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MTEL General Curriculum Mathematics Practice

 Question 1

Which of the following is an irrational number?

 A $$\large \sqrt[3]{8}$$Hint: This answer is the cube root of 8. Since 2 x 2 x 2 =8, this is equal to 2, which is rational because 2 = 2/1. B $$\large \sqrt{8}$$Hint: It is not trivial to prove that this is irrational, but you can get this answer by eliminating the other choices. C $$\large \dfrac{1}{8}$$Hint: 1/8 is the RATIO of two integers, so it is rational. D $$\large -8$$Hint: Negative integers are also rational, -8 = -8/1, a ratio of integers.
Question 1 Explanation:
Topic: Identifying rational and irrational numbers (Objective 0016).
 Question 2

0 years

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

12 years

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

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.

42 years

Hint:
Question 2 Explanation:
Topic: Compare different data sets (Objective 0025).
 Question 3

Which of the lists below is in order from least to greatest value?

 A $$\large \dfrac{1}{2},\quad \dfrac{1}{3},\quad \dfrac{1}{4},\quad \dfrac{1}{5}$$Hint: This is ordered from greatest to least. B $$\large \dfrac{1}{3},\quad \dfrac{2}{7},\quad \dfrac{3}{8},\quad \dfrac{4}{11}$$Hint: 1/3 = 2/6 is bigger than 2/7. C $$\large \dfrac{1}{4},\quad \dfrac{2}{5},\quad \dfrac{2}{3},\quad \dfrac{4}{5}$$Hint: One way to look at this: 1/4 and 2/5 are both less than 1/2, and 2/3 and 4/5 are both greater than 1/2. 1/4 is 25% and 2/5 is 40%, so 2/5 is greater. The distance from 2/3 to 1 is 1/3 and from 4/5 to 1 is 1/5, and 1/5 is less than 1/3, so 4/5 is bigger. D $$\large \dfrac{7}{8},\quad \dfrac{6}{7},\quad \dfrac{5}{6},\quad \dfrac{4}{5}$$Hint: This is in order from greatest to least.
Question 3 Explanation:
Topic: Ordering Fractions (Objective 0017)
 Question 4

Solve for x: $$\large 4-\dfrac{2}{3}x=2x$$

 A $$\large x=3$$Hint: Try plugging x=3 into the equation. B $$\large x=-3$$Hint: Left side is positive, right side is negative when you plug this in for x. C $$\large x=\dfrac{3}{2}$$Hint: One way to solve: $$4=\dfrac{2}{3}x+2x$$ $$=\dfrac{8}{3}x$$.$$x=\dfrac{3 \times 4}{8}=\dfrac{3}{2}$$. Another way is to just plug x=3/2 into the equation and see that each side equals 3 -- on a multiple choice test, you almost never have to actually solve for x. D $$\large x=-\dfrac{3}{2}$$Hint: Left side is positive, right side is negative when you plug this in for x.
Question 4 Explanation:
Topic: Solve linear equations (Objective 0020).
 Question 5

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 5 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 6

21 cm

Hint:
How many miles would correspond to 24 cm on the map? Try adjusting from there.

22 cm

Hint:
How many miles would correspond to 24 cm on the map? Try adjusting from there.

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.

24 cm

Hint:
4 groups of 6 cm is over 2800 miles on the map, which is too much.
Question 6 Explanation:
Topic: Apply proportional thinking to estimate quantities in real world situations (Objective 0019).
 Question 7

A

Hint:
Rise is more than 30 inches.

B

Hint:
Run is almost 24 feet, so rise can be almost 2 feet.

C

Hint:
Run is 12 feet, so rise can be at most 1 foot.

D

Hint:
Slope is 1:10 -- too steep.
Question 7 Explanation:
Topic: Interpret meaning of slope in a real world situation (Objective 0022).
 Question 8

The student‘s solution is correct.

Hint:
Try plugging into the original solution.

The student did not correctly use properties of equality.

Hint:
After $$x=-2x+10$$, the student subtracted 2x on the left and added 2x on the right.

The student did not correctly use the distributive property.

Hint:
Distributive property is $$a(b+c)=ab+ac$$.

The student did not correctly use the commutative property.

Hint:
Commutative property is $$a+b=b+a$$ or $$ab=ba$$.
Question 8 Explanation:
Topic: Justify algebraic manipulations by application of the properties of equality, the order of operations, the number properties, and the order properties (Objective 0020).
 Question 9

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 9 Explanation:
Topic: Select the linear equation that best models a real-world situation (Objective 0022).
 Question 10

30

Hint:
The floor is 120 sq feet, and the tiles are smaller than 1 sq foot. Also, remember that 1 sq foot is 12 $$\times$$ 12=144 sq inches.

