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## MTEL General Curriculum Mathematics Practice

 Question 1

#### 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 1 Explanation:
Topic: Converting between fraction and decimal representations (Objective 0017)
 Question 2

#### 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 2 Explanation:
Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic) (Objective 0021)
 Question 3

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

Hint:
All views would be circles.

#### A cone

Hint:
Two views would be triangles, not rectangles.

#### A pyramid

Hint:
How would one view be a circle?
Question 3 Explanation:
Topic: Match three-dimensional figures and their two-dimensional representations (e.g., nets, projections, perspective drawings) (Objective 0024).
 Question 4

#### The "houses" below are made of toothpicks and gum drops. #### 212

Hint:
Can the number of toothpicks be even?

#### 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.

#### 217

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

#### 265

Hint:
Remember that the "houses" overlap some walls.
Question 4 Explanation:
Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic). (Objective 0021).
 Question 5

#### 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 5 Explanation:
Topics: Laws of exponents, order of operations, interpret absolute value (Objective 0019).
 Question 6

#### 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 6 Explanation:
Topic: Derive and use formulas for calculating the lengths, perimeters, areas, volumes, and surface areas of geometric shapes and figures (Objective 0023).
 Question 7

#### The expression $$\large{{8}^{3}}\cdot {{2}^{-10}}$$ is equal to which of the following?

 A $$\large 2$$Hint: Write $$8^3$$ as a power of 2. B $$\large \dfrac{1}{2}$$Hint: $$8^3 \cdot {2}^{-10}={(2^3)}^3 \cdot {2}^{-10}$$ =$$2^9 \cdot {2}^{-10} =2^{-1}$$ C $$\large 16$$Hint: Write $$8^3$$ as a power of 2. D $$\large \dfrac{1}{16}$$Hint: Write $$8^3$$ as a power of 2.
Question 7 Explanation:
Topic: Laws of Exponents (Objective 0019).
 Question 8

#### The function d(x) gives the result when 12 is divided by x.  Which of the following is a graph of d(x)?

 A Hint: d(x) is 12 divided by x, not x divided by 12. B Hint: When x=2, what should d(x) be? C Hint: When x=2, what should d(x) be? D Question 8 Explanation:
Topic: Identify and analyze direct and inverse relationships in tables, graphs, algebraic expressions and real-world situations (Objective 0021)
 Question 9

#### 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 9 Explanation:
Topic: Apply knowledge of combinations and permutations to the computation of probabilities (Objective 0026).
 Question 10

#### Commutative Property.

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

#### Associative Property.

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

#### 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.

#### 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 10 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.
There are 10 questions to complete.

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