Praxis- Math and Science
A student uses cubes to build a rectangular solid. Each of the cubes has a side length of one half1212 inch. The rectangular solid has a length of 4 inches, a width of 3 inches, and a height of 2 and one half212212 inches. How many cubes does the student use to build the rectangular solid?
240 cubes. The number of cubes placed along the length of the solid is 4 divided by 1/2, which equals 8. The number of cubes placed along the width of the solid is 3 divided 1/2, which equals 6. The number of cubes placed along the height of the solid is 2 1/2 divided by 1/2, which equals 5. The total number of cubes the student uses is 8x6x5, which equals 240.
Ms. Robinson is working with her students on mixed-number subtraction, and she wonders how well her students understand that each whole in a mixed number is equivalent to the fraction n/n equal parts. She wants to construct a subtraction problem for her students that will assess their understanding of the concept. Which of the following problems will give Ms. Robinson the most information about her students' understanding that each whole in a mixed number is equivalent to the fraction n over n/n equal parts?
4 1/3 - 1 5/6 Because 1/3 is less that 5/6, students can use regrouping to rewrite 4 1/3 as 3+3/3=1/3=3+4/3 and then subtract 4 1/3-1 5/6 as (3-1) + (4/3-5/6)
Ms. Egbuniwe wants her students to analyze information and provide evidence that individuals inherit traits from their parents but with variation between the individuals and their parents. She would like to provide them with resources that contain appropriate evidence for an upper-elementary student. Of the following, which TWO would provide the best evidence?
A. A picture of a female cocker spaniel with her litter of eight puppies, some black, some golden, some brown D. A diagram that shows a corn plant with red kernels and a corn plant with yellow kernels that have produced offspring containing both red and yellow kernels These two resources clearly show a parent, its offspring from sexual reproduction, and both similarities between and variation among all the individuals.
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. D. A square is a rhombus that is also a rectangle. 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.
In a classroom activity using images of the night sky, a teacher helps the students observe that the positions of stars seen in the summer at 11 P.M. are different than those seen in the winter at the same time. Which of the following statements best helps explain the observation?
A. Earth revolves around the Sun. The view of stars from Earth changes as Earth revolves around the Sun.
Students are studying various aspects of magnetism. Which of the following questions should upper-elementary students be able to answer experimentally with classroom resources?
A. What would cause a compass to not point to the north? This question can be investigated empirically inside the classroom with compasses, magnets, and other objects.
Ms. Vargas asked her students to write an expression equivalent to 4 times, open parenthesis, x minus y, close parenthesis4(x−y). A student named Andrew tried two substitutions of values for the ordered pair x comma y(x,y) in the expression, then incorrectly wrote that 4x-y is equivalent to 4(x-y), based on the substitutions he tried. Which TWO of the following substitutions of values for the ordered pair x comma y(x,y) could be ones that Andrew tried?
A. x=0 and y=0 C. x=1 and y=0 Since 4(x-y)= 4x-4y, the expression 4x-y is equivalent to 4(x-y) only when 4x-y=4x-4y. The last equation can be rewritten as 3y=0, pr more simply, y=0. The two substitutions Andrew tried must have used the value 0 for the variable y.
On a class field trip, students look at a profile of a local canyon wall. Sandy says, "I didn't realize this area was once an ocean." The teacher asks the other students to explain why they think Sandy may have come to this conclusion. Which student response indicates a lack of understanding about the evidence and geologic history?
B. "The layers of rock are in different colors in the shape of an ocean wave." The student lacks an understanding of the evidence because the appearance of different-colored layers of rocks is not evidence that the area was once covered by an ocean.
A student investigated whether matter is conserved during a change of state. The student weighed an empty glass beaker and then filled it with water before weighing it again to determine the weight of the water. The student then poured the water from the beaker into an ice-cube tray and placed it in a freezer. The student later removed the ice cubes from the tray and weighed them. After repeating the procedure five times, the student found that the weight of the ice cubes was slightly less than the initial weight of the water. The student concluded that freezing the water reduced its mass. Which of the following modifications to the procedure would best remove the error that affected the student's data?
B. Weighing the water after it is placed in the tray and the ice cubes before they are removed from the tray Modifying the procedure by weighing the water after it is placed in the tray and weighing the ice cubes before they are removed from the tray would most likely eliminate the error. Transferring the water twice for weighing probably led to a systematic loss of mass.
Ms. Marley is working with her students on fraction addition. She wants to highlight how the commutative property of addition can be used to make landmark numbers when adding fractions. Which of the following expressions best highlights the use of the commutative property of addition for this purpose?
B.1/3 +1/4+2/3 Option (B) is correct.
