7813 Math Praxis
A student incorrectly answered the problem 305.74×100. The student's answer is represented in the work shown. 305.74 x 100 = 305.7400 Which of the following student work samples shows incorrect work that is most similar to the preceding work? A. 246.7 x 100 = 2,467 B. 13.05 x 100 = 13,500 C. 46.13 x 10 = 460.130 D. 94.03 x 10 = 94.030
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.
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?
The correct answer is 280 minutes. Since 600 meters is equivalent to 0.6 kilometers, the third trail is 3.6+3.7−0.6=6.7 kilometers long. Therefore, Joshua walked a total distance of 3.6+3.7+6.7=14 kilometers on his hike. Since Joshua walks at an average speed of 3 kilometers per hour and there are 60 minutes in an hour, the proportion 3 kilometers/60 minutes= 14 kilometers/x minutes can be used to find how many minutes it takes Joshua to complete his hike. Based on the proportion, 3x=(14)(60), and since (14)(60)=840, x=840/3=280, which means that it takes Joshua 280 minutes to complete his hike.
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÷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. 1. 15 chocolate chip cookies will be evenly divided between 2 children. How many cookies will each child get? 2. A group of 15 people booked rooms in a hotel, and up to 2 people stayed in each room. What is the minimum number of rooms that the group could have booked? 3. A company wants to equip each new workstation with 2 computer monitors. The company has 15 monitors. How many new workstations can be equipped with 2 monitors? 4. John wants to buy a new toy car that costs $15, and he saves $2 at the end of each week for the car. At the end of how many weeks will John have enough money to buy the car?
1. 15 chocolate chip cookies will be evenly divided between 2 children. How many cookies will each child get? 2. A group of 15 people booked rooms in a hotel, and up to 2 people stayed in each room. What is the minimum number of rooms that the group could have booked? 3. A company wants to equip each new workstation with 2 computer monitors. The company has 15 monitors. How many new workstations can be equipped with 2 monitors? 4. John wants to buy a new toy car that costs $15, and he saves $2 at the end of each week for the car. At the end of how many weeks will John have enough money to buy the car? Correct Answer: 2, 6, 7, 12 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.
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. 1. In Ayana's recipe there are 3 bananas for 2 loaves, so 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 in 3. So in Ayana's loaf there are 1 and a half bananas, and in Natalie's there are 1 and a third, and a half is more than a third. 2. 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. 3. 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.
Correct Answer: 1, 4, 5 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.
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.
Correct Answer: A 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.
Match each fraction with its equivalent decimal number. 20/100, 2/10, 20/10, 200/100, 2/100
Correct Answer: First column: E; Second column: A, B; Third column: C, D The correct answers are that 2/100 is equivalent to 0.02, 20/100 and 2/10 are equivalent to 0.2, and 20/10 and 200/100 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.
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.
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."
Ms. Vargas asked her students to write an expression equivalent to 4(x−y)4(x−y). After substituting some values for x and y, a student named Andrew rewrote the expression as 4x−y4x−y. Andrew's expression is not equivalent to 4(x−y)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=0 and y≠0 C.x≠0 and 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)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. 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.
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.
Which of the following fractions has a value between the values of the fractions 7/9 and 8/11 ?
Option (C) is correct. Since 7/9≈0.78≈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. Duchamp asked her students to write explanations of how they found the answer to the problem 24×1524×15. 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.
Option (C) is correct. Sergio first uses the distributive property to think of 24×15 as 24×(10+5), or 24×10+24×5. After Sergio multiplies 24 and 10 to get 240, he multiplies 24 and 5 using a doubling and halving strategy. Since 24=12×2, 24×5=(12×2)×5=12×(2×5)=12×10, so the product of 24 and 5 is equal to the product of 12 and 10, which is 120. The explanation in option (C) also uses the distributive property but in a different way. This student thinks of 24×15 as (20+4)×15, or 20×15+4×15. After the student multiplies 20 and 15, the student uses the doubling and halving strategy to find the product of 4 and 15. Therefore, this explanation uses reasoning that is most mathematically similar to Sergio's reasoning. The explanations in options (A) and (B) do not use the doubling and halving strategy, and the explanation in option (D) does not use the distributive property.
Last Tuesday, a group of 5 researchers in a laboratory recorded observations during a 24-hour period. The day was broken into 5 non overlapping 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 1/2 hours C.Between 4 1/2 and 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. 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. 2/5 x 4/3 = 6/20 = 3/10 Which of the following student work samples shows incorrect work that is most similar to the preceding work? A. 5/4 x 1/2 = 6/8 = 3/4 B. 5/3 x 12/15 = 60/45 = 3/2 C. 7/4 x 7/3 = 28/21 = 4/3 D. 1/2 x 9/10 = 10/18 = 5/9
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 28/21 rather than 21/28.
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? A.9d and 5 B.8xy and xy C.5a4 and 2a4 D.4h2 and 7h3
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.
n the partitive model of division, the quotient is the size of each group. In the measurement model of division, the quotient is the number of groups. Which of the following problems illustrates the measurement model of division? Select all that apply. A.Joe is making chocolate fudge and the recipe calls for 3 1/4 cups of sugar. Joe uses a 1/4 cup measuring cup to measure the sugar. How many times does Joe need to fill the measuring cup to measure the sugar needed for the recipe? B.3 1/4 cups of soup fill 1/4 of a container. How many cups of soup will it take to fill the whole container? C.A trail is 3 1/4 miles long and trail markers are placed at 1/4 mile intervals along the trail. How many trail markers are placed along the trail?
Options (A) and (C) are correct. In the problem in option (A), the size of the measuring cup (the group) is given, and the problem asks for the number of times the measuring cup needs to be filled to measure the sugar. Similarly, in the problem in option (C), the size of the intervals (the group) is given, and the problem asks for the number of trail markers along the trail, which is determined by finding the number of 1/4 mile intervals along the trail. In contrast, the problem in option (B) asks for the number of cups of soup in the whole, or the one, container, so the problem is asking for the size of the whole group and is, therefore, using the partitive model of division.
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.
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=□32÷8=□, the word problem in option (C) can be represented by the equation 45÷5=□45÷5=□, and the word problem in option (E) can be represented by the equation 63÷7=□63÷7=□. The word problem in option (B) can be represented by the equation 7×6=□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.
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 1/10 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?
Options (B) and (C) are correct. The goal of the activity is to help students understand the relationship between place values depending on their relative position rather than relying on the intrinsic value of each base-ten block. If the rod is chosen to represent the unit, then the little cube represents 1/10 of the value of the rod (or 0.1 times the value of the rod), the flat represents 10 times the value of the rod, and the big cube represents 10 times the value of the flat (or 100 times the value of the rod). The number 32.6 can then be represented by using 3 flats, 2 rods, and 6 little cubes, so option (B) is correct. If the flat is chosen to represent the unit, then the rod represents 1/10 of the value of the flat, the little cube represents 1/10 of the value of the rod (or 1/100 of the value of the flat), and the big cube represents 10 times the value of the flat. The number 32.6 can then be represented by using 3 big cubes, 2 flats, and 6 rods, so option (C) is correct. If the little cube is chosen to represent the unit, then none of the base-ten blocks shown can be used to represent tenths, so option (A) is incorrect. Similarly, if the big cube is chosen to represent the unit, then none of the base-ten blocks shown can be used to represent tens, so option (D) is incorrect.
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.
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.
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. 1. Determine the elapsed time from 7:05 am to 1:45 pm. 7-6=6 45-5=40 6 hours 40 minutes passed 2. Determined the elapsed time from 11:45 pm to 3:20 am. 11-3=8 45-20=25 8 hours 25 minutes 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.
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. Student Explanation 1. I started with 10 cubes and 5 cubes—that is 15 cubes—and then I took away the extra 2 cubes and got 13 cubes. 2. I took 2 cubes from the 5 and put them with the 8, and then I knew 10 and 3 is 13. 3. I started with 8. Then I counted 9, 10, 11, 12, 13. I separated 3 cubes from the 8 so I have 5 and 5, which is 10, and 10 and 3 add up to 13.
Student Explanation 1. I started with 10 cubes and 5 cubes—that is 15 cubes—and then I took away the extra 2 cubes and got 13 cubes. (does not use make ten strategy) 2. I took 2 cubes from the 5 and put them with the 8, and then I knew 10 and 3 is 13. (Uses make ten strategy) 3. I started with 8. Then I counted 9, 10, 11, 12, 13. (does not use make ten strategy) I separated 3 cubes from the 8 so I have 5 and 5, which is 10, and 10 and 3 add up to 13. (Uses make ten strategy) Correct Answer: 2, 3, 6, 7 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+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.
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.