120

Hint:
The floor is 120 sq feet, and the tiles are smaller than 1 sq foot.

Hint:

360

Hint:
One way to do this is to note that 6 inches = 1/2 foot and 8 inches = 2/3 foot, so the area of each tile is 1/2 $$\times$$ 2/3=1/3 sq foot, or each square foot of floor requires 3 tiles. The area of the floor is 120 square feet. Note that the tiles would fit evenly oriented in either direction, parallel to the walls.
Question 10 Explanation:
Topic: Estimate and calculate measurements, use unit conversions to solve measurement problems, solve measurement problems in real-world situations (Objective 0023).
 Question 11

58 x 22

Hint:
This problem involves regrouping, which the student does not do correctly.

16 x 24

Hint:
This problem involves regrouping, which the student does not do correctly.

31 x 23

Hint:
There is no regrouping with this problem.

141 x 32

Hint:
This problem involves regrouping, which the student does not do correctly.
Question 11 Explanation:
Topic: Analyze computational algorithms (Objective 0019).
 Question 12

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 12 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 13

A sales companies pays its representatives $2 for each item sold, plus 40% of the price of the item. The rest of the money that the representatives collect goes to the company. All transactions are in cash, and all items cost$4 or more.   If the price of an item in dollars is p, which expression represents the amount of money the company collects when the item is sold?

 A $$\large \dfrac{3}{5}p-2$$Hint: The company gets 3/5=60% of the price, minus the $2 per item. B $$\large \dfrac{3}{5}\left( p-2 \right)$$Hint: This is sensible, but not what the problem states. C $$\large \dfrac{2}{5}p+2$$Hint: The company pays the extra$2; it doesn't collect it. D $$\large \dfrac{2}{5}p-2$$Hint: This has the company getting 2/5 = 40% of the price of each item, but that's what the representative gets.
Question 13 Explanation:
Topic: Use algebra to solve word problems involving fractions, ratios, proportions, and percents (Objective 0020).
 Question 14

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.

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.

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 14 Explanation:
Topic:Analyze and apply geometric transformations (e.g., translations, rotations, reflections, dilations) (Objective 0024).
 Question 15

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 15 Explanation:
Topic: Find the least common multiple of a set of numbers (Objective 0018).
 Question 16

4 congruent sides

Hint:
The most common definition of a rhombus is a quadrilateral with 4 congruent sides.

A center of rotational symmetry

Hint:
The diagonal of a rhombus separates it into two congruent isosceles triangles. The center of this line is a center of 180 degree rotational symmetry that switches the triangles.

4 congruent angles

Hint:
Unless the rhombus is a square, it does not have 4 congruent angles.

2 sets of parallel sides

Hint:
All rhombi are parallelograms.
Question 16 Explanation:
Topic: Classify and analyze polygons using attributes of sides and angles, and symmetry (Objective 0024).
 Question 17

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

Which equation can be used to solve for the number of minutes, m (with m>400) that a person would have to spend on the phone each month in order for the bills for plan A and plan B to be equal?

 A $$\large 3.10m=400+0.2m$$Hint: These are the numbers in the problem, but this equation doesn't make sense. If you don't know how to make an equation, try plugging in an easy number like m=500 minutes to see if each side equals what it should. B $$\large 3+0.1m=29.99+.20m$$Hint: Doesn't account for the 400 free minutes. C $$\large 3+0.1m=400+29.99+.20(m-400)$$Hint: Why would you add 400 minutes and $29.99? If you don't know how to make an equation, try plugging in an easy number like m=500 minutes to see if each side equals what it should. D $$\large 3+0.1m=29.99+.20(m-400)$$Hint: The left side is$3 plus $0.10 times the number of minutes. The right is$29.99 plus \$0.20 times the number of minutes over 400.
Question 18 Explanation:
Identify variables and derive algebraic expressions that represent real-world situations (Objective 0020).
 Question 19

Which of the numbers below is not equivalent to 4%?

 A $$\large \dfrac{1}{25}$$Hint: 1/25=4/100, so this is equal to 4% (be sure you read the question correctly). B $$\large \dfrac{4}{100}$$Hint: 4/100=4% (be sure you read the question correctly). C $$\large 0.4$$Hint: 0.4=40% so this is not equal to 4% D $$\large 0.04$$Hint: 0.04=4/100, so this is equal to 4% (be sure you read the question correctly).
Question 19 Explanation:
Converting between fractions, decimals, and percents (Objective 0017).
 Question 20

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 20 Explanation:
Topic: Recognize and apply the concept of conditional probability (Objective 0026).
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