While Mr. Lynch's students were working on finding the answer to the problem 5/6 divided by 1/3, one student, Maggie, said the following. "I think that fraction division works just like fraction multiplication. You can divide the numerators and the denominators to find the answer. Like in this problem, 5 divided by 1 is 5, and 6 divided by 3 is 2, so the answer is 5 over 2." Which of the following is true of Maggie's conjecture about fraction division?
C Maggie's conjecture is true for all fraction division problems, but in some problems, the resulting quotients from the numerators and denominators will not be whole numbers. Maggie's method works for all fraction division problems but in some problems the result will be a "complex fraction" where one or both of the numerator and the denominator are fractions; as an example, 1/2 divided by 3/4= 1/3 divided by 2/4 is a true equation but the fraction on the right side of the equation is not written in a form where both the numerator and the denominator are whole numbers.
A third-grade teacher wants to assess students' mastery of providing explanations. The teacher asks the class to select a species that lives in groups and to write an explanation of how certain interactions within a group will help increase the likelihood of an individual's survival, growth, and reproduction. Which of the following explanations best meets the objective?
C. "Fish live in schools. By being one of a large number of fish, each individual has less chance of being eaten." In this explanation, the student describes a specific behavior to support the claim and makes a connection to how the school provides protection for all its members.
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?
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 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 (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 (C) is correct.
A student made the preceding graph to represent the distribution of water on Earth. The student concluded that there is no difference in the amounts of freshwater stored as surface water, groundwater, or water in glaciers and ice caps. Which of the following suggestions from a teacher would best help the student overcome this misinterpretation of the data?
C. Graph only the data for freshwater and set the y-axis range to go from 0 to 5 percent. Graphing only the freshwater data would help the student observe the difference in the amounts of freshwater.
Mr. Bennett's class was discussing strategies for adding whole numbers. One student, Katie, said, "When I add two numbers, I get the same answer as when the numbers switch places." Mr. Bennett asked his students to explain why Katie's claim is true when adding any two whole numbers. After giving the class time to work, he asked another student, Joel, to present his explanation. Joel said, "If the answer is a little number like 4, then 2 plus 2 is the same either way. And if the answer is a big number like 400, then 200 plus 200 is the same either way." Which of the following statements best characterizes Joel's explanation?
C. It neither gives useful examples for showing that Katie's claim is true, nor does it establish in general why her claim is true. In each of Joel's examples, he chose two addends that are equal. Joel's examples are not useful for showing that Katie's claim is true because changing the order of the addends does not change the appearance of the sums. Joel's explanation does not establish in general why Katie's claim is true because the examples cannot establish that something is true in general. In particular, Joel's examples do not address any case in which the addends are two different numbers.
Marina explained how she found the difference 35 minus 1835−18, 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 following figure.
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. When Marina adds 2 to 18 and 2 to 35, she shifts the problem from 35−1835−18 35 minus 18 to 37−2037−2037 minus 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. 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.
Mr. Aslanian's class watched a video about a species of mosquito that is found only at relatively high altitudes on a particular mountain in Africa. Mr. Aslanian then asked the students to use the information presented in the video to propose an explanation for the mosquitoes' limited geographical range. Which of the following responses indicates the most accurate understanding of the factors that influence the natural range of a species?
C. LaShawn wrote that the mosquitoes are better adapted to the cooler, damper climate at higher altitudes on the mountain than to the hotter, dryer climate at lower altitudes. The question tests your ability to critique students' explanations of natural phenomena on the basis of whether the explanations are consistent with existing knowledge. The response presented in option (C) provides the most accurate explanation for the limited geographical range of the mosquito species.
Ms. Lussier asked one of her students, Matthew, to explain how he found the answer to the subtraction problem 54-37. Matthew explained his answer as follows. "I didn't want to have to trade, so I added 3 to 54 so I would have 7 like in 37. Then I added 3 to 37 to keep the problem the same, but now I have 57 minus 40 so I just do 7 minus 0 and 5 minus 4." Which of the following statements best characterizes Matthew's strategy for finding the answer to two-digit subtraction problems?
C. Matthew's strategy is valid, but it will not always provide an equivalent subtraction problem where the ones digit of the number to be subtracted is less than the ones digit of the number from which to subtract. This strategy is valid and is similar to the strategy known as shifting the problem. The strategy will not always provide an equivalent subtraction problem, but the method works as long as the difference between the values in the ones place of the subtrahend and the minuend is greater than the difference between 10 and the value of the ones place of the subtrahend.
Mr. Wilson's upper elementary class is researching the positive and negative effects of pesticide and herbicide use in local agriculture. The class has already spent many hours searching on the Internet but has collected very little information. He would like to provide two additional opportunities for the students to collect information that would be the most useful. Which TWO activities would provide the best opportunities to collect useful information?
