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MTEL General Curriculum Mathematics Practice


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Question 1

A biology class requires a lab fee, which is a whole number of dollars, and the same amount for all students. On Monday the instructor collected $70 in fees, on Tuesday she collected $126, and on Wednesday she collected $266. What is the largest possible amount the fee could be?

A

$2

Hint:
A possible fee, but not the largest possible fee. Check the other choices to see which are factors of all three numbers.
B

$7

Hint:
A possible fee, but not the largest possible fee. Check the other choices to see which are factors of all three numbers.
C

$14

Hint:
This is the greatest common factor of 70, 126, and 266.
D

$70

Hint:
Not a factor of 126 or 266, so couldn't be correct.
Question 1 Explanation: 
Topic: Use GCF in real-world context (Objective 0018)
Question 2

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 2 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 3

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 3 Explanation: 
Topic: Integers (Objective 0016)
Question 4

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 4 Explanation: 
Topic: Match three-dimensional figures and their two-dimensional representations (e.g., nets, projections, perspective drawings) (Objective 0024).
Question 5

The "houses" below are made of toothpicks and gum drops.

How many toothpicks are there in a row of 53 houses?

A

212

Hint:
Can the number of toothpicks be even?
B

213

Hint:
One way to see this is that every new "house" adds 4 toothpicks to the leftmost vertical toothpick -- so the total number is 1 plus 4 times the number of "houses." There are many other ways to look at the problem too.
C

217

Hint:
Try your strategy with a smaller number of "houses" so you can count and find your mistake.
D

265

Hint:
Remember that the "houses" overlap some walls.
Question 5 Explanation: 
Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic). (Objective 0021).
Question 6

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 6 Explanation: 
Topic: Models of Fractions (Objective 0017)
Question 7

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 7 Explanation: 
Topics: Estimation, simplifying fractions (Objective 0016, overlaps with other objectives).
Question 8

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 8 Explanation: 
Topic:Analyze and apply geometric transformations (e.g., translations, rotations, reflections, dilations) (Objective 0024).
Question 9

The chart below gives percentiles for the number of sit-ups that boys of various ages can do in 60 seconds (source , June 24, 2011)

 

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

A

95% of 12 year old boys can do 56 sit-ups in 60 seconds.

Hint:
The 95th percentile means that 95% of scores are less than or equal to 56, and 5% are greater than or equal to 56.
B

At most 25% of 7 year old boys can do 19 or more sit-ups in 60 seconds.

Hint:
The 25th percentile means that 25% of scores are less than or equal to 19, and 75% are greater than or equal to 19.
C

Half of all 13 year old boys can do less than 41 sit-ups in 60 seconds and half can do more than 41 sit-ups in 60 seconds.

Hint:
Close, but not quite. There's no accounting for boys who can do exactly 41 sit ups. Look at these data: 10, 20, 41, 41, 41, 41, 50, 60, 90. The median is 41, but more than half can do 41 or more.
D

At least 75% of 16 year old boys can only do 51 or fewer sit-ups in 60 seconds.

Hint:
The "at least" is necessary due to duplicates. Suppose the data were 10, 20, 51, 51. The 75th percentile is 51, but 100% of the boys can only do 51 or fewer situps.
Question 9 Explanation: 
Topic: Analyze and interpret various graphic and nongraphic data representations (e.g., frequency distributions, percentiles) (Objective 0025).
Question 10

There are six gumballs in a bag — two red and four green.  Six children take turns picking a gumball out of the bag without looking.   They do not return any gumballs to the bag.  What is the probability that the first two children to pick from the bag pick the red gumballs?

A
\( \large \dfrac{1}{3}\)
Hint:
This is the probability that the first child picks a red gumball, but not that the first two children pick red gumballs.
B
\( \large \dfrac{1}{8}\)
Hint:
Are you adding things that you should be multiplying?
C
\( \large \dfrac{1}{9}\)
Hint:
This would be the probability if the gumballs were returned to the bag.
D
\( \large \dfrac{1}{15}\)
Hint:
The probability that the first child picks red is 2/6 = 1/3. Then there are 5 gumballs in the bag, one red, so the probability that the second child picks red is 1/5. Thus 1/5 of the time, after the first child picks red, the second does too, so the probability is 1/5 x 1/3 = 1/15.
Question 10 Explanation: 
Topic: Calculate the probabilities of simple and compound events and of independent and dependent events (Objective 0026).
Question 11

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 11 Explanation: 
Topic: Using number lines (Objective 0017)
Question 12

Which of the lists below contains only irrational numbers?

A
\( \large\pi , \quad \sqrt{6},\quad \sqrt{\dfrac{1}{2}}\)
B
\( \large\pi , \quad \sqrt{9}, \quad \pi +1\)
Hint:
\( \sqrt{9}=3\)
C
\( \large\dfrac{1}{3},\quad \dfrac{5}{4},\quad \dfrac{2}{9}\)
Hint:
These are all rational.
D
\( \large-3,\quad 14,\quad 0\)
Hint:
These are all rational.
Question 12 Explanation: 
Topic: Identifying rational and irrational numbers (Objective 0016).
Question 13

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 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 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 14 Explanation: 
Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic) (Objective 0021)
Question 15

Four children randomly line up, single file.  What is the probability that they are in height order, with the shortest child in front?   All of the children are different heights.

A
\( \large \dfrac{1}{4}\)
Hint:
Try a simpler question with 3 children -- call them big, medium, and small -- and list all the ways they could line up. Then see how to extend your logic to the problem with 4 children.
B
\( \large \dfrac{1}{256} \)
Hint:
Try a simpler question with 3 children -- call them big, medium, and small -- and list all the ways they could line up. Then see how to extend your logic to the problem with 4 children.
C
\( \large \dfrac{1}{16}\)
Hint:
Try a simpler question with 3 children -- call them big, medium, and small -- and list all the ways they could line up. Then see how to extend your logic to the problem with 4 children.
D
\( \large \dfrac{1}{24}\)
Hint:
The number of ways for the children to line up is \(4!=4 \times 3 \times 2 \times 1 =24\) -- there are 4 choices for who is first in line, then 3 for who is second, etc. Only one of these lines has the children in the order specified.
Question 15 Explanation: 
Topic: Apply knowledge of combinations and permutations to the computation of probabilities (Objective 0026).
Question 16

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 16 Explanation: 
Topic: Classify and analyze three-dimensional figures using attributes of faces, edges, and vertices (Objective 0024).
Question 17

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 17 Explanation: 
Topic: Venn Diagrams (Objective 0025)
Question 18

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 18 Explanation: 
Topics: Place value, scientific notation, estimation (Objective 0016)
Question 19

A family on vacation drove the first 200 miles in 4 hours and the second 200 miles in 5 hours.  Which expression below gives their average speed for the entire trip?

A
\( \large \dfrac{200+200}{4+5}\)
Hint:
Average speed is total distance divided by total time.
B
\( \large \left( \dfrac{200}{4}+\dfrac{200}{5} \right)\div 2\)
Hint:
This seems logical, but the problem is that it weights the first 4 hours and the second 5 hours equally, when each hour should get the same weight in computing the average speed.
C
\( \large \dfrac{200}{4}+\dfrac{200}{5} \)
Hint:
This would be an average of 90 miles per hour!
D
\( \large \dfrac{400}{4}+\dfrac{400}{5} \)
Hint:
This would be an average of 180 miles per hour! Even a family of race car drivers probably doesn't have that average speed on a vacation!
Question 19 Explanation: 
Topic: Solve a variety of measurement problems (e.g., time, temperature, rates, average rates of change) in real-world situations (Objective 0023).
Question 20

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