# Classroom Case Studies, 3-5 Part A: Observing a Case Study (25 minutes)

To begin the exploration of what topics in number and operations look like in a classroom at your grade level, watch a video segment of a teacher who took the Number and Operations course and then adapted the mathematics to her own teaching situation.

In the video segment, Ms. Donnell introduces students to That’s Logical! puzzles, which can be solved by using logic, spatial clues, and number theory clues. Each puzzle consists of a three-by-three grid and a set of clues to help students decide where to place the numbers 1 through 9 on the grid. When the numbers are placed correctly, all the clues are true. Read the information on clue grids and the clues below before watching the video segment.

The Clue Grids
Each clue grid consists of several cells from the puzