have rotational symmetry, but not reflective symmetry.
C
\( \large 3\)
D
\( \large 4\)
Hint:
All four have rotational symmetry, but not reflective symmetry.
Question 1 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 2

The window glass below has the shape of a semi-circle on top of a square, where the side of the square has length x.  It was cut from one piece of glass.

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 2 Explanation: 
Topic: Derive and use formulas for calculating the lengths, perimeters, areas, volumes, and surface areas of geometric shapes and figures (Objective 0023).
Question 3

Elena is going to use a calculator to check whether or not 267 is prime. She will pick certain divisors, and then find 267 divided by each, and see if she gets a whole number. If she never gets a whole number, then she€™s found a prime. Which numbers does Elena NEED to check before she can stop checking and be sure she has a prime?

A

All natural numbers from 2 to 266.

Hint:
She only needs to check primes -- checking the prime factors of any composite is enough to look for divisors. As a test taking strategy, the other three choices involve primes, so worth thinking about.
B

All primes from 2 to 266 .

Hint:
Remember, factors come in pairs (except for square root factors), so she would first find the smaller of the pair and wouldn't need to check the larger.
C

All primes from 2 to 133 .

Hint:
She doesn't need to check this high. Factors come in pairs, and something over 100 is going to be paired with something less than 3, so she will find that earlier.
D

All primes from \( \large 2\) to \( \large \sqrt{267}\).

Hint:
\(\sqrt{267} \times \sqrt{267}=267\). Any other pair of factors will have one factor less than \( \sqrt{267}\) and one greater, so she only needs to check up to \( \sqrt{267}\).
Question 3 Explanation: 
Topic: Identify prime and composite numbers (Objective 0018).
Question 4

Which property is not shared by all rhombi?

A

4 congruent sides

Hint:
The most common definition of a rhombus is a quadrilateral with 4 congruent sides.
B

A center of rotational symmetry

Hint:
The diagonal of a rhombus separates it into two congruent isosceles triangles. The center of this line is a center of 180 degree rotational symmetry that switches the triangles.
C

4 congruent angles

Hint:
Unless the rhombus is a square, it does not have 4 congruent angles.
D

2 sets of parallel sides

Hint:
All rhombi are parallelograms.
Question 4 Explanation: 
Topic: Classify and analyze polygons using attributes of sides and angles, and symmetry (Objective 0024).
Question 5

Use the expression below to answer the question that follows.

      \( \large 3\times {{10}^{4}}+2.2\times {{10}^{2}}\)

Which of the following is closest to the expression above?

A

Five million

Hint:
Pay attention to the exponents. Adding 3 and 2 doesn't work because they have different place values.
B

Fifty thousand

Hint:
Pay attention to the exponents. Adding 3 and 2 doesn't work because they have different place values.
C

Three million

Hint:
Don't add the exponents.
D

Thirty thousand

Hint:
\( 3\times {{10}^{4}} = 30,000;\) the other term is much smaller and doesn't change the estimate.
Question 5 Explanation: 
Topics: Place value, scientific notation, estimation (Objective 0016)
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

At a school fundraising event, people can buy a ticket to spin a spinner like the one below.  The region that the spinner lands in tells which, if any, prize the person wins.

If 240 people buy tickets to spin the spinner, what is the best estimate of the number of keychains that will be given away?

A

40

Hint:
"Keychain" appears on the spinner twice.
B

80

Hint:
The probability of getting a keychain is 1/3, and so about 1/3 of the time the spinner will win.
C

100

Hint:
What is the probability of winning a keychain?
D

120

Hint:
That would be the answer for getting any prize, not a keychain specifically.
Question 7 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 8

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 8 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 9

Use the samples of a student€™s work below to answer the question that follows:

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

\( \large \dfrac{2}{5}\times \dfrac{7}{7}=\dfrac{7\times 2}{5\times 7}=\dfrac{2}{5}\)

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

Which of the following best describes the mathematical validity of the algorithm the student is using?

A

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

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

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.
C

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

Hint:
Lowest terms is irrelevant.
D

It is valid for all rational numbers.

Hint:
Can't be correct as the first and last examples have the wrong answers.
Question 9 Explanation: 
Topic: Analyze Non-Standard Computational Algorithms (Objective 0019).
Question 10

Here is a number trick:

 1) Pick a whole number

 2) Double your number.

 3) Add 20 to the above result.

