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MTEL General Curriculum Mathematics Practice

Question 1

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 1 Explanation: 
Topic: Identifying rational and irrational numbers (Objective 0016).
Question 2

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 2 Explanation: 
Topics: Place value, scientific notation, estimation (Objective 0016)
Question 3

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 3 Explanation: 
Topic: Analyze the relationships among proportions, constant rates, and linear functions (Objective 0022).
Question 4

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 4 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 5

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 5 Explanation: 
Topic: Venn Diagrams (Objective 0025)
Question 6

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 6 Explanation: 
Topic: Analyze and justify standard and non-standard computational techniques (Objective 0019).
Question 7

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

A
\(\large1.3\times {{10}^{6}}\)
B
\(\large1.3\times {{10}^{9}}\)
Hint:
That's one billion three hundred million.
C
\(\large1.03\times {{10}^{6}}\)
Hint:
That's one million thirty thousand.
D
\(\large1.03\times {{10}^{9}}\)
Hint:
That's one billion thirty million
Question 7 Explanation: 
Topic: Scientific Notation (Objective 0016)
Question 8

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 8 Explanation: 
Topic: Calculate the probabilities of simple and compound events and of independent and dependent events (Objective 0026).
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

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).
There are 10 questions to complete.

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