MSM PRAXIS
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If h and k are nonzero real numbers, which of the following expressions is equivalent to (h^-3/k^-2)^5 ? Answer the question by selecting the correct response. A. h2 + k3 B. k3 + h2 C. h15+ k10 D. k10+ h15
Option (D) is correct. When a base with a power is raised to another power, the base remains the same and the exponents are multiplied.
A teacher wants to use algebra tiles like those represented in the following figure to help students understand the concept of completing the square. The teacher wants to identify a quadratic expression for which a visual representation using algebra tiles will be useful in demonstrating the concept of completing the square. IMAGE Which of the following expressions best serves the teacher's purpose? Answer the question by selecting the correct response. A. x^2-3x-4 B. x^2+6x+11 C. x^2-8x+7 D. ^x2+9x+16
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For the past 10 years, Bill's retirement account has increased in value by 8% each year. If the pattern of increase continues for the next 3 years, which of the following is closest to the total percent increase in the value of Bill's retirement account over the 3 years? Answer the question by selecting the correct response. A. 24% B. 25% C. 26% D. 27%
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Four students correctly stated that the slope of the line in the following is 1, but they gave different explanations of how they found their answers. IMAGE Which of the following student explanations provide evidence of a mathematically valid strategy for determining the slope of the line? Select ALL that apply. Answer the question by selecting the correct responses. IMAGE
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In the following triangle, the length of line segment EC=25, the length of line segment EF=12, the length of line segment BD=33, and line segment BD is parallel to line segment EF. IMAGE What is the length of line segment BE ?
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The point with coordinates (-4, t) lies on line ℓ in the xy-plane, as shown in the following figure. IMAGE What is the value of t ? Answer the question by selecting the correct response. A. 4 B. 5 C. 9 D. 14
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Which of the following functions increases over its entire domain? Answer the question by selecting the correct response. IMAGE
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The length of a right rectangular prism is twice its height, and the height of the prism is twice its width. If the volume of the prism is 1,000 cubic inches, what is the height, in inches, of the prism? Answer the question by selecting the correct response. A. 5 B. 10 C. 15 D. 20
Option (B) is correct. Let ℓ be the length of the prism, let h be the height of the prism, and let w be the width of the prism.
If f(x)=-x2-2x+4, what is the value of f(-2) ? Answer the question by selecting the correct response. A. -4 B. -2 C. 4 D. 12
Option (C) is correct. To find the value f(-2), substitute -2 for x and then follow the order of operations.
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Option (D) is correct. 36, 54, 63
In the following figure, line segment K M intersects line segment J L at point K. What is the value of x ? _____
The correct answer is 12. SET THEM EQUAL TO 180
Company F charges a one‑time fee of $85.00 to rent a truck plus $0.60 for every mile driven. Company G charges a one‑time fee of $61.00 to rent a truck plus $0.90 for every mile driven. For what number of miles is the cost to rent a truck the same at both companies? ____Miles
The correct answer is 80 miles
An athlete's average rate of speed during a race was 5 meters per second. Which of the following values best approximates the athlete's average rate of speed during the race, in miles per hour? Answer the question by selecting the correct response. A. 2 B. 7 C. 11 D. 29
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A light display for a festival consists of a string of 1,000 lightbulbs in the colors red, yellow, green, and blue repeated consecutively in that order for the entire display. The 543rd lightbulb needs to be replaced. If the first lightbulb is red, what is the color of the lightbulb that needs to be replaced? Answer the question by selecting the correct response. A. Red B. Yellow C. Green D. Blue
Option (C) is correct.
Ms. Fisher's students are working on identifying like terms in algebraic expressions. When Ms. Fisher asks them how they know when terms are like terms, one student, Coleman, says, "Like terms have to have the same variable in them." Ms. Fisher wants to use a pair of terms to show Coleman that his description of like terms is incomplete and needs to be refined. Which of the following pairs of terms is best for Ms. Fisher to use for this purpose? Answer the question by selecting the correct response. A. 9m and 5 B. 8xy and xy C. 5n^4 and 2n^4 D. 4h^2 and 7h^3
Option (D) is correct. The best pair of terms for Ms. Fisher's purpose should contain the same variable but should not be like terms. The only option that shows such a pair is option (D), in which the variables are the same but the terms are not like terms because they have different exponents.
The mean of 40, 80, and m is 50. The mean of 40, 80, m, and v is 60. What is the value of v ? Enter your answer as an integer or a decimal in the answer box. Backspace to erase. v equals
90. guess and check
A vase has 3 red roses, 4 pink roses, and 5 white roses. Another vase has 2 red roses, 2 pink roses, and 2 white roses. If one rose is to be selected at random from each vase, what is the probability that both roses will be the same color? Give your answer as a fraction.
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Which of the following is an equation of the line in the xy-plane that passes through the points with coordinates (−2,−2) and (7,11) ? Answer the question by selecting the correct response. A. 5x−9y−8=0 B. 9x−5y+8=0 C. 9x−13y−8=0 D. 13x−9y+8=0
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TriangleBCD in the following xy-plane is rotated 90° counterclockwise about the origin and then reflected across the y-axis to form triangle B′C′D′ , where B′ corresponds to the transformation of point B. IMAGE What are the coordinates of B′ ? Answer the question by selecting the correct response. A. (5,1) B. (3,5) C. (1,3) D. (1,5)
Option (A) is correct. A 90° counterclockwise rotation about the origin transforms a point with coordinates (x,y) to the point with coordinates(-y,x), so the coordinates of B are transformed from (1,5) to (-5,1). To determine whether this is correct, remember that when a point is rotated 90° about the origin, the line segment from the origin to the point and the line segment from the origin to the image of the point form a 90° angle. Also, remember that a point in the first quadrant that is rotated 90° counterclockwise about the origin will have its image in the second quadrant, and the point with coordinates (-5,1) is in the second quadrant. Then, when a point is reflected over the y-axis, the x-coordinate of the point changes to its opposite, but the y-coordinate remains the same, so the point with coordinates (-5,1) is then transformed to the point with coordinates(5,1), which means the coordinates of B′ are (5,1).
Ms. Carswell asks her students to order 5/8, 7/12, and 2/3 from least to greatest. One student, Jacob, correctly answers that 7/12 is less than 5/8 and 5/8 is less than 2/3, and Ms. Carswell asks him to explain his reasoning. Jacob says, "The number on the bottom tells you how many pieces there are, and the number on top tells you how many of them are shaded. The smallest fraction is the one that has the biggest number of its pieces not shaded." Ms. Carswell wants to provide a counterexample for Jacob to help him realize his method does not always work. Which of the following pairs of fractions provides a counterexample to Jacob's method? Answer the question by selecting the correct response. A. 1/3 and 2/5 B. 1/4 and 1/3 C. 3/10 and 2/5 D. 7/20 and 5/12
Option (A) is correct. For a counterexample to Jacob's method, the smallest fraction is the one that does not have the biggest number of its pieces unshaded. In this option, the smallest fraction, 1/3, has 2 pieces unshaded, and the largest fraction, 2/5, has 3 pieces unshaded.
n order to estimate the population of snails in a certain woodland, a biologist captured and marked 84 snails that were then released back into the woodland. Fifteen days later, the biologist captured 90 snails from the woodland, 12 of which bore the markings of the previously captured snails. If all of the marked snails were still active in the woodland when the second group of snails was captured, what should the biologist estimate the snail population to be, based on the probabilities suggested by this experiment? Answer the question by selecting the correct response. A. 630 B. 1,010 C. 1,040 D. 1,080
Option (A) is correct. Given the conditions of the experiment, it is reasonable to assume that the 90 snails captured by the biologist 15 days after the markings were made represent a random sample of the snail population. Thus, the the fraction 12/90=84/x can be used to estimate the snail population in the woodland, which is represented by x. Therefore, 12x=7,560 , x=630, which means the biologist should estimate that there are 630 snails in the woodland.
Marie rode her bicycle 5 miles in 30 minutes, and Naomi rode her bicycle 4 miles in 20 minutes. What was the ratio of Marie's average speed to Naomi's average speed? Answer the question by selecting the correct response. A. 5 to 6 B. 4 to 5 C. 3 to 4 D. 2 to 3
Option (A) is correct. Since average speed equals, distance over time Marie's average speed is the fraction 5/30 miles per minute, and Naomi's average speed is the fraction 4/20 miles per minute. Therefore, the ratio of Marie's average speed to Naomi's average speed is the fraction 5/30 to the fraction 4/20, which is equivalent to the ratios 1/6 to 1/5, 30(1/6) to 30(1/5), and 5 to 6.
A farmer wants to double the area of a square pen for sheep. The length of each side of the pen is currently 12 meters. Which of the following values is closest to the increase in the length of each side, in meters, needed to double the area of the pen? Answer the question by selecting the correct response. A. 5 B. 8 C. 9 D. 12
Option (A) is correct. The area of the original square pen is 12×12=144 square meters. Doubling the area of the square pen results in an area of 288 square meters. A square pen with side lengths of 17 meters would result in a pen of 17×17=289 square meters. Since 289 is slightly more than twice as big as 144, the original area, 5 meters is closest to the increase in length of each side of the pen, since 17-12=5.
he function f is defined by f(x)=−x^2−3 for all real numbers x. Which of the following intervals represents the range of f ? Answer the question by selecting the correct response. A. (−∞,−3] B. (−∞,3] C. [−3,∞) D. [3,∞)
Option (A) is correct. The domain of a function is the set of all input values for which the function is defined, and the range of a function is the set of all output values based on the given input values. When a function is graphed on the xy-plane, the domain is shown on the x-axis and the range is shown on the y-axis. Since the domain of f(x)=-x^2-3 is all real numbers and the graph of the function opens downward and has a maximum at the point with coordinates (0,-3), the range of the function expressed in interval notation is (−∞,−3], which means the range of the function is all real numbers from negative infinity up to and including -3.
Ms. Patel's students are working on summarizing data collected from measurements on a single variable. She asks them to make observations about the following list of the numbers of students in each mathematics class at Franklin Middle School. 16, 20, 21, 18, 28, 29, 22, 21, 26, 28, 22, 19, 25, 22, 28 Which of the following student statements provides evidence of statistically appropriate reasoning about the data? Answer the question by selecting the correct response. A. There are two numbers that show up three times in the list, so that means the list has two modes. B. The outliers in the list are 16 and 29 because those are the smallest and largest numbers in the list. C. The median number of students in a math class is 21 because there are 7 numbers before it and 7 numbers after it in the list you showed us. D. Since there are 6 numbers in the list that are less than 22 and 6 numbers that are greater than 22, the mean number of students in a math class has to be 22.
Option (A) is correct. The mode is the number that appears most often in a set of data
During a unit on functions, Mr. Thompson asks his students to create a profit model of the following scenario. Nora makes bracelets to sell to her classmates. She spent $60 on supplies for the bracelets, and she sells each bracelet for $2. One student, Brianna, creates the following model to relate the number of bracelets Nora sold and the amount of money Nora made from selling the bracelets. image Mr. Thompson says that Brianna did a good job with her model overall, but there is something about it that is not exactly right. He asks the class to describe what the problem with the model is. Which of the following student responses best describes the problem with Brianna's model? Answer the question by selecting the correct response. A. The graph is on both sides of the y-axis. The graph should start at the y-axis and go to the right. B. The graph should not go below the x-axis. Graphs like this are always above the x-axis, never below. C. Brianna put the number of bracelets on the x-axis and the money earned on the y-axis. It should be the other way around. D. You can't go up by tens on the x-axis while you go up by twenties on the y-axis. You need to go up by the same amount on both of them.
Option (A) is correct. The number of bracelets that Nora sells cannot be negative, so the domain of the function in this context does not include negative numbers. Therefore, the model would be more accurate if it started at the y-axis and moved up and to the right from there. The other aspects of the model that were described by the students in options (B), (C), and (D) are correct as they are.
At a fair, a large container holds 500 Ping-Pong balls of various colors, as summarized in the following table. IMAGE If one of the Ping-Pong balls is to be selected at random from the container, what is the probability that the selected ball will be either orange or white? Answer the question by selecting the correct response. A. 1/5 B. 1/4 C. 1/3 D. 2/5
Option (B) is correct
A certain brand of paper plates can be purchased only in packages of 15 plates or in packages of 25 plates. Tanya purchased 8 packages of the paper plates, and there was a total of 150 plates in the packages. Of the packages Tanya purchased, how many were packages of 25 plates? Answer the question by selecting the correct response. A. 2 B. 3 C. 4 D. 6
Option (B) is correct.
Andre has driven 98 miles, which is 35% of the total distance he must drive on his trip. How many more miles does Andre have to drive on his trip? Answer the question by selecting the correct response. A. 34 B. 182 C. 280 D. 378
Option (B) is correct. IMAGE
Leah wants to estimate the number of students in her eighth-grade class whose favorite color is green. She wants to survey 25 students who are likely to be representative of her entire eighth‑grade class. From which of the following populations should Leah randomly select her sample of 25 students? Answer the question by selecting the correct response. A. Students at a basketball game B. Students at an eighth-grade assembly C. Students arriving at her middle school one morning D. Students in her eighth-grade social studies class who are wearing green
Option (B) is correct. A random sample of students at an eighth-grade assembly is likely to be representative of the entire eighth-grade class.
During a two-week period, a library reported that 6,590 people used the library. The following table shows the numbers of people in various categories of library use for that two-week period. IMAGE Approximately what percentage of the total number of people who used the library used it during the afternoon? Answer the question by selecting the correct response. A. 15% B. 20% C. 25% D. 30%
Option (B) is correct. In order to calculate the percentage, you must know two things: the total number of people who used the library during the two-week period and the total number of people who used the library during the afternoon.
