MTTC 3-6 math 2024

D

1. At the beginning of each new thematic unit, a fifth-grade mathematics teacher provides parents/guardians with a list of skills to be presented in that unit and a survey to gather information about how parents/guardians relate these skills to the everyday lives of their children. The teacher uses this input to inform decisions about the context and scope of activities within the unit. This strategy fosters students' success in which of the following ways? A.eliciting advice by parents/guardians on individualized instructional strategies B. keeping parents/guardians informed regarding emerging instructional practices C. encouraging parents/guardians to reteach these skills to their children as necessary D. helping the teacher deliver lessons that are culturally sensitive and relevant to students

C

10. A fifth-grade teacher presents to the class a scenario with a series of related problems, as follows. Which of the following questions should the teacher ask to support students' thinking about how to solve each problem? A. "How can we write this problem using mathematical symbols?" B. "How can we use estimation to check our solution for this problem?" C. "What's happening in this problem, and what will the answer tell us?" D. "What are the key words in this problem, and which operations should we use to solve it?"

D

27. Two students compare the decimal values 0.1071 and 0.1701 . The teacher overhears students debating the answer. Student A: I wrote 0.1071 is greater than 0.1701 because 71 is greater than 01. Student B: I wrote 0.1071 is equal to 0.1701 because zeros don't mean anything. Based on these explanations, the students would benefit most from additional review on which of the following strategies? A. rewriting decimal numbers in expanded form B. representing comparisons using the symbols <less than, >greater than, and = C. using place value to estimate and round values to thousandths D. comparing two whole numbers when both numbers have digits of 0

D

11. A fourth-grade teacher begins an introductory lesson on multidigit division by asking students to solve the following problem. Which of the following observations by the teacher best describes the key mathematical difference in the two students' approaches to solving this problem? A. Student A demonstrates fluency in composing and decomposing whole numbers, while Student B uses a trial-and-error method. B. Student A's approach demonstrates an understanding of place value, while Student B's approach indicates an understanding of division as repeated subtraction. C. Student A uses a direct modeling approach that can be applied to any division problem involving multidigit numbers, while Student B uses an invented strategy that is not reproducible. D. Student A's approach mirrors the concept of partitioning used in the standard algorithm for division, while Student B's approach demonstrates an understanding of inverse relationships.

B

12. Students examine tallied data from a recent poll about which board game students would prefer to play at an upcoming class party. One student says, "I know that 23 students want to play mancala because in the last row, there are 3 tally marks." Which of the following questions can the teacher ask to best elicit this student's thinking? A. "How many more students voted for mancala than checkers?" B. "Can you explain how you got the twenty in your answer of twenty-three?" C. "Can you skip-count by a number other than 5 to find the totals in each column?" D. "How many students would want to play mancala if the last line had 9 tally marks?"

B

13. A fourth-grade teacher presents a division task to students. The teacher tells the students to think of arranging 20 chairs for a concert and then to draw a model for what 20 ÷divided by 4 could represent. The work from two students is shown. Which of the following quantities is different for these two solutions? A. the remainder B. the number of chairs per row C. the total amount being divided D. the numerical value of the quotient

A

14. A fourth-grade student uses the division algorithm to solve the following problem. Which of the following questions can the teacher use to clarify the difference between the answer shown in the algorithm and the student's response? A. "Can you explain how you got an answer of 18 from your calculations?" B. "Can you explain why you show 63 minus 4 in your computation?" C. "How could you check your answer using repeated subtraction?" D. "How could you check your answer using multiplication?"

C

15. A fourth-grade student asks, "I understand that 2400 ÷divided by 2 is 1200 because that makes sense to me. I can picture taking 2400 tokens and dividing them into two piles, and that I would have 1200 in each pile. I know the numbers are big, but I promise I can picture it. I also know that if I use long division, I get the right answer. But why does long division work?" Which of the following replies answers the student's question? "When you do long division, you get the same answer as you do when you split the tokens into the two piles, so you have proven that long division is a successful mathematical method." "Imagine that you have 2400 tokens and place them into the dividend location. Then write the number of piles where the divisor goes. When you then follow the long division steps, you get the quotient." "With long division, first you divide 2000 into 2 pieces to get 1000, and then 400 into 2 to get 200. Then you add those results together to get 1200. It's like your pile method, just done in smaller sections." "When you do long division, you end up with a quotient, which in this case is 1200. If you take that number and multiply it by the divisor, so by 2, you get the dividend back. That shows that process is correct."

D

16. A student explains how they solve the problem 423 −317 = __ Step 1: First, I subtract 300, so 423 -300 = 123. Step 2: Then I look at 23 and 17 and figure out 17 + __ = 23. Step 3: The blank is 6. Step 4: I get a final answer of 106. Which of the following statements explains the student's actions in Steps 2 and 3? A. The student uses place-value strategies to subtract. B. The student considers subtraction in the form of "change unknown." C. The student uses compensation to calculate the remaining difference. D. The student recognizes the relationship between addition and subtraction.

B

17. A teacher overhears a third-grade student make this argument: "I think it is silly that 12 divided by 3 has to be 4. Imagine we have a rectangle that is split into 12 pieces. There are so many ways to divide the rectangle into 3! What if I want to color in 7 pieces? There is still some left for me to split into two pieces. Look at my drawing. Some pieces are light gray, some pieces are dark gray, and there are still white pieces left over." Based on this drawing and argument, the teacher can conclude that the student interprets division in which of the following ways? A. It is a set of memorized facts. B. It allows for unequal partitions. C. It must be solved with a drawing. D. It requires whole-number solutions.

C

18. A teacher compares the following manipulatives for an upcoming activity. Compared with base-ten blocks, which of the following statements describes a limitation of using paper clips to model operations? A. Paper clips can be chained at any length. B. Paper clips have a flat shape, which makes them hard to count. C. Paper clips are difficult to regroup into chains of recognizable length. D. Paper clips are less familiar objects to students than base-ten blocks.

C

19. A teacher sees this tip in a textbook: Pennies are great for helping students understand how to divide small (<less than 100) values by separating the total value into piles of equal size. However, if you want students to practice partitioning larger values, you should incorporate dimes and dollars, or move to using base-ten blocks. Which of the following statements provides the best rationale for this suggestion? A. For students to develop place-value knowledge when dividing, they need to use manipulatives designed for this goal. B. Pennies are exciting for younger students, but as students learn to manipulate larger values, pennies lose their appeal. C. With larger values, the division process in small units becomes tedious, which detracts from the learning goal and slows down calculations. D. Bringing in coins of various values increases the complexity of the problem and helps students recognize different ways to break down a value.

A

2. A fifth-grade teacher designs a puzzle activity that has many different possible solutions. Which of the following strategies can the teacher use to encourage broad participation for this activity? A. encouraging students to determine their own rules and then describe how their rules work with a classmate B. presenting a solution to the second puzzle to students who have yet to identify any rules after a specific length of time C. creating a chart where students volunteer to describe their ideas on how to calculate the numbers in the white regions D. defining rules used to generate the numbers in the white regions of the first puzzle and then instructing students to complete the remaining puzzles

B

20. A student explains how they think about the expression 7 + (12 ×times 3): "I start at seven, and then I go up by 12 three times." The teacher draws a representation on the board so other students can follow along. Which of the following drawings best captures this student's description? (Note: Drawings are not to scale.)

B

21. A third-grade teacher asks a student to solve the following problem. Based on this work, the teacher can draw which of the following conclusions about the student's mathematical thinking? A. The student has memorized the basic addition facts at least up to 7. B. The student most likely relies on counting on to solve addition problems. C. The student understands the concept of place value but has trouble applying it to addition of multidigit numbers. D. The student understands the concept of regrouping but only when adding a single-digit number to a multidigit number.

A

22. During a lesson on solving problems involving whole-number operations, a fourth-grade teacher presents to the class the following problem. Which of the following responses from the teacher most effectively clarifies the student's thinking? A. "First tell us how you know to divide, and then let's think about what that 6 stands for." B. "I understand why you divided 46 by 10, but what do we do with that extra 6? Can the waiter fill a glass with those 6 ounces?" C. "46 divided by 10 is 4, and then you have a remainder of 6, so the server can fill 4 people's glasses with a little extra water left over." D. "46 ounces divided by 10 ounces is 4 full glasses, and we can ignore the extra since we only want full glasses, so the server can fill 4 glasses."

