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MTEL General Curriculum Mathematics Practice
Your answers are highlighted below.
Question 1 |
Which property is not shared by all rhombi?
4 congruent sidesHint: The most common definition of a rhombus is a quadrilateral with 4 congruent sides. | |
A center of rotational symmetryHint: 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 anglesHint: Unless the rhombus is a square, it does not have 4 congruent angles. | |
2 sets of parallel sidesHint: All rhombi are parallelograms. |
Question 1 Explanation:
Topic: Classify and analyze polygons using attributes of sides and angles,
and symmetry (Objective 0024).
Question 2 |
Aya and Kendra want to estimate the height of a tree. On a sunny day, Aya measures Kendra's shadow as 3 meters long, and Kendra measures the tree's shadow as 15 meters long. Kendra is 1.5 meters tall. How tall is the tree?
7.5 metersHint: 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. | |
22.5 metersHint: Draw a picture. | |
30 metersHint: Draw a picture. | |
45 metersHint: Draw a picture. |
Question 2 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 3 |
Elena is going to use a calculator to check whether or not 267 is prime. She will pick certain divisors, and then find 267 divided by each, and see if she gets a whole number. If she never gets a whole number, then she's found a prime. Which numbers does Elena NEED to check before she can stop checking and be sure she has a prime?
All natural numbers from 2 to 266.Hint: She only needs to check primes -- checking the prime factors of any composite is enough to look for divisors. As a test taking strategy, the other three choices involve primes, so worth thinking about. | |
All primes from 2 to 266 .Hint: Remember, factors come in pairs (except for square root factors), so she would first find the smaller of the pair and wouldn't need to check the larger. | |
All primes from 2 to 133 .Hint: She doesn't need to check this high. Factors come in pairs, and something over 100 is going to be paired with something less than 3, so she will find that earlier. | |
All primes from \( \large 2\) to \( \large \sqrt{267}\).Hint: \(\sqrt{267} \times \sqrt{267}=267\). Any other pair of factors will have one factor less than \( \sqrt{267}\) and one greater, so she only needs to check up to \( \sqrt{267}\). |
Question 3 Explanation:
Topic: Identify prime and composite numbers (Objective 0018).
Question 4 |
What fraction of the area of the picture below is shaded?
\( \large \dfrac{17}{24}\) 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) . | |
\( \large \dfrac{3}{4}\) Hint: Be sure you're taking into account the different sizes of the pieces. | |
\( \large \dfrac{2}{3}\) 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. | |
\( \large \dfrac{17}{6} \) 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 4 Explanation:
Topic: Models of Fractions (Objective 0017)
Question 5 |
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.
\( \large W=-5\) Hint: When y is even, W is even. | |
\( \large W=4\) 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. | |
\( \large W=6\) Hint: Try fixing x or y at 0, and start by finding W for x=0 y=14 or x=9, y=0. | |
\( \large W=32\) Hint: Try fixing x or y at 0, and start by finding W for x=0 y=14 or x=9, y=0. |
Question 5 Explanation:
Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic) (Objective 0021)
Question 6 |
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)
Which of the following statements can be inferred from the above chart?
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 6 Explanation:
Topic: Analyze and interpret various graphic and nongraphic data
representations (e.g., frequency distributions, percentiles) (Objective 0025).
Question 7 |
If x is an integer, which of the following must also be an integer?
\( \large \dfrac{x}{2}\) Hint: If x is odd, then \( \dfrac{x}{2} \) is not an integer, e.g. 3/2 = 1.5. | |
\( \large \dfrac{2}{x}\) Hint: Only an integer if x = -2, -1, 1, or 2. | |
\( \large-x\) Hint: -1 times any integer is still an integer. | |
\(\large\sqrt{x}\) Hint: Usually not an integer, e.g. \( \sqrt{2} \approx 1.414 \). |
Question 7 Explanation:
Topic: Integers (Objective 0016)
Question 8 |
Which of the following is closest to the height of a college student in centimeters?
