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

P is a prime number that divides 240.  Which of the following must be true?

A

P divides 30

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

P divides 48

Hint:
P=5 doesn't work.
C

P divides 75

Hint:
P=2 doesn't work.
D

P divides 80

Hint:
P=3 doesn't work.
Question 1 Explanation: 
Topic: Find the prime factorization of a number and recognize its uses (Objective 0018).
Question 2

Use the table below to answer the question that follows:

Gordon wants to buy three pounds of nuts.  Each of the stores above ordinarily sells the nuts for $4.99 a pound, but is offering a discount this week.  At which store can he buy the nuts for the least amount of money?

A

Store A

Hint:
This would save about $2.50. You can quickly see that D saves more.
B

Store B

Hint:
This saves 15% and C saves 25%.
C

Store C

D

Store D

Hint:
This is about 20% off, which is less of a discount than C.
Question 2 Explanation: 
Topic: Understand the meanings and models of integers, fractions, decimals,percents, and mixed numbers and apply them to the solution of word problems (Objective 0017).
Question 3

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 3 Explanation: 
Topic: Demonstrate knowledge of divisibility rules (Objective 0018).
Question 4

The histogram below shows the number of pairs of footware owned by a group of college students.

Which of the following statements can be inferred from the graph above?

A

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.
B

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....
C

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.
D

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 4 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 5

Use the problem below to answer the question that follows:

T shirts are on sale for 20% off. Tasha paid $8.73 for a shirt.  What is the regular price of the shirt? There is no tax on clothing purchases under $175.

Let p represent the regular price of these t-shirt. Which of the following equations is correct?

A
\( \large 0.8p=\$8.73\)
Hint:
80% of the regular price = $8.73.
B
\( \large \$8.73+0.2*\$8.73=p\)
Hint:
The 20% off was off of the ORIGINAL price, not off the $8.73 (a lot of people make this mistake). Plus this is the same equation as in choice c.
C
\( \large 1.2*\$8.73=p\)
Hint:
The 20% off was off of the ORIGINAL price, not off the $8.73 (a lot of people make this mistake). Plus this is the same equation as in choice b.
D
\( \large p-0.2*\$8.73=p\)
Hint:
Subtract p from both sides of this equation, and you have -.2 x 8.73 =0.
Question 5 Explanation: 
Topics: Use algebra to solve word problems involving percents and identify variables, and derive algebraic expressions that represent real-world situations (Objective 0020).
Question 6

What set of transformations will transform the leftmost image into the rightmost image?

 
A

A 90 degree clockwise rotation about (2,1) followed by a translation of two units to the right.

Hint:
Part of the figure would move below the x-axis with these transformations.
B

A translation 3 units up, followed by a reflection about the line y=x.

Hint:
See what happens to the point (5,1) under this set of transformations.
C

A 90 degree clockwise rotation about (5,1), followed by a translation of 2 units up.

D

A 90 degree clockwise rotation about (2,1) followed by a translation of 2 units to the right.

Hint:
See what happens to the point (3,3) under this set of transformations.
Question 6 Explanation: 
Topic:Analyze and apply geometric transformations (e.g., translations, rotations, reflections, dilations) (Objective 0024).
Question 7

If two fair coins are flipped, what is the probability that one will come up heads and the other tails?

A
\( \large \dfrac{1}{4}\)
Hint:
Think of the coins as a penny and a dime, and list all possibilities.
B
\( \large \dfrac{1}{3} \)
Hint:
This is a very common misconception. There are three possible outcomes -- both heads, both tails, and one of each -- but they are not equally likely. Think of the coins as a penny and a dime, and list all possibilities.
C
\( \large \dfrac{1}{2}\)
Hint:
The possibilities are HH, HT, TH, TT, and all are equally likely. Two of the four have one of each coin, so the probability is 2/4=1/2.
D
\( \large \dfrac{3}{4}\)
Hint:
Think of the coins as a penny and a dime, and list all possibilities.
Question 7 Explanation: 
Topic: Calculate the probabilities of simple and compound events and of independent and dependent events (Objective 0026).
Question 8

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 8 Explanation: 
Topics: Laws of exponents, order of operations, interpret absolute value (Objective 0019).
Question 9

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 9 Explanation: 
Topics: Understand various representations of functions, and translate among different representations of functional relationships (Objective 0021).
Question 10

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 10 Explanation: 
Topic: Using Number Lines (Objective 0017)
Question 11

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

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

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

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

A

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

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

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

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

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

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

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

Hint:
When we have common denominators, the problem is in the form a/b divided by c/b, and the answer is a/c, as the student's algorithm predicts.
Question 11 Explanation: 
Topic: Analyze Non-Standard Computational Algorithms (Objective 0019).
Question 12

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 12 Explanation: 
Topic: Derive and use formulas for calculating the lengths, perimeters, areas, volumes, and surface areas of geometric shapes and figures (Objective 0023).
Question 13

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 13 Explanation: 
Topic: Recognize and apply connections between algebra and geometry (e.g., the use of coordinate systems, the Pythagorean theorem) (Objective 0024).
Question 14

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 14 Explanation: 
Topics: Place value, scientific notation, estimation (Objective 0016)
Question 15

Use the graph below to answer the question that follows.

