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# Classroom Case Studies, 3-5 Part B: Reasoning About Number and Operations (40 minutes)

In This Part: Exploring Standards
The National Council of Teachers of Mathematics (NCTM, 2000) has identified number and operations as a strand in its Principles and Standards for School Mathematics. In grades pre-K through 12, instructional programs should enable all students to do the following:

• Understand numbers, ways of representing numbers, relationships among numbers, and number systems
• Understand the meaning of operations and how they relate to one another
• Compute fluently and make reasonable estimates

In grades 3-5, students are expected to do the following:
• Describe classes of numbers according to characteristics such as the nature of their factors
• Develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers
• Develop and use strategies to estimate the results of whole-number computations and to judge the reasonableness of such results

“Throughout their study of numbers, students in grades 3-5 should identify classes of numbers and examine their properties. For example, integers that are divisible by 2 are called even numbers, and numbers that are produced by multiplying a number by itself are called square numbers. Students should recognize that different types of numbers have particular characteristics; for example, square numbers have an odd number of factors, and prime numbers have only two factors” (NCTM, 2000, p. 151).

Problem B1

Watch another video segment from Ms. Donnell’s class, and think about how the students are developing an understanding of number and operations.

Video Segment
In this segment, Ms. Donnell prepares the students to create their own puzzles by discussing how to categorize all of the possible numbers.

You can find this segment on the session video approximately 21 minutes and 2 seconds after the Annenberg Media logo.

Problem B2
How does this activity deepen the students’ sense of number?

Problem B3

Problem B4
How would you help students figure out which pair of clues would identify a particular number — for example, 6?

Problem B5
What are some ways that you see the NCTM Standards being incorporated into Ms. Donnell’s lesson?

##### Standards are listed with the permission of the National Council of Teachers of Mathematics (NCTM). NCTM does not endorse the content or validity of these alignments.

In This Part: Examining Students’ Reasoning
Here are scenarios from two different teachers’ classrooms, each involving young children’s developing ideas about number and operations. Snippets of students’ discussions are given for each scenario. For each student, consider the following:

• Understanding or Misunderstanding: What does the statement reveal about the student’s understanding or misunderstanding of number and operations ideas? Which ideas are embedded in the student’s observations?
• Next Instructional Moves: If you were the teacher, how would you respond to this student? What questions might you ask so that the student would ground his or her comments in the context? What further tasks and situations might you present for the student to investigate? See Note 3 below.

Problem B6
Nicole and Photina were working together on a puzzle. Here are the two clues they were trying to put together:

Below is a snippet of their conversation:

Nicole: I think we could put them together in three ways. We could slide the left one over so that the E is below the cube, or slide it over more so that the square is on top of the E, or keep going so that the P is underneath the square.

Photina: I don’t think that they all can work.

Nicole: Well, the first one has to work, because nothing overlaps.

Photina: In the second one, you can have a square number that is even. That’s 4.

Nicole: Then the P is underneath the cube. That’s okay, because 1 is a cube.

Photina: And the third one works, too, because 1 is a prime that is square.

Nicole: I guess now we have to look at another clue.

a. What does this conversation tell you about how the students are thinking about the problem?
b. How would you help them deal with any misconceptions they have?

Problem B7
Shauna and Tony were working together on a puzzle. Here are the two clues they were trying to put together:

Below is a snippet of their conversation:

Shauna: I see that these two pieces could be put together in two ways. You could slide the right one over so that the top P is on top of the triangle, or slide it over one more square so that the bottom P is on top of the E.

Tony: Okay, for the first one, the only overlap is the P on top of the triangle. That works, because 3 is a prime and a triangular number.

Shauna: The second one works, too, because the bottom P is on top of an E, and 2 is an even prime number.

Tony: Are there any more overlaps for that one?

Shauna: I don’t think so.

Tony: I think that the top P is on top of something. Let’s cut it out and try.

Shauna: Yes, that P is on top of an E. But we said that was okay before.

a. What does this conversation tell you about how the students are thinking about the problem?
b. How would you help them deal with any misconceptions they have?

### Notes

Note 3
You may wish to make a two-column chart, labeled “Understanding or Misunderstanding” and “Next Instructional Moves,” to help you organize your thinking for each problem. If you are working in a group, these charts could be the basis for a meaningful discussion on how to assess students’ understanding of number sense and spatial reasoning.

### Solutions

Problem B1
Answers will vary. One idea is to create the following solution grid:

You would then need to consider the characteristics of each of the numbers to create the appropriate clues. Here are some possible clues for this puzzle:

Problem B2
The lesson helps students solidify the knowledge they already had or that they gained during the lesson, such as definitions and various characteristics of numbers. It also helps them develop a better understanding of the relationships that exist between different numbers. This is particularly emphasized in the part of the activity where students have to think “in reverse” and catalog all the different characteristics that make each number unique — that is, distinguishable from other numbers. Lastly, the activity allows students to deepen their flexibility with numbers and to clear up some misconceptions they may have.

Problem B3
Students may be confused about whether 1 is a prime number. They may also need help understanding the meaning of “unique” in the context of giving clues that uniquely describe a number. The teacher can help students decide when the clues they have are sufficient to uniquely describe a particular number.

Problem B4
Students need to think of ways to pair clues to identify particular numbers — for example, the number 6 is even and triangular. They also need to check that their clues uniquely describe a number — in other words, that no other number will fit this description.

Problem B5
Ms. Donnell’s lesson is structured around understanding numbers, ways of representing numbers, and explorations of relationships among those numbers. As students solve and design their own puzzles, they work on describing classes of numbers according to their characteristics, and look for shared characteristics among the numbers. Both skills will help them in the future as they work to understand the complexity of the number system.

Problem B6
a. These students seem to have a good grasp of the number theory and spatial reasoning needed for this problem. They both give correct assessments for almost every possibility. However, they have one mistaken notion: They believe that 1 is a prime number.
b. Believing that 1 is a prime number is a common error, brought on by the idea that a prime number is only divisible by itself and 1. But in the case of the number 1, “itself” is 1! The class should be reminded that prime numbers have exactly two factors. They can then discuss why the number 1 does not fit this criterion, and thus is not a prime number.

Keep in mind, and perhaps suggest to students, that 1 is not a prime number because of a mathematical convention. One is excluded because it makes other processes, such as unique prime factorization, work.

Problem B7
a.
Shauna and Tony correctly assess the first option: The P on top of a triangle will make the number 3. However, they do not realize that the second option will not work. In that case, they will have two different squares that need an even prime. There is only one even prime: 2. They say that the even and prime worked before, not realizing that they cannot use that combination twice in the same grid. It is also possible that Shauna and Tony simply don’t understand the rules of the game (that you can’t use the 2 more than once in a given grid). Unlike Nicole and Photina’s error, this confusion is not mathematical in nature.
b. Shauna and Tony need more instruction about these concepts, and they should think about these concepts more deeply. They need to be sure that they understand how the clues fit together. They should solve more problems of this type. It would help if they made lists of which numbers would fit various sets of two or more clues. They may also need to review the rules of the game.