Math Praxis

One of Mr Terry's students, Yvonne, found the answer to the problem 3⁄8 x 2/9 as represented in the work shown: 3⁄8 x 2/9 = 2/8 x 3/9 = 1⁄4 x 1⁄3 = 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? 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.

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.

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.

D. The strategy is valid and the unit of measurement is note cards.

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 7:05am to 1:45pm. 7-1 =6 45-5 = 40 6 hours 40 minutes passed 2. Determine the elapsed time from 11:45pm to 3:20am. 11-3=845-20=258 hours 25 minutes passed If the student continues to use the same method, for which 2 of the following time intervals will the student give the correct elapsed time? a. 6:45am to 9:15am b. 10:10am to 3:35pm c. 2:30pm to 7:50pm d. 8:55pm to 4:20am e. 11:25pm to 5:30am

c. 2:30pm to 7:50pm e. 11:25pm to 5:30am

Which 2 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

C. 0.56>0.506 E. 0.506<0.65 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's incorrect solutions to two equations are represented in the work shown. 7x = 70 x = 0 4x = 44 x = 4 If the student continues to use the same strategy, what will be the student's solution to the equation 2x=24?

x = 4 The work shown indicates that the student likely thinks that x is a placeholder for the digit in the units place and that the coefficient of x represents the digit in the tens place. The student likely does not understand that 7x means 7 times x or that 4xmeans 4 times x. Therefore, if the student continues to use the same strategy, the student will indicate that the solution to the equation 2x=24 is x=4 since 4 is the digit in the units place in 24.

2Match each fraction with its equivalent decimal number. 2/100 2/10 20/10 200/100 2/100 0.02 = 0.2 = 2 =

0.02 = 2/100 0.2 = 20/100, 2/10 2 = 200/100, 20/10

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:

1/4 The chef uses 1/2 / 2 = 1/4 liter of marinade per kilogram of salmon.

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?

280 minutes 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 mins = 14 kilo / 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=8403=280, which means that it takes Joshua 280 minutes to complete his hike.

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=0 and y≠0 C. x≠0 and y=0 D. x≠0 and x=y E. x≠0 and x=−y

A. x=0 and y = 0 C. x≠0 and y=0

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 greater

B. Asking students to represent the numbers 35 and 53 using base-ten blocks

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?

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?

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.

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.

Which of the following word problems can be represented by the equation 4 x 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?

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? 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 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.

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 rectangular 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.

a. A square is a rectangle that has 4 sides of equal length. d. A square is a rhombus that is also a rectangle.

In 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 used a 1/4 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 fills 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?

a. Joe is making chocolate fudge and the recipe calls for 3 1/4 cups of sugar. Joe used a 1/4 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? 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?

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? a. Littlecube b. Rod(10cubes) c. Flat (10x10 cubes) d. Bigcube(cubemadeupofcubes)

b. Rod (10cubes) c. Flat (10x10 cubes)

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

b. Showing students 10 pencils and asking them to get enough erasers for all the pencils

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? One square is broken into fourths by drawing diagonal lines from each corner, making 4 triangle-like shapes. One square is broken into fourths by drawing two lines down the middle of each side, making 4 equal smaller squares within the big square. 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.

b. The students think that the areas are not equal because the shaded regions are different shapes.

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

c. 3/4 7/9 = 0/78 and 8/11 = 0/73, the fraction 3/4=0/75 has a value between the values of 7/9 and 8/11.

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. 4/3 _____ 5/7 3 + 7/4 + 7 ____ 5 + 4/7 + 4 10/11 > 9/11 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

c. 3/7 and 2/5

One of Mr. Spilker's students, Vanessa, incorrectly answered the addition problem 457+138 as represented in the work shown. 457 + 138 = 585 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+214 b. 555+134 c. 394 + 182 d. 871 + 225

c. 394 + 182

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

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

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 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.

c. The area of the bulletin board is twice the area of the message board. 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=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=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 square inches. Thus, the only true statement is that the bulletin board has a width of 48 inches.

Which 3 of the following word problems can be represented by a division equation that has an unknown quotient? a. Mr Bronson works the same number f 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 f 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?

a. Mr Bronson works the same number f hours each day. After 8 days of work, she had worked 32 hours. How many hours does Ms. Bronson work each day? 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? 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?

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

a. eleven The number name "eleven" does not follow any pattern of number-name construction with reference to the tens and ones.

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+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. a. "I started with 10 cubes and 5 cubes - that is 15 cubes - and then I took away the extra 2 cubes and I got 13 cubes" b. "I took 2 cubes from the 5 and put them with the 8, and then I knew 10 and 3 is 13" c. "I started with 8. Then I counted 9, 10, 11, 12, 13" d. "I separated 3 cubes from the 8 so I have 5 and 5, which is 10 and 10 and 3 adds up to 13"

b. "I took 2 cubes away from the 5 and put them with the 8, and then I knew 10 and 3 is 13" d. "I separated 3 cubes from the 8 so I have 5 and 5, which is 10, and 10 and 3 adds up to 13"

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 d. A place-value strategy

b. A benchmark strategy

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 pends 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?

b. How many packages of pens and how many package of pencils are needed to have the same number of pens as pencils? 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.

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.

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. 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.

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 x 2.89 - 0.75 b. (1.5 + 2.25) x 2.89 - 0.75 c. 1.5 + 2.25 x (2.89 - 0.75) d. (1.5 + 2.25) x (2.89 - 0.75)

d. (1.5+2.25) x (2.89 - 0.75) 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) x (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.

A student found an incorrect answer to the problem 2/5 x 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

d. 1/2 x 9/10 = 10/18 = 5/9

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 i s 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. 4h^2 and 7h^3

d. 4h^2 and 7h^3 The best pair of terms for Ms. Fischer's purpose should obtain 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 student incorrectly answered the problem 305.74 x 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. 246.7x100=2,467 b. 13.05x100=13,500 c. 46.13 x 10 = 460.130 d. 94.03 x 10 = 94.030

d. 94.03 x 10 = 94.030 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. Between4and41⁄4hours b. Between41⁄4and41⁄2hours c. Between 4 1⁄2 and 4 3⁄4 hours d. Between 4 3/4 and 5 hours

d. Between 4 3/4 and 5 hours Since 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. 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

d. Knowing that each previous number names refers to a quantity which is one less