A group of 41 students were asked to record the number of books each of them read while on a break from school. The following histogram shows the number of students who read from 0 to 6 books over the break. image Based on the histogram, what was the median number of books read by the students? Answer the question by selecting the correct response. A. 1 B. 2 C. 3 D. 4
Option (B) is correct. Since there are 20 entries below the 21st entry in this set of data and 20 entries above the 21st entry in this set of data, the median is determined by the 21st, or middle, entry in this set of data. There were 4 students who read zero books, 16 who read 1 book, 10 who read 2 books, 5 who read 3 books, 3 who read 4 books, 2 who read 5 books, and 1 who read 6 books, so the 21st entry is a 2, since each entry from the 21st entry to the 30th entry is a 2.
Set X consists of all integers that are prime numbers. Set Y consists of all 2-digit positive integers that are less than 70 and have a units digit of 7. How many integers are members of both set X and set Y ? Answer the question by selecting the correct response. A. 3 B. 4 C. 5 D. 6
Option (B) is correct. The members of set Y, the 2-digit positive numbers that are less than 70 and have a units digit of 7, are 17, 27, 37, 47, 57, and 67. Of these numbers, 17, 37, 47, and 67 are prime numbers, so four numbers are members of both set X and set Y. Note that in most multiple-choice questions that ask for numerical values, the exact answer should be found, as it should be in this question. If a multiple-choice question includes a word like "approximately," it generally indicates that the correct option will not be an exact value.
Which of the following is an equation of a line in the xy-plane that is perpendicular to the line y=2x+13 ? Answer the question by selecting the correct response. A. y=-2x+6 B. y=-12x+3 C. y=12x+3 D. y=2x+6
Option (B) is correct. The product of the slopes of two perpendicular lines is -1. When a linear equation is written in the form y=mx+b, the slope of the line is m. As a result, the slope of the line y=2x+13 is 2. Thus the slope of a line that is perpendicular to this line has to be -1/2, which satisfies the requirement that 2(-1/2)=-1, and the equation of the line among the options that has a slope of -1/2 is y=-12x+3.
A student is finding the perimeter of the following three rectangles. The student incorrectly states that the perimeter of the first rectangle is 8 units and the perimeter of the second rectangle is 12 units. The figure shows three shaded rectangles drawn on a grid. IMAGE If the student continues to use the same strategy to find the perimeter of the third rectangle, what will the student's answer be? Answer the question by selecting the correct response. A. 14 units B. 16 units C. 20 units D. 24 units
Option (B) is correct. The student appears to be finding the perimeter of each rectangle by counting the unit squares along the sides and on the interior of each rectangle
For inventory purposes, a manufacturing company assigns a 3‑character code to each different item produced. The first character of the code must be a letter from the 26‑letter English alphabet. The second and third characters must each be a digit from 0 to 9, but the digit cannot be repeated. How many different 3-character codes can be assigned? Answer the question by selecting the correct response. A. 720 B. 2,340 C. 2,600 D. 15,600
Option (B) is correct. There are 26 possibilities for the first character, 10 possibilities for the second character, and 9 possibilities for the third (since the second digit cannot be repeated). To find the total number of different 3-character codes, multiply 26 times 10, times 9 26×10×9, which is equal to 2,340.
The following partially completed table shows the yearly snowfall in a certain location. For the four years shown, the total snowfall, rounded to the nearest inch, was 327 inches. Image Which of the following values could have been the snowfall, in inches, in 2019 ? Answer the question by selecting the correct response. A. 68.0 B. 69.4 C. 70.9 D. 75.1
Option (C) is correct
Which of the following equations is equivalent to the equation 9x-4y=12 ? Answer the question by selecting the correct response. A. y=−9/4x−3 B. y=−9/4x+3 C. y=9/4x−3 D. y=9/4x+3
Option (C) is correct.
Following an introductory lesson on multiplying two binomials, Ms. Davis wants to follow up with a quick problem at the end of class to assess student proficiency. The students will write their final answers on slips of paper and hand them to Ms. Davis as they exit the class. Which of the following expressions would be LEAST useful for assessing student proficiency in multiplying two binomials? Answer the question by selecting the correct response. A. (x-3)(x-5) B. (x+3)(x-5) C. (x-5)(x+5) D. (x+5)(x+5)
Option (C) is correct. A common error that students make when multiplying binomials is to multiply only the x terms and the constant terms, forgetting to multiply each of these by one another. Therefore, a useful assessment problem would be one where this common error would lead to an incorrect answer so that Ms. Davis would know whether a student is using incorrect reasoning. A student who makes this error when multiplying the binomials in option (C) will get an answer of x2-25, which is the same as the correct answer, so Ms. Davis would have no way of knowing whether the student is using this incorrect reasoning; consequently, this is not a useful problem for assessing student proficiency in this situation. A student who makes the same error when multiplying the binomials in options (A), (B), and (D) will get an incorrect answer, so these problems would be useful for assessing student proficiency in this situation.
