MATH (ALL SUBJECT) PRAXIS
Mr. Kirk asked his students to compare 0.196 and 0.15. Four of his students correctly answered that 0.196 is greater than 0.15, but they gave different explanations when asked to describe their strategies to the class. Indicate whether each of the following student explanations provides evidence of a mathematically valid strategy for comparing decimal numbers. Student ExplanationProvides EvidenceDoes Not Provide Evidence0.196 is larger because there is one in the tenths, and then nine hundredths is more than five hundredths. And then I'm done.1.2.0.196 is greater because in the thousandths place six is greater than five, and in the hundredths place nine is greater than one.3.4.0.196 is bigger than 0.15 because if it is three numbers long, it will always be bigger than if it is two numbers long.5.6.0.196 is more than 0.15 because nineteen hundredths is bigger than fifteen hundredths.7.8.
Correct Answer: 1, 4, 6, 7 The first and fourth explanations provide evidence of a mathematically valid strategy for comparing decimal numbers, but the second and third explanations do not. The most efficient strategy for comparing the values of any two decimal numbers is based on the place values of their digits, which is what the first and the fourth students do, although in different ways. The first student considers the digits from left to right, starting from the tenths. The student first notices that there is a 1 in the tenths place in both numbers, and because the number is the same, the student moves on to the hundredths place and observes that 9 hundredths is greater than 5 hundredths. At this point, the student states, "I'm done," meaning that no matter what digits follow, they will not change how the decimal numbers compare to each other. The student's imprecise language does not affect the validity of the strategy. The fourth student likely compares the numbers to the hundredths place in one step since the values in the tenths place are the same, so this explanation also provides evidence of a mathematically valid strategy. The second student compares the digits of the numbers from right to left, rather than from left to right, and the third student compares the decimal numbers as if they were whole numbers, so neither of these explanations provides evidence of a mathematically valid strategy for comparing decimal numbers.
Match each fraction with its equivalent decimal number. (picture)
Correct Answer: First column: E; Second column: A, B; Third column: C, D The correct answers are that 21002100 is equivalent to 0.02, 2010020100 and 210210 are equivalent to 0.2, and 20102010 and 200100200100 are equivalent to 2. The decimal equivalent of any fraction can be found by dividing the numerator of the fraction by its denominator. Please note that credit for the correct answer is given regardless of the boxes in which the fractions are placed when matched with their equivalent decimal numbers.
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IMAGE student adds the fractions 3/4, 5/6, and 1/10 as represented in the work shown. The teacher notices that there is an error in the student's work that keeps the work from being mathematically correct. Which of the following aspects of the work does the student need to revise for the work to be mathematically correct? A.The student needs to simplify answers to lowest terms. B.The student needs to use the least common denominator. C.The student needs to write the final answer as a mixed number. D.The student needs to use the equal sign properly.
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Mr. Schroeder asked his students to find the product 36×536×5. Four students shared their strategies with the class, and Mr. Schroeder recorded their methods on the board. Indicate whether each of the following representations of student methods demonstrates that the student used reasoning based on the distributive property or reasoning based on the associative property. IMAGE
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Ms. Duchamp asked her students to write explanations of how they found the answer to the problem 24×15. One student, Sergio, wrote, "I did 24 times 10 and got 240, then I did 24 times 5 and that's the same as 12 times 10 or 120, and then I put together 240 and 120 and got 360." Ms. Duchamp noticed that four other students found the same answer to the problem but explained their strategies differently. Which of the following student explanations uses reasoning that is most mathematically similar to Sergio's reasoning? A.Since 24 is the same as 12 times 2 and 15 is the same as 5 times 3, I did 12 times 5 and got 60, then I did 2 times 3 and got 6, and 60 times 6 is 360. B.To get 24 times 5, I did 20 times 5 and 4 times 5, which is 120 altogether, and then I needed 3 of that, and 120 times 3 is 360. C.15 times 20 is the same as 30 times 10, and that gave me 300, and then I did 15 times 4 to get 60, and 300 plus 60 is 360. D.24 divided by 2 is 12, and 15 times 2 is 30, so 24 times 15 is the same as 12 times 30, and so my answer is 360.
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Ms. Giansante asked her students to compare 3/4 and 5/7 and show their work. A student named Timothy compared the fractions as represented in the work shown. Ms. Giansante wants to provide a counterexample for Timothy to help him realize his method is not valid and will not always give the correct comparison. Which of the following pairs of fractions provides a counterexample to Timothy's method? A.2/3 and 3/5 B.4/7 and 5/9 C.3/7 and 2/5 D.2/3 and 4/7
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One of Mr. Terry's students, Yvonne, found the answer to the problem 38×2938×29 as represented in the work shown. IMAGE When Mr. Terry asked Yvonne to explain her work, she said, "It's easier to just switch the numerators to make simpler fractions." Which of the following statements is true of Yvonne's strategy? A.Yvonne's strategy can only be used to rewrite products of fractions where the difference between the numerators is 1 and the difference between the denominators is 1. B.Yvonne's strategy can only be used to rewrite products of fractions where both fractions are less than 1. C.Yvonne's strategy can be used to rewrite any product of two fractions, but it will not always result in fractions that can be simplified. D.Yvonne's strategy can be used to rewrite any product of two fractions, and it will always result in fractions that can be simplified.