1.6 cmHint: This is more the height of a Lego toy college student -- less than an inch! | |
16 cmHint: Less than knee high on most college students. | |
160 cmHint: Remember, a meter stick (a little bigger than a yard stick) is 100 cm. Also good to know is that 4 inches is approximately 10 cm. | |
1600 cmHint: This college student might be taller than some campus buildings! |
Question 8 Explanation:
Topic: Estimate and calculate measurements using customary, metric, and
nonstandard units of measurement (Objective 0023).
Question 9 |
On a map the distance from Boston to Detroit is 6 cm, and these two cities are 702 miles away from each other. Assuming the scale of the map is the same throughout, which answer below is closest to the distance between Boston and San Francisco on the map, given that they are 2,708 miles away from each other?
21 cmHint: How many miles would correspond to 24 cm on the map? Try adjusting from there. | |
22 cmHint: How many miles would correspond to 24 cm on the map? Try adjusting from there. | |
23 cmHint: 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 cmHint: 4 groups of 6 cm is over 2800 miles on the map, which is too much. |
Question 9 Explanation:
Topic: Apply proportional thinking to estimate quantities in real world situations (Objective 0019).
Question 10 |
A class is using base-ten block to represent numbers. A large cube represents 1000, a flat represents 100, a rod represents 10, and a little cube represents 1. Which of these is not a correct representation for 2,347?
23 flats, 4 rods, 7 little cubesHint: Be sure you read the question carefully: 2300+40+7=2347 | |
2 large cubes, 3 flats, 47 rodsHint: 2000+300+470 \( \neq\) 2347 | |
2 large cubes, 34 rods, 7 little cubesHint: Be sure you read the question carefully: 2000+340+7=2347 | |
2 large cubes, 3 flats, 4 rods, 7 little cubesHint: Be sure you read the question carefully: 2000+300+40+7=2347 |
Question 10 Explanation:
Topic: Place Value (Objective 0016)
Question 11 |
Use the table below to answer the question that follows:
Gordon wants to buy three pounds of nuts. Each of the stores above ordinarily sells the nuts for $4.99 a pound, but is offering a discount this week. At which store can he buy the nuts for the least amount of money?
Store AHint: This would save about $2.50. You can quickly see that D saves more. | |
Store BHint: This saves 15% and C saves 25%. | |
Store C | |
Store DHint: This is about 20% off, which is less of a discount than C. |
Question 11 Explanation:
Topic: Understand the meanings and models of integers, fractions, decimals,percents, and mixed numbers and apply them to the solution of word
problems (Objective 0017).
Question 12 |
A homeowner is planning to tile the kitchen floor with tiles that measure 6 inches by 8 inches. The kitchen floor is a rectangle that measures 10 ft by 12 ft, and there are no gaps between the tiles. How many tiles does the homeowner need?
30Hint: 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. | |
120Hint: The floor is 120 sq feet, and the tiles are smaller than 1 sq foot. | |
300Hint: Recheck your calculations. | |
360Hint: 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 12 Explanation:
Topic: Estimate and calculate measurements,
use unit conversions to solve measurement
problems, solve measurement problems in real-world situations (Objective 0023).
Question 13 |
The table below gives the result of a survey at a college, asking students whether they were residents or commuters:
Based on the above data, what is the probability that a randomly chosen commuter student is a junior or a senior?
\( \large \dfrac{34}{43}\) | |
\( \large \dfrac{34}{71}\) Hint: This is the probability that a randomly chosen junior or senior is a commuter student. | |
\( \large \dfrac{34}{147}\) Hint: This is the probability that a randomly chosen student is a junior or senior who is a commuter. | |
\( \large \dfrac{71}{147}\) Hint: This is the probability that a randomly chosen student is a junior or a senior. |
Question 13 Explanation:
Topic: Recognize and apply the concept of conditional probability (Objective 0026).