D

23. A third-grade student solves the problem "8 ×7 = ?" as follows. Which of the following solutions of the problem "80 ×70 = ?" represents an extension of this student's method for interpreting multiplication problems? A. (8 ×7) × (10 × 10) B. (10 ×8) × (10 × 7) C. (16 × 5) + (14 × 5) D. (80 × 50) + (80 × 20)

C

24. Fourth-grade students explain how they think of the problem 5 × 1/4= __. Which of the four students' interpretations of the problem defines multiplying by a fraction as division? Student A Student B Student C Student D

A

25. A third-grade student models the fraction 2/5 as shown. Which of the following questions most effectively draws attention to the student's misconception and affords the student the opportunity to self-correct? A. "How many parts make a whole for this fraction?" B. "Where is the numerator and denominator in your model?" C. "Why do you have two light gray squares in your drawing?" D. "How would your answer change if you used apples instead?"

B

26. Two students debate how to write a fraction given 15 marbles, where 5 are red and 10 are black. Student A: I think we should write 5 fifteenths because 5 of the fifteen marbles are red. Student B: I think we should write 5 tenths since there are 5 red and 10 black. Which of the following interpretations of a fraction are present in this discussion? A. relative measures versus part-to-part ratios B. part-to-whole ratios versus part-to-part ratios C. division of two numbers versus relative measures D. part-to-whole ratios versus division of two numbers

C

28. A student asks, "Why does one-half equal three-sixths? One-half means one out of two. So if I have two apples, and eat half, I eat one apple. But for three-sixths, I eat three out of six possible apples. Three is more than one. So how can they be equal?" Which of the following explanations best attends to the student's concern? "Let's change the story. Divide a circle into two and color in one section. It is the same total area as dividing the circle into six and coloring in three sections." "Three-sixths is the same as one-sixth plus one-sixth plus one-sixth. So, it is just eating one apple three times. So, three-sixths is not any more than one-half." "Recall that half is relative to the total value, or half of all the possible apples. If you have two apples, half is one. Now, if you split six in half, you get three, right?" "If you divide both fractions, they come to the same value. Try it! One divided by two equals 0.5. Three divided by six also equals 0.5. Therefore, they are the same."

D

29. A student solves the problem 5.17+ 0.1= __. Which of the following reminders would best help the student understand their mistake? A. "Remember to write in extra zeros at the end of short values so your digits line up." B. "Remember to put the greater value in the top row in order to keep it all organized." C. "Remember that you need to align your decimal points before adding or subtracting." D. "Remember that 0.10 point 1 is one-tenth, so you need to add that to the tenths value in 5.175 point 1 7."

C

3. To help sixth-grade students extend their knowledge of the number system, a teacher provides a table displaying the daily high temperatures over a week in Saginaw, as shown. Students order the temperatures from coldest to warmest. To which of the following learning goals does this activity most align? A. Interpret statements of inequality between rational or whole numbers. B. Use absolute value to determine and compare values' distances from zero. C. Write and explain statements ordering rational numbers in real-world contexts. D. Recognize that inequalities of the form x <is less than a or x >is greater than a have infinitely many solutions.

D

30. A student writes the following explanation of how they rewrote 2 3/8 as a fraction. The teacher plans to use this student's thought process and apply it to a different modeling tool to represent this fraction. Which of the following tools is most appropriate for this purpose? A. dimes and pennies B. x-yX Y coordinate pegboards C. base-ten blocks and unit cubes D. number lines divided into eighths

C

31. A fifth-grade student attempts to evaluate 9/10+ 2/3 by drawing a model, as shown. The student explains: "First I drew nine-tenths, then I noticed that nine is three sets of three, so I colored in two sets of three because that's two-thirds. The part that is colored in twice is six out of ten, so the answer is six-tenths." The teacher should address this student's misconception by providing activities focusing on which of the following concepts? A. accurately representing various fractions using visuals B. recognizing common denominators among benchmark fractions C. accurately modeling various operations (e.g., addition vs. multiplication) with fractions D. choosing appropriate models (e.g., rectangles vs. circles) to represent a math problem

D

32. A teacher models a fraction on the number line, as shown. The teacher asks a student to identify the fraction represented by the number line, and the student incorrectly indicates "five-eighths."

