Hints will display for most wrong answers; explanations for most right answers.   You can attempt a question multiple times; it will only be scored correct if you get it right the first time.

I used the official objectives and sample test to construct these questions, but cannot promise that they accurately reflect what’s on the real test.   Some of the sample questions were more convoluted than I could bear to write.   See terms of use.   See the MTEL Practice Test main page to view questions on a particular topic or to download paper practice tests.

## MTEL General Curriculum Mathematics Practice

 Question 1

#### 23 flats, 4 rods, 7 little cubes

Hint:
Be sure you read the question carefully: 2300+40+7=2347

#### 2 large cubes, 3 flats, 47 rods

Hint:
2000+300+470 $$\neq$$ 2347

#### 2 large cubes, 34 rods, 7 little cubes

Hint:
Be sure you read the question carefully: 2000+340+7=2347

#### 2 large cubes, 3 flats, 4 rods, 7 little cubes

Hint:
Be sure you read the question carefully: 2000+300+40+7=2347
Question 1 Explanation:
Topic: Place Value (Objective 0016)
 Question 2

#### Use the graph below to answer the question that follows: #### George left home at 10:00 and drove to work on a crooked path. He was stopped in traffic at 10:30 and 10:45. He drove 30 miles total.

Hint:
Just because he ended up 30 miles from home doesn't mean he drove 30 miles total.

#### George drove to work. On the way to work there is a little hill and a big hill. He slowed down for them. He made it to work at 11:15.

Hint:
The graph is not a picture of the roads.

#### George left home at 10:15. He drove 10 miles, then realized he‘d forgotten something at home. He turned back and got what he‘d forgotten. Then he drove in a straight line, at many different speeds, until he got to work around 11:15.

Hint:
A straight line on a distance versus time graph means constant speed.

#### George left home at 10:15. He drove 10 miles, then realized he‘d forgotten something at home. He turned back and got what he‘d forgotten. Then he drove at a constant speed until he got to work around 11:15.

Question 2 Explanation:
Topic: Use qualitative graphs to represent functional relationships in the real world (Objective 0021).
 Question 3

#### A cylindrical soup can has diameter 7 cm and height 11 cm. The can holds g grams of soup.   How many grams of the same soup could a cylindrical can with diameter 14 cm and height 33 cm hold?

 A $$\large 6g$$Hint: You must scale in all three dimensions. B $$\large 12g$$Hint: Height is multiplied by 3, and diameter and radius are multiplied by 2. Since the radius is squared, final result is multiplied by $$2^2\times 3=12$$. C $$\large 18g$$Hint: Don't square the height scale factor. D $$\large 36g$$Hint: Don't square the height scale factor.
Question 3 Explanation:
Topic: Determine how the characteristics (e.g., area, volume) of geometric figures and shapes are affected by changes in their dimensions (Objective 0023).
 Question 4

#### Use the samples of a student's work below to answer the question that follows:

$$\large \dfrac{2}{3}\times \dfrac{3}{4}=\dfrac{4\times 2}{3\times 3}=\dfrac{8}{9}$$ $$\large \dfrac{2}{5}\times \dfrac{7}{7}=\dfrac{7\times 2}{5\times 7}=\dfrac{2}{5}$$ $$\large \dfrac{7}{6}\times \dfrac{3}{4}=\dfrac{4\times 7}{6\times 3}=\dfrac{28}{18}=\dfrac{14}{9}$$

#### It is not valid. It never produces the correct answer.

Hint:
In the middle example,the answer is correct.

#### It is not valid. It produces the correct answer in a few special cases, but it‘s still not a valid algorithm.

Hint:
Note that this algorithm gives a/b divided by c/d, not a/b x c/d, but some students confuse multiplication and cross-multiplication. If a=0 or if c/d =1, division and multiplication give the same answer.

#### It is valid if the rational numbers in the multiplication problem are in lowest terms.

Hint:
Lowest terms is irrelevant.