 4) Multiply the above by 5

 5) Subtract 100

 6) Divide by 10

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 10 Explanation: 
Topic: Recognize and apply the concepts of variable, function, equality, and equation to express relationships algebraically (Objective 0020).
Question 11

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 11 Explanation: 
Topic: Converting between fraction and decimal representations (Objective 0017)
Question 12

Which of the following sets of polygons can be assembled to form a pentagonal pyramid?

A

2 pentagons and 5 rectangles.

Hint:
These can be assembled to form a pentagonal prism, not a pentagonal pyramid.
B

1 square and 5 equilateral triangles.

Hint:
You need a pentagon for a pentagonal pyramid.
C

1 pentagon and 5 isosceles triangles.

D

1 pentagon and 10 isosceles triangles.

Question 12 Explanation: 
Topic:Classify and analyze three-dimensional figures using attributes of faces, edges, and vertices (Objective 0024).
Question 13

Below is a portion of a number line:

 Point B is halfway between two tick marks.  What number is represented by Point B?

 
A
\( \large 0.645\)
Hint:
That point is marked on the line, to the right.
B
\( \large 0.6421\)
Hint:
That point is to the left of point B.
C
\( \large 0.6422\)
Hint:
That point is to the left of point B.
D
\( \large 0.6425\)
Question 13 Explanation: 
Topic: Using Number Lines (Objective 0017)
Question 14

Kendra is trying to decide which fraction is greater, \(  \dfrac{4}{7}\) or \(  \dfrac{5}{8}\). Which of the following answers shows the best reasoning?

A

\( \dfrac{4}{7}\) is \( \dfrac{3}{7}\)away from 1, and \( \dfrac{5}{8}\) is \( \dfrac{3}{8}\)away from 1. Since eighth‘s are smaller than seventh‘s, \( \dfrac{5}{8}\) is closer to 1, and is the greater of the two fractions.

B

\( 7-4=3\) and \( 8-5=3\), so the fractions are equal.

Hint:
Not how to compare fractions. By this logic, 1/2 and 3/4 are equal, but 1/2 and 2/4 are not.
C

\( 4\times 8=32\) and \( 7\times 5=35\). Since \( 32<35\) , \( \dfrac{5}{8}<\dfrac{4}{7}\)

Hint:
Starts out as something that works, but the conclusion is wrong. 4/7 = 32/56 and 5/8 = 35/56. The cross multiplication gives the numerators, and 35/56 is bigger.
D

\( 4<5\) and \( 7<8\), so \( \dfrac{4}{7}<\dfrac{5}{8}\)

Hint:
Conclusion is correct, logic is wrong. With this reasoning, 1/2 would be less than 2/100,000.
Question 14 Explanation: 
Topics: Comparing fractions, and understanding the meaning of fractions (Objective 0017).
Question 15

Use the table below to answer the question that follows:

Each number in the table above represents a value W that is determined by the values of x and y.  For example, when x=3 and y=1, W=5.  What is the value of W when x=9 and y=14?  Assume that the patterns in the table continue as shown.

A
\( \large W=-5\)
Hint:
When y is even, W is even.
B
\( \large W=4\)
Hint:
Note that when x increases by 1, W increases by 2, and when y increases by 1, W decreases by 1. At x=y=0, W=0, so at x=9, y=14, W has increased by \(9 \times 2\) and decreased by 14, or W=18-14=4.
C
\( \large W=6\)
Hint:
Try fixing x or y at 0, and start by finding W for x=0 y=14 or x=9, y=0.
D
\( \large W=32\)
Hint:
Try fixing x or y at 0, and start by finding W for x=0 y=14 or x=9, y=0.
Question 15 Explanation: 
Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic) (Objective 0021)
Question 16

The Venn Diagram below gives data on the number of seniors, athletes, and vegetarians in the student body at a college:

How many students at the college are seniors who are not vegetarians?

A
\( \large 137\)
Hint:
Doesn't include the senior athletes who are not vegetarians.
B
\( \large 167\)
C
\( \large 197\)
Hint:
That's all seniors, including vegetarians.
D
\( \large 279\)
Hint:
Includes all athletes who are not vegetarians, some of whom are not seniors.
Question 16 Explanation: 
Topic: Venn Diagrams (Objective 0025)
Question 17

How many lines of reflective symmetry and how many centers of rotational symmetry does the parallelogram depicted below have?