C

33.Which of the following activities would best support fifth-grade students' understanding of multiplication of decimals? A. Students use a number line to visualize multiplication of decimals as repeated addition. B. Students use calculators to multiply pairs of decimal numbers and look for patterns in the results. C. The teacher uses base-ten blocks to illustrate the multiplication of decimal numbers by finding the area of a rectangle. D. The teacher demonstrates the rule for counting the number of decimal places in each factor to determine where the decimal belongs in the answer.

D

34. In a lesson on proportional reasoning, a sixth-grade teacher presents the following problem. Which of the following explanations by the teacher most clearly expresses how the double number line represents the relationship in the table? A. "The double number line helps us see that to find the number of bees needed to make 48 teaspoons of honey, we multiply 12 times 48." B. "The double number line shows us that there is a multiplicative relationship between the number of bees and the number of teaspoons of honey." C. "The double number line shows us that the number of bees needed to make 1 teaspoon of honey is one-twelfth the number needed to make 48 teaspoons." D. "The double number line shows us that for every 1 teaspoon, we need 12 bees, so if we want 48 times as many teaspoons, we need 48 times as many bees."

B

35. A teacher asks students to solve the following problem. Which of the following responses by the teacher would best help the students think about computation with fractions? A. "Why did you shade 2 boxes in one color and 3 boxes in another color?" B. "Can you explain how the equation matches up with the model?" C. "Did anyone solve the problem by creating a different model?" D. "Remind us how to add fractions with like denominators."

D.

36. A fourth-grade teacher asks students to determine which of the fractions 5/6 and 3/8 is closer to 4/5. One student says, "Five-sixths is closer to four-fifths" and shares their reasoning on the board. A second student responds, "There's an easier way! Five-sixths is closer to four-fifths because sixths and fifths are almost the same size." The teacher can conclude that both students would most benefit from which of the following activities? A. reinforcing the basic understanding that a fraction is a distinct number B. using number lines to teach students to recognize benchmark fractions C. reinforcing the algorithm of finding common denominators for comparing fractions D. using visual models to develop conceptual understanding of the relative size of fractions

C

37. A sixth-grade teacher asks a student how to solve the following problem. "How many 3/4 cup servings are in 2/3 of a cup of cereal?" The student replies, "You do two-thirds divided into three-fourths." Which of the following responses from the teacher most effectively clarifies and records the student's thinking? A. "Good." (writes 2/3 ÷ 3/4 on the board) B. "You mean two-thirds divided by three-fourths, right?" (writes 2/3 ÷ 3/4 on the board) C. "How would you write 'two-thirds divided into three-fourths' using a number sentence?" D. "Do you mean two-thirds divided by three-fourths or two-thirds divided into three-fourths?"

C

38. A student multiplies 1/4 × 5/8 as follows. Student: I need to find common denominators and then multiply the numerators, like this. Which of the following teacher responses demonstrates an understanding of the student's incorrect reasoning and most effectively supports the student's understanding of fraction multiplication? A. "Think of the problem like this: 1/4 × 5/8 is the same as 1/4 of 5/8, which tells us that the answer should be smaller than 5/8." B. "Remember that we only need to find a common denominator when combining fractions by addition or subtraction, not when we multiply fractions." C. "Let's draw a model to represent this multiplication problem, and one to represent the addition problem 1/4 + 5/8. In what ways are the models different?" D. "Let's use the two fractions in the problem to estimate what we expect the answer to be. Does your answer seem right when you compare it to this estimate?"

C

39. During an interdisciplinary unit on nutrition, fifth-grade students learn that 0.75 cup of yogurt is a serving size. The teacher puts the students into groups and each group must determine a generalized method for calculating how much yogurt is needed to serve different numbers of students and then present their solution to the class. Two groups' methods are as follows. The approaches used by both groups are mathematically consistent because both groups: A. used a mathematical tool to justify their reasoning. B. created an equation to determine the total cups of yogurt needed. C. applied the proportional relationship by utilizing repeated addition. D. concluded that multiplication can be used to determine the serving amounts.

