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MTEL General Curriculum Mathematics Practice


Your answers are highlighted below.
Question 1

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 1 Explanation: 
Topic: Demonstrate knowledge of divisibility rules (Objective 0018).
Question 2

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 2 Explanation: 
Topic:Classify and analyze three-dimensional figures using attributes of faces, edges, and vertices (Objective 0024).
Question 3

Which of the following is the equation of a linear function?

A
\( \large y={{x}^{2}}+2x+7\)
Hint:
This is a quadratic function.
B
\( \large y={{2}^{x}}\)
Hint:
This is an exponential function.
C
\( \large y=\dfrac{15}{x}\)
Hint:
This is an inverse function.
D
\( \large y=x+(x+4)\)
Hint:
This is a linear function, y=2x+4, it's graph is a straight line with slope 2 and y-intercept 4.
Question 3 Explanation: 
Topic: Distinguish between linear and nonlinear functions (Objective 0022).
Question 4

A map has a scale of 3 inches = 100 miles.  Cities A and B are 753 miles apart.  Let d be the distance between the two cities on the map.  Which of the following is not correct?

A
\( \large \dfrac{3}{100}=\dfrac{d}{753}\)
Hint:
Units on both side are inches/mile, and both numerators and denominators correspond -- this one is correct.
B
\( \large \dfrac{3}{100}=\dfrac{753}{d}\)
Hint:
Unit on the left is inches per mile, and on the right is miles per inch. The proportion is set up incorrectly (which is what we wanted). Another strategy is to notice that one of A or B has to be the answer because they cannot both be correct proportions. Then check that cross multiplying on A gives part D, so B is the one that is different from the other 3.
C
\( \large \dfrac{3}{d}=\dfrac{100}{753}\)
Hint:
Unitless on each side, as inches cancel on the left and miles on the right. Numerators correspond to the map, and denominators to the real life distances -- this one is correct.
D
\( \large 100d=3\cdot 753\)
Hint:
This is equivalent to part A.
Question 4 Explanation: 
Topic: Analyze the relationships among proportions, constant rates, and linear functions (Objective 0022).
Question 5

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 5 Explanation: 
Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic) (Objective 0021)
Question 6

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 6 Explanation: 
Topic: Interpret meaning of slope in a real world situation (Objective 0022).
Question 7

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 7 Explanation: 
Topic: Interpret the meaning of the slope and the intercepts of a linear equation that models a real-world situation (Objective 0022).
Question 8

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 8 Explanation: 
Topics: Place value, scientific notation, estimation (Objective 0016)
Question 9

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 9 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 10

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 10 Explanation: 
Topics: Estimation, simplifying fractions (Objective 0016, overlaps with other objectives).
Question 11

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 11 Explanation: 
Topics: Calculate measurements and derive and use formulas for calculating the areas of geometric shapes and figures (Objective 0023).
Question 12

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 12 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 13

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 13 Explanation: 
Topics: Laws of exponents, order of operations, interpret absolute value (Objective 0019).
Question 14

A publisher prints a series of books with covers made of identical material and using the same thickness of paper for each page.  The covers of the book together are 0.4 cm thick, and 125 pieces of the paper used together are 1 cm thick.

The publisher uses a linear function to determine the total thickness, T(n) of a book made with n sheets of paper.   What are the slope and intercept of T(n)?

A

Intercept = 0.4 cm, Slope = 125 cm/page

Hint:
This would mean that each page of the book was 125 cm thick.
B

Intercept =0.4 cm, Slope = \(\dfrac{1}{125}\)cm/page

Hint:
The intercept is how thick the book would be with no pages in it. The slope is how much 1 extra page adds to the thickness of the book.
C

Intercept = 125 cm, Slope = 0.4 cm

Hint:
This would mean that with no pages in the book, it would be 125 cm thick.
D

Intercept = \(\dfrac{1}{125}\)cm, Slope = 0.4 pages/cm

Hint:
This would mean that each new page of the book made it 0.4 cm thicker.
Question 14 Explanation: 
Topic: Interpret the meaning of the slope and the intercepts of a linear equation that models a real-world situation (Objective 0022).
Question 15

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 15 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 16

M is a multiple of 26.  Which of the following cannot be true?

A

M is odd.

Hint:
All multiples of 26 are also multiples of 2, so they must be even.
B

M is a multiple of 3.

Hint:
3 x 26 is a multiple of both 3 and 26.
C

M is 26.

