MTTC Test (Lower Elementary Test #119)

A kindergarten teacher conferences with a student. The teacher tells the student that there are four chips in the frame and asks the student to count all the chips in the following arrangement. A diagram of a partially filled in ten frame is shown. Four boxes in the first row of the ten frame are filled with circles that represent counters. Three additional counters are shown outside of the ten frame. The student touches each chip one at a time and counts from one to seven. The student's response shows evidence to support which of the following claims? A. The student can correctly count and shows mastery of kindergarten counting standards. B. The student can subitize as part of a strategy for counting and demonstrates efficiency as well as accuracy. C. The student understands one-to-one correspondence and would benefit from opportunities to engage in counting on. D. The student shows a limited understanding of one-to-one correspondence and would benefit from a strategy group or intervention on cardinality.

A. Although the student counted accurately, that is not enough evidence to demonstrate mastery of counting and cardinality kindergarten mathematical standards. B. The student did not demonstrate evidence of recognizing the number of chips present anywhere in the model; instead, they counted each chip individually. C. CORRECT. The student shows evidence of understanding that each chip corresponds to one number while counting but does not show efficiency by starting to count from the known number. D. The student demonstrates grade-level-appropriate understanding that each chip is one number and identifies the correct number of elements in the set.

A teacher asks students to count by ones from 3 to 10. Four student solutions are represented. Student Method A points to 3 on the number line, and then points and says aloud every number 3 through 10 B quietly says "1, 2" under their breath, and then more loudly states, "3, 4, 5, 6, 7, 8, 9, 10" C puts 3 tokens on the table, says "3," and then, while adding tokens one at time to the table, says the corresponding total number D holds up 3 fingers, and then counts up to 10, lifting an additional finger for each number to punctuate the counting Based on the student responses, which student would benefit most from additional instruction for "counting on"? A. Student A B. Student B C. Student C D. Student D

A. By starting at 3 on the number line, Student A demonstrates a strategy for counting on, without beginning at 1. B. CORRECT. Student B needs to start at 1 to count up to 10. C. Student C starts with 3 tokens, demonstrating that they are able to count on without starting at 1. D. Student D uses a similar strategy to Student C, by beginning directly with 3 fingers and then counting up.

A third-grade teacher considers the following task. UnderlineThe word Underline is underlined. the triangles. Circle The word Circle is circled. the quadrilaterals. Put an X through the gray shapes. A row of polygons is shown. Some polygons are shaded, and others are not. The row is composed of an unshaded square, a shaded rectangle, an unshaded hexagon, an unshaded parallelogram, a shaded triangle, an unshaded rhombus, and a shaded pentagon. The teacher uses the same set of shapes but rewrites the task for their students. Which of the following rewritten versions makes the task more cognitively demanding and creates multiple entry points for students? A. List the ways these shapes can be sorted. B. How many gray shapes are quadrilaterals? C. Cross out the shapes that are not quadrilaterals. D. What is the number of gray non-triangle shapes?

A. CORRECT. Students have an opportunity to engage with the task by using all the attributes of the presented shapes, and it increases cognitive demand by allowing students to do the sorting and explanation. B. This task only has one entry point and allows for limited engagement with the shape attributes. C. This task only has one correct answer and offers less demand than the original task. D. The task only has one correct answer and does not allow for multiple points of entry.

10. During a lesson on subtraction within 20, first-grade students engage with a series of word problems about lost items (e.g., books, toys, mittens). Some students have difficulty solving the subtraction problems without drawing out the entire scenario. The teacher can support these students as they think about subtraction by: A. encouraging the students to try a count-back or a count-up strategy. B. explaining how to use context to determine which operation to use. C. reminding the students of their prior learning regarding operations. D. explaining how addition and subtraction are related operations.

A. CORRECT. Students who consistently solve subtraction problems by drawing (i.e., by creating lines or pictures to represent objects and then crossing them out) should be encouraged to practice other subtraction strategies, such as counting-back or counting-up, to promote their conceptual and procedural understanding. B. The students demonstrate that they know to use subtraction, so an explanation about determining an operation from a situational context is not necessary. C. This response does not specifically target students' demonstrated need to practice subtraction strategies that do not involve drawing them out. D. An explanation of how subtraction and addition are related operations (e.g., by explaining how the equations a − c = b and a + b = c are related) does not directly guide the students to apply subtraction strategies that do not require drawing.

20. A first-grade student solves 8 + 5 and shows their work: Two ten-frames are shown above a student work sample. The top ten-frame shows the number 8. The bottom ten-frame shows the number 5.The student work sample shows the equation 8 plus 5 equals 13. Two arrows extend below the 5 and connect to the numbers 2 and 3, which are shown within circles. Two equations are shown below this work.The first equation reads 8 plus 2 plus 3 equals 13. The second equation reads 10 plus 3 equals 13. The student's work provides evidence that the student integrates addition with which of the following skills? A. decomposing and composing numbers into place-value components B. applying the associative and commutative properties C. integrating inverse operations to confirm a solution D. understanding early fraction representations

A. CORRECT. The student decomposes "5" into "2 + 3" for the purpose of creating a unit of ten from the 8 and 2. B. The student does not rearrange terms or change the order for operations in any way, so the associative and commutative properties are not used in the solution. C. The student does not use subtraction in their solution, so they are not applying inverse operations. D. The problem involves adding whole numbers, and the student does not use fractional reasoning in their solution.

18. A teacher asks a student to solve the following problem. Cup A contains 7 marbles. Cup B contains 3 marbles. How many more marbles are in Cup A than Cup B? The student explains how they recognized it was subtraction because of the phrase "how many more." The student then shows their solution with a drawing. The student draws 7 marbles in Cup A and 3 marbles in Cup B. The student shades circles in Cup A until the cups match and then counts the shaded marbles to get an answer of 4. Two rectangles, each containing circles, is shown. The left rectangle is labeled Cup A. It contains 4 shaded circles and 3 unshaded circles. The right rectangle is labeled Cub B. it comes 3 unshaded circles. The student's model is consistent with which of the following approaches? A. calculating a difference by comparison B. calculating a difference by adding or counting on C. calculating a difference by decomposing into anchor numbers D. calculating a difference by rewriting it as a more familiar expression

A. CORRECT. The student understands that "how many more" implies a difference, so they compared the number of marbles in Cup A to the number of marbles in Cup B by shading the greater set of marbles until both sets were equal. B. The student did not solve the problem by adding or counting on because they did not attempt to add marbles to Cup A until the quantities of marbles were equivalent. C. The student shows no evidence of decomposing either set of marbles into sums of different quantities, so this is not consistent with the student's solution. D. The student's solution is consistent with the subtraction problem as it was originally presented (i.e., 7 - 3) without rewriting it as a more familiar expression.

A second-grade teacher meets with a small group of students about this story problem. One piece of rope is 15 feet long. Another piece of rope is 18 feet long. How much longer is the second piece of rope than the first? Teacher: What is an equation we could use to solve this problem? Student A: I know! You could do 18 −minus 15 = 3. Student B: I don't agree, because subtraction means taking away. Teacher: Yes, subtraction can be used to solve problems that involve taking away, but it also can be used to compare and to find difference between numbers, too. The teacher draws a model of the two lengths next to each other on a number line and shows how the difference between the two lengths is three feet. The teacher's demonstration affords students the opportunity to understand which of the following concepts? A. Computations can correctly model different mathematical situations. B. A number line is the most efficient strategy for solving subtraction problems. C. Answers to different operations have specific names like "sum" or "difference." D. Students should not say the phrase "take away" when talking about subtraction.

A. CORRECT. The teacher expands the definition and meaning of subtraction for students. B. The teacher uses the number line to model the operation, not to show the most efficient strategy. C. The teacher explains the concept of difference, but does not address it as the name of subtraction problems. D. The teacher does not dismiss the student's vocabulary, but instead expands their definition of how subtraction operation is used.