Cameron wants to create a display that shows how much a gallon of regular unleaded gasoline at a certain gas station costs at 8 A.M. each day over a two-week period. Which of the following is the most appropriate type of graph for him to use to display the data? Answer the question by selecting the correct response. A. Bar graph B. Circle graph C. Line graph D. Pictograph
Option (C) is correct. A line graph is used to display change over time, so of the choices given, a line graph is the most appropriate type of graph to use to display the data. The other types of graphs listed in the choices are not appropriate to use to display change over time. Multiple-choice question with one correct answer choice. Additional information needed to answer the question is provided in the reference region.
A teacher is selecting examples to illustrate linear functions. Which of the following examples best illustrates f(x)=mx, where m is a constant? Answer the question by selecting the correct response. A. The volume of water in a tank after being drained at a constant rate for x minutes B. The amount of material needed for a circular parachute that has a radius of x feet C. The distance traveled by light in x seconds when traveling at 186,282 miles per second D. The number of bacteria in a culture after x hours when the number of bacteria is increasing at a constant percent rate per hour
Option (C) is correct. Since f(x)=mx has a linear term but no constant term, an example that illustrates the function needs to be a proportional relationship. In the example in option (C), m would be 186,282 miles per second and f(x) would be the distance traveled by light in x seconds, so this example illustrates the function f(x)=mx. Option (A) is an example of a linear function, but it contains a constant term since there is an initial amount of water in the pool, so this example does not illustrate the function f(x)=mx. Option (B) is an example of a quadratic function, and option (D) is an example of an exponential function.
Four congruent circles are inscribed in a square with sides of length 4 units, as shown in the following figure. The circles are tangent to each other and to the sides of the square. IMAGE Which of the following values best approximates the percent of the area of the square that is shaded? Answer the question by selecting the correct response. A. 25.0% B. 22.5% C. 21.5% D. 20.3%
Option (C) is correct. The total area of the square is 4 squared, which equals 16 square units. Since each of the four circles has a radius of 1 unit, the area of each circle is pi times 1 squared, which equals pi square units. Therefore, the total area of the square that is unshaded is 4 pi square units, which means the total area of the square that is shaded is 16 minus 4 pi square units. Since 16−4π/16≈0.215, the option that best approximates the area of the square that is shaded is 21.5%.
A teacher wants to give students an example that highlights how using the distributive property can simplify computations. Which of the following examples best serves the teacher's purpose? Answer the question by selecting the correct response. A. 20×(25+35) B. 20×(23-16) C. 24×(712+58) D. 24×(67-45)
Option (C) is correct. To determine how useful the distributive property will be in simplifying the computations in each example, one should consider whether the fractions in the parentheses are easy to combine and whether the denominators of the fractions are factors of the number to be distributed. In the example in option (C), adding the fractions would require students to find a common denominator, which would take some computational effort. Distributing the 24, however, allows students to eliminate both denominators, which leads to a whole-number addition problem that is likely to feel much easier to students and will serve as a convincing example of how using the distributive property can simplify computations. Option (A) is not the best example because the sum of the fractions is 1, and students could likely find the sum quickly in their heads and then multiply the sum by 20. Options (B) and (D) are not the best examples because the denominators of the fractions are not factors of the number to be distributed, so these examples will not provide a compelling reason to distribute first.
The manager of a music store recorded the number of customers who entered the store each day over a 14-day period. A different number of customers entered the store on each of the days. The results are summarized in the following boxplot. IMAGE Which of the following conclusions can be made based on the data given in the boxplot? Answer the question by selecting the correct response. A. Between 30 and 55 customers entered the store on half of the days. B. Fewer than 55 customers entered the store on more than half of the days. C. The mean number of customers who entered the store each day was greater than 55. D. The number of days that 30 or fewer customers entered the store was equal to the number of days that 65 or more customers entered the store.
Option (D) is correct
A large right circular cylinder has a radius and height that are each twice the size of the radius and height of a small right circular cylinder. The volume of the large cylinder is how many times the volume of the small cylinder? Answer the question by selecting the correct response. A. 2 B. 4 C. 6 D. 8
Option (D) is correct.