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Recognize that in a multidigit number, a digit in one place represents 10 times as much as it represents in the place to its right and 110110 of what it represents in the place to its left. Mr. Johansen and his class are working on the mathematical objective shown. To assess what his students understand about the objective, Mr. Johansen plans to use base-ten blocks in a nonconventional way. He plans to first tell students which block will represent the unit and then ask students to determine what number is represented with the given base-ten blocks. Mr. Johansen wants to identify numbers that can be represented using only little cubes, rods, flats, and big cubes. If Mr. Johansen wants students to identify the number 32.6 from his representation, which two of the following base-ten blocks can he choose to represent the unit? IMAGE A.Little cube B.Rod C.Flat D.Big cube
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Joshua walks the length of each of three trails on a hike. The first trail is 3.6 kilometers long. The second trail is 3.7 kilometers long. The third trail is 600 meters shorter than the sum of the lengths of the first two trails. Joshua walks at an average speed of 3 kilometers per hour over the course of the entire hike. How many minutes does it take Joshua to complete his hike?
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Ms. Palmeri has worked with her students on the concepts of area and perimeter of rectangles. She is concerned that students may have confused the concepts because both concepts were taught in the same unit and lessons for both included drawing rectangles on grid paper. She wants to assess students on only the concept of perimeter at this time. To do this, she will show her students pictures of rectangles drawn on grids and then have students find the perimeters of the rectangles and write their final answers on individual marker boards so that she can check them quickly. Which of the following pictures would be LEAST useful for Ms. Palmeri to show as she assesses her students' understanding of perimeter of rectangles in this way? A.IMAGE B. C. D.
Option (A) is correct. Based on the information given, a picture in which the numerical value of the perimeter is equal to the numerical value of the area will not be useful for Ms. Palmeri at this time. In looking at the final answers only, Ms. Palmeri would not be able to determine whether students were thinking about the perimeter of the figure or about the area of the figure. The perimeter of the rectangle shown in option (A) is 18 units, and the area of the rectangle is 18 square units. In the rectangles shown in the other options, students may find the area of the rectangles instead of the perimeter, but the answer will be easily identified as incorrect because it will be different from the perimeter value.
Which of the following word problems can be represented by the equation 4×n+8=16 ? A.A set of 5 baskets holds a total of 16 apples. The first basket has 8 apples and the other baskets each hold an equal number of apples. How many apples are in each of the other baskets? B.There are 12 baskets, 8 of which are empty. There are 16 apples, with an equal number of apples in each of the other 4 baskets. How many apples are in each of the 4 baskets? C.There are 16 baskets, 8 of which are empty. Each of the other baskets contains 4 apples. How many apples are there in all? D.There are 8 baskets with 4 apples in each basket and 16 apples that are not in a basket. How many apples are there in all?
Option (A) is correct. If there are 5 baskets and one basket holds 8 apples, the rest of the apples are split evenly among the other 4 baskets. Therefore, to find the number of apples in each of the 4 baskets, the equation 4×n+8=16 can be set up, where n is the number of apples in each of the 4 baskets. The problem in option (B) can be represented by the equation 4×n+8×0=16, the problem in option (C) can be represented by the equation 4×(16−8)=n, and the problem in option (D) can be represented by the equation 8×4+16=n.
1/4 of Anne's hot chocolate is made of milk. What is the amount of milk in 8 cups of Anne's hot chocolate? Devon explained how he found the answer to the problem shown, saying, "If 1/4 is milk, it's like the hot chocolate is 4 cups and 1 of them is the milk. Then if you double the hot chocolate you have to double the milk, so in 8 cups there are 2 cups of milk." Devon's partner, Milena, represented Devon's strategy using cubes as represented in the figure shown. IMAGE Which of the following statements best characterizes how Milena's work represents Devon's strategy? A.Milena's work accurately represents Devon's strategy because it shows 1/4 of 4 and how that result was doubled. B.Milena's work accurately represents Devon's strategy, but an area model for multiplication would better represent the different units of measure involved in the problem. C.Milena's work does not accurately represent Devon's strategy because unit cubes cannot be used to represent parts of a whole. D.Milena's work does not accurately represent Devon's strategy because Devon's strategy used a different whole than the whole Milena used in her work.
Option (A) is correct. In Milena's work, the first set of 4 cubes represents 4 cups of hot chocolate, and the 1 yellow cube represents the 1 cup of milk. The set of cubes is then doubled, just as Devon explained in his strategy. Milena's work makes use of a set model of fractions in which 4 cubes represent the whole, which is a valid way to represent the fractions in the problem.
A rectangular message board in Aleyah's dormitory room has a length of 30 inches and a perimeter of 108 inches. A rectangular bulletin board in the hallway outside Aleyah's room is twice as long and twice as wide as the message board. Which of the following statements about the bulletin board is true? A.The bulletin board has a width of 48 inches. B.The bulletin board has a length of 96 inches. C.The area of the bulletin board is twice the area of the message board. D.The perimeter of the bulletin board is four times the perimeter of the message board.
Option (A) is correct. Since the message board has a length of 30 inches and a perimeter of 108 inches, the width of the message board can be found by solving the equation 2(30)+2w=1082(30)+2w=108 for w. To solve the equation for w, subtract 60 from both sides of the equation and then divide both sides of the equation by 2 to find that w=24w=24. This means that the length and width of the bulletin board are 60 inches and 48 inches, respectively, and it can be concluded that the area of the message board is 720 square inches, the perimeter of the bulletin board is 216 inches, and the area of the bulletin board is 2,880 square inches. Thus, the only true statement is that the bulletin board has a width of 48 inches.