3. First-grade students consider the following equations. 7 = 10 − 37 = 5 + 210 − 3 = 5 + 2 Most students state that the last equation is incorrect. In order to address the students' misconception, the teacher should plan a review of which of the following concepts? A. meaning and function of the equal sign B. how addition and subtraction are related C. the use of benchmark equations to find the answer D. the concepts of "greater than," "less than," and "equal to"

A. CORRECT. The teacher should review the meaning and function of the equal sign because students who agree that only the first two equations are correct may be interpreting the equal sign to be a symbol that indicates the result of the last operation (i.e., they would likely believe that the third equation should be written as 10 − 3 = 7 + 2 or 10 − 7 = 5 + 2). B. None of the equations shown makes use of addition and subtraction as inverse operations. C. A review of benchmark equations (e.g., sums and differences involving 5 and 10) is not necessary because students have previously agreed that 7 = 10 − 3 and 7 = 5 + 2. D. There is evidence that the students are interpreting "=" to mean "the result of the last operation," and reviewing the concepts of "greater than" and "less than" would not address this misconception directly or efficiently.

A teacher meets with a small group of kindergarten students using ten frames. Teacher: Show me how you build the number six. Student: I can fill the top row with five, and put one at the bottom. Teacher: Can you show me six a different way? Student: I could put three on the top and three on the bottom row. Teacher: Can you think of any other ways to show six? The teacher's questions afford the students which of the following opportunities? A. composing a number different ways B. building fact fluency of a specific number C. extending conversations about equations D. solving addition equations using ten frames

A. CORRECT. The teacher's questions prompt the students to specifically compose a number in different ways. B. Fact fluency involves computing answers with automaticity, which differs from the thoughtful composition of 6 that occurs in this activity. C. The teacher should mention operations and equality in a lesson about equations. D. While students may use addition implicitly to compose 6, the teacher does not refer to an equation or solution strategies in this activity.

A kindergarten teacher plans instruction for a small group of students. The learning target for the group is to compose numbers 11 through 19 and record them as a ten plus a group of ones. Which of the following materials would be most appropriate for the learning target of this activity? A. a set of two ten frames, counters, and writing materials B. a collection of random dominos with pips and a whiteboard C. addition flash cards of equations with sums of 11 through 19 D. a math game in which students roll six-sided dice and add the results

A. CORRECT. The two ten frames are an appropriate material set to model specifically the composition of numbers by filling a ten frame and adding a set of ones to the other. B. These materials are better suited to modeling a skill like subitizing or the fluency of mathematical facts, not the composition of numbers as a ten and a group of ones. C. The learning target focuses on composition of numbers, not on fact fluency. D. Rolling six-sided dice and adding the quantities shown on the dice does not produce the circumstance for students to compose a set of ten, and then some more.

15. After a lesson on the progression of fractions from 0 to 1 using fourths, a teacher presents this problem to the class. A question is shown. It reads, "What is the unknown value in the bottom number line?"Below this question, a diagram of three number lines is shown. The first number line displays the numbers 0, one-fourth, two-fourths, three-fourths, 1, and one and one-fourth.The second number line shows 0, one-third, two-thirds, 1, and one and one-third.The third number line shows 0, 1 and a question mark. The first tick mark is labeled 0, the fifth tick mark is labeled 1, and the sixth tick mark is labeled with a question mark. One student responds, "The question mark is after the 1, so the answer is 1 12and 1 half". Which of the following teacher responses would best prompt a discussion about unit fractions and iteration? A. "Which patterns do you notice between 0 and 1 in the first two number lines?" B. "Does the unknown mark on the number line look halfway between 1 and 2?" C. "Where does the fraction 121 half appear on the bottom number line?" D. "Should the missing value be greater or less than 1 13and 1 third?"

A. CORRECT. This question encourages the student to make connections between the number of tick marks between 0 and 1 and the denominators of the fractions on the number line. B. The teacher would do better to assist the student with interpreting details that can be observed directly rather than by asking them to visualize features that are not present, such as visualizing where 2 would be if the number line were extended. C. The bottom number line is divided into fifths, which can make it difficult for the student to identify one-half when they demonstrate a misconception about how the tick marks relate to denominator of a fraction. D. This question does not address the student's misconception about the relationship between the tick marks in the number line and the denominator of a fraction.

12. First-grade students explore representations of two-digit numbers using base-ten blocks. The teacher asks one student to show the number 33. The student presents the following arrangement. The number 33 is shown represented with base-ten blocks. There are three ten-rods and three unit squares in this diagram. When asked to explain their answer, the student says, "I know I need three of both since it's 3 and 3." Which of the following questions should the teacher ask to facilitate a clearer understanding of composing numbers greater than 10? A. "How would you explain the difference between a rod and a block?" B. "How would you show a number like 44 with rods and blocks?" C. "How can you use tally marks to represent the number 33?" D. "How can you explain the role of 3 in 333?"

A. CORRECT. This question helps to clarify whether the student interprets rods and blocks to represent different positions in the number (i.e., the left 3 means the number of rods and the right 3 means the number of blocks) or different units (i.e., the left 3 means 30, which is represented by 3 rods, and the right 3 means 3, which is represented by 3 blocks). B. The student would likely answer this question by saying, "I know I need four of both since it's 4 and 4," which doesn't offer any deeper insight into the student's thinking than the original answer. C. The student demonstrates at least a partial understanding of how to represent 33 with base-ten blocks, and the teacher should attempt to build upon the student's response rather than switch to a different representation—especially when the alternative representation does not demonstrate place-value concepts effectively. D. Expanding the number of place value positions is unlikely to help the student explain what value each digit of 3 represents.

A third-grade teacher checks exit tickets after a lesson on multiplication. The teacher discovers that students rely on different strategies to solve the multiplication equations on the exit ticket. Some students draw circles and marks to show equal groups, some use open number lines, and others show repeated addition with numbers. The teacher plans for group work the next day. Which of the following approaches affords students the opportunity to participate equitably through their various mathematical affinities and capacities? A. Mix students randomly. Allow students to use their exit ticket to solve the day's work. B. Group students together by the strategy they used. Visit each group to demonstrate other methods. C. Mix students based on the strategies they used. Have each student explain their strategy to the small group. D. Group students by correct and incorrect answers. Explicitly teach strategies for students who did not answer the exit ticket correctly.

A. Grouping is unintentional and does not plan for participation of all students. B. Students have a passive role in this set of groupings, and the teacher has made no plan to encourage student participation. C. CORRECT. The teacher creates a learning opportunity that intentionally groups students in a way that fosters participation of all students and takes stock of the mathematical capacities they bring. D. There is no evidence of a plan for students to participate based on their mathematical capacities.

1. A kindergarten teacher observes as a small group of students practice comparing numbers and quantities using manipulatives. Each student has four counters. One student's counters are spaced farther apart than the other students' counters, and several members of the group claim that student has more counters than everyone else. The teacher can build on the students' understanding of counting and cardinality by: A. encouraging the one student to count their counters for the group. B. identifying the error and moving the one student's counters closer together. C. asking probing questions about the total number of counters each student has. D. prompting the group to combine their counters and count how many they have in all.

A. Having one student count their own set of counters aloud does not address the misconception that the number of counters in a set depends on how they are arranged. B. This manner of addressing the misconception does not necessarily build upon the students' understanding because the teacher did not check whether they grasped the explanation or provide them with opportunities to explain in their own words why the total number of objects in each group is the same. C. CORRECT. By asking probing questions about the total number of counters each student has, the teacher can help students move beyond a naïve conception that bigger equals more and deepen their conceptual understanding of counting and cardinality. D. The affordances created by combining the counters into a large group do not offer these students more substantive insights into understanding counting and cardinality concepts than the affordances created by using smaller groups of counters.

11. First-grade students play with tiles. One table group counts their tiles and finds there are 24. The teacher asks the students if they can group the tiles so they can identify the total number easily. A student creates the following representation: A diagram of an arrangement of 24 squares is shown. The squares are positioned in rows. There are 10 squares in each of the first two rows and 4 squares in the third row. Presenting this representation to the class demonstrates the use of manipulatives to: A. critique the reasoning of others. B. support reasoning with evidence. C. create a basis for the concept of grouping. D. develop understanding of the base-ten system.