C

4. On a pretest, students compare decimal values. The teacher notices that many students exhibit the misconception that the numeral with the most digits always has the greatest value, which is shown by the students' answers. 0.065 < 0.0072 0.65 > 0.7 0.65 < 0.700 Which of the following classroom activities most effectively addresses this misconception? A. utilizing play money to represent various decimal values B. comparing a set of decimal values that all contain the same number of digits C. shading area models to represent the worth of different decimal place values D. comparing whole numbers with the same number of digits and discussing patterns

C

40. During an interdisciplinary unit on nutrition, fifth-grade students learn that 0.75 cup of yogurt is a serving size. The teacher puts the students into groups and each group must determine a generalized method for calculating how much yogurt is needed to serve different numbers of students and then present their solution to the class. Two groups' methods are as follows. In which of the following ways does this activity promote broad participation from students? A. providing a problem that uses familiar decimal values B. engaging students by using culturally relevant contexts C. allowing students to develop solutions using multiple entry points D. establishing and applying norms for participation in group activities

D

5. A fourth-grade teacher poses the following question as an exit slip for students. Which of the following statements can the teacher conclude from the student's response? A. The student understands how to find a common denominator. B. The student is unsure of what to do when a quotient has a remainder. C. The student is unsure of how to apply the standard division algorithm. D. The student conceptualizes how to use area models to compare fractions.

C

6.Two students discuss how to solve the problem 2/5 + 1/3 Student A: I got an answer of three-eighths. Student B: That's the wrong answer. The right answer is eleven-fifteenths. Student A: I hate this work! Eleven doesn't make sense. I'll never get it because I'm not a math person. Which of the following teacher responses most appropriately promotes a positive identity as a mathematical thinker for Student A? A. "I am available to meet with you any day after school if that will help you to understand." B. "Getting frustrated only makes the situation worse. Let's try to compare these answers and figure it out." C. "I understand your frustration. We can compare these answers so that we all better understand the problem." D. "Of course you can be a math person; it just takes practice. Let's review your answer to figure out where you made an error.

A

7. Students in a sixth-grade math class work in groups to solve the multistep word problem shown A teacher listens to the following discussion between two students. Student A: A 2 to 1 ratio means half the plants are pepper plants and half are tomato plants, so there are 12 of each. Student B: No, I think the 2 to 1 ratio means that one of the numbers should be two times as big as the other. The group stops making progress toward a solution because of this disagreement. Which of the following instructional supports should the teacher provide to best promote the group's ability to engage in productive struggle? encouraging the students in the group to compare their different interpretations of the ratio using a model displaying a reminder at the front of the room that shows how to convert between part-to-part ratios and fractions asking a group that correctly interpreted the ratio of tomato plants to pepper plants to present their solution to the class guiding the students in the group through an easier version of the word problem with the same ratio using only 6 plants total

C

8. A fifth-grade teacher instructs students to compare two numerical patterns: Start at 0 and repeatedly add 3. Start at 0 and repeatedly add 6. Which of the following affordances for this table guides the student to observe that the terms in Pattern 2 have twice the value of the corresponding terms in Pattern 1? A. The pattern rules are written directly to the left of each pattern. B. Both patterns in the table have the same number of terms. C. There is a clear alignment of the terms in each pattern. D. Both patterns in the table start with the number 0.

D

9. A third-grade teacher plans a lesson to support students' fluency in composing and decomposing whole numbers. The teacher gives each student a set of base-ten blocks including singles, tens, and hundreds. Which of the following prompts from the teacher best supports the students in their thinking about composition and decomposition? A. "Show me 124 with the blocks. How many hundreds, tens, and singles did you use?" B. "Show me 62 with the blocks. Show me 31 with the blocks. How can we represent 124 with the blocks?" C. "Show me 62 with the blocks. Show me 31 with the blocks. How much do you get if you combine all the blocks?" D. "Show me 124 with the blocks. Show me 124 with the blocks in another way. How would we show 1000 using the blocks?"