Early elementary ed mttc math

B

17. 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.

B

32. 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

33. 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

C

34. A third-grade teacher asks students to add 57 + 34 on their whiteboards. A student shows their work: 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

35. A second-grade teacher meets with a small group of students about this story problem. 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.

D

36. A third-grade teacher reviews a student's work from an assessment, as shown. 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.

B

37. A second-grade teacher checks one student's work on an assessment. 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

38. 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

C

39. A second-grade teacher gives students this word problem involving a number line. One student's response is shown. Which of the following claims can be made about the student's work? The student understands how to regroup to perform subtraction. The student understands how to interpret subtraction as taking away. The student understands how to compare the difference between two numbers by using an unknown-change model. The student understands how to compute a difference by combining the strategies of skip-counting and counting on.

D

4. 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. 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.

B

40. A third-grade teacher asks students to draw a representation to solve the problem 4×3. Four student solutions are shown. 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

C

5. 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

B

6. 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

D

7. 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

C

8. 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

9. A third-grade teacher considers the following task. UnderlineThe word Underline is underlined. the triangles. CircleThe word Circle is circled. the quadrilaterals. Put an X through the gray shapes. 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?

B

22. 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

D

23. 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. 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

D

1. 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.

C

10. 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. 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.

C

11. A student answers an assessment item the following way. Which of the following numbers are even? 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?"

D

12. 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

C

13. A teacher draws three images on the board and asks the students to say what each shape is and how they know. 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)?"

C

14. 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?"

D

15. 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?"

C

16. 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. 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.

C

18. 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.

B

19. 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?"

B

2. 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?"

D

20. 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."

B

21. 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

24. Which of the following models shows hierarchical inclusion? A. A series of nested ellipses. The innermost ellipse contains a square and the number 1. A second, larger ellipse contains the first ellipse, as well as another square and the number 2. This pattern repeats until the largest ellipse contains a total of 5 squares and the digits 1 through 5. B. A set of 5 squares in a row. Each square is labeled with a digit beneath, 1 through 5, from left to right. C. The digits 1 through 5 in ascending order, from left to right, with the written name for each digit shown beneath them. D. Dots are stacked in five columns. The first column contains one dot, and the label 1. Each successive column adds one dot to the previous stack and increases the label by 1. The final column has five dots and is labeled with a five.

C

25. Which of the following questions can help students to think about comparing the fractions 3 quarters and 7 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 3/4 of a pizza or 7/8 of a pizza?"

D

26. In a department meeting, a teacher shares a poster on how they represent "one-third" to second-grade students three different ways. 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 1/3 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.

D

27. A third-grade teacher draws this image on the board: Along with the image, which of the following descriptions would help students understand what it means to find 3/4 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."

C

28. 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 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?"

D

29. 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

C

3. 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.

D

30. A teacher draws these number lines on the board as part of an explanation for how to understand that 1/3 is less than 3/4 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."

D

31. Two students discuss how 2/3= 4/6 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."