A. Opportunities for the class to engage in the constructive criticism of a student's reasoning, such as by participating in a discussion about the student's thought processes or suggesting possible refinements to their representation, have not taken place at this point in the learning activity. B. The student's reasons for grouping the tiles as shown have not yet been shared with the class. C. The ones unit is the only unit represented in the representation (i.e., each tile represents 1), and the basis of grouping, where a collection of 10 ones may be interchanged with a single unit of ten, is not illustrated here. D. CORRECT. The student arranged their representation for easy counting by forming two rows of 10 and one row of 4, and presenting this to the class helps them develop their understanding of the base-ten system.

A third-grade teacher selects manipulatives to promote students' ability to understand how fractions can describe parts of a set. Which of the following manipulatives best promotes this interpretation? A. paper strips B. number lines C. grid-paper regions D. two-color counters

A. Paper strips are better suited for describing fractions as a subdivision of a length. B. Number lines are better suited for describing fractions as a length or distance. C. Grid paper is an appropriate model for describing fractions as an area of a shaded region relative to a whole shape. D. CORRECT. A part of a set my be modeled by counting the number of counters of a certain color relative to the total number of counters.

Which of the following questions can help students to think about comparing the fractions 343 quarters and 787 eighths? A. "Are fourths or eighths larger pieces?" B. "Which fraction has a larger denominator?" C. "If you draw both, which one is closer to a whole?" D. "Would you rather eat 343 quarters of a pizza or 787 eighths of a pizza?"

A. Realizing that fourths are bigger than eighths does not necessarily lead students to conclude that 343 quarters <is less than 787 eighths. B. To understand the difference between fractions, students need to be able to explain the meaning of fractions, not just know the terms. C. CORRECT. This question gives initial direction to the students to think about what each fraction looks like out of one whole and then to compare. D. Connecting the value of each fraction to a portion of a whole pizza does not directly help students compare the fractions.

A second-grade teacher gives students this word problem involving a number line. One student's response is shown. Two children compare the number of shells they found on the beach. The first child found 37 shells. The second child found 18 shells. How many more shells did the first child find than the second? A number line that starts at zero. The first labeled tick mark to the right of zero is 18, and there are tick marks for all whole numbers from 18 to 37. An arrow is drawn to indicate plus 2 the right of 18, which points to the tick mark labeled 20. A second arrow is drawn to indicate plus ten the right of 20, which points to the tick mark labeled 30. An arrow is drawn to indicate plus 7 to the right of 30, which points to the tick mark labeled 37. The sum of the distances for the arrows plus 2, plus 10, and plus 7 is circled and labeled "plus 19 more shells" in the student's handwriting. Which of the following claims can be made about the student's work? A. The student understands how to regroup to perform subtraction. B. The student understands how to interpret subtraction as taking away. C. The student understands how to compare the difference between two numbers by using an unknown-change model. D. The student understands how to compute a difference by combining the strategies of skip-counting and counting on.

A. Regrouping refers to the exchanging of value between different units (place values); the circled values are not an example of the regrouping performed in subtraction. B. Subtraction is to be interpreted as comparison in this context and not taking away. C. CORRECT. The direction of the arrows from 18 to 37 indicates that the student found the difference by adding an unknown amount to 18: 18 +plus ?unknown quantity = 37. D. The student's work shows several different addends were uses to count from 18 to 37, which is different from skip-counting by a constant addend.

A third-grade teacher asks students to draw a representation to solve the problem 4 ×times 3. Four student solutions are shown. StudentRepresentationABCD Which student's work sample is the most accurate representation for expressing multiplication as repeated addition? A. Student A B. Student B C. Student C D. Student D

A. Student A drew dots to represent the factors, but shows no understanding of multiplication as repeated addition. B. CORRECT. Student B represented three repetitions of four units, which is most closely related to the concept of repeated addition. C. Student C attempted to solve the problem by grouping, but their work does not show repetition of equal groups. D. Student D showed the correct number of blocks, but there is no evidence in their work that they represented multiplication as repeated addition.

14. A third-grade teacher asks students how they would determine which fraction is greater when the fractions have the same numerator and a different denominator, like the fractions 232 thirds and 262 sixths, initiating the following conversation: Student A: If you score two goals out of three attempts in soccer, that's better than two out of six. So 232 thirds is greater than 262 sixths. Student B: But you still only scored two goals. Student A: Yeah, but you scored more of the goals you attempted. Which of the following interpretations best compares both students' explanations? A. Student A explains that a fraction is greater if the numerator is closer to the denominator, while Student B interprets only the denominator. B. Student A compares the fractions using a common number of attempts, while Student B compares only the number of goals scored. C. Student A uses equivalent fractions to compare magnitudes, while Student B compares the numerators as parts of the same whole. D. Student A interprets the number of attempts as the whole, while Student B compares only the number of goals scored.

A. Student B does not consider the denominator when they interpret the fractions. B. Student A does not attempt to compare the two fractions by representing them with a common denominator. C. Student A does not use the concept of equivalent fractions in the discussion. D. CORRECT. Student A refers to the whole for each fraction when they say "three attempts" and "out of six," and Student B only compares the numerators when they say "only scored two goals."

4. A first-grade teacher uses an activity involving dice to help students make the jump from counting to addition. Students roll two dice, then determine the sum of the dots that are face up. On a piece of paper, students draw their dice as an addition problem and write the problem using numbers. One student's work is shown. Two dice are shown above an equation. The left die shows 3 pips, the right die shows 4 pips, and the equation reads 3 plus 4 equals 7. The teacher can increase students' success by taking which of the following actions before explaining the activity? A. teaching students how to add without using a counting strategy B. providing context by describing games in which dice may be used C. giving students the opportunity to become familiar with dice and their dots D. posting addition tables at the front of the room and on each student's desk

A. Students should explore the relationship between counting and addition before they learn to add without a counting strategy. B. Describing games in which dice may be used does not direct students' attention to attributes of dice that makes them useful manipulatives for learning addition: the neat arrangement of dots on the faces, the unique numbers of dots on each side, and the many different outcomes that can be formed by rolling two dice. C. CORRECT. Providing students with opportunities to explore manipulatives on their own stimulates their curiosity, interest, and comfort with them and prepares students to explore how they may be used as mathematical tools. D. Posting addition tables does not support students' ability to understand how the mathematical concepts used to count a single collection of objects can be extended to count two combined collections of objects through addition.

A first-grade teacher presents the following ten frame to a small group of students. The teacher asks students to write an equation that represents the mathematics shown. A diagram of a partially filled in ten frame is shown. The first row of the ten frame is filled with circles that represent counters. Four additional counters are shown outside of the ten frame. Which of the following qualities of this activity makes it particularly effective for promoting broad participation? A. It encourages debate and discussion among students about correct answers. B. It provides students with concise steps to using manipulatives for problem solving. C. It affords students the opportunity to build models of addition and subtraction equations. D. It encourages multiple entry points and ways for students to be mathematically successful.

A. The activity is focused on generating different equations and perspectives rather than debating or evaluating any one claim. B. The teacher does not focus on process-based problem solving during this lesson. C. The activity is focused on observing a model and generating math thinking, not manipulating a model to solve problems. D. CORRECT. Student thinking is central in the discussion and activity, and the students have many ways of showing their thinking.

5. The dramatic play area of a prekindergarten classroom is set up a like a kitchen, with three-dimensional blocks standing in for plates and cups. While children work in stations pretending to make and serve food to each other, the teacher can help the children extend their understanding of three-dimensional attributes by asking them: A. to find the yellow cup. B. which plate is their favorite. C. to bring a certain object to the table. D. how the plate and the cup are the same and different.

A. The color yellow and the object name "cup" are not used to describe attributes of three-dimensional shapes. B. This question does not direct students to focus on the three-dimensional attributes of different plates. C. The name of an object is not a three-dimensional attribute. D. CORRECT. Students answer this question by describing the similarities and differences of the three-dimensional attributes of the plate and the cup (e.g., "The plate is short and flat, and the cup is tall", "The plate and the cup both have round edges").

A teacher overhears a student talking to their peers: "I saw my older cousin doing fractions, and it's so easy! Look, I'll show you. If you add three-sixths and two-fourths, you get five-tenths! So cool, right? Adding fractions is just counting. Look!" The student draws this model: A model of an equation consisting of 3 arrays with handwritten text. A 2 by 3 rectangular array appears first, with 3 of the sections shaded to indicate three sixths. A plus sign is used to indicate addition with a second array. The second array has dimensions 2 by 2, where half of the sections are shaded to show two fourths. The sum of the fractions is indicated by an equal sign and a 2 by 5 array, where 5 of the 10 regions are shaded to indicate five tenths. The teacher decides to join the conversation to help direct the learning of fractions. Which of the following remarks would help steer the conversation into more accurate mathematics? A. "Every fraction in the problem is equal to one-half, and one-half plus one-half equals one whole." B. "Why can you combine the six blocks and the four blocks to create the ten blocks in your answer?" C. "That's a great start! Now, let's make all three shapes the same size to begin with, and then see what happens." D. "That's really cool! Did you know the top number is called the numerator and the bottom is called the denominator?"

A. The reasoning provided in this response is too high-level for students who are new to fractions. B. This question encourages misconceptions about adding fractions, and it is clear from the context that the student does not have the conceptual understanding necessary to answer the question. C. CORRECT. This remark builds from the student's example and provides clear instruction for how the students can self-correct the misconception and learn how to add fractions. D. This remark brings in new information about fractions, but it does not address the misconception presented by the student.

16. A teacher assesses students' understanding of fractions with the following question on an exit slip: Which fraction is greater: 343 fourths or 464 sixths? A student responds that 464 sixths > 343 fourths and provides the following explanation. A diagram of student work representing the fractions three-fourths and four-sixths is shown.Three-fourths is shown as three shaded parts and 1 unshaded part, with all parts arranged in a row. The fraction three-fourths is written to the right of this row.Four-sixths is shown as four shaded parts and two unshaded parts, with all parts arranged in a row. The fraction four-sixths is written to the right of this row.The two rows are vertically aligned and extend from left to right. The student's response supports which of the following claims? A. The student understands that the value represented by the numerator depends on the value of the denominator. B. The student understands how to compare unit fractions with different denominators. C. The student understands how to interpret an individual fraction as a part of a whole. D. The student understands how to form different partitions of two equivalent wholes.

A. The student's drawing shows evidence that they understand fractions as parts of a set and numerators as iterations of unit fractions, but this claim would only be supported if their drawing showed evidence that the unit fractions 141 fourth and 161 sixth were different, which it does not. B. For this to be true, the student would need to represent and compare 141 fourth and 161 sixth using a common whole; instead, they use unit fractions with equal sizes and then draw unequal wholes from them. C. CORRECT. The student's drawing accurately represents each fraction as parts of a whole, and this claim is not weakened or contradicted by the unequal sizes the student uses to represent the fraction wholes. D. The student does not show evidence of dividing the same whole into different amounts of equal parts (e.g., dividing a rectangle into fourths and dividing the same rectangle into sixths).

A third-grade teacher asks students to add 57 + 34 on their whiteboards. A student shows their work: A number line that starts with zero and has a tick mark at 57. An arrow is drawn to indicate a jump of plus 30 to the right, and the arrow points to a tick mark labeled 87. Another jump to the right of plus 3 is shown to the tick mark 90. One final jump of plus 1 to the right is drawn to a tick mark for 91, which is the last labeled tick mark on the number line. Which of the following descriptions is an appropriate narrative of the student's mathematical work in solving the equation? A. The student took hops of ten along the number line to get to an anchor number, then split the ones to arrive at 91. B. The student used the standard algorithm to combine the ones and tens separately, regrouping the ten made when adding the ones to arrive at 91. C. The student added the tens from 34, then added three of the four ones to jump to an anchor number, then added the remaining one to arrive at 91. D. The student split the tens and ones, then used three of the ones to jump to an anchor number, then added the remaining 30 and one separately to arrive at 91.

A. The student's number line does not show using hops of tens to arrive at an anchor number. B. The student uses an open number line strategy; the diagram does not show the standard algorithm for addition. C. CORRECT. The student first adds 30 (all the tens from 34) to 57 on the number line; then uses three of the four ones to jump to 90, an anchor number; then adds the remaining one to get to the correct sum. D. The student does not jump to an anchor number until after adding the tens.35

9. A teacher introduces a paired activity involving index cards that the teacher has numbered from 0 through 9. For the activity, the teacher gives each pair of students a set of index cards. Each student draws two cards, and together they use the numerals on the cards to make 2 two-digit numbers. They determine which of the two numbers is greater, recording their thinking as an inequality. Then, the students build both numbers using base-ten blocks to check if their thinking is correct. An appropriate learning target for this activity is for students to be able to: A. identify reasonable numbers as solutions. B. compare two numbers using place values. C. use mathematical tools to explain their thinking. D. construct numbers to solve mathematics problems.

A. The students check their work by modeling the numbers with base-ten blocks and not by determining whether the numbers are reasonable solutions. B. CORRECT. Students compare numbers when they record their thinking as an inequality, and they use place values concepts to create representations of two-digit numbers to check their work. C. The use of mathematical tools in this activity, such as the use of base-ten blocks, supports students' understanding of the learning target but is not the learning target itself. D. This learning target does not specify the types of mathematics problems that are solved in this learning activity.

A first-grade teacher meets with a small group of students and asks them to build the number 16 on a set of two ten frames. The teacher asks students to explain their strategy for composing the number. Student A: I filled the first ten frame, then the first row of the second frame, then I added one more. Student B: I filled the first ten frame, then added six more to the bottom. Student C: I kept adding one at a time until I got to 16. Teacher: Let's see if we can use the first shared strategy of starting with tens and fives to build the number 18. Which of the following statements best justifies the teacher's instructional approach? A. Students can compose the number accurately, so it is appropriate to try a greater number. B. Students show different strategies, so it is important that the group members all understand each other's strategy. C. Students show varying levels of sophistication in composing numbers, so it is appropriate to unpack an efficient strategy. D. Students can explain valid strategies, so it is important to recognize how mathematicians compose different quantities in different ways.

A. The teacher asks the students to repeat the strategies for which they have already shown their competence. B. The teacher recognizes varying levels of sophistication in the student responses, but chooses only the most efficient strategy shared to explore more deeply. C. CORRECT. The teacher engages with a more sophisticated strategy with the purpose of building mathematical competence for the students. D. The teacher doesn't take time to engage with each strategy equally, and chooses to only investigate the most sophisticated approach.

6. A teacher shares this geometric pattern with the class. A pattern formed from geometric shapes arranged in a row is shown. The shapes, from left to right, are a triangle, a square, a pentagon, a hexagon, a triangle, and a square. The teacher asks students to explain the order of the shapes in the pattern. As students share their explanations, the teacher writes these student explanations on the board. "The shapes go from smallest to biggest. Then they start over." "They are in order by the number of corners." "The pattern is the least number of sides to the greatest number of sides, then it starts again." By writing these explanations on the board, the teacher: A. highlights the most efficient ways to explain the geometric pattern. B. encourages students to revise their explanations if they misinterpreted the pattern. C. offers students an entry point for collaboratively refining explanations of the geometric pattern. D. allows students to recognize explanations that are accurate and dismiss inaccurate explanations.

A. The teacher conducts the conversation about the pattern in a way that solicits a wide range of input from students without commenting about whether one description is more efficient than another. B. All students—not only those who misinterpreted the pattern—may wish to revise their explanations when they consider the other students' responses. C. CORRECT. By recording a wide range of student responses, the teacher exposes students to a rich variety of word choices and mathematical details that students can draw upon to refine their own explanation of the pattern. D. Dismissing students' responses creates a counterproductive learning dynamic, and while students may judge details in the responses to be accurate or inaccurate, the teacher is not likely recording the responses with the intention of having the students identify which should be removed.

A kindergarten teacher conferences with a small group of students after a lesson on addition to check in with the students. The teacher takes notes on each student's response to the following prompt. "Today we learned about addition at the carpet. What are some ways you can already use what we learned about today in your homes or outside of school?" Which of the following statements most effectively describes the rationale for this question? A. The teacher assesses students' knowledge of mathematical vocabulary. B. The teacher checks for accurate solutions to real-world addition examples. C. The teacher measures students' prior mathematical instruction outside of the school day. D. The teacher builds rapport with students and listens for student interests to inform future instruction.

A. The teacher did not frame the question in a way that requires students to respond with mathematical vocabulary. B. The teacher does not provide any mathematical problems for students to complete and does not check to see if they are correct. C. The teacher does not provide any assessment of prior knowledge or past instruction. D. CORRECT . The teacher asks a question to elicit student-generated connections to addition in their lives. This information is important to creating future relevant and meaningful instruction.

A first-grade teacher meets with a small group of students and has the following conversation. Teacher: Tell me all about triangles. Student A: They have three sides. Student B: The sides are straight! Teacher: OK, thank you. The teacher then presents a set that fits the students' given criteria and asks if all of the following are triangles. A row of shapes is shown. These shapes, from left to right are:An equilateral triangle that has one side aligned with the bottom of the row and a vertex pointing up, an equilateral triangle that has one side aligned to the top of the row and a vertex pointing down, a 3 sided shape that looks like a square without a top edge, a right triangle, and a tall, skinny isosceles triangle. The teacher presents these examples and non-examples of triangles in order to: A. introduce students to named triangles based on their angles. B. show that triangles can look different than an equilateral triangle. C. afford students the opportunity to build a more complete definition of triangles. D. allow students opportunities to sort triangles based on attributes such as side length.

A. The teacher doesn't provide any scaffolds that would lead to conversation about named triangles. B. The teacher does not showcase a set of equilateral triangles in comparison to other types of triangles. C. CORRECT. The teacher chooses to include an open non-example to build the students' definition of triangles to include being closed. D. The teacher's question and examples encourage the analysis and comparison of triangles and non-triangles, not different types of triangles.

A first-grade teacher conferences with a student about addition within 20. The teacher asks the student to use ten frames to show the addition problems. The student accurately places counters on the ten frames for each number, then counts the total counters one by one to find the answer. Which of the following tools would best support the student's understanding of addition? A. a timed assessment to ensure that the student can solve the problems quickly B. a sequence of tasks that helps the student construct more efficient strategies for adding C. a pair of addition problems in which the order of the two numbers being added is reversed D. a series of equations of addition within 20 that increase in difficulty to challenge the student

A. The teacher has enough evidence from the conference observations to make instructional decisions, so a timed assessment is unnecessary. B. CORRECT. The student solves problems accurately, and the next step forward is working toward more sophisticated, efficient addition strategies. C. There is no evidence from the observation that the student needs reinforcement of the commutative property. D. The learning goal is to work with addition within 20, and the student shows evidence of accurate problem solving.

A small group in a kindergarten classroom is building two-dimensional shapes using sticks and clay. Teacher: What could we do to change a square into a triangle? Student: We could just take away a side! Then it would have three sides. Which of the following responses should the teacher use to clarify and accurately understand the student's thinking? A. "What do you have to do after you take away the side?" B. "Is there another way you can think of to make a triangle?" C. "Does everyone else agree? Anything else you might add?" D. "Will you please show me your strategy with sticks and clay?"

A. The teacher leads the student to a next step instead of seeking clarification from the student. B. The teacher encourages the student to think of alternative approaches but doesn't take the time to evaluate and understand the initial claim. C. The teacher appeals to the group looking for additional details but does not allow for the student to explain the answer in greater detail. D. CORRECT. Having the student show their strategy using manipulatives ensures that the teacher sees a clear representation of the student's thinking.

3. An elementary school teacher places a pitcher of water on a table in the classroom. The teacher then initiates this discussion: Teacher: How could this water be shared between 4 people equally? Student A: You could pour it into 4 cups. Student B: You should measure it, though. Teacher: Why should you measure it? Student B: To make sure everyone gets the same amount. Teacher: So, the idea is to measure the water and split it into 4 cups, with everyone getting an equal amount. By restating the students' thinking, the teacher can: A. motivate them to explore measurement skills independently. B. challenge connections they make between measurement and fractions. C. identify the mathematical competency they demonstrate in their approach. D. prompt them to engage in higher-level questioning during mathematics discussions.

A. The teacher restates the students' ideas to clarify their conversation, not to promote different measurement skills. B. The teacher's summary does not emphasize reasoning with fractions. C. CORRECT. The teacher can point out the students' mathematical understanding of the situation by restating their thinking. D. The teacher restates the students' thinking to focus on the points they have discussed, not to raise higher-level questions.

A teacher acts out and gives this explanation to compare addition and subtraction: "If I have 5 blocks, and add 3 more blocks to the pile, I have 8 blocks in total. This shows us that 5 plus 3 equals 8. "If instead, I start with 5 blocks but I take away 3 blocks, I have 2 left. This shows us that 5 minus 3 equals 2. "While addition makes a big number, subtraction makes a small number." A limitation of representing the relationship between addition and subtraction in this way is that it risks leaving students with which of the following misconceptions? A. There is not a clear way to represent addition and subtraction mathematically. B. The difference between addition and subtraction is the size of the end results. C. To add or subtract, you need to have blocks to help you model the equation. D. Adding and subtracting always starts with the number 5.

A. The teacher's explanation does a good job of expressing the equation with both objects and mathematics. B. CORRECT. The teacher suggests that it's the size of the solution that is the difference between the operations, rather than the relative size of the solution compared to the starting value. C. Because the teacher's explanation also includes the mathematical equation, it is unlikely that a student would think that blocks are a necessary component of adding or subtracting. D. Because of the summary statement in the teacher's explanation, it is unlikely that students would assume that an example needs to start with the number 5.

A third-grade teacher reviews a student's work from an assessment, as shown. Solve the following equation. Show your work. The subtraction problem 67 minus 38 solved using the standard algorithm. There is a slash through the 6, with a 5 above it, and a slash through the 7 with a 17 above it, to indicate regrouping. The correct answer of 29 is shown. Explain your strategy below: I borrowed a one from the six. Which of the following conclusions can the teacher draw from the student's work and explanation? A. The student relies on manipulatives or physical models of subtraction. B. The student relies on an accurate strategy, but may lack an understanding of efficiency. C. The student relies on a strong understanding of place value to solve subtraction problems. D. The student relies on a procedural strategy, but may lack an understanding of place value.

A. There is no evidence of using manipulatives or pictures drawn in the student's response. B. The standard algorithm is an appropriately efficient strategy for this equation. C. The student's explanation does not mention place value, and in fact refers to the ten that was regrouped as a one. D. CORRECT. The student shows a correct application of the standard algorithm, but the student's explanation leaves out key place-value information about what happens in the solution.

A first-grade mathematics teacher plans a formative assessment to evaluate student understanding of subtraction within 20. The teacher can most effectively identify evidence of student understanding by assigning which of the following student activities? A. solving a word problem that requires subtraction with more than two numbers B. creating a series of subtraction equations in which the unknown value is the difference C. creating a subtraction equation in which students represent the equation as a picture and a word problem D. solving a word problem involving subtraction that provides some information that is unnecessary to solve the problem

A. This activity allows the teacher to identify student proficiency with one type of subtraction problem, but it doesn't showcase broad understanding. B. This activity involves straightforward computations and doesn't encourage students to think about solution strategies in different forms. C. CORRECT. This activity encourages students to think about subtraction from multiple perspectives and offers insight into their overall understanding. D. This activity is unnecessarily convoluted if the goal is to assess understanding of subtraction.

A new student transfers into a math class. The other students struggle to pronounce and remember the new student's name. As a result, the students avoid talking to the new student, and the student feels left out. Which of the following activities would help the students get practice with the new name and meet the math goal of comparing measurable attributes? A. Students write down the new name 15 times and then come up with pictures that help them break down and remember the syllables. B. Students play a game in which they pair up and answer questions such as "Whose name is longer?" and "Whose name has more/fewer vowels?" C. The teacher asks each student to tell the class the story behind their name (e.g., who named them, if it has a special meaning) while taking notes on the board. D. The teacher conducts the students in a "repeat after me x times" challenge with the new student's name, for example, "Can you say it one time? Can you say it three times?"