 
A

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?
B

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?
C

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.
D

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 17 Explanation: 
Topic: Analyze geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to concepts of symmetry (Objective 0024).
Question 18

Below are four inputs and outputs for a function machine representing the function A:

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 18 Explanation: 
Topics: Understand various representations of functions, and translate among different representations of functional relationships (Objective 0021).
Question 19

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 19 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 20

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 20 Explanation: 
Topic: Use algebra to solve word problems involving fractions, ratios, proportions, and percents (Objective 0020).
Question 21

The picture below shows identical circles drawn on a piece of paper.  The rectangle represents an index card that is blocking your view of \( \dfrac{3}{5}\) of the circles on the paper.  How many circles are covered by the rectangle?

A

4

Hint:
The card blocks more than half of the circles, so this number is too small.
B

5

Hint:
The card blocks more than half of the circles, so this number is too small.
C

8

Hint:
The card blocks more than half of the circles, so this number is too small.
D

12

Hint:
2/5 of the circles or 8 circles are showing. Thus 4 circles represent 1/5 of the circles, and \(4 \times 5=20\) circles represent 5/5 or all the circles. Thus 12 circles are hidden.
Question 21 Explanation: 
Topic: Models of Fractions (Objective 0017)
Question 22

Use the graph below to answer the question that follows:

 

The graph above best matches which of the following scenarios:

A

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.
B

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.
C

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.
D

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 22 Explanation: 
Topic: Use qualitative graphs to represent functional relationships in the real world (Objective 0021).
Question 23

On a map the distance from Boston to Detroit is 6 cm, and these two cities are 702 miles away from each other. Assuming the scale of the map is the same throughout, which answer below is closest to the distance between Boston and San Francisco on the map, given that they are 2,708 miles away from each other?

A

21 cm

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

22 cm

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

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.
D

24 cm

Hint:
4 groups of 6 cm is over 2800 miles on the map, which is too much.
Question 23 Explanation: 
Topic: Apply proportional thinking to estimate quantities in real world situations (Objective 0019).
Question 24

Use the expression below to answer the question that follows:

                 \( \large \dfrac{\left( 7,154 \right)\times \left( 896 \right)}{216}\)

Which of the following is the best estimate of the expression above?

A

2,000

Hint:
The answer is bigger than 7,000.
B

20,000

Hint:
Estimate 896/216 first.
C

3,000

Hint:
The answer is bigger than 7,000.
D

30,000

Hint:
\( \dfrac{896}{216} \approx 4\) and \(7154 \times 4\) is over 28,000, so this answer is closest.
Question 24 Explanation: 
Topics: Estimation, simplifying fractions (Objective 0016, overlaps with other objectives).
Question 25

Which of the following is equivalent to

\( \large A-B+C\div D\times E\)?

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 25 Explanation: 
Topic: Justify algebraic manipulations by application of the properties of order of operations (Objective 0020).
Question 26

Aya and Kendra want to estimate the height of a tree. On a sunny day, Aya measures Kendra€™s shadow as 3 meters long, and Kendra measures the tree€™s shadow as 15 meters long. Kendra is 1.5 meters tall. How tall is the tree?

A

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.
B

22.5 meters

Hint:
Draw a picture.
C

30 meters

Hint:
Draw a picture.
D

45 meters

Hint:
Draw a picture.
Question 26 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 27

Solve for x: \(\large 4-\dfrac{2}{3}x=2x\)

A
\( \large x=3\)
Hint:
Try plugging x=3 into the equation.
B
\( \large x=-3\)
Hint:
Left side is positive, right side is negative when you plug this in for x.
C
\( \large x=\dfrac{3}{2}\)
Hint:
One way to solve: \(4=\dfrac{2}{3}x+2x\) \(=\dfrac{8}{3}x\).\(x=\dfrac{3 \times 4}{8}=\dfrac{3}{2}\). Another way is to just plug x=3/2 into the equation and see that each side equals 3 -- on a multiple choice test, you almost never have to actually solve for x.
D
\( \large x=-\dfrac{3}{2}\)
Hint:
Left side is positive, right side is negative when you plug this in for x.
Question 27 Explanation: 
Topic: Solve linear equations (Objective 0020).
Question 28

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 28 Explanation: 
Topics: Laws of exponents, order of operations, interpret absolute value (Objective 0019).
Question 29

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 29 Explanation: 
Topics: Laws of exponents (Objective 0019).
Question 30

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 30 Explanation: 
Topics: Calculate measurements and derive and use formulas for calculating the areas of geometric shapes and figures (Objective 0023).
Question 31

What is the least common multiple of 540 and 216?