Elizabeth owns x books. Kierra owns 5 more books than Elizabeth and 8 fewer than Ana. Which of the following expressions represents the total number of books the three girls own? Answer the question by selecting the correct response. A. 3x-3 B. 3x+3 C. 3x+13 D. 3x+18
Option (D) is correct. IMAGE
The number of bacteria in a culture doubled every hour. The culture started with 10 bacteria. Which of the following functions models the number of bacteria in the culture after t hours? Answer the question by selecting the correct response. A. f(t)=10t B. f(t)=10+2t C. f(t)=10t D. f(t)=10(2)t
Option (D) is correct. image
IMAGE Which of the following is an equation of the graph of the quadratic function shown in the preceding x y plane? Answer the question by selecting the correct response. A. y=x^2-4x+1 B. y=(x-1)^2-4 C. y=-x^2+4x+1 D. y=-(x-1)^2+4
Option (D) is correct. From the graph, the vertex is the point with coordinates (h,k)=(1,4) and a point on the graph is the point with coordinates (-1,0). The equation of the graph can be determined by substituting these two points into the vertex form y=a(x-h)^2+k, which yields 0=a(-1-1)^2+4, so 0=4a+4 and a=-1. Therefore, the equation of the graph is y=-(x-1)^2+4.
Ms. Chambers writes the following equation on the board. 5 X19-40 = 40- 5x 19 Ms. Chambers asks her students to determine if the equation is true. One student says, "The equation is true because on both sides of the equation, you subtract after you multiply 5 times 19." Which of the following statements most likely gives the reason underlying the student's error? Answer the question by selecting the correct response. A. The student disregarded the order of operations. B. The student incorrectly applied the associative property. C. The student does not understand the meaning of the equal sign. D. The student has overgeneralized by applying the commutative property to subtraction.
Option (D) is correct. The student applied the order of operations to determine that multiplication is performed before subtraction in the expression on either side of the equation but misapplied the commutative property, which works only with addition or multiplication. Using the student's reasoning, the equation would simplify to 95-40=40-95 or 55=-55.
The figure shows a right triangle and student work. The horizontal side of the triangle is labeled 3, the vertical side of the triangle is labeled x, and the hypotenuse is labeled 5. There is a right angle symbol where the horizontal side and vertical side meet. The student work is as follows: IMAGE Based on the information given, which of the following skills is the most important to review with this student? Answer the question by selecting the correct response. A. Solving for a variable in a quadratic equation B. Finding the approximate value of a square root C. Determining whether a triangle is a right triangle D. Identifying which side of a right triangle is represented by each variable in the Pythagorean theorem
Option (D) is correct. The student mistakenly switched the hypotenuse with the unknown leg length when using the Pythagorean theorem.
Ms. Adams tells her students that the x in the equation 3(2x+5)=39 is called a variable. She then asks her students to write down what they think the word "variable" means in this context. One student wrote, "The variable is a letter that stands for something we don't know." Which of the following revisions to the student's definition most improves the precision of the definition as it relates to this use of the word "variable" in this context? Answer the question by selecting the correct response. A. The variable is a letter that stands for any number that is unknown. B. The variable is a letter that stands for a number that is unknown in the equation. C. The variable is a letter that stands for an unknown number that can take on different values. D. The variable is a letter that stands for the unknown number that will make the equation true.
Option (D) is correct. This revision of the definition paraphrases the student's use of "something we don't know" as an unknown and adds the information that the unknown stands for the single unknown number that makes the equation true.
A certain recipe calls for 1/4 cup of white sugar, 1/2 cup of brown sugar, 2 1/4 cups of flour, 1/4 teaspoon of salt, and 2 tablespoons of cornstarch. The amount of cornstarch called for is what fraction of the total amount of sugar called for?(Note: 1 cup equals= 16 tablespoons and 1 tablespoon equals= 3 teaspoons) Answer the question by selecting the correct response. A. 1/24 B. 1/12 C. 1/8 D. 1/6
Option (D) is correct. To compare the amount of cornstarch to the total amount of sugar, you have to note the amount of each used and the unit that each is given in and make sure that both are in the same units.
Which of the following represents the x-intercept and the y-intercept of the graph of the equation 3x-4y=12 in the xy-plane? Answer the question by selecting the correct response. A. the point with coordinates (-4,0) and the point with coordinates(0,3) B. the point with coordinates (-3,0) and the point with coordinates (0,4) C. the point with coordinates (3,0) and the point with coordinates (0,-4) D. the point with coordinates (4,0) and the point with coordinates (0,-3)
Option (D) is correct. To find the x-intercept of the graph of the equation, let y=0 in the equation and solve for x. Substituting zero for y gives 3x-4(0)=12, so 3x=12, and therefore x=4, which means the x-intercept of the graph of the equation is the point with coordinates (4,0). To find the y-intercept of the graph of the equation, let x=0 in the equation and solve for y. Substituting zero for x gives 3(0)-4y=12, so -4y=12, and therefore y=-3, which means the y-intercept of the graph of the equation is the point with coordinates (0,-3).