Hint:
1 x 26 is a multiple of 26.
D

M is 0.

Hint:
0 x 26 is a multiple of 26.
Question 16 Explanation: 
Topic: Characteristics of composite numbers (Objective 0018).
Question 17

What is the mathematical name of the three-dimensional polyhedron depicted below?

A

Tetrahedron

Hint:
All the faces of a tetrahedron are triangles.
B

Triangular Prism

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

Triangular Pyramid

Hint:
A pyramid has one base, not two.
D

Trigon

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

Taxicab fares in Boston (Spring 2012) are $2.60 for the first \(\dfrac{1}{7}\) of a mile or less and $0.40 for each \(\dfrac{1}{7}\) of a mile after that.

Let d represent the distance a passenger travels in miles (with \(d>\dfrac{1}{7}\)). Which of the following expressions represents the total fare?

A
\( \large \$2.60+\$0.40d\)
Hint:
It's 40 cents for 1/7 of a mile, not per mile.
B
\( \large \$2.60+\$0.40\dfrac{d}{7}\)
Hint:
According to this equation, going 7 miles would cost $3; does that make sense?
C
\( \large \$2.20+\$2.80d\)
Hint:
You can think of the fare as $2.20 to enter the cab, and then $0.40 for each 1/7 of a mile, including the first 1/7 of a mile (or $2.80 per mile).

Alternatively, you pay $2.60 for the first 1/7 of a mile, and then $2.80 per mile for d-1/7 miles. The total is 2.60+2.80(d-1/7) = 2.60+ 2.80d -.40 = 2.20+2.80d.
D
\( \large \$2.60+\$2.80d\)
Hint:
Don't count the first 1/7 of a mile twice.
Question 18 Explanation: 
Topic: Identify variables and derive algebraic expressions that represent real-world situations (Objective 0020), and select the linear equation that best models a real-world situation (Objective 0022).
Question 19

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 19 Explanation: 
Topic: Apply proportional thinking to estimate quantities in real world situations (Objective 0019).
Question 20

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 20 Explanation: 
Topic: Justify algebraic manipulations by application of the properties of order of operations (Objective 0020).
Question 21

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 21 Explanation: 
Topic: Select the linear equation that best models a real-world situation (Objective 0022).
Question 22

If  x  is an integer, which of the following must also be an integer?

A
\( \large \dfrac{x}{2}\)
Hint:
If x is odd, then \( \dfrac{x}{2} \) is not an integer, e.g. 3/2 = 1.5.
B
\( \large \dfrac{2}{x}\)
Hint:
Only an integer if x = -2, -1, 1, or 2.
C
\( \large-x\)
Hint:
-1 times any integer is still an integer.
D
\(\large\sqrt{x}\)
Hint:
Usually not an integer, e.g. \( \sqrt{2} \approx 1.414 \).
Question 22 Explanation: 
Topic: Integers (Objective 0016)
Question 23

Use the expression below to answer the question that follows.

                 \( \large \dfrac{\left( 4\times {{10}^{3}} \right)\times \left( 3\times {{10}^{4}} \right)}{6\times {{10}^{6}}}\)

Which of the following is equivalent to the expression above?

A

2

Hint:
\(10^3 \times 10^4=10^7\), and note that if you're guessing when the answers are so closely related, you're generally better off guessing one of the middle numbers.
B

20

Hint:
\( \dfrac{\left( 4\times {{10}^{3}} \right)\times \left( 3\times {{10}^{4}} \right)}{6\times {{10}^{6}}}=\dfrac {12 \times {{10}^{7}}}{6\times {{10}^{6}}}=\)\(2 \times {{10}^{1}}=20 \)
C

200

Hint:
\(10^3 \times 10^4=10^7\)
D

2000

Hint:
\(10^3 \times 10^4=10^7\), and note that if you're guessing when the answers are so closely related, you're generally better off guessing one of the middle numbers.
Question 23 Explanation: 
Topics: Scientific notation, exponents, simplifying fractions (Objective 0016, although overlaps with other objectives too).
Question 24

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 24 Explanation: 
Topics: Estimation, Scientific Notation in the real world (Objective 0016).
Question 25

Which of the following nets will not fold into a cube?