#### It is valid for all rational numbers.

Hint:
Can't be correct as the first and last examples have the wrong answers.
Question 4 Explanation:
Topic: Analyze Non-Standard Computational Algorithms (Objective 0019).
 Question 5

#### 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 5 Explanation:
Topic: Select the linear equation that best models a real-world situation (Objective 0022).
 Question 6

#### 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 6 Explanation:
Topic: Solve a variety of measurement problems (e.g., time, temperature, rates, average rates of change) in real-world situations (Objective 0023).
 Question 7
 I. $$\large \dfrac{1}{2}+\dfrac{1}{3}$$ II. $$\large .400000$$ III. $$\large\dfrac{1}{5}+\dfrac{1}{5}$$ IV. $$\large 40\%$$ V. $$\large 0.25$$ VI. $$\large\dfrac{14}{35}$$

#### I, III, V, VI

Hint:
I and V are not at all how fractions and decimals work.

#### III, VI

Hint:
These are right, but there are more.

#### II, III, VI

Hint:
These are right, but there are more.

#### II, III, IV, VI

Question 7 Explanation:
Topic: Converting between fractions, decimals, and percents (Objective 0017)
 Question 8

#### 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 8 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 9

#### The picture below represents a board with pegs on it, where the closest distance between two pegs is 1 cm.  What is the area of the pentagon shown? A $$\large 8\text{ c}{{\text{m}}^{2}}$$Hint: Don't just count the dots inside, that doesn't give the area. Try adding segments so that the slanted lines become the diagonals of rectangles. B $$\large 11\text{ c}{{\text{m}}^{2}}$$Hint: Try adding segments so that the slanted lines become the diagonals of rectangles. C $$\large 11.5\text{ c}{{\text{m}}^{2}}$$Hint: An easy way to do this problem is to use Pick's Theorem (of course, it's better if you understand why Pick's theorem works): area = # pegs inside + half # pegs on the border - 1. In this case 8+9/2-1=11.5. A more appropriate strategy for elementary classrooms is to add segments; here's one way. There are 20 1x1 squares enclosed, and the total area of the triangles that need to be subtracted is 8.5 D $$\large 12.5\text{ c}{{\text{m}}^{2}}$$Hint: Try adding segments so that the slanted lines become the diagonals of rectangles.
Question 9 Explanation:
Topics: Calculate measurements and derive and use formulas for calculating the areas of geometric shapes and figures (Objective 0023).
 Question 10

#### 0.38

Hint:
If you are just writing the numerator next to the denominator then your technique is way off, but by coincidence your answer is close; try with 2/3 and 0.23 is nowhere near correct.

#### 0.125

Hint:
This is 1/8, not 3/8.

#### 0.83

Hint:
3/8 is less than a half, and 0.83 is more than a half, so they can't be equal.
Question 10 Explanation:
Topic: Converting between fractions and decimals (Objective 0017)
 Question 11

#### The pattern below consists of a row of black squares surrounded by white squares. #### 314

Hint:
Try your procedure on a smaller number that you can count to see where you made a mistake.

#### 317

Hint:
Are there ever an odd number of white squares?

#### 320

Hint:
One way to see this is that there are 6 tiles on the left and right ends, and the rest of the white tiles are twice the number of black tiles (there are many other ways to look at it too).

#### 322

Hint:
Try your procedure on a smaller number that you can count to see where you made a mistake.
Question 11 Explanation:
Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic) (Objective 0021).
 Question 12

#### The result is always the number that you started with! Suppose you start by picking N. Which of the equations below best demonstrates that the result after Step 6 is also N?

 A $$\large N*2+20*5-100\div 10=N$$Hint: Use parentheses or else order of operations is off. B $$\large \left( \left( 2*N+20 \right)*5-100 \right)\div 10=N$$ C $$\large \left( N+N+20 \right)*5-100\div 10=N$$Hint: With this answer you would subtract 10, instead of subtracting 100 and then dividing by 10. D $$\large \left( \left( \left( N\div 10 \right)-100 \right)*5+20 \right)*2=N$$Hint: This answer is quite backwards.
Question 12 Explanation:
Topic: Recognize and apply the concepts of variable, function, equality, and equation to express relationships algebraically (Objective 0020).
 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

#### Below is a portion of a number line: #### Point B is halfway between two tick marks.  What number is represented by Point B?

