MTTC #119 Mathematics
Two students discuss how 2/3 = 4/6. 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) Then, I cut them in half again. The two gray pieces become four pieces. (shows drawing) 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?
"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 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?
"What do you notice about the shapes you are able to compose out of cubes?"
A student answers an assessment item the following way. Directions: Which of the following numbers are even? Student circles: 8, 18, 21 Student does not circle: 5 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?
"What pattern can we use to check if a number is even?"
Which of the following teacher explanations best supports a first-grade student's ability to identify, name, and use an equal sign?
"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."
After a lesson on the progression of fractions from 0 to 1 using fourths, a teacher presents this problem to the class. Number line: 0, blank, blank, blank, blank, 1, ? One student responds, "The question mark is after the 1, so the answer is 1 1/2". Which of the following teacher responses would best prompt a discussion about unit fractions and iteration?
"Which patterns do you notice between 0 and 1 in the first two number lines?"
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?
"Will you please show me your strategy with sticks and clay?"
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?
"Can you show or tell your friend how the block is bigger?"
A teacher draws three images on the board and asks the students to say what each shape is and how they know. A: square B: triangle C: pentagon 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?
"Do you mean four sides (points to sides) or four angles (points to angles)?"
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?
"How did you figure out that there are six cups?"
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. three long blocks 3 cubed blocks 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?
"How would you explain the difference between a rod and a block?"
A third-grade teacher draws this image on the board: *a number line hopping from 0 to 3/4, labeling each hop as 1/4* Along with the image, which of the following descriptions would help students understand what it means to find 3/4 on a number line?
"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."
Which of the following questions can help students to think about comparing the fractions 3/4 and 7/8?
"If you draw both, which one is closer to a whole?"
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?
"One-third is less than three-fourths because three-fourths is farther to the right than one-third."
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: 6 boxes with 3 shaded in (3/6) + 4 boxes with 2 shaded in (2/4) = 10 boxes with 5 shaded in (5/10) 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?
"That's a great start! Now, let's make all three shapes the same size to begin with, and then see what happens."
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:
compare two numbers using place values
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?
composing a number different ways
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?
creating a game in which students show the same amount of money using different combinations of coins and bills
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?
creating a subtraction equation in which students represent the equation as a picture and a word problem
A first-grade student solves 8 + 5 and shows their work: 8 + 5 = 13 5 -> 2 and 3 8 + 2 + 3 = 13 10 + 3 = 13 The student's work provides evidence that the student integrates addition with which of the following skills?
decomposing and composing numbers into place-value components
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: 10 tiles 10 tiles 4 tiles Presenting this representation to the class demonstrates the use of manipulatives to:
develop understanding of the base-ten system
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:
encouraging the students to try a count-back or a count-up strategy
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. *dice drawing* 3 + 4 = 7 The teacher can increase students' success by taking which of the following actions before explaining the activity?
giving students the opportunity to become familiar with dice and their dots
The dramatic play area of a prekindergarten classroom is set up 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:
how the plate and the cup are the same and different
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:
identify the mathematical competency they demonstrate in their approach
First-grade students consider the following equations. 7 = 10 - 3 7 = 5 + 2 10 - 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?
meaning and function of the equal sign
A teacher asks students to count by ones from 3 to 10. Four student solutions are represented. Student A: points to 3 on the number line, and then points and says aloud every number 3 through 10 Student B: quietly says "1, 2" under their breath, and then more loudly states, "3, 4, 5, 6, 7, 8, 9, 10" Student 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 Student 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"?
Student B
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: Group A: places the tokens in a straight line, and then counts them one by one Group B: assembles the tokens into piles of 2 tokens each, and then skip-counts by twos Group C: groups the tokens into equal piles of 8 tokens each, and then adds 8 + 8 + 8 + 8 on a calculator Group 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?
Group D
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. 5 dots in a tens frame 4 dots outside of a tens frame Which of the following qualities of this activity makes it particularly effective for promoting broad participation?
It encourages multiple entry points and ways for students to be mathematically successful.
A third-grade teacher considers the following task. Directions: Underline the triangles Circle 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?
List the ways these shapes can be sorted.
A third-grade teacher asks students to draw a representation to solve the problem 4 ×times 3. Four student solutions are shown. Student A: an array with 4 across on top and 3 down the side Student B: a number line hopping 3 times to the number 4 Student C: a circle with 3 dots inside and 1 dot outside Student D: 10 squares drawn on top and 2 squares drawn below Which student's work sample is the most accurate representation for expressing multiplication as repeated addition?
Student B
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?
Students count on a number line to find the distance between numbers.
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?
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?"
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?
Mix students based on the strategies they used. Have each student explain their strategy to the small group.
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?
Since each gray partition is a different shape, it shows that what matters for a fraction is equal area, not identical shape.
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?
Some ducks fly down to the farm. Now there are 25 animals on the farm. How many ducks are there?
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 2/3 and 2/6, 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 2/3 is greater than 2/6. 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?
Student A interprets the number of attempts as the whole, while Student B compares only the number of goals scored.
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?
The student relies on a procedural strategy, but may lack an understanding of place value.
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 starting at 18 and hopping to 20 by +2, hopping to 30 by +10, and hopping to 37 by +7* Which of the following claims can be made about the student's work?
The student understands how to compare the difference between two numbers by using an unknown-change model.
A teacher assesses students' understanding of fractions with the following question on an exit slip: Which fraction is greater 3/4 or 4/6? A student responds that 4/6 > 3/4 and provides the following explanation. 3 shaded rocks, 1 unshaded rock = 3/4 4 shaded rocks, 2 unshaded rocks = 4/6 The student's response supports which of the following claims?
The student understands how to interpret an individual fraction as a part of a whole.
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. 4 dots in a tens frame 3 dots outside of a tens 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?
The student understands one-to-one correspondence and would benefit from opportunities to engage in counting on.
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 sequence of tasks that helps the student construct more efficient strategies for adding
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 set of two ten frames, counters, and writing materials
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:
afford students the opportunity to build a more complete definition of triangles.
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:
asking probing questions about the total number of counters each student has
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?
assigning roles once the students are in groups to ensure that each student makes a meaningful contribution
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. The student's model is consistent with which of the following approaches?
calculating a difference by comparison
A teacher shares this geometric pattern with the class. triangle, square, pentagon, hexagon, triangle, 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:
offers students an entry point for collaboratively refining explanations of the geometric pattern
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?
providing two-dimensional shapes for students to compare with the three-dimensional models
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?
two-color counters
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?
using assessment data to form heterogeneous groups, then implementing a group activity that connects the two units
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?
Computations can correctly model different mathematical situations.
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?
Students show varying levels of sophistication in composing numbers, so it is appropriate to unpack an efficient strategy.
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?
Students stand between two different quantities and arrange their arms into a greater-than or less-than symbol
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?
The difference between addition and subtraction is the size of the end results.
A teacher considers using the image shown of a hanger to represent an example of triangles in the real world. Which of the following statements best explains why a hanger is not a good example of a triangle?
The exterior of the hanger is not composed of straight lines
A third-grade teacher asks students to add 57 + 34 on their whiteboards. A student shows their work: *a number line starting at 57, hopping to 87 by +30, hopping to 90 by +3, and hopping to 91 by +1* Which of the following descriptions is an appropriate narrative of the student's mathematical work in solving the equation?
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.
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?
The student's drawing appropriately models the operation of subtraction by finding the difference between the two numbers.
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?
The teacher builds rapport with students and listens for student interests to inform future instruction.