 

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

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 16 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 17

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 17 Explanation: 
Topic: Use algebra to solve word problems involving fractions, ratios, proportions, and percents (Objective 0020).
Question 18

Below are front, side, and top views of a three-dimensional solid.

Which of the following could be the solid shown above?

A

A sphere

Hint:
All views would be circles.
B

A cylinder

C

A cone

Hint:
Two views would be triangles, not rectangles.
D

A pyramid

Hint:
How would one view be a circle?
Question 18 Explanation: 
Topic: Match three-dimensional figures and their two-dimensional representations (e.g., nets, projections, perspective drawings) (Objective 0024).
Question 19

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 19 Explanation: 
Topic: Venn Diagrams (Objective 0025)
Question 20

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 20 Explanation: 
Topics: Comparing fractions, and understanding the meaning of fractions (Objective 0017).
Question 21

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 21 Explanation: 
Topic: Identifying rational and irrational numbers (Objective 0016).
Question 22

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 22 Explanation: 
Topic: Analyze geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to concepts of symmetry (Objective 0024).
Question 23

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 23 Explanation: 
Converting between fractions, decimals, and percents (Objective 0017).
Question 24

Which of the lists below is in order from least to greatest value?

A
\( \large -0.044,\quad -0.04,\quad 0.04,\quad 0.044\)
Hint:
These are easier to compare if you add trailing zeroes (this is finding a common denominator) -- all in thousandths, -0.044, -0.040,0 .040, 0.044. The middle two numbers, -0.040 and 0.040 can be modeled as owing 4 cents and having 4 cents. The outer two numbers are owing or having a bit more.
B
\( \large -0.04,\quad -0.044,\quad 0.044,\quad 0.04\)
Hint:
0.04=0.040, which is less than 0.044.
C
\( \large -0.04,\quad -0.044,\quad 0.04,\quad 0.044\)
Hint:
-0.04=-0.040, which is greater than \(-0.044\).
D
\( \large -0.044,\quad -0.04,\quad 0.044,\quad 0.04\)
Hint:
0.04=0.040, which is less than 0.044.
Question 24 Explanation: 
Topic: Ordering decimals and integers (Objective 0017).
Question 25

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 25 Explanation: 
Topic: Recognize and apply connections between algebra and geometry (e.g., the use of coordinate systems, the Pythagorean theorem) (Objective 0024).
Question 26

Which of the numbers below is the decimal equivalent of \( \dfrac{3}{8}?\)

A

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.
B

0.125

Hint:
This is 1/8, not 3/8.
C

0.375

D

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 26 Explanation: 
Topic: Converting between fractions and decimals (Objective 0017)
Question 27

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 27 Explanation: 
Topic: Use unit conversions and dimensional analysis to solve measurement problems (Objective 0023).
Question 28

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 28 Explanation: 
Topic: Apply proportional thinking to estimate quantities in real world situations (Objective 0019).
Question 29

Here is a method that a student used for subtraction:

Which of the following is correct?

A

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).
B

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.
C

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.
D

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 29 Explanation: 
Topic: Analyze and justify standard and non-standard computational techniques (Objective 0019).
Question 30

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 30 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 31

A family went on a long car trip.  Below is a graph of how far they had driven at each hour.

Which of the following is closest to their average speed driving on the trip?

 
A
\( \large d=20t\)
Hint:
Try plugging t=7 into the equation, and see how it matches the graph.
B
\( \large d=30t\)
Hint:
Try plugging t=7 into the equation, and see how it matches the graph.
C
\( \large d=40t\)
D
\( \large d=50t\)
Hint:
Try plugging t=7 into the equation, and see how it matches the graph.
Question 31 Explanation: 
Topic: Select the linear equation that best models a real-world situation (Objective 0022).
Question 32

Use the graph below to answer the question that follows:

The graph above represents the equation \( \large 3x+Ay=B\), where A and B are integers.  What are the values of A and B?

A
\( \large A = -2, B= 6\)
Hint:
Plug in (2,0) to get B=6, then plug in (0,-3) to get A=-2.
B
\( \large A = 2, B = 6\)
Hint:
Try plugging (0,-3) into this equation.
C
\( \large A = -1.5, B=-3\)
Hint:
The problem said that A and B were integers and -1.5 is not an integer. Don't try to use slope-intercept form.
D
\( \large A = 2, B = -3\)
Hint:
Try plugging (2,0) into this equation.
Question 32 Explanation: 
Topic: Find a linear equation that represents a graph (Objective 0022).
Question 33

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 33 Explanation: 
Topic: Recognize and apply the concepts of variable, function, equality, and equation to express relationships algebraically (Objective 0020).
Question 34

In January 2011, the national debt was about 14 trillion dollars and the US population was about 300 million people.  Someone reading these figures estimated that the national debt was about $5,000 per person.   Which of these statements best describes the reasonableness of this estimate?