 A $$\large 0.645$$Hint: That point is marked on the line, to the right. B $$\large 0.6421$$Hint: That point is to the left of point B. C $$\large 0.6422$$Hint: That point is to the left of point B. D $$\large 0.6425$$
Question 14 Explanation:
Topic: Using Number Lines (Objective 0017)
 Question 15

#### Five million

Hint:
Pay attention to the exponents. Adding 3 and 2 doesn't work because they have different place values.

#### Fifty thousand

Hint:
Pay attention to the exponents. Adding 3 and 2 doesn't work because they have different place values.

Hint:

#### Thirty thousand

Hint:
$$3\times {{10}^{4}} = 30,000;$$ the other term is much smaller and doesn't change the estimate.
Question 15 Explanation:
Topics: Place value, scientific notation, estimation (Objective 0016)
 Question 16

#### 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 16 Explanation:
Topic: Ordering Fractions (Objective 0017)
 Question 17

#### In each expression below  N represents a negative integer. Which expression could have a negative value?

 A $$\large {{N}^{2}}$$Hint: Squaring always gives a non-negative value. B $$\large 6-N$$Hint: A story problem for this expression is, if it was 6 degrees out at noon and N degrees out at sunrise, by how many degrees did the temperature rise by noon? Since N is negative, the answer to this question has to be positive, and more than 6. C $$\large -N$$Hint: If N is negative, then -N is positive D $$\large 6+N$$Hint: For example, if $$N=-10$$, then $$6+N = -4$$
Question 17 Explanation:
If you are stuck on a question like this, try a few examples to eliminate some choices and to help you understand what the question means. Topic: Characteristics of integers (Objective 0016).
 Question 18

#### The table below gives data from various years on how many young girls drank milk. #### 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 18 Explanation:
Topic: Recognize and apply the concept of conditional probability (Objective 0026).
 Question 19

#### 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 19 Explanation:
Topic: Identifying rational and irrational numbers (Objective 0016).
 Question 20

#### 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 20 Explanation:
Topic: Converting between fraction and decimal representations (Objective 0017)
 Question 21

#### The teacher can be sure that the mean and median will be the same without doing any computation.

Hint:
Does this make sense? How likely is it that the mean and median of any large data set will be the same?

#### The teacher can be sure that the mean is bigger than the median without doing any computation.

Hint:
This is a skewed distribution, and very large countries like China and India contribute huge numbers to the mean, but are counted the same as small countries like Luxembourg in the median (the same thing happens w/data on salaries, where a few very high income people tilt the mean -- that's why such data is usually reported as medians).

#### The teacher can be sure that the median is bigger than the mean without doing any computation.

Hint:
Think about a set of numbers like 1, 2, 3, 4, 10,000 -- how do the mean/median compare? How might that relate to countries of the world?

#### There is no way for the teacher to know the relative size of the mean and median without computing them.

Hint:
Knowing the shape of the distribution of populations does give us enough info to know the relative size of the mean and median, even without computing them.
Question 21 Explanation:
Topic: Use measures of central tendency (e.g., mean, median, mode) and spread to describe and interpret real-world data (Objective 0025).
 Question 22

#### Which of the following is the equation of a linear function?

 A $$\large y={{x}^{2}}+2x+7$$Hint: This is a quadratic function. B $$\large y={{2}^{x}}$$Hint: This is an exponential function. C $$\large y=\dfrac{15}{x}$$Hint: This is an inverse function. D $$\large y=x+(x+4)$$Hint: This is a linear function, y=2x+4, it's graph is a straight line with slope 2 and y-intercept 4.
Question 22 Explanation:
Topic: Distinguish between linear and nonlinear functions (Objective 0022).
 Question 23

#### An equiangular triangle that is not equilateral.

Hint:
The AAA property of triangles states that all triangles with corresponding angles congruent are similar. Thus all triangles with three equal angles are similar, and are equilateral.