Question 14 |
The pattern below consists of a row of black squares surrounded by white squares.
How many white squares would surround a row of 157 black squares?
314Hint: Try your procedure on a smaller number that you can count to see where you made a mistake. | |
317Hint: Are there ever an odd number of white squares? | |
320Hint: 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). | |
322Hint: Try your procedure on a smaller number that you can count to see where you made a mistake. |
Question 14 Explanation:
Topic: Recognize and extend patterns using a variety of representations (e.g., verbal, numeric, pictorial, algebraic) (Objective 0021).
Question 15 |
In each expression below N represents a negative integer. Which expression could have a negative value?
\( \large {{N}^{2}}\) Hint: Squaring always gives a non-negative value. | |
\( \large 6-N\) 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. | |
\( \large -N\) Hint: If N is negative, then -N is positive | |
\( \large 6+N\) Hint: For example, if \(N=-10\), then \(6+N = -4\) |
Question 15 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 16 |
Use the samples of a student's work below to answer the question that follows:
\( \large \dfrac{2}{3}\times \dfrac{3}{4}=\dfrac{4\times 2}{3\times 3}=\dfrac{8}{9}\) \( \large \dfrac{2}{5}\times \dfrac{7}{7}=\dfrac{7\times 2}{5\times 7}=\dfrac{2}{5}\) \( \large \dfrac{7}{6}\times \dfrac{3}{4}=\dfrac{4\times 7}{6\times 3}=\dfrac{28}{18}=\dfrac{14}{9}\)Which of the following best describes the mathematical validity of the algorithm the student is using?
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 16 Explanation:
Topic: Analyze Non-Standard Computational Algorithms (Objective 0019).
Question 17 |
The equation \( \large F=\frac{9}{5}C+32\) is used to convert a temperature measured in Celsius to the equivalent Farentheit temperature.
A patient's temperature increased by 1.5° Celcius. By how many degrees Fahrenheit did her temperature increase?
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 17 Explanation:
Topic: Interpret the meaning of the slope and the intercepts of a linear equation
that models a real-world situation (Objective 0022).
Question 18 |
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.
If 240 people buy tickets to spin the spinner, what is the best estimate of the number of keychains that will be given away?
40Hint: "Keychain" appears on the spinner twice. | |
80Hint: The probability of getting a keychain is 1/3, and so about 1/3 of the time the spinner will win. | |
100Hint: What is the probability of winning a keychain? | |
120Hint: That would be the answer for getting any prize, not a keychain specifically. |
Question 18 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 19 |
Here is a number trick:
1) Pick a whole number
2) Double your number.
3) Add 20 to the above result.
4) Multiply the above by 5
5) Subtract 100
6) Divide by 10
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?
\( \large N*2+20*5-100\div 10=N\) Hint: Use parentheses or else order of operations is off. | |
\( \large \left( \left( 2*N+20 \right)*5-100 \right)\div 10=N\) | |
\( \large \left( N+N+20 \right)*5-100\div 10=N\) Hint: With this answer you would subtract 10, instead of subtracting 100 and then dividing by 10. | |
\( \large \left( \left( \left( N\div 10 \right)-100 \right)*5+20 \right)*2=N\) Hint: This answer is quite backwards. |
Question 19 Explanation:
Topic: Recognize and apply the concepts of variable, function, equality, and equation to express relationships algebraically (Objective 0020).
Question 20 |
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.
\( \large \dfrac{1}{7}\) 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. | |
\( \large \dfrac{1}{14}\) Hint: What would be the sample space here? Ie, how would you list 14 things that you pick one from? | |
\( \large \dfrac{1}{42}\) 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. | |
\( \large \dfrac{1}{49}\) Hint: This is the probability that they are both born on a particular day, e.g. Sunday. |
Question 20 Explanation:
Topic: Calculate the probabilities of simple and compound events and of
independent and dependent events (Objective 0026).
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