Ms. Garrett has been working on verbal counting with her students. She wants them to be more aware of patterns in the way number names are typically constructed. Which of the following number names LEAST reflects the typical pattern in the way number names are constructed in the base ten system? A.Eleven B.Sixteen C.Twenty-five D.Ninety
Option (A) is correct. The number name "eleven" does not follow any pattern of number-name construction with reference to the tens and ones. Option (C) is not correct because "twenty-five" follows the most typical structure of how number names are constructed for whole numbers, since the number of tens in the number is referred to first, followed by the number of ones. Although the numbers in options (B) and (D) do not follow the most typical structure like "twenty-five" does, where the tens are called out specifically, the numbers in these options do follow a structure of the number of ones being named, followed by "teen," which refers to the ten in the number. Therefore, these numbers follow a pattern, unlike "eleven.
Mr. Walters asked his students to order 89, 708, 37, and 93 from least to greatest, and to be ready to explain the process they used to order the numbers. One student, Brianna, ordered the numbers correctly, and when Mr. Walters asked her to explain her process, she said, "The numbers 89, 37, and 93 are less than 100, so they are all less than 708, since that is greater than 100. Also, 37 is the least because it comes before 50 and the other two numbers are close to 100. Then 89 is less than 90, but 93 is greater than 90." Which of the following best describes the strategy on which Brianna's explanation is based? A.A counting strategy B.A benchmarking strategy C.An estimation strategy D.A place-value strategy
Option (B) is correct. Brianna first indicates that 708 is the greatest number because it is greater than 100, while 37, 89, and 93 are all less than 100. Next, Brianna indicates that 37 is the least number because it is less than 50, while 89 and 93 are greater than 50. Finally, Brianna recognizes that 89 is less than 93 because 89 is less than 90, while 93 is greater than 90. Thus, over the course of her explanation, Brianna used 100, then 50, and then 90 as points of reference for comparisons, which is exactly what benchmark numbers are—points of reference for comparison. Brianna did not count between any of the numbers, estimate the numbers, or use the place values in any of the numbers to make her comparisons, so the other options do not describe the strategy on which Brianna's explanation is based.
Ms. Roderick asked her lunch helper in her kindergarten class to get one paper plate for each student in the class. Which of the following counting tasks assesses the same mathematical counting work as this task? A.Having students line up according to the number of the day of the month in which they were born B.Showing students 10 pencils and asking them to get enough erasers for all the pencils C.Showing students a row of 12 buttons and asking them to make a pile of 8 buttons D.Asking students to count the number of triangles printed on the classroom rug
Option (B) is correct. Getting one paper plate for each student in the class assesses whether students can determine when the number of objects in one set is equal to the number of objects in another set, and the task described in option (B) involves a similar determination. The task in option (A) assesses whether students can compare and order numbers. The task in option (C) assesses whether students can count a subset of objects from a larger set. The task in option (D) assesses whether students can count the number of objects in a set.
Ms. Simeone is working with her first-grade students on writing two-digit numerals. She wants to use an activity to assess whether her students are attending to the left-to-right directionality of the number system. Which of the following activities is best aligned with Ms. Simeone's purpose? A.Asking students to read the numbers 20 through 29 B.Asking students to represent the numbers 35 and 53 using base-ten blocks C.Asking students how many tens and how many ones are in the number 33 D.Showing students 23 cubes and 32 cubes and asking them which quantity is grea
Option (B) is correct. Having the students represent 35 and 53 using base-ten blocks will help Ms. Simeone assess whether students know which place is the tens place and which place is the ones place or whether students have reversed the ones place and the tens place, thinking the ones place is on the left and the tens place is on the right. Representing the numbers provides more information about students' understanding of place value than just reading numbers.
In word problems that have a multiplicative comparison problem structure, two different sets are compared, and one of the sets consists of multiple copies of the other set. Which of the following best illustrates a word problem that has a multiplicative comparison problem structure? A.There are 4 shelves in Joaquin's bookcase, and there are 28 books on each shelf. How many books are in Joaquin's bookcase? B.Marcus drives 3 times as many miles to get to work as Hannah does. Hannah drives 16 miles to get to work. How many miles does Marcus drive to get to work? C.A football field is 360 feet long and 160 feet wide. A soccer field is 300 feet long and 150 feet wide. The area of the football field is how many square feet greater than the area of the soccer field? D.An ice cream parlor sells 29 different flavors of ice cream and 4 different types of cones. How many different combinations consisting of an ice cream flavor and a type of cone are available at the ice cream parlor?
Option (B) is correct. In the problem in option (B), the two values being compared are the number of miles that Marcus drives to get to work and the number of miles that Hannah drives to get to work, and the number of miles that Marcus drives is 3 times the number of miles that Hannah drives. The problem in option (A) has an equal-groups problem structure, the problem in option (C) has a product-of-measures problem structure (since the product is a different type of unit from the factors in the problem), and the problem in option (D) has a combinations problem structure.
During a lesson in her second-grade class, Ms. Costa draws two squares of the same size, each representing the same whole. She then divides and shades the squares as represented in the figure. Her students consistently identify the area of each shaded region as one-fourth, but when they are asked if the areas are equal, some students say no. Which of the following statements most likely explains why the students see the areas as not being equal? A.The students think that the areas are not equal because the wholes are different sizes. B.The students think that the areas are not equal because the shaded regions are different shapes. C.The students have difficulty determining the size of geometric figures that include diagonal lines. D.The students have difficulty determining the part-to-whole relationship when working with visual models of fractions.
Option (B) is correct. One misconception that students often have when first beginning to work with area models of fractions is that the parts of the whole must be congruent for the areas of the parts to be equal, and this misconception explains the responses described in the question. The statement in option (A) is not correct because students at this level do not normally attend to the size of the whole when working with visual models of fractions, and the question states that Ms. Costa draws two squares of the same size. The statement in option (C) is not correct because students will make a similar error even when figures do not include diagonal lines. The statement in option (D) is not correct because it does not explain the situation presented, since the students consistently identify the area of each shaded region as one-fourth.
The scenario in a word problem states that an office supply store sells pens in packages of 12 and pencils in packages of 20. Which of the following questions about the scenario involves finding a common multiple of 12 and 20 ? A.In one package each of pens and pencils, what is the ratio of pens to pencils? B.How many packages of pens and how many packages of pencils are needed to have the same number of pens as pencils? C.If the store sells 4 packages each of pens and pencils, what is the total number of pens and pencils sold in the packages altogether? D.How many gift sets can be made from one package each of pens and pencils if there are the same number of pens in each set, the same number of pencils in each set, and all the pens and pencils are used?
Option (B) is correct. The least common multiple of 12 and 20 is 60, and 5 packages of pens and 3 packages of pencils are needed to have 60 of each writing utensil. The question in option (A) uses factors, not multiples, since 12 pens/20 pencils=3/5. The question in option (C) is best answered by calculating 4×12+4×20=48+80=128, which does not involve finding either a common factor or a common multiple of 12 and 20. The question in option (D) is best answered by finding that the greatest common factor of 12 and 20 is 4, which means that 4 gift sets can be made, each containing 3 pens and 5 pencils.
1/3= 3/9 =6/18 1/4=4/16=3/12 Ms. White's students are working on generating equivalent fractions like the ones shown. She asks her students to write a set of instructions for how to generate equivalent fractions. One student writes, "You have to multiply the bottom and the top of the fraction by a number." Which of the following revisions most improves the student statement in terms of validity and generalizability? A.You have to multiply both denominator and numerator by the same number. B.You have to multiply both denominator and numerator by the same nonzero number. C.You have to multiply both denominator and numerator by the same whole number. D.You have to multiply both denominator and numerator by the same positive whole number.
Option (B) is correct. To generate an equivalent fraction, it is not necessary to multiply the numerator and denominator of the original fraction by a whole number, but it is necessary to multiply the numerator and the denominator by the same number and for that number to be a number other than zero. The revision in option (B) is the only sentence that restates the student conjecture, makes it valid, and generalizes it by including all fractions.
Ms. Rodriguez is working with her kindergarten students to develop the skill of counting on. Which of the following tasks is best aligned with the goal of having students count on? A.The teacher gives each student a number book with a different number on each page. The students must count out and glue the same number of pictures to match the given number on each page. B.The teacher gives each student a 10-piece puzzle, disassembled with a single number written on each piece. The students must put the puzzle together with the numbers in order. C.The teacher gives each student a shuffled deck of 10 cards, each with a single number from 1 to 10. When the students draw a number card, they must count to 20, starting from the number on the card they drew. D.The teacher gives each student 8 blocks and a number cube, with the sides of the number cube numbered from 3 to 8. When the students roll the number cube, they must count out the same number of blocks as the number rolled and create a tower with that number of blocks.
Option (C) is correct. A student would begin with the number drawn and count on from that number until 20 is reached. For example, if the student draws a card with 15 on it, the student would count on from 15, saying, "15, 16, 17, 18, 19, 20." The other tasks described do not require students to count on.
One of Mr. Spilker's students, Vanessa, incorrectly answered the addition problem 457+138457+138 as represented in the work shown. Mr. Spilker wants to give Vanessa another problem to check whether she misunderstands the standard addition algorithm or whether she simply made a careless error. Which of the following problems will be most useful for Mr. Spilker's purpose? A.784+214784+214 B.555+134555+134 C.394+182394+182 D.871+225
Option (C) is correct. In the work shown, after adding the ones and recording the 5 in the ones place, Vanessa did not record that the additional 10 ones were 1 ten, nor did she add the regrouped ten in the tens place. The problem in option (C) will be most useful for Mr. Spilker's purpose because it requires regrouping from the tens place to the hundreds place. The problems in options (A) and (B) do not require any regrouping, and Vanessa may just record 10 hundreds without thinking about regrouping when answering the problem in option (D).
Ms. Cook's class was discussing strategies to compare two fractions. One student, Levi, said, "When the top numbers are the same, you know that the one with the smaller number on bottom is bigger." Ms. Cook asked her students to explain why Levi's claim is true. After giving the class time to work, she asked another student, Maria, to present her explanation. Maria said, "It's just like Levi said. For 1/4 and 1/2, they both have ones on top, and 4 is greater than 2, so 1/4 is less, just like 1/4 of a pizza is less than 1/2 of a pizza." Which of the following statements best characterizes Maria's explanation? A.It clearly explains why Levi's claim is true. B.It clearly explains why the converse of Levi's claim is true, but it does not explain why his actual claim is true. C.It shows that Levi's claim is true for one example, but it does not establish why his claim is true in general. D.It assumes that Levi's claim is true, but it does not establish why his claim is true in general.
Option (C) is correct. Maria explains why 1/4 is less than 1/2, which provides one example for which Levi's claim is true, but it does not explain why whenever two fractions have the same numerator, the fraction with the smaller denominator will always be the greater fraction. A general explanation would point out that when a whole is broken into a greater number of pieces of equal size, then each of those pieces will be smaller than the pieces when the whole is broken into fewer pieces of equal size.
Which of the following fractions has a value between the values of the fractions 7/9 and 8/11 ? A.1/2 B.2/3 C.3/4 D.4/5
Option (C) is correct. Since 7/9≈0.78 and 8/11≈0.73 , the fraction 3/4 has a value between the values of 7/9 and 8/11 because 3/4=0.75.
Ms. Chamberlain's students are discussing the following quadrilaterals and their diagonals. IMAGE Ms. Chamberlain asks her students what they notice about the diagonals in the quadrilaterals. One student says, "I noticed that the diagonals of the quadrilaterals always cross at a right angle." Of the following sets of quadrilaterals, for which set is the student's conjecture always true? A.Quadrilaterals with one pair of congruent opposite angles B.Quadrilaterals with one pair of parallel opposite sides C.Quadrilaterals with two pairs of congruent adjacent sides D.Quadrilaterals with two pairs of parallel opposite sides
Option (C) is correct. The quadrilaterals Ms. Chamberlain's students are discussing are, from left to right, a square, a rhombus, and a kite. All kites have perpendicular diagonals, and kites are defined as quadrilaterals with two pairs of congruent adjacent sides. Squares and rhombuses have two pairs of congruent adjacent sides as well, since they are kites with additional restrictions on their characteristics. Quadrilaterals with one pair of congruent opposite angles could be scalene quadrilaterals, which do not necessarily have perpendicular diagonals, so option (A) is incorrect. Quadrilaterals with one pair of parallel opposite sides are trapezoids, which do not necessarily have perpendicular diagonals, so option (B) is incorrect. Quadrilaterals with two pairs of opposite parallel sides are parallelograms, which do not necessarily have perpendicular diagonals, so option (D) is incorrect.
Marina explained how she found the difference 35−18, saying, "I knew that 18 plus 2 is 20, and 35 plus 2 is 37, so 35 minus 18 is the same as 37 minus 20, which is 17. So 35 minus 18 is 17." Marina's partner, Jeremy, represented Marina's strategy using a number line, as shown in the figure. IMAGE Which of the following statements best characterizes how Jeremy's work represents Marina's strategy? A.Jeremy's work accurately represents Marina's strategy because it shows that she correctly found the difference between 35 and 18. B.Jeremy's work accurately represents the part of Marina's strategy in which she considered 20 instead of 18 as the subtrahend, but it does not accurately represent how she took 20 away from 37. C.Jeremy's work does not accurately represent Marina's strategy because Marina's strategy involved shifting the problem, but Jeremy's work shows a counting-up strategy. D.Jeremy's work does not accurately represent Marina's strategy because Marina used a comparison interpretation of subtraction, but Jeremy's work shows a takeaway interpretation of subtraction.
Option (C) is correct. When Marina adds 2 to 18 and 2 to 35, she shifts the problem from 35−18 to 37−20, which has the same difference, and then she subtracts. In contrast, Jeremy's work shows how a student would start from 18 and count up 2 to get to 20, then count up 10 more to get to 30, and then count up 5 more to get to 35, after which the 2, 10, and 5 would be added to find the difference of 17. Note that option (D) is incorrect because it reverses the students' interpretations of subtraction in their work. Marina's strategy uses a takeaway interpretation of subtraction because she subtracted 20 from 37 to find the difference, but Jeremy's work shows a comparison interpretation of subtraction because he found the distance between 18 and 35 on the number line.
Ms. Shaughnessy is working with her class on measuring area using nonstandard units. While the students are finding the area of the surface of their desks using rectangular note cards, one student says, "I can just measure the long side of the desk with the long side of the card, then measure the short side of the desk with the short side of the card, and multiply them." Which of the following best describes the validity of the student's strategy? A.The strategy is not valid because the same unit must be used to measure each side of the desk. B.The strategy is valid only if the note cards are squares. C.The strategy is valid and the unit of measurement is square units. D.The strategy is valid and the unit of measurement is note cards.
Option (D) is correct. Area can be measured using any two-dimensional unit that covers a surface, but the label of the area must reflect that unit. In this case the student has used note cards as the unit to measure the area of the desk. When using square units, one counts how many times the side of the square unit fits on each side of the rectangle whose area is to be measured. When using a unit that is not a square, like a note card, it is important to keep the orientation of the unit constant to cover the area without overlapping. This method results in one dimension of the rectangle being measured with the long side of the note card and the other dimension of the rectangle being measured with the short side of the note card.
Mr. Aronson noticed that one of his students, Wesley, incorrectly solved a multiplication problem, as represented in the work shown. IMAGE Which of the following most likely describes the reason for Wesley's error? A.Wesley did not regroup correctly. B.Wesley did not add the regrouped 2 correctly. C.Wesley did not apply his multiplication facts correctly. D.Wesley did not attend to the place value of each digit in the factors correctly.
Option (D) is correct. In his work, Wesley correctly multiplied 34×2 to get 68 for the first partial product, but in the second partial product, he did not attend to the place value of the 5 and calculated 34×5instead of 34×50.
Ms. Howe's students are learning how to use models to help them answer word problems. The models use bars to represent the relationships between the given quantities and the unknown quantity. In each model, the unknown quantity is represented with a question mark. The quantities given in the word problem occupy the other boxes. Ms. Howe shows the following model to her students. Which of the following word problems best corresponds to the model shown? A.Max had $24. He gave $18 to Olivia and the rest to Sarah. How much money did Max give to Sarah? B.Max had $24. He gave 1/3 of his money to Sarah and the rest to Olivia. How much money did Max give to Olivia? C.Max gave $24 to his friend Sarah and $18 to his friend Olivia. What is the total amount of money Max gave to his two friends? D.Max has $24 in his piggy bank, which is 2/3 of the amount of money that Max has altogether. How much money does Max have altogether?
Option (D) is correct. In the model shown, the total amount is the unknown quantity, and the quantity of $24 given in the problem is 2/3 of the total quantity. Since the problem in (D) asks for the total amount of money that Max has and states that $24 is 2/3 of the total, it is the problem that best corresponds to the model.
Ms. Carter shows one of her students, Brandon, a set of cubes. She tells Brandon that there are 13 cubes in the set and asks him to take 1 cube away from the set. Ms. Carter then asks Brandon, "How many cubes do you think are in the set now?" Brandon quickly answers, "Twelve." Brandon has demonstrated evidence of understanding which of the following mathematical ideas or skills? A.Using numerals to describe quantities B.Counting with one-to-one correspondence C.Recognizing a small quantity without counting D.Knowing that each previous number name refers to a quantity which is one less
Option (D) is correct. In the scenario, Ms. Carter shows Brandon a set of cubes, explicitly tells him how many cubes are in the set, and asks him to take one cube away from the set. This process allows Ms. Carter to ensure that Brandon knows that there is now one less cube in the set. When Ms. Carter asks how many cubes are in the set after one cube is removed, Brandon readily states, without counting the cubes, that there are 12 cubes. This provides evidence that Brandon knows that 12 is the number name that precedes 13 and that 12 refers to a quantity that is one less than 13; it can also be assumed that Brandon has the same understanding for other whole numbers. Brandon did not use written numerals in the scenario, so option (A) is not correct. Also, Brandon is told how many cubes are in the set, so there is no evidence that he can count with one-to-one correspondence or recognize a small quantity without counting, so options (B) and (C) are not correct
A student found an incorrect answer to the problem 2/5×4/3. The student's answer is represented in the work shown. IMAGE Which of the following student work samples shows incorrect work that is most similar to the preceding work? A. B. C. IMAGE D.
Option (D) is correct. In the work shown, the student multiplies the numerator of the first fraction by the denominator of the second to get the numerator of the resulting fraction, and multiplies the denominator of the first fraction by the numerator of the second to get the denominator of the resulting fraction. The same process is shown in option (D). The error in option (A) is that the numerators are added instead of being multiplied. The error in option (B) is that the product is simplified incorrectly. The error in option (C) is similar to the error in the work shown, but the resulting fraction is 28/211 rather than 21/28.
A student incorrectly answered the problem 305.74×100305.74×100. The student's answer is represented in the work shown. Which of the following student work samples shows incorrect work that is most similar to the preceding work? A. B. C. D. IMAGE
Option (D) is correct. In the work shown, when the student multiplied 305.74 by 100, the student rewrote 305.74 and added two zeros at the end. The work sample that is most similar to this is the sample in option (D), since this sample shows that when the student multiplied 94.03 by 10, the student rewrote 94.03 and added one zero at the end.
Last Tuesday, a group of 5 researchers in a laboratory recorded observations during a 24-hour period. The day was broken into 5 nonoverlapping shifts of equal length, and each researcher recorded observations during one of the shifts. Which of the following best represents the amount of time each researcher spent recording observations last Tuesday? A.Between 4 and 4 1/4 hours B.Between 4 1/4 and 4 1/2hours C.Between 4 1/2and 4 3/4 hours D.Between 4 3/4 and 5 hours
Option (D) is correct. Since the 24-hour period is broken into 5 overlapping shifts of equal length, the problem is solved by finding 24/5, which is equivalent to 4 4/5. Since 4 4/5 is greater than 4 3/4 and less than 5, each researcher spent between 4 3/4 and 5 hours recording observations last Tuesday.
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? IMAGE
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.
A grocery store sells both green grapes and red grapes for a regular price of $2.89 per pound. Nelson buys 1.5 pounds of green grapes and 2.25 pounds of red grapes at the store on a day when the regular price is reduced by $0.75 per pound. Which of the following expressions represents the amount, in dollars, that Nelson will pay for the grapes? A.1.5+2.25×2.89−0.75 B.(1.5+2.25)×2.89−0.75 C.1.5+2.25×(2.89−0.75) D.(1.5+2.25)×(2.89−0.75)
Option (D) is correct. To find the amount, in dollars, that Nelson will pay for the grapes, the total weight of the grapes, in pounds, needs to be multiplied by the reduced price of the grapes, in dollars. The total weight of the grapes, in pounds, is 1.5+2.25, and the reduced price of the grapes, in dollars, is 2.89−0.75, so the amount, in dollars, that Nelson will pay for the grapes is (1.5+2.25)×(2.89−0.75). The parentheses must be included in the expression as shown so that the total weight of the grapes will be multiplied by the reduced price of the grapes.
Ms. Vargas asked her students to write an expression equivalent to 4(x−y). After substituting some values for x and y, a student named Andrew rewrote the expression as 4x−y. Andrew's expression is not equivalent to 4(x−y), but he thought his work was correct based on the substitutions he tried. For which of the following integer values of x and y would Andrew's expression appear to be correct? Select two choices. A.x=0 and y=0 B.x=0and y≠0 C.x≠0and y=0 D.x≠0 and x=y E.x≠0 and x=−y
Options (A) and (C) are correct. Since 4(x−y)=4x−4y, the expression 4(x−y)is equivalent to 4x−y only when 4y=y, and 4y=y only when y=0. The only options where y=0 are options (A) and (C).