IMAGE A. −9 B. −3 C. 0 D. 3
Option (D) is correct. When functions f and g are graphed in the xy-plane, the x-coordinate of their point of intersection will be the solution to the equation f(x)=g(x). The graphs of the functions intersect at the point with approximate coordinates (3.02,0.12), so of the values of x given in the options, the value that best approximates the solution to the equation f(x)=g(x) is 3.
Three students correctly answered the following word problem by concluding that Ayana's recipe results in a greater amount of banana in each loaf, but they gave different explanations when describing their strategies to the class. Ayana's banana bread recipe uses 3 bananas to make 2 loaves of banana bread. Natalie's banana bread recipe uses 4 bananas to make 3 loaves of banana bread. Whose recipe results in a greater amount of banana in each loaf of banana bread? Which of the following student explanations provides evidence of a mathematically valid strategy for determining whose recipe results in a greater amount of banana in each loaf? Select ALL that apply. Answer the question by selecting the correct response or responses. A. In Ayana's recipe there is a whole banana for each loaf and you split the last banana in half. In Natalie's recipe there is one banana for each loaf and the fourth banana is split into thirds. So in Ayana's loaf there are one and a half bananas, and in Natalie's there are one and a third, and a half is more than a third. B. In Ayana's recipe the bananas are split between only 2 loaves, while in Natalie's recipe the bananas are split between 3 loaves. If I have to split a cookie, I would rather split it in two because I get more, so Ayana's loaves contain more bananas. C. Ayana makes only 2 loaves and Natalie makes 3 loaves. If they made the same number of loaves, like 6, then Ayana would use 9 bananas and Natalie would use 8. So Ayana's loaves have more because 9 is more than 8.
Options (A) and (C) are correct. In option (A), the student finds the unit rate of bananas per loaf in each recipe and then correctly compares the unit rates. In option (C), the student finds a number of loaves, 6, that is a common multiple of the number of loaves made in each recipe and then multiplies the numbers in each recipe proportionally to be able to compare the number of bananas used in each recipe to make the common number of loaves. However, in option (B), the student considers only the number of loaves, not the number of bananas, so this explanation does not provide evidence of a mathematically valid strategy.
Mr. Varela asked his students to define a square in terms of other two-dimensional geometric figures. Which TWO of the following student definitions precisely define a square? Answer the question by selecting the correct responses. A. A square is a rectangle that has 4 sides of equal length. B. A square is a parallelogram that has 4 angles of equal measure. C. A square is a parallelogram that has 4 sides of equal length. D. A square is a rhombus that is also a rectangle. E. A square is a rectangle that is not a rhombus.
Options (A) and (D) are correct. A square is a quadrilateral with 4 sides of equal length and 4 angles of equal measure, whereas a rectangle is a quadrilateral with 4 angles of equal measure, a rhombus is a quadrilateral with 4 sides of equal length, and a parallelogram is a quadrilateral where opposite sides are parallel. Therefore, a rectangle that has 4 sides of equal length is a square, and a rhombus that is also a rectangle is a square, so options (A) and (D) are both precise definitions of a square.
Three students correctly answered the following word problem, saying that there are 15 juniors in the class, but they gave different explanations when describing their strategies to the class. There are a total of 25 seniors and juniors in a class, and the ratio of seniors to juniors in the class is 4 to 6. How many juniors are in the class? Which of the following student explanations provides evidence of mathematically valid reasoning about the problem? Select ALL that apply. Answer the question by selecting the correct response or responses. A. The ratio 4 to 6 equals the fraction 4/6. I know the fraction 8/12 represents 20 students and the fraction 12/18 represents 30 students. Because 25 is halfway between 20 and 30 and 15 is halfway between 12 and 18, the answer is 15. B. For every 10 students, there are 4 seniors and 6 juniors. So 20 students is 8 seniors and 12 juniors. I only need 5 more students. If there are 5 students, 2 are seniors and 3 are juniors. Since 12 plus 3 equals 15, the answer is 15. C. In the ratio 4 to 6, 6 is one and a half times 4. If I use 12 and 13, it's too many seniors because 12 plus 6 is 18. If I use 11 and 14, it's still too many seniors because adding half of 11 would make over 16. If I use 10 and 15, it works because 10 plus 5 is 15, so the answer is 15.