A
Hint:
If you have trouble visualizing, cut them out and fold (during the test, you can tear paper to approximate).
B
C
Hint:
If you have trouble visualizing, cut them out and fold (during the test, you can tear paper to approximate).
D
Hint:
If you have trouble visualizing, cut them out and fold (during the test, you can tear paper to approximate).
Question 25 Explanation: 
Topic: Match three-dimensional figures and their two-dimensional representations (e.g., nets, projections, perspective drawings) (Objective 0024).
Question 26

In the triangle below, \(\overline{AC}\cong \overline{AD}\cong \overline{DE}\) and \(m\angle CAD=100{}^\circ \).  What is \(m\angle DAE\)?

A
\( \large 20{}^\circ \)
Hint:
Angles ACD and ADC are congruent since they are base angles of an isosceles triangle. Since the angles of a triangle sum to 180, they sum to 80, and they are 40 deg each. Thus angle ADE is 140 deg, since it makes a straight line with angle ADC. Angles DAE and DEA are base angles of an isosceles triangle and thus congruent-- they sum to 40 deg, so are 20 deg each.
B
\( \large 25{}^\circ \)
Hint:
If two sides of a triangle are congruent, then it's isosceles, and the base angles of an isosceles triangle are equal.
C
\( \large 30{}^\circ \)
Hint:
If two sides of a triangle are congruent, then it's isosceles, and the base angles of an isosceles triangle are equal.
D
\( \large 40{}^\circ \)
Hint:
Make sure you're calculating the correct angle.
Question 26 Explanation: 
Topic: Classify and analyze polygons using attributes of sides and angles, including real-world applications. (Objective 0024).
Question 27

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 27 Explanation: 
Topic: Find the least common multiple of a set of numbers (Objective 0018).
Question 28

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 28 Explanation: 
Topic: Understand models of operations on numbers (Objective 0019).
Question 29

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 29 Explanation: 
Topic: Estimate and calculate measurements using customary, metric, and nonstandard units of measurement (Objective 0023).
Question 30

The histogram below shows the frequency of a class€™s scores on a 4 question quiz.

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 30 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 31

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 31 Explanation: 
Topic: Converting between fraction and decimal representations (Objective 0017)
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

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

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 34 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 35

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 35 Explanation: 
Topic: Select the linear equation that best models a real-world situation (Objective 0022).
Question 36

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

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 37 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 38

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 38 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 39

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 39 Explanation: 
Topic: Analyze geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to concepts of symmetry (Objective 0024).
Question 40

Which of the lines depicted below is a graph of \( \large y=2x-5\)?

A

a

Hint:
The slope of line a is negative.
B

b

Hint:
Wrong slope and wrong intercept.
C

c

Hint:
The intercept of line c is positive.
D

d

Hint:
Slope is 2 -- for every increase of 1 in x, y increases by 2. Intercept is -5 -- the point (0,-5) is on the line.
Question 40 Explanation: 
Topic: Find a linear equation that represents a graph (Objective 0022).
Question 41

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 41 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 42

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 42 Explanation: 
Topic: Analyze and justify standard and non-standard computational techniques (Objective 0019).
Question 43

Below is a portion of a number line.

Point A is one-quarter of the distance from 0.26 to 0.28.  What number is represented by point A?

A
\( \large0.26\)
Hint:
Please reread the question.
B
\( \large0.2625\)
Hint:
This is one-quarter of the distance between 0.26 and 0.27, which is not what the question asked.
C
\( \large0.265\)
D
\( \large0.27\)
Hint:
Please read the question more carefully. This answer would be correct if Point A were halfway between the tick marks, but it's not.
Question 43 Explanation: 
Topic: Using number lines (Objective 0017)
Question 44

The table below gives data from various years on how many young girls drank milk.

Based on the data given above, what was the probability that a randomly chosen girl in 1990 drank milk?

A
\( \large \dfrac{502}{1222}\)
Hint:
This is the probability that a randomly chosen girl who drinks milk was in the 1989-1991 food survey.
B
\( \large \dfrac{502}{2149}\)
Hint:
This is the probability that a randomly chosen girl from the whole survey drank milk and was also surveyed in 1989-1991.
C
\( \large \dfrac{502}{837}\)
D
\( \large \dfrac{1222}{2149}\)
Hint:
This is the probability that a randomly chosen girl from any year of the survey drank milk.
Question 44 Explanation: 
Topic: Recognize and apply the concept of conditional probability (Objective 0026).
Question 45

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 45 Explanation: 
Topic: Using Number Lines (Objective 0017)
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