A
\( \large{{2}^{5}}\cdot {{3}^{6}}\cdot 5\)
Hint:
This is the product of the numbers, not the LCM.
B
\( \large{{2}^{3}}\cdot {{3}^{3}}\cdot 5\)
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 for each prime that's a factor of either number, use the largest exponent that appears in one of the factorizations. You can also take the product of the two numbers divided by their GCD.
C
\( \large{{2}^{2}}\cdot {{3}^{3}}\cdot 5\)
Hint:
216 is a multiple of 8.
D
\( \large{{2}^{2}}\cdot {{3}^{2}}\cdot {{5}^{2}}\)
Hint:
Not a multiple of 216 and not a multiple of 540.
Question 31 Explanation: 
Topic: Find the least common multiple of a set of numbers (Objective 0018).
Question 32

The American€™s with Disabilties Act (ADA) regulations state that the maximum slope for a wheelchair ramp in new construction is 1:12, although slopes between 1:16 and 1:20 are preferred.  The maximum rise for any run is 30 inches.   The graph below shows the rise and runs of four different wheelchair ramps.  Which ramp is in compliance with the ADA regulations for new construction?

A

A

Hint:
Rise is more than 30 inches.
B

B

Hint:
Run is almost 24 feet, so rise can be almost 2 feet.
C

C

Hint:
Run is 12 feet, so rise can be at most 1 foot.
D

D

Hint:
Slope is 1:10 -- too steep.
Question 32 Explanation: 
Topic: Interpret meaning of slope in a real world situation (Objective 0022).
Question 33

Which of the following is closest to the height of a college student in centimeters?

A

1.6 cm

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

16 cm

Hint:
Less than knee high on most college students.
C

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.
D

1600 cm

Hint:
This college student might be taller than some campus buildings!
Question 33 Explanation: 
Topic: Estimate and calculate measurements using customary, metric, and nonstandard units of measurement (Objective 0023).
Question 34

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 34 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 35

Here is a mental math strategy for computing 26 x 16:

Step 1: 100 x 16 = 1600

Step 2: 25 x 16 = 1600 ÷· 4 = 400

Step 3: 26 x 16 = 400 + 16 = 416

Which property best justifies Step 3 in this strategy?

A

Commutative Property.

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

Associative Property.

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

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 35 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 36

Which of the following is equal to one million three hundred thousand?

A
\(\large1.3\times {{10}^{6}}\)
B
\(\large1.3\times {{10}^{9}}\)
Hint:
That's one billion three hundred million.
C
\(\large1.03\times {{10}^{6}}\)
Hint:
That's one million thirty thousand.
D
\(\large1.03\times {{10}^{9}}\)
Hint:
That's one billion thirty million
Question 36 Explanation: 
Topic: Scientific Notation (Objective 0016)
Question 37

Exactly one of the numbers below is a prime number.  Which one is it?

A
\( \large511 \)
Hint:
Divisible by 7.
B
\( \large517\)
Hint:
Divisible by 11.
C
\( \large519\)
Hint:
Divisible by 3.
D
\( \large521\)
Question 37 Explanation: 
Topics: Identify prime and composite numbers and demonstrate knowledge of divisibility rules (Objective 0018).
Question 38

The polygon depicted below is drawn on dot paper, with the dots spaced 1 unit apart.  What is the perimeter of the polygon?

A
\( \large 18+\sqrt{2} \text{ units}\)
Hint:
Be careful with the Pythagorean Theorem.
B
\( \large 18+2\sqrt{2}\text{ units}\)
Hint:
There are 13 horizontal or vertical 1 unit segments. The longer diagonal is the hypotenuse of a 3-4-5 right triangle, so its length is 5 units. The shorter diagonal is the hypotenuse of a 45-45-90 right triangle with side 2, so its hypotenuse has length \(2 \sqrt{2}\).
C
\( \large 18 \text{ units} \)
Hint:
Use the Pythagorean Theorem to find the lengths of the diagonal segments.
D
\( \large 20 \text{ units}\)
Hint:
Use the Pythagorean Theorem to find the lengths of the diagonal segments.
Question 38 Explanation: 
Topic: Recognize and apply connections between algebra and geometry (e.g., the use of coordinate systems, the Pythagorean theorem) (Objective 0024).
Question 39

The table below gives the result of a survey at a college, asking students whether they were residents or commuters:

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 39 Explanation: 
Topic: Recognize and apply the concept of conditional probability (Objective 0026).
Question 40

What is the perimeter of a right triangle with legs of lengths x and 2x?