A

It is too low by a factor of 10

Hint:
14 trillion \( \approx 15 \times {{10}^{12}} \) and 300 million \( \approx 3 \times {{10}^{8}}\), so the true answer is about \( 5 \times {{10}^{4}} \) or $50,000.
B

It is too low by a factor of 100

C

It is too high by a factor of 10

D

It is too high by a factor of 100

Question 34 Explanation: 
Topics: Estimation, Scientific Notation in the real world (Objective 0016).
Question 35

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 35 Explanation: 
Topic: Estimate and calculate measurements using customary, metric, and nonstandard units of measurement (Objective 0023).
Question 36

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 36 Explanation: 
Topic: Analyze Non-Standard Computational Algorithms (Objective 0019).
Question 37

The following story situations model \( 12\div 3\):

I)  Jack has 12 cookies, which he wants to share equally between himself and two friends.  How many cookies does each person get?

II) Trent has 12 cookies, which he wants to put into bags of 3 cookies each.  How many bags can he make?

III) Cicely has $12.  Cookies cost $3 each.  How many cookies can she buy?

Which of these questions illustrate the same model of division, either partitive (partioning) or measurement (quotative)?

A

I and II

B

I and III

C

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.
D

All three problems model the same meaning of division

Question 37 Explanation: 
Topic: Understand models of operations on numbers (Objective 0019).
Question 38

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 38 Explanation: 
Topic: Recognize uses of prime factorization of a number (Objective 0018).
Question 39

Use the graph below to answer the question that follows.

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 39 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 40

The equation \( \large F=\frac{9}{5}C+32\) is used to convert a temperature measured in Celsius to the equivalent Farentheit temperature.

A patient's temperature increased by 1.5° Celcius.  By how many degrees Fahrenheit did her temperature increase?

A

1.5°

Hint:
Celsius and Fahrenheit don't increase at the same rate.
B

1.8°

Hint:
That's how much the Fahrenheit temp increases when the Celsius temp goes up by 1 degree.
C

2.7°

Hint:
Each degree increase in Celsius corresponds to a \(\dfrac{9}{5}=1.8\) degree increase in Fahrenheit. Thus the increase is 1.8+0.9=2.7.
D

Not enough information.

Hint:
A linear equation has constant slope, which means that every increase of the same amount in one variable, gives a constant increase in the other variable. It doesn't matter what temperature the patient started out at.
Question 40 Explanation: 
Topic: Interpret the meaning of the slope and the intercepts of a linear equation that models a real-world situation (Objective 0022).
Question 41

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 41 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 42

The pattern below consists of a row of black squares surrounded by white squares.

 How many white squares would surround a row of 157 black squares?

A

314

Hint:
Try your procedure on a smaller number that you can count to see where you made a mistake.
B

317

Hint:
Are there ever an odd number of white squares?
C

320

Hint:
One way to see this is that there are 6 tiles on the left and right ends, and the rest of the white tiles are twice the number of black tiles (there are many other ways to look at it too).
D

322

Hint:
Try your procedure on a smaller number that you can count to see where you made a mistake.
Question 42 Explanation: 
Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic) (Objective 0021).
Question 43

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 43 Explanation: 
Topic: Estimate and calculate measurements, use unit conversions to solve measurement problems, solve measurement problems in real-world situations (Objective 0023).
Question 44

How many factors does 80 have?

A
\( \large8\)
Hint:
Don't forget 1 and 80.
B
\( \large9\)
Hint:
Only perfect squares have an odd number of factors -- otherwise factors come in pairs.
C
\( \large10\)
Hint:
1,2,4,5,8,10,16,20,40,80
D
\( \large12\)
Hint:
Did you count a number twice? Include a number that isn't a factor?
Question 44 Explanation: 
Topic: Understand and apply principles of number theory (Objective 0018).
Question 45

Cell phone plan A charges $3 per month plus $0.10 per minute. Cell phone plan B charges $29.99 per month, with no fee for the first 400 minutes and then $0.20 for each additional minute.

Which equation can be used to solve for the number of minutes, m (with m>400) that a person would have to spend on the phone each month in order for the bills for plan A and plan B to be equal?

A
\( \large 3.10m=400+0.2m\)
Hint:
These are the numbers in the problem, but this equation doesn't make sense. If you don't know how to make an equation, try plugging in an easy number like m=500 minutes to see if each side equals what it should.
B
\( \large 3+0.1m=29.99+.20m\)
Hint:
Doesn't account for the 400 free minutes.
C
\( \large 3+0.1m=400+29.99+.20(m-400)\)
Hint:
Why would you add 400 minutes and $29.99? If you don't know how to make an equation, try plugging in an easy number like m=500 minutes to see if each side equals what it should.
D
\( \large 3+0.1m=29.99+.20(m-400)\)
Hint:
The left side is $3 plus $0.10 times the number of minutes. The right is $29.99 plus $0.20 times the number of minutes over 400.
Question 45 Explanation: 
Identify variables and derive algebraic expressions that represent real-world situations (Objective 0020).
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