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? 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. Option (B) describes a rectangle that is not necessarily a square, option (C) describes a rhombus that is not necessarily a square, and option (E) describes a rectangle that is not a square.
Ms. Stockton and her students are working on naming different polygons using their geometric attributes. She finds a suggestion online to use pattern blocks for this work. A set of pattern blocks is shown. IMAGE Which two of the following statements describe limitations of using pattern blocks for the purpose of naming polygons based on their attributes? A.The blocks could support the conclusion that all polygons are convex. B.The blocks do not clearly show how polygons can be composed of other polygons. C.The blocks could support the conclusion that all polygons can be composed of equilateral triangles. D.The blocks do not show which polygons have the defining attribute of having at least one set of parallel lines. E.The blocks could support the conclusion that an attribute of polygons is that they must have at least two sides of equal length.
Options (A) and (E) are correct. All the pattern blocks are convex polygons, and not all polygons are convex. Also, all the pattern blocks have at least two sides of equal length, and this is not an attribute of polygons. Option (B) is not correct because, for example, the yellow hexagon can be composed of two red trapezoids, and the blue rhombus can be composed of two green triangles. Option (C) is not correct because the tan rhombus and the orange square cannot be composed of equilateral triangles. Option (D) is not correct because the set of pattern blocks includes polygons that have the defining attribute of having at least one set of parallel lines.
Which three of the following word problems can be represented by a division equation that has an unknown quotient? A.Ms. Bronson works the same number of hours each day. After 8 days of work, she had worked 32 hours. How many hours does Ms. Bronson work each day? B.Mr. Kanagaki put tape around 6 windows before painting a room. He used 7 feet of tape for each window. How many feet of tape did he use? C.Micah used the same number of sheets of paper in each of 5 notebooks. He used 45 sheets of paper in all. How many sheets of paper did Micah use in each notebook? D.Each shelf in a school supply store has 8 packs of markers on it. Each pack has 12 markers in it. How many markers are on each shelf in the store? E.Trina gave each of 7 friends an equal number of beads to use to make a bracelet. She gave the friends a total of 63 beads. How many beads did she give to each friend?
Options (A), (C), and (E) are correct. The word problem in option (A) can be represented by the equation 32÷8=□, the word problem in option (C) can be represented by the equation 45÷5=□, and the word problem in option (E) can be represented by the equation 63÷7=□. The word problem in option (B) can be represented by the equation 7×6=□ or the equation □÷6=7, and the word problem in option (D) can be represented by the equation 12×8=□ or the equation □÷8=12; however, neither of these word problems can be represented by a division equation that has an unknown quotient.
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. 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. Options (A) and (E) are incorrect because 15 and 27 are not even numbers, and options (D) and (F) are incorrect because 24 and 30 are both multiples of 6.
Mr. Benner places a row of 5 cubes on a student's desk and asks the student, Chanel, how many cubes are on the desk. As Chanel points at the cubes one by one from left to right, she counts, saying, "One, two, three, four, five." Then she says, "There are five cubes!" Mr. Benner then asks Chanel to pick up the third cube in the row. As Chanel points at three cubes one by one from left to right, she counts, saying, "One, two, three." She stops, then picks up the three cubes, and gives them to Mr. Benner. Chanel has demonstrated evidence of understanding which two of the following mathematical ideas or skills? A.Using numerals to describe quantities B.Recognizing a small quantity by sight C.Counting out a particular quantity from a larger set D.Understanding that the last word count indicates the amount of objects in the set E.Understanding that ordinal numbers refer to the position of an object in an ordered set
Options (C) and (D) are correct. Chanel first counts the cubes one by one and then she states that there are 5 cubes. Her work demonstrates that she understands that when counting a set of objects, the last word count indicates the cardinality of the set (that is, the number of objects in the set), so option (D) is correct. When she is asked to pick the third cube in the row, she is able to count and stop at three, thus demonstrating that she can count out a quantity from a larger set, so option (C) is correct. However, since she does not pick up the third cube counted, but instead picks up the three cubes she counted, she shows that she does not yet understand that the ordinal number "third" refers to the position of third counted cube, which means that option (E) is incorrect. Also, Chanel does not demonstrate the ability to recognize a small quantity by sight since she counts one by one, so option (B) is incorrect. Finally, she is not asked to record the numeral that describes the number of cubes on her desk, so there is no evidence that Chanel can use numerals to describe quantities, which means that option (A) is incorrect.
A student answered two elapsed-time problems using the same method, as represented in the work shown. The student's answer to the first problem was correct, but the student's answer to the second problem was incorrect. IMAGE If the student continues to use the same method, for which two of the following time intervals will the student give the correct elapsed time? A.6:45 A.M. to 9:15 A.M. B.10:10 A.M. to 3:35 P.M. C.2:30 P.M. to 7:50 P.M. D.8:55 P.M. to 4:20 A.M. E.11:25 P.M. to 5:30 A.M.
Options (C) and (E) are correct. The student's error is subtracting smaller values from larger values in both the minutes portion of the time and the hours portion of the time, without considering which value should be the subtrahend and which should be the minuend. In doing so, the student will get an elapsed time of 5 hours and 20 minutes for the interval in option (C) and an elapsed time of 6 hours and 5 minutes for the interval in option (E), both of which are correct answers achieved using a method based on flawed reasoning. The intervals in options (A), (B), and (D) will result in wrong answers when the flawed reasoning is applied. The interval in option (A) will result in an incorrect elapsed time of 3 hours and 30 minutes rather than the correct elapsed time of 2 hours and 30 minutes. The interval in option (B) will result in an incorrect elapsed time of 7 hours and 25 minutes rather than the correct elapsed time of 5 hours and 25 minutes. The interval in option (D) will result in an incorrect elapsed time of 4 hours and 35 minutes rather than the correct elapsed time of 7 hours and 25 minutes.