A
\( \large 6x\)
Hint:
Use the Pythagorean Theorem.
B
\( \large 3x+5{{x}^{2}}\)
Hint:
Don't forget to take square roots when you use the Pythagorean Theorem.
C
\( \large 3x+\sqrt{5}{{x}^{2}}\)
Hint:
\(\sqrt {5 x^2}\) is not \(\sqrt {5}x^2\).
D
\( \large 3x+\sqrt{5}{{x}^{{}}}\)
Hint:
To find the hypotenuse, h, use the Pythagorean Theorem: \(x^2+(2x)^2=h^2.\) \(5x^2=h^2,h=\sqrt{5}x\). The perimeter is this plus x plus 2x.
Question 40 Explanation: 
Topic: Recognize and apply connections between algebra and geometry (e.g., the use of coordinate systems, the Pythagorean theorem) (Objective 0024).
Question 41

A homeowner is planning to tile the kitchen floor with tiles that measure 6 inches by 8 inches.  The kitchen floor is a rectangle that measures 10 ft by 12 ft, and there are no gaps between the tiles.  How many tiles does the homeowner need?

A

30

Hint:
The floor is 120 sq feet, and the tiles are smaller than 1 sq foot. Also, remember that 1 sq foot is 12 \(\times\) 12=144 sq inches.
B

120

Hint:
The floor is 120 sq feet, and the tiles are smaller than 1 sq foot.
C

300

Hint:
Recheck your calculations.
D

360

Hint:
One way to do this is to note that 6 inches = 1/2 foot and 8 inches = 2/3 foot, so the area of each tile is 1/2 \(\times\) 2/3=1/3 sq foot, or each square foot of floor requires 3 tiles. The area of the floor is 120 square feet. Note that the tiles would fit evenly oriented in either direction, parallel to the walls.
Question 41 Explanation: 
Topic: Estimate and calculate measurements, use unit conversions to solve measurement problems, solve measurement problems in real-world situations (Objective 0023).
Question 42

Here is a student€™s work on several multiplication problems:

For which of the following problems is this student most likely to get the correct solution, even though he is using an incorrect algorithm?

A

58 x 22

Hint:
This problem involves regrouping, which the student does not do correctly.
B

16 x 24

Hint:
This problem involves regrouping, which the student does not do correctly.
C

31 x 23

Hint:
There is no regrouping with this problem.
D

141 x 32

Hint:
This problem involves regrouping, which the student does not do correctly.
Question 42 Explanation: 
Topic: Analyze computational algorithms (Objective 0019).
Question 43

A teacher has a list of all the countries in the world and their populations in March 2012.  She is going to have her students use technology to compute the mean and median of the numbers on the list.   Which of the following statements is true?

A

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?
B

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).
C

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?
D

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 43 Explanation: 
Topic: Use measures of central tendency (e.g., mean, median, mode) and spread to describe and interpret real-world data (Objective 0025).
Question 44

Use the solution procedure below to answer the question that follows:

\( \large {\left( x+3 \right)}^{2}=10\)

\( \large \left( x+3 \right)\left( x+3 \right)=10\)

\( \large {x}^{2}+9=10\)

\( \large {x}^{2}+9-9=10-9\)

\( \large {x}^{2}=1\)

\( \large x=1\text{ or }x=-1\)

Which of the following is incorrect in the procedure shown above?

A

The commutative property is used incorrectly.

Hint:
The commutative property is \(a+b=b+a\) or \(ab=ba\).
B

The associative property is used incorrectly.

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

Order of operations is done incorrectly.

D

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

Some children explored the diagonals in 2 x 2 squares on pages of a calendar (where all four squares have numbers in them).  They conjectured that the sum of the diagonals is always equal; in the example below, 8+16=9+15.

 

Which of the equations below could best be used to explain why the children€™s conjecture is correct?

A
\( \large 8x+16x=9x+15x\)
Hint:
What would x represent in this case? Make sure you can describe in words what x represents.
B
\( \large x+(x+2)=(x+1)+(x+1)\)
Hint:
What would x represent in this case? Make sure you can describe in words what x represents.
C
\( \large x+(x+8)=(x+1)+(x+7)\)
Hint:
x is the number in the top left square, x+8 is one below and to the right, x+1 is to the right of x, and x+7 is below x.
D
\( \large x+8+16=x+9+15\)
Hint:
What would x represent in this case? Make sure you can describe in words what x represents.
Question 45 Explanation: 
Topic: Recognize and apply the concepts of variable, equality, and equation to express relationships algebraically (Objective 0020).
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