C. Mr. Wilson can meet with the school librarian, outlining his objectives and requirements. He can then bring his class to the library for several research periods after the librarian has had a chance to collect and lay out relevant journals, newspapers, and books. D. Mr. Wilson can schedule a panel discussion with representatives from the Environmental Protection Agency, United States Department of Agriculture, and the local extension service. They can provide materials that may not be available on the Internet, provide suggestions for additional resources, and answer student questions. They describe good sources of unbiased, authoritative information. The other options would present limited, possibly biased, information.
Which TWO of the following word problems can be answered by dividing 4 by 1/5?
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? 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.
A teacher poses the following problem to the class. Helena has 20 red beads for making bracelets. She has 12 more red beads than yellow beads. How many yellow beads does she have? Which of the following problems has the same mathematical structure as the problem the teacher poses?
Chanelle has 13 crayons. Chanelle has 7 more crayons than Chuck has. How many crayons does Chuck have? Both the problem in option (B) and the problem the teacher poses have a "comparison/start unknown" structure, in which a larger quantity is given, the amount by which the larger quantity differs from a smaller quantity is also given, and the problem asks for the smaller quantity.
Mr. Marquez and his students are working on naming fractions with denominators of 10 or 100 using decimal notation. Mr. Marquez finds an activity online that involves having students shade some parts of a tenths grid and some parts of a hundredths grid, name the fractions that have been shaded, and write the fractions in decimal notation. The following figure shows an example from the activity.
D. It does not make the relative value of the wholes in the two representations clear. The activity assumes that both the tenths grid and the hundredths grid represent a whole, but some students may interpret the tenths grid as representing 10 hundredths because the tenths grid looks like a single column of the hundredths grid. In the example, some students may misinterpret the 7 shaded squares in the tenths grid as7/100, which would be the correct interpretation of 7 shaded squares in a single column of the hundredths grid.
Ms. Keane had her students plant seeds in pots of soil. The students then placed some of the pots on a sunny windowsill and some on a table that was out of direct sunlight. As the seeds began to sprout, the students cared for them and observed their growth. After a weeklong break, the students found that some of their plants were wilted and some were dead. Ms. Keane asked the students to suggest a way to investigate why some of the plants had died and others had not. Which of the following plans will best allow the students to collect data that can be used to determine why some of the plants died?
D. Place new groups of plants in the original locations and water half of the plants in each group daily and water the other half weekly. This plan will allow students to test only the water variable while keeping the plants alive and allowing a comparison between groups of plants in different locations.
In Ms. Werner's classroom, students are measuring the length of objects around the room. Shaun measures the length of the rug using a ruler and gets 108 inches. Tim measures the length of the same rug using a yardstick and gets 3 yards. Ms. Werner overhears them discuss their results. Shaun says that inches are bigger than yards because there are more inches in the length of the rug, but Tim says that yards are bigger than inches because there are fewer yards. Which of the following statements best expresses Shaun's misconception about measurement?
D. The number of units of measure increases as the size of the unit of measure increases. Shaun's misconception is that the number of units of measure increases with the size of the unit, so 108 inches is bigger than 3 yards because 108 is bigger than 3. Shaun may be overgeneralizing his conception of counting numbers to his understanding of measurement units.
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?
D. The strategy is valid and the unit of measurement is note cards. 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.
In an activity using an illustration of a plant, a teacher guides students through a lesson about the different parts of a plant. When discussing the structure and function of the roots, the teacher points to small projections that are shown on the plant's roots. The teacher tells the students that the projections are called root hairs and asks the students to predict what would happen if all the root hairs were suddenly gone. Which of the following is likely to be the most direct effect on the plant of the sudden loss of all its root hairs?
D. Water and minerals will not be absorbed from the soil. The root hairs provide the roots of plants with the surface area needed to absorb water and minerals.
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?
Option (B) is correct. Having the students represent 35 and 53 using base-10 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 that 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 does.
Ms. Karp gives each of her students 24 counters. She asks the students to count the number of counters each of them has and to write the number on their worksheets. All the students write "24" on their worksheets. Next, she asks the students to explain the meaning of the 2 and the 4 in "24." Which of the following student explanations provides the most evidence of place-value understanding?
The 2 is two groups of ten counters and the 4 is four counters.
One of Mr. Terry's students, Yvonne, found the answer to the problem 3/8 x 2/9 as represented in the following work. 3/8x2/9=2/8x3/9=1/4x1/3=1/12 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?
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. Yvonne's strategy is valid because it is based on the definition of the product of two fractions and the commutative property of multiplication. For any two fractions a/b and c/d, the product of the fractions is defined as a/bxc/d=axc/b/d. Based on the commutative property of multiplication, axc/bxd=cxa/bxd, which means that a/bxc/d=c/dxa/d. However, Yvonne's strategy does not always result in fractions that can be simplified.