Which two of the following inequalities are true? A.0.56>0.605 B.0.065>0.56 C.0.56>0.506 D.0.605<0.056 E.0.506<0.65 F.0.65<0.605
Options (C) and (E) are correct. To compare these decimal numbers, first compare the digits in the tenths place—the decimal number with the greater digit in the tenths place will be the greater number. If the digits in the tenths place are the same, compare the digits in the hundredths place to determine which decimal number is greater. This process can be continued as needed. Another way to compare these decimal numbers is to write each number to the thousandths place to make the comparison easier. For example, in option (C), 0.560>0.506, so 0.56>0.506.
Ms. Celantano found the following representation of the decimal multiplication problem 0.7×0.4=0.280.7×0.4=0.28 in her curricular materials. The representation uses a 10-by-10 grid of small squares. IMAGE Which three of the following statements about the representation are true? A.The area of the four small gray squares in the top row represents 0.4, and the area of the seven small gray squares in the left column represents 0.7. B.The representation shows a repeated-addition interpretation of decimal multiplication. C.In the representation, each of the small squares represents one hundredth. D.In the representation, the outer square represents one. E.The representation can be used to show that 0.7×0.4=0.4×0.7.
Options (C), (D), and (E) are correct. Option (C) is correct since there are 100 congruent small squares in the grid. Option (D) is correct since the length of each side of the outer square is 1 unit, so the area of the outer square is 1 square unit. Option (E) is correct since the representation can be reflected across a diagonal line from top left to bottom right to show the product 0.4×0.7. Option (A) is incorrect because the area of the four small gray squares in the top row represents 0.04, and the area of the seven small gray squares in the left column represents 0.07. Option (B) is incorrect because the representation shows an area interpretation of decimal multiplication, not a repeated-addition interpretation.
Mr. Rasche wants his students to understand that, depending on the context of a division word problem that has a remainder, the answer to the word problem will be found by ignoring the remainder, dividing the remainder into equal shares, or using the least whole number that is greater than the quotient. Mr. Rasche wants to illustrate these cases with word problems that involve the quotient 15÷215÷2. Indicate whether the answer to each of the following word problems is found by ignoring the remainder, dividing the remainder into equal shares, or using the least whole number that is greater than the quotient. IMAGE
The answer to the first problem is found by dividing the remainder into equal shares, the answers to the second and fourth problems are found by using the least whole number that is greater than the quotient, and the answer to the third problem is found by ignoring the remainder. In the first problem, each child will get 7 whole cookies, and the remaining cookie can be divided into equal shares, so each child will get 7.5 cookies. In the second problem, there will be 7 rooms with 2 people in each room, but the remaining person also needs a room, so the minimum number of rooms the group could have booked is 8. In the third problem, there will be 7 workstations, each equipped with 2 monitors, but since the leftover monitor cannot be paired with another monitor, only 7 workstations can be equipped. In the fourth problem, John will have saved only $14 at the end of the seventh week, so he has to wait until the end of the eighth week, when he will have $16, to buy the car.
A chef at a restaurant uses 1/5 liter of lemon juice and 3/10 liter of teriyaki sauce to make a marinade for 2 kilograms of salmon. How many liters of marinade does the chef use per kilogram of salmon? Give your answer as a fraction.
The correct answer is 1/4. The chef uses 1/5+3/10=2/10+3/10=5/10=1/2 liter of marinade for 2 kilograms of salmon. This means that the chef uses 1/2÷2=1/4 liter of marinade per kilogram of salmon.
A student's incorrect solutions to two equations are represented in the work shown. IMAGE If the student continues to use the same strategy, what will be the student's solution to the equation 2x=24?
The correct answer is 4.
Ms. Gibbs' students have been using interlocking cubes to help them represent and solve single-digit addition problems. Ms. Gibbs asked her students to use their interlocking cubes to find the sum 8+58+5. Four of her students found the correct sum of 13, but they gave different explanations when asked to describe their strategies to the class. Ms. Gibbs wants to use their explanations to highlight the making-ten strategy. Indicate whether each of the following student explanations makes use of the making-ten strategy. IMAGE
The first and third explanations do not make use of the making-ten strategy, but the second and fourth explanations do. Adding two numbers by making ten is an addition strategy that uses the associative property of addition to add two numbers by making an equivalent sum of some tens and some ones, which facilitates mental computations. For the sum 8+58+5, the making-ten strategy can be applied in one of the following two ways: 8+5=8+(2+3)=(8+2)+3=10+3=13, as explained by the second student, or 8+5=(3+5)+5=3+(5+5)=3+10=13, as explained by the fourth student. The first explanation is an example of a compensation strategy, in which the student uses a number that is easier to add, 10, instead of 8, and then compensates for it by taking 2 away from the total. The third explanation is an example of a counting-on strategy, in which the student starts with 8 and counts up 5 to 13.
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? Mr. Ma asked his class to solve the word problem shown. Three students correctly answered 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. Indicate whether each 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. IMAGE
The first and third explanations provide evidence of a mathematically valid strategy for determining whose recipe results in a greater amount of banana in each loaf, but the second explanation does not. In the first explanation, the student finds the unit rate of bananas per loaf in each recipe and then correctly compares the unit rates. In the third explanation, 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 the second explanation, 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.