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MTEL General Curriculum Mathematics Practice


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Question 1

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 1 Explanation: 
Topic: Analyze and apply geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to concepts of symmetry, similarity, and congruence; and use these concepts to solve problems (Objective 0024).
Question 2

The speed of sound in dry air at 68 degrees F is 343.2 meters per second.  Which of the expressions below could be used to compute the number of kilometers that a sound wave travels in 10 minutes (in dry air at 68 degrees F)?

A
\( \large 343.2\times 60\times 10\)
Hint:
In kilometers, not meters.
B
\( \large 343.2\times 60\times 10\times \dfrac{1}{1000}\)
Hint:
Units are meters/sec \(\times\) seconds/minute \(\times\) minutes \(\times\) kilometers/meter, and the answer is in kilometers.
C
\( \large 343.2\times \dfrac{1}{60}\times 10\)
Hint:
Include units and make sure answer is in kilometers.
D
\( \large 343.2\times \dfrac{1}{60}\times 10\times \dfrac{1}{1000}\)
Hint:
Include units and make sure answer is in kilometers.
Question 2 Explanation: 
Topic: Use unit conversions and dimensional analysis to solve measurement problems (Objective 0023).
Question 3

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 3 Explanation: 
Topic: Find the prime factorization of a number and recognize its uses (Objective 0018).
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

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 5 Explanation: 
Topic: Justify algebraic manipulations by application of the properties of order of operations (Objective 0020).
Question 6

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 6 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 7

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 7 Explanation: 
Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic). (Objective 0021).
Question 8

The letters A, B, and C represent digits (possibly equal) in the twelve digit number x=111,111,111,ABC.  For which values of A, B, and C is x divisible by 40?

A
\( \large A = 3, B = 2, C=0\)
Hint:
Note that it doesn't matter what the first 9 digits are, since 1000 is divisible by 40, so DEF,GHI,JKL,000 is divisible by 40 - we need to check the last 3.
B
\( \large A = 0, B = 0, C=4\)
Hint:
Not divisible by 10, since it doesn't end in 0.
C
\( \large A = 4, B = 2, C=0\)
Hint:
Divisible by 10 and by 4, but not by 40, as it's not divisible by 8. Look at 40 as the product of powers of primes -- 8 x 5, and check each. To check 8, either check whether 420 is divisible by 8, or take ones place + twice tens place + 4 * hundreds place = 18, which is not divisible by 8.
D
\( \large A =1, B=0, C=0\)
Hint:
Divisible by 10 and by 4, but not by 40, as it's not divisible by 8. Look at 40 as the product of powers of primes -- 8 x 5, and check each. To check 8, either check whether 100 is divisible by 8, or take ones place + twice tens place + 4 * hundreds place = 4, which is not divisible by 8.
Question 8 Explanation: 
Topic: Understand divisibility rules and why they work (Objective 018).
Question 9

What is the probability that two randomly selected people were born on the same day of the week?  Assume that all days are equally probable.

A
\( \large \dfrac{1}{7}\)
Hint:
It doesn't matter what day the first person was born on. The probability that the second person will match is 1/7 (just designate one person the first and the other the second). Another way to look at it is that if you list the sample space of all possible pairs, e.g. (Wed, Sun), there are 49 such pairs, and 7 of them are repeats of the same day, and 7/49=1/7.
B
\( \large \dfrac{1}{14}\)
Hint:
What would be the sample space here? Ie, how would you list 14 things that you pick one from?
C
\( \large \dfrac{1}{42}\)
Hint:
If you wrote the seven days of the week on pieces of paper and put the papers in a jar, this would be the probability that the first person picked Sunday and the second picked Monday from the jar -- not the same situation.
D
\( \large \dfrac{1}{49}\)
Hint:
This is the probability that they are both born on a particular day, e.g. Sunday.
Question 9 Explanation: 
Topic: Calculate the probabilities of simple and compound events and of independent and dependent events (Objective 0026).
Question 10

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 10 Explanation: 
Topic: Identifying rational and irrational numbers (Objective 0016).
Question 11

Which of the graphs below represent functions?

I. II. III. IV.   
A

I and IV only.

Hint:
There are vertical lines that go through 2 points in IV .
B

I and III only.

Hint:
Even though III is not continuous, it's still a function (assuming that vertical lines between the "steps" do not go through 2 points).
C

II and III only.

Hint:
Learn about the vertical line test.
D

I, II, and IV only.

Hint:
There are vertical lines that go through 2 points in II.
Question 11 Explanation: 
Understand the definition of function and various representations of functions (e.g., input/output machines, tables, graphs, mapping diagrams, formulas). (Objective 0021).
Question 12

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

Step 1: 100 x 16 = 1600

Step 2: 25 x 16 = 1600 ÷· 4 = 400

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

Which property best justifies Step 3 in this strategy?

A

Commutative Property.

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

Associative Property.

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

Identity Property.

Hint:
0 is the additive identity, because \( a+0=a\) and 1 is the multiplicative identity because \(a \times 1=a\). The phrase "identity property" is not standard.
D

Distributive Property.

Hint:
\( (25+1) \times 16 = 25 \times 16 + 1 \times 16 \). This is an example of the distributive property of multiplication over addition.
Question 12 Explanation: 
Topic: Analyze and justify mental math techniques, by applying arithmetic properties such as commutative, distributive, and associative (Objective 0019). Note that it's hard to write a question like this as a multiple choice question -- worthwhile to understand why the other steps work too.
Question 13

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 13 Explanation: 
Topic: Calculate the probabilities of simple and compound events and of independent and dependent events (Objective 0026).
Question 14

The first histogram shows the average life expectancies for women in different countries in Africa in 1998; the second histogram gives similar data for Europe:

  

How much bigger is the range of the data for Africa than the range of the data for Europe?

A

0 years

Hint:
Range is the maximum life expectancy minus the minimum life expectancy.
B

12 years

Hint:
Are you subtracting frequencies? Range is about values of the data, not frequency.
C

18 years

Hint:
It's a little hard to read the graph, but it doesn't matter if you're consistent. It looks like the range for Africa is 80-38= 42 years and for Europe is 88-64 = 24; 42-24=18.
D

42 years

Hint:
Read the question more carefully.
Question 14 Explanation: 
Topic: Compare different data sets (Objective 0025).
Question 15

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 15 Explanation: 
Topics: Understand various representations of functions, and translate among different representations of functional relationships (Objective 0021).
Question 16

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 16 Explanation: 
Topic: Scientific Notation (Objective 0016)
Question 17

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 17 Explanation: 
Topic: Find a linear equation that represents a graph (Objective 0022).
Question 18

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

Which of the following is equal to eleven billion four hundred thousand?

A
\( \large 11,400,000\)
Hint:
That's eleven million four hundred thousand.
B
\(\large11,000,400,000\)
C
\( \large11,000,000,400,000\)
Hint:
That's eleven trillion four hundred thousand (although with British conventions; this answer is correct, but in the US, it isn't).
D
\( \large 11,400,000,000\)
Hint:
That's eleven billion four hundred million
Question 19 Explanation: 
Topic: Place Value (Objective 0016)
Question 20

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