#### An equiangular quadrilateral that is not equilateral.

Hint:
A rectangle is equiangular (all angles the same measure), but if it's not a square, it's not equilateral (all sides the same length).

#### An equilateral quadrilateral that is not equiangular.

Hint:
This rhombus has equal sides, but it doesn't have equal angles: #### An equiangular hexagon that is not equilateral.

Hint:
This hexagon has equal angles, but it doesn't have equal sides: Question 23 Explanation:
Topic: Classify and analyze polygons using attributes of sides and angles (Objective 0024).
 Question 24

#### 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 24 Explanation:
Topic: Identify and analyze direct and inverse relationships in tables, graphs, algebraic expressions and real-world situations (Objective 0021)
 Question 25

#### What fraction of the area of the picture below is shaded? A $$\large \dfrac{17}{24}$$Hint: You might try adding segments so each quadrant is divided into 6 pieces with equal area -- there will be 24 regions, not all the same shape, but all the same area, with 17 of them shaded (for the top left quarter, you could also first change the diagonal line to a horizontal or vertical line that divides the square in two equal pieces and shade one) . B $$\large \dfrac{3}{4}$$Hint: Be sure you're taking into account the different sizes of the pieces. C $$\large \dfrac{2}{3}$$Hint: The bottom half of the picture is 2/3 shaded, and the top half is more than 2/3 shaded, so this answer is too small. D $$\large \dfrac{17}{6}$$Hint: This answer is bigger than 1, so doesn't make any sense. Be sure you are using the whole picture, not one quadrant, as the unit.
Question 25 Explanation:
Topic: Models of Fractions (Objective 0017)
 Question 26

#### 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 26 Explanation:
Topic: Calculate the probabilities of simple and compound events and of independent and dependent events (Objective 0026).
 Question 27

#### 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 27 Explanation:
Topic: Match three-dimensional figures and their two-dimensional representations (e.g., nets, projections, perspective drawings) (Objective 0024).
 Question 28

#### A family went on a long car trip.  Below is a graph of how far they had driven at each hour. #### Which of the following is closest to their average speed driving on the trip?

 A $$\large d=20t$$Hint: Try plugging t=7 into the equation, and see how it matches the graph. B $$\large d=30t$$Hint: Try plugging t=7 into the equation, and see how it matches the graph. C $$\large d=40t$$ D $$\large d=50t$$Hint: Try plugging t=7 into the equation, and see how it matches the graph.
Question 28 Explanation:
Topic: Select the linear equation that best models a real-world situation (Objective 0022).
 Question 29

#### 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 29 Explanation:
Topic: Classify and analyze polygons using attributes of sides and angles, and symmetry (Objective 0024).
 Question 30

#### 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 30 Explanation:
Topic: Integers (Objective 0016)
 Question 31

#### 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 31 Explanation:
Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic) (Objective 0021)
 Question 32

#### Which of the lines depicted below is a graph of $$\large y=2x-5$$? #### a

Hint:
The slope of line a is negative.

#### b

Hint:
Wrong slope and wrong intercept.

#### c

Hint:
The intercept of line c is positive.

#### 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 32 Explanation:
Topic: Find a linear equation that represents a graph (Objective 0022).
 Question 33

#### 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 33 Explanation:
Identify variables and derive algebraic expressions that represent real-world situations (Objective 0020).
 Question 34

#### 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 34 Explanation:
Topic: Apply proportional thinking to estimate quantities in real world situations (Objective 0019).
 Question 35

#### 1.5°

Hint:
Celsius and Fahrenheit don't increase at the same rate.

#### 1.8°

Hint:
That's how much the Fahrenheit temp increases when the Celsius temp goes up by 1 degree.