Options (A), (B), and (C) are correct. In option (A), the student finds ratios that are equivalent to the ratio 4 to 64:6, determines the total number of students represented by each ratio, and then determines the equivalent ratio that gives the correct total number of students. Since this student used an acceptable method correctly, this student's explanation provides evidence of mathematically valid reasoning. In option (B), the student finds two ratios that are equivalent to the ratio 4:6, such that the total number of students represented by the ratios is equal to 25, and then adds the number of juniors from each ratio together. Since this student used an acceptable method correctly, this student's explanation provides evidence of mathematically valid reasoning. In option (C), the student uses a guess-and-check strategy based on numbers that add up to 25 and realizes that the number of juniors in the class is one and a half times the number of seniors. Since this student used an acceptable method correctly, this student's explanation provides evidence of mathematically valid reasoning.
The following figure shows line ℓ in the xy-plane. Line k (not shown) is perpendicular to line ℓ. IMAGE Which of the following statements could be true? Select ALL that apply. Answer the question by selecting the correct response or responses. A. The slope of line k is less than 0. B. The x-intercept of line k is less than 0. C. The y-intercept of line k is less than 0.
Options (B) and (C) are correct. Line lℓℓ has a negative slope. When two nonhorizontal, nonvertical lines are perpendicular, the product of their slopes is −1, which requires one of the slopes to be positive and the other to be negative. Since the slope of line ℓ is negative, the slope of line k must be positive, so option (A) cannot be true. Line ℓ and line k may intersect at any point on line ℓ. Consequently, the value of the x-intercept of line k could be positive, negative, or zero, and the value of the y-intercept of line k could be positive, negative, or zero. Therefore, both options (B) and (C) could be true.
Mr. French's students are working on finding numbers less than 100 that are multiples of given one-digit numbers. When Mr. French asks them how they know when a number is a multiple of 6, one student, Crystal, says, "Even numbers are multiples of 6!" Mr. French wants to use two numbers to show Crystal that her description of multiples of 6 is incomplete and needs to be refined. Which of the following numbers are best for Mr. French to use for this purpose? Select TWO numbers. Answer the question by selecting the correct responses. A. 15 B. 16 C. 20 D. 24 E. 27 F. 30
Options (B) and (C) are correct. The best numbers for Mr. French's purpose are even numbers that are not multiples of 6, and 16 and 20 are both even numbers, but they are not multiples of 6.
Which TWO of the following word problems can be answered by dividing 4 by 1/5 ? Answer the question by selecting the correct responses. A. A quilt requires one fifth15 square meter of purple cloth. How many square meters of purple cloth are needed for 4 of these quilts? B. A machine working at a constant rate takes an hour to sort 4 bins of mail. How many bins of mail does the machine sort in 1/5 hour? C. Each person at a dinner was served 1/5 of a fish fillet. A total of 4 fish fillets were served. How many people were served at the dinner? D. A roll of copper wire that was 4 meters long was cut into sections that were each 1/5 meter long. How many sections were made from the roll of copper wire? E. Each of 4 people in a study group read 1/5 of the total chapters in a book. No student read the same chapter as any other student. What fraction of the total chapters in the book did the study group read?
Options (C) and (D) are correct. Problem (C) can be answered by dividing the number of fillets, 4, by the fraction of a fillet served per person, 1/5 fillet per person, to determine that 20 people were served at the dinner. Problem (D) can be answered by dividing the length of the roll of wire, 4 meters, by the length per section, 1/5 meter per section, to determine that 20 sections were made from the roll.
Which of the following values can be represented by the preceding area model? IMAGE Select ALL that apply. Answer the question by selecting the correct responses. A. 0.3% B. 3% C. 30% D. 0.3/10 E. 3/10 F. 30/10
Options (C) and (E) are correct. The area model is composed of 10 groups of 10 units. The shaded region can be represented as 3 out of 10 units, which is equivalent to the fraction 3/10, 0.3, or 30%, which is represented in options (C) and (E).
In a collection of writing utensils consisting of pens and pencils, 64% of the writing utensils are pens. What is the ratio of pens to pencils in the collection? Give your answer as a fraction. -----
The correct answer is the fraction 16/9. The collection consists of pens and pencils, and 64% of the writing utensils are pens, so 36% of the writing utensils are pencils. Therefore the ratio of pens to pencils in the collection, written as a fraction, is the fraction 64/36, which simplifies to the fraction 16/9.
In the x y plane, what is the distance, in units, between the (0,0)and the point with coordinates (-5,12)? Answer the question by selecting the correct response. A. 10 B. 11 C. 12 D. 13
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