A. This activity could put the new student uncomfortably in the spotlight, and the activity focuses on breaking down a long item into components, rather than comparison of attributes. B. CORRECT. This activity engages the students in talking with the new student and provides the students with practice with the new name, while also meeting the math goal of thinking about and comparing attributes (e.g., length, quantity). C. This activity helps the students get to know each other better, but it does not include mathematical elements to meet the desired math goal. D. This activity could put the new student uncomfortably in the spotlight, and saying the name several times helps promote counting, not measuring attributes.

A second-grade teacher plans to conclude a unit on solving word problems involving money. Which of the following approaches can the teacher use to reinforce the concepts while also encouraging students' enjoyment of the mathematics? A. creating an activity in which students use a computer to look up items that can be purchased for a given amount of money B. creating a game in which students compete to earn the greatest amount of money by answering questions correctly C. creating a game in which students show the same amount of money using different combinations of coins and bills D. creating an activity in which students count amounts of money using different currency

A. This activity does not involve counting money, and it does not utilize the technology in a way that is conducive to learning mathematics. B. The competitive element may encourage some students, but may discourage others, and the mathematical content isn't central to this activity. C. Correct. A game allows students to interact in a low-stakes environment that values mathematical understanding and provides a physical reward. D. The addition of new currency can confuse students and distract from the learning goal.

2. A first-grade teacher plans initial lessons on comparing number values. Which of the following activities would be developmentally appropriate and engaging when introducing this concept? A. Students form multiple-digit numbers using index cards labeled with the digits 1, 2, 3, and 4. B. Students measure the lengths of classmates' shoes and then sort the shoes from smallest to largest. C. Students discuss the values of different piles of coins, such as a pile of 5 quarters and a pile of 5 pennies. D. Students stand between two different quantities and arrange their arms into a greater-than or less-than symbol.

A. This activity does not require students to compare numbers. B. The skills required to measure and sort rational numbers—the numbers that would be used to describe shoe lengths—are too advanced to be included in a first-grade lesson activity about comparing number values. C. This activity is not developmentally appropriate because the concepts of number comparison should be introduced to students without requiring them to also apply additional mathematical knowledge that does not directly support their understanding of these concepts. D. CORRECT. The alignment and rigor of the activity is developmentally appropriate for introducing first-grade students to the concept of comparing number values and the kinesthetic activity promotes their engagement.

A second-grade teacher plans a lesson on understanding subtraction as an unknown-addend problem. Which of the following tasks affords students the opportunity to strengthen their understandings of this concept? A. Students decompose numbers from 11 to 20 using place value. B. Students count on a number line to find the distance between numbers. C. Students check to see if equations involving addition or subtraction are true or false. D. Students simplify differences by using easier numbers, such as expressing 13 −minus 4 as 13 −minus 3 −minus 1.

A. This decomposition task does not involve the use of an unknown addend. B. CORRECT. Counting on a number line allows students to visualize a related addition problem for any difference. C. This task serves to promote students' knowledge of an equal sign, but does not emphasize subtraction as an operation. D. While this is an effective subtraction strategy, this does not utilize an unknown addend to subtract.

A teacher draws these number lines on the board as part of an explanation for how to understand that 131 third is less than 343 fourths. Two number lines, one on top of the other so that their zero and 1 tick marks are vertically aligned. The first number line shows from zero to 1, with tick marks and labels for one fourth, two fourths, and three fourths. The second number line shows from zero to 1, with tick marks and labels at one third and two thirds. Which of the following explanations uses the drawing to support the comparison? A. "One-third is less than three-fourths because three-fourths is bigger than one-third." B. "One-third is less than three-fourths because it has a smaller value in the numerator." C. "One-third is less than three-fourths because four is a greater denominator than three." D. "One-third is less than three-fourths because three-fourths is farther to the right than one-third."

A. This explanation does not refer to the drawing or provide justification for why 343 fourths >is greater than 131 third. B. This explanation uses a rule that is not always true (e.g., 131 third >is greater than 292 ninths). C. This explanation uses a rule that is not always true (e.g., 191 ninth <is less than 383 eighths). D. CORRECT. The fraction that is positioned farther to the right on these number lines is greater in value because the number lines use equal, vertically aligned distances to represent value in the range from 0 to 1.

Which of the following teacher explanations best supports a first-grade student's ability to identify, name, and use an equal sign? A. "An equal sign looks like two dashes written on top of each other, like this (draws an equal sign). It means that both sides of an equation are equal." B. "Notice than when we do math, we often put this symbol (points to an equal sign) on our papers. This symbol indicates that the values on both sides of it are equivalent." C. "This symbol (points to an equal sign) is called an equal sign. Unlike the plus and minus symbols, which tell us how to combine values, the equal sign tells us how the two values relate." D. "When you have two values that are the same, like 5 and 1 + 4, you can put this symbol between them (draws an equal sign), which is called an equal sign, to show that they are the same."

A. This explanation helps with identifying and naming the symbol, but it does not define what it means to be equal. B. This explanation does not name the symbol, and the language that describes what it means to be equal (i.e., "equivalent") may be too complex for first-grade students. C. This explanation shows and names the equal sign, but instead of defining it, explains how the equal sign is used in general. D. CORRECT. This explanation shows the symbol, names it, and explains the meaning of equivalence with a clear definition and example.

7. A teacher considers using the image shown of a hanger to represent an example of triangles in the real world. A flat wooden hanger is shown. The part of the hanger that attaches to the hook is flat on the bottom and has a varying thickness along its top. This part is arranged like an upside-down v, and the arms of this shape extend past a flat bar that attaches to it to provide horizontal support. An isosceles triangle appears along edges facing the interior of the hanger. Which of the following statements best explains why a hanger is not a good example of a triangle? A. Not all triangles are isosceles. B. Not all triangles are made of wood. C. Students may not have schema for a hanger. D. The exterior of the hanger is not composed of straight lines.

A. This explanation is not sufficient because it implies that the teacher must find a triangle that can represent all triangles, and such a triangle does not exist. B. The material used to create a triangle is not one of its defining attributes, and students should be exposed to triangles constructed out of a variety of materials, including wood, so they can understand why this true. C. Students should be presented with examples of real-world objects and situations that relate to the mathematics they are studying, regardless of whether they have been exposed to them before. D. CORRECT. A triangle has three straight sides, and this picture may introduce the misconception that a triangle may have curvy lines or rounded corners.

A teacher gives each group of students 32 tokens. The teacher asks the groups to count the total number of tokens. Four groups' methods are shown in the following table. Group Method A places the tokens in a straight line, and then counts them one by one B assembles the tokens into piles of 2 tokens each, and then skip-counts by twos C groups the tokens into equal piles of 8 tokens each, and then adds 8 + 8 + 8 + 8 on a calculator D arranges the tokens into piles of 10, skip-counts by tens, and then counts the last 2 tokens by ones Which group's method best supports an understanding of place value? A. Group A B. Group B C. Group C D. Group D

A. This group simply counts the tokens, which does not indicate a deeper understanding of how to think about numbers. B. This group's strength is in skip-counting, which is a counting strategy, not a place-value understanding. C. This group's method is a good precursor to more complex addition problems or multiplication, but it does not use place value. D. CORRECT. Because this group uses piles of tens and ones, this shows that the students have an early understanding of the base-ten system, which can be used to support place-value knowledge.

A second-grade teacher analyzes student performance on an assessment and concludes that 75% of the class meets or exceeds the standards for proficiency. Which of the following approaches can the teacher plan to provide the most effective instruction? A. designing an out-of-class assignment for students to complete with parents/guardians that provides additional practice on the last unit B. using assessment data to form heterogeneous groups, then implementing a group activity that connects the two units C. delaying the start of the new unit indefinitely and reviewing the concepts until all students have achieved proficiency D. allowing students who have yet to reach proficiency the option to retake the end-of-unit assessment the next day

A. This is a blanket solution that does not consider the individual strengths and weaknesses of students and places too much of the onus on parents/guardians. B. Correct. This method utilizes the assessment results in a thoughtful way alongside an activity meant to prepare students for the new unit. C. This approach is likely to cause proficient students to disengage. D. The teacher does not address the gaps in the students' learning before offering a new assessment.

A third-grade teacher draws this image on the board: A number line from 0 to 1. The number line has tick marks in increments of one fourth. Three arrows are drawn, each labeled with one fourth, to show three "hops" from zero to three fourths. Along with the image, which of the following descriptions would help students understand what it means to find 343 fourths on a number line? A. "When you want to find three-fourths on a number line, you should draw three curved arrows. The third arrow will point to the answer. See?" B. "To find the value of three-fourths, you look at the number line. When you see the three written on top of a four, with a horizontal line in between, you have found it." C."First you will need to divide your number line into sections, and label the sections with their appropriate fractions. Then, look at the numerators in your fractions. Add them up and that's your answer." D."If you divide the space on number line between zero and one into four sections of equal length, each section is one-fourth. If you jump this one-fourth distance three times, the distance from zero to this new spot is three-fourths."

A. This is a procedural explanation that does not go into detail about the meaning (i.e., why drawing three arrows would lead to 343 fourths). B. This is a procedural explanation that does not incorporate the meaning of 343 fourths and only works if the number line already has the desired value written on it. C. This explanation does not require that the student divide the number line into equal sections, nor does it explain how to label the sections. D. CORRECT. This explanation gives clear instruction on how to set up the number line and expresses in lay terms that 141 fourth three times is 343 fourths.

Two students discuss how 232 thirds = 464 sixths. Student A: If I cut my sandwich into three pieces and eat two of them, it is the same as if I cut my sandwich into six pieces and eat four of them. Student B: How can that be true? Eating two pieces cannot be equal to eating four pieces. Don't try to trick me! Student A: No look! I'll show you. First, I cut my sandwich into three, like this. (shows drawing) A rectangular array. The array is divided, using vertical lines, into three equally sized sections. The sections are numbered 1 to 3. Sections 1 and 2 are shaded. Then, I cut them in half again. The two gray pieces become four pieces. (shows drawing) The previous array is divided in half with a horizontal line, and the sections are relabeled 1 through 6. The previously shaded sections, 1 and 2, are now represented by the four shaded sections 1, 2, 3, and 4. Student B: Hmm, maybe. It still seems confusing to me. The teacher can use which of the following explanations to most effectively clarify Student A's reasoning? A. "Student A's explanation works because both sandwiches are the same size. If the sandwiches were different sizes, you would be right." B. "What happens if we draw both fractions on a number line? If you can find them both, you will see that they are written in the exact same spot! So cool, right?" C. "Let's renumber the sandwich pieces so they are written in order. That should help you understand how the two pieces in the first picture equal the four pieces in the second." D. "The pieces of a sandwich cut into six equal parts will be smaller than the pieces of a sandwich cut into three equal parts, so you can eat 'more' of them without eating more total food."

A. This is only partially correct. It is true that the equivalence depends on equivalent wholes, but if the second sandwich were smaller, eating 4 pieces would end up being a smaller amount of food. B. While technically true, this explanation does not pay attention to Student A's reasoning, nor does it address for Student B why the two fractions are marked in the same spot on the number line. C. While Student A did number the pieces strangely in the second picture, it is unlikely that this is why Student B is confused. D. CORRECT. Student A shows that the two fractions are equivalent since the four pieces in the second drawing make up the same area as the two pieces in the first drawing.

17. A teacher gives a small group of students a result-unknown word problem. A farm has 12 chickens and 9 pigs. How many chickens and pigs are on the farm all together? After the students arrive at the correct answer, the teacher asks a change-unknown word problem that builds on the information from the first word problem. Which of the following questions could be the question that the teacher asks? A. How many more chickens are on the farm than pigs? B. Each chicken lays one egg per day. How many total eggs are laid each day? C. Some ducks fly down to the farm. Now there are 25 animals on the farm. How many ducks are there? D. On the farm, 6 chickens run away. Then 5 more pigs join the farm. How many animals are there all together now?

A. This problem can be described by the result-unknown equation 12 − 9 = ?result unknown. B. This problem can be described with the result-unknown equation 12 × 1 = ?result unknown. C. CORRECT. This problem can be described by an equation where a change is unknown: 9 + 12 + ?unknown = 25 or 21 + ?unknown = 25. D. This problem can be described by the result-unknown equation (12 − 6) + (9 + 5) = ?result unknown.

A kindergarten teacher meets with a small group of students. The teacher shows them models of three-dimensional shapes. One student explains that the cube is flat compared to the round cone and sphere, so it must be two-dimensional. The teacher can most effectively clarify the student's partial understanding of two- and three-dimensional shapes by using which of the following instructional strategies? A. Reviewing a list of common two-dimensional shapes B. explaining that they are only using three-dimensional shapes today C. formatively assessing students on the names of the three-dimensional shape models D. providing two-dimensional shapes for students to compare with the three-dimensional models

A. This provides the student with more information about two-dimensional shapes, but it does not afford the opportunity to compare shapes to foster a strong mathematical definition. B. This explanation tells the student that they have a partial understanding, but it does not afford them the opportunity to discover the core misunderstanding of "flat" in their justification. C. The student provides evidence of knowing the names of shapes already, and this does not address the student's partial understanding in their justification. D. CORRECT. The teacher identifies an error in the student's understanding of two- and three-dimensional shapes, then challenges the student's understanding by comparing the types of shapes through examples.

A student answers an assessment item the following way. Which of the following numbers are even? The numbers 8, 5, 18, and 21 are shown. Each of those numbers except for 5 is circled. The teacher conferences with the student about the item. Which of the following teacher questions investigates the student's understanding of the definition of even numbers? A. "What do all your answers have in common?" B. "Do you think 21 is even because there is a two in it?" C. "What pattern can we use to check if a number is even?" D. "What do you notice about the number that you did not choose?"

A. This question allows the student to explain their strategy but doesn't ask them to look for a mathematical definition or pattern dealing with even and odd numbers. B. This question speculates about a misunderstanding and only leads to a "yes" or "no" answer. C. CORRECT. By asking the student to explain the mathematical pattern of even and odd numbers, the teacher can investigate the degree to which the student understands the mathematical definitions for even and odd numbers. D. This question affords the student the opportunity to explain the attributes about the number not selected and does not ask them to think about how it fits into the pattern or definition of even and odd numbers.

A kindergarten teacher places five cups on a table and asks a group of students, "How many cups are here?" After a minute talking quietly with each other, the students announce, "There are five cups.'" The teacher then adds a sixth cup to the table and asks the students, "How many cups are here now?" Again, the students talk quietly together and then announce, "There are six cups." Which of the following questions can the teacher ask to determine whether the students counted up from five, recounted all the cups, or subitized to come up with their answer? A. "Can you draw for me how you count to six?" B. "How did you figure out that there are six cups?" C. "How many cups will be on the table if I take away two cups?" D. "Did you count on, count all, or subitize to get the answer?"

A. This question does not provide enough instruction, and so students may not make the connection between this question and how they figured out the number of cups, and may demonstrate a different method. B. CORRECT. This question asks students to explain their thinking process, and provides the teacher with the feedback necessary to categorize the students' thinking. C. If the students again talk among themselves before providing an answer, the teacher can only guess the students' methods. D. This is a vocabulary question, and it is unlikely that a kindergarten student would be able to speak about their thinking process in such terms.

A first-grade teacher distributes bins of small cubes to their class. The teacher expects students to engage in discussion with each other and discover which three-dimensional shapes can be composed by iterating the cubes. Which of the following questions is most effective to elicit student thinking and encourage discussion? A. "Can you make a sphere out of the cubes?" B. "Is it possible to make a big cube using eight smaller cubes?" C. "What do you notice about the shapes you are able to compose out of cubes?" D. "How many cubes does it take to make a rectangular prism that is two cubes wide, six cubes long, and three cubes high?"

A. This question is closed and leads students to a "yes" or "no" answer. B. This question encourages limited discussion beyond identification of the answer. C. CORRECT. This question elicits thinking and conversation based on students' analysis of their composite shapes compared with other three-dimensional shapes. D. This question leads students to a correct answer and does not invite students to interpret their findings from the activity.

8. Two students play in the block area. One student builds a tower using five blocks. A second student tries to build a similar tower with the same type of blocks, but it keeps falling over. The teacher hears the students have the following conversation. Student A: You have to put this kind on the bottom. It's the biggest one. Student B: No, this one is the biggest. Student A: Not the biggest this way, the biggest that way. The teacher wants to facilitate the students' thinking and language about shapes. Which of the following questions should the teacher use to achieve this goal? A. "What can you do to help your friend?" B. "Do you know how many sides your block has?" C. "Do you mean the base block needs to be bigger?" D. "Can you show or tell your friend how the block is bigger?"

A. This question is vague and does not help facilitate the students' discussion. B. The students are discussing an attribute of the block that does not depend on its number of sides. C. With this question, the teacher clarifies that the block can be called the "base block," but this information does not help students establish a common understanding of the attribute they are discussing. D. CORRECT. This question helps students establish a common understanding about how the word "biggest" applies to the shape.

19. A third-grade teacher asks a student to solve 33 − 7 = ?, initiating the following conversation. Student: The answer is 26. Teacher: How did you get your answer? Student: First, I took a 10 from 33, so I made it into 10 and 23. Then I took away 7 ones from the 10 and counted what was left to get 3 + 23, which is 26. Teacher: Why did you make the 33 into 10 and 23? Student: It was easier to subtract the 7. Which of the following representations with base-ten blocks should the teacher use to model the student's approach? A. A diagram shows base-10 blocks being manipulated in 3 steps. The first step shows 2 unshaded rods, 1 shaded rod, and 3 unshaded squares. The second step shows 2 unshaded rods and 1 shaded rod and 3 shaded squares. The shaded rod and squares are circled. The third step shows 2 unshaded rods and 6 unshaded squares. B. A diagram shows base-10 blocks being manipulated in 3 steps. The first step shows 1 shaded rod, 2 unshaded rods, and 3 unshaded squares. The second step shows 10 shaded square, 2 unshaded rods and 3 unshaded squares. Six of shaded squares are circled. The third step shows 2 unshaded rods and 6 unshaded squares. C. A diagram shows base-10 blocks being manipulated in 3 steps. The first step shows 1 unshaded rod and 1 shaded rod alongside 13 unshaded squares. The second step shows 1 unshaded rod, and 10 shaded squares, and 13 unshaded squares. Six of the shaded squares are circled. The third step shows 1 unshaded rod and 16 unshaded squares.

A. This representation shows the initial decomposition of 33 into 3 tens and 3 ones, its arrangement into a group of 2 tens and a group of 1 ten and 3 ones, and then the exchange of 1 ten and 3 ones for 6 ones; this does not model the student's description. B. CORRECT. This representation shows the decomposition of 33 into 3 tens and 3 ones, the grouping of 1 ten into 10 ones, and the removal of 7 ones from that group. C. This representation shows the decomposition of 33 into 2 tens and 13 ones, a regrouping of one 10 to create 1 ten and 23 ones, and then the removal of 7 ones; this does not match the student's description. D. This representation shows all the tens, an initial decomposition of 33 being traded for 30 ones, and then 7 ones removed from the 33 ones; this does not match the student's description.

A teacher draws three images on the board and asks the students to say what each shape is and how they know. A row of 3 shapes is shown. Each shape has a letter at its center. The letter A is shown within a square on the left of the row, the letter B is shown within an equilateral triangle at the center of the row, and the letter C is shown within a pentagon on the right side of the row. A student says, "I know that A is a square because it has four things." Which of the following teacher responses would help the student clarify their response? A. "Can you come up and draw another square on the board?" B. "What is the definition of a square and how did you use that to help you?" C. "Do you mean four sides (points to sides) or four angles (points to angles)?" D. "Yes! Using that reasoning, what is the name of Shape B (points to Shape B)?"

A. This response may help clarify that the student can represent a square, but it does not help the student develop more precise language. B. This response may indirectly help the student think about the terms needed to define a square, but the student may just as likely repeat that "A square has four things." C. CORRECT. This response combines vocabulary and gestures to help the student develop the terminology needed to explain their thinking, and encourages the student to use more precise language. D. This response applauds the student for the correct shape name, but it does not attend to the imprecise language before moving on

A second-grade teacher checks one student's work on an assessment. There are two strings. One string is 21 inches long.The other string is 28 inches long. How much longer is the second string? A number line that starts at zero. A horizontal line is shown above the number line that extends from zero to the next labeled tick mark of 21. Another horizontal line is drawn above that from 0 to 28. There are tick marks to indicate every whole number from 21 to 28. An arrow extends from the end of the shorter horizontal line, 21, to the end of the longer horizontal line, 28, to show a difference of plus 7. Which of the following descriptions of the student's work is most accurate? A. The student's drawing incorporates "taking away" to appropriately model subtraction. B. The student's drawing appropriately models the operation of subtraction by finding the difference between the two numbers. C. The student's operation is subtraction, but they show addition on the number line, so the model is mathematically inconsistent. D. The student's operation is a correct representation of the assessment question, but their drawing is not consistent with the definition of subtraction.

A. This solution is not consistent with solving by "taking away." B. CORRECT. The student solves the problem accurately, and their model represents the problem and operation of subtraction meaningfully. C. The student shows a counting-up strategy on the number line, and it is an appropriate model to find the difference between two numbers and show subtraction. D. Counting up on a number line to find the difference between two numbers is mathematically consistent with the operation of subtraction, and is particularly relevant in this problem.

A first-grade teacher plans a lesson to get students talking about the attributes of composite shapes. For the activity, students work independently with tangrams to create a composite shape, trace their shape, list its attributes, and give their shape a name. Then the students work in small groups to create an image or story that uses the composite shapes of all group members. Which of the following modifications would most effectively promote participation? A. substituting tracing plastic tangrams with gluing construction paper cutouts onto a piece of posterboard to create the composite shapes B. directing students to work independently so each student gets more say in how their shape is used in the final product C. offering an alternative computational activity for students who are not interested in designing a composite shape D. assigning roles once the students are in groups to ensure that each student makes a meaningful contribution

A. This substitution does not change the task in a way that would promote broad participation. B. Some students may prefer this, but it does not foster cooperation and does not allow for students to discuss the attributes of shapes. C. If computation is added to the task, it would detract from the learning goal of discussing attributes of composite shapes. D. CORRECT. Assigning students roles for their group work promotes equitable participation by specifying how they will collaborate, by clarifying which responsibilities they must fulfill, and by preventing over-participation.

In a department meeting, a teacher shares a poster on how they represent "one-third" to second-grade students three different ways. Three rectangular arrays. Each array is divided into thirds. The first array is divided into thirds vertically, and one region is shaded and labeled one third. The second array is divided into thirds horizontally and one region is shaded and labeled one third. The last rectangle is divided into thirds using a combination of vertical and horizontal pieces. One of the vertical regions is labeled one third. Which of the following statements explains the purpose of showing these three representations? A. Each drawing uses a rectangular whole, which is easier for students to analyze than a circle. B. By writing 131 third on each of the three different pictures, it reinforces the numerical representation of the fraction. C. As long as a whole is split into three sections, regardless of partition size, each partition will be one-third of the whole. D. Since each gray partition is a different shape, it shows that what matters for a fraction is equal area, not identical shape.

A. While it is true that it is easier for students to partition a rectangle, that does not explain why the representation of "one-third" varies in each image. B. Since the drawings are all different, the purpose of using them is likely to focus on comparing shape, rather than writing a fraction. C. Since each partition is actually equal in size, these drawings are not attempting to teach or address this misconception. D. CORRECT. Each representation has the same total area grayed out, but the partitions are equal in size.