#### 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.

#### 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 35 Explanation:
Topic: Interpret the meaning of the slope and the intercepts of a linear equation that models a real-world situation (Objective 0022).
 Question 36

#### A car is traveling at 60 miles per hour.  Which of the expressions below could be used to compute how many feet the car travels in 1 second?  Note that 1 mile = 5,280 feet.

 A $$\large 60\dfrac{\text{miles}}{\text{hour}}\cdot 5280\dfrac{\text{feet}}{\text{mile}}\cdot 60\dfrac{\text{minutes}}{\text{hour}}\cdot 60\dfrac{\text{seconds}}{\text{minute}}$$Hint: This answer is not in feet/second. B $$\large 60\dfrac{\text{miles}}{\text{hour}}\cdot 5280\dfrac{\text{feet}}{\text{mile}}\cdot \dfrac{1}{60}\dfrac{\text{hour}}{\text{minutes}}\cdot \dfrac{1}{60}\dfrac{\text{minute}}{\text{seconds}}$$Hint: This is the only choice where the answer is in feet per second and the unit conversions are correct. C $$\large 60\dfrac{\text{miles}}{\text{hour}}\cdot \dfrac{1}{5280}\dfrac{\text{foot}}{\text{miles}}\cdot 60\dfrac{\text{hours}}{\text{minute}}\cdot \dfrac{1}{60}\dfrac{\text{minute}}{\text{seconds}}$$Hint: Are there really 60 hours in a minute? D $$\large 60\dfrac{\text{miles}}{\text{hour}}\cdot \dfrac{1}{5280}\dfrac{\text{mile}}{\text{feet}}\cdot 60\dfrac{\text{minutes}}{\text{hour}}\cdot \dfrac{1}{60}\dfrac{\text{minute}}{\text{seconds}}$$Hint: This answer is not in feet/second.
Question 36 Explanation:
Topic: Use unit conversions and dimensional analysis to solve measurement problems (Objective 0023).
 Question 37

#### Use the four figures below to answer the question that follows: #### How many of the figures pictured above have at least one line of reflective symmetry?

 A $$\large 1$$ B $$\large 2$$Hint: The ellipse has 2 lines of reflective symmetry (horizontal and vertical, through the center) and the triangle has 3. The other two figures have rotational symmetry, but not reflective symmetry. C $$\large 3$$ D $$\large 4$$Hint: All four have rotational symmetry, but not reflective symmetry.
Question 37 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 38

#### 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 38 Explanation:
Topic: Place Value (Objective 0016)
 Question 39

#### At a school fundraising event, people can buy a ticket to spin a spinner like the one below.  The region that the spinner lands in tells which, if any, prize the person wins. #### 40

Hint:
"Keychain" appears on the spinner twice.

#### 80

Hint:
The probability of getting a keychain is 1/3, and so about 1/3 of the time the spinner will win.

#### 100

Hint:
What is the probability of winning a keychain?

#### 120

Hint:
That would be the answer for getting any prize, not a keychain specifically.
Question 39 Explanation:
Topic: I would call this topic expected value, which is not listed on the objectives. This question is very similar to one on the sample test. It's not a good question in that it's oversimplified (a more difficult and interesting question would be something like, "The school bought 100 keychains for prizes, what is the probability that they will run out before 240 people play?"). In any case, I believe the objective this is meant for is, "Recognize the difference between experimentally and theoretically determined probabilities in real-world situations. (Objective 0026)." This is not something easily assessed with multiple choice .
 Question 40

#### 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) #### 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.

#### 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.

#### 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.

#### 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 40 Explanation:
Topic: Analyze and interpret various graphic and nongraphic data representations (e.g., frequency distributions, percentiles) (Objective 0025).
 Question 41

#### The least common multiple of 60 and N is 1260. Which of the following could be the prime factorization of N?

 A $$\large2\cdot 5\cdot 7$$Hint: 1260 is divisible by 9 and 60 is not, so N must be divisible by 9 for 1260 to be the LCM. B $$\large{{2}^{3}}\cdot {{3}^{2}}\cdot 5 \cdot 7$$Hint: 1260 is not divisible by 8, so it isn't a multiple of this N. C $$\large3 \cdot 5 \cdot 7$$Hint: 1260 is divisible by 9 and 60 is not, so N must be divisible by 9 for 1260 to be the LCM. D $$\large{{3}^{2}}\cdot 5\cdot 7$$Hint: $$1260=2^2 \cdot 3^2 \cdot 5 \cdot 7$$ and $$60=2^2 \cdot 3 \cdot 5$$. In order for 1260 to be the LCM, N has to be a multiple of $$3^2$$ and of 7 (because 60 is not a multiple of either of these). N also cannot introduce a factor that would require the LCM to be larger (as in choice b).
Question 41 Explanation:
Topic: Least Common Multiple (Objective 0018)
 Question 42

#### 7.5 meters

Hint:
Here is a picture, note that the large and small right triangles are similar: One way to do the problem is to note that there is a dilation (scale) factor of 5 on the shadows, so there must be that factor on the heights too. Another way is to note that the shadows are twice as long as the heights.

Hint:
Draw a picture.

Hint:
Draw a picture.

#### 45 meters

Hint:
Draw a picture.
Question 42 Explanation:
Topic: Apply geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to similarity, ; and use these concepts to solve problems (Objective 0024) . Fits in other places too.
 Question 43

#### Exactly one of the numbers below is a prime number.  Which one is it?

 A $$\large511$$Hint: Divisible by 7. B $$\large517$$Hint: Divisible by 11. C $$\large519$$Hint: Divisible by 3. D $$\large521$$
Question 43 Explanation:
Topics: Identify prime and composite numbers and demonstrate knowledge of divisibility rules (Objective 0018).
 Question 44

#### In the triangle below, $$\overline{AC}\cong \overline{AD}\cong \overline{DE}$$ and $$m\angle CAD=100{}^\circ$$.  What is $$m\angle DAE$$? A $$\large 20{}^\circ$$Hint: Angles ACD and ADC are congruent since they are base angles of an isosceles triangle. Since the angles of a triangle sum to 180, they sum to 80, and they are 40 deg each. Thus angle ADE is 140 deg, since it makes a straight line with angle ADC. Angles DAE and DEA are base angles of an isosceles triangle and thus congruent-- they sum to 40 deg, so are 20 deg each. B $$\large 25{}^\circ$$Hint: If two sides of a triangle are congruent, then it's isosceles, and the base angles of an isosceles triangle are equal. C $$\large 30{}^\circ$$Hint: If two sides of a triangle are congruent, then it's isosceles, and the base angles of an isosceles triangle are equal. D $$\large 40{}^\circ$$Hint: Make sure you're calculating the correct angle.
Question 44 Explanation:
Topic: Classify and analyze polygons using attributes of sides and angles, including real-world applications. (Objective 0024).
 Question 45

#### $$7-4=3$$ and $$8-5=3$$, so the fractions are equal.

Hint:
Not how to compare fractions. By this logic, 1/2 and 3/4 are equal, but 1/2 and 2/4 are not.

#### $$4\times 8=32$$ and $$7\times 5=35$$. Since $$32<35$$ , $$\dfrac{5}{8}<\dfrac{4}{7}$$

Hint:
Starts out as something that works, but the conclusion is wrong. 4/7 = 32/56 and 5/8 = 35/56. The cross multiplication gives the numerators, and 35/56 is bigger.

#### $$4<5$$ and $$7<8$$, so $$\dfrac{4}{7}<\dfrac{5}{8}$$

Hint:
Conclusion is correct, logic is wrong. With this reasoning, 1/2 would be less than 2/100,000.
Question 45 Explanation:
Topics: Comparing fractions, and understanding the meaning of fractions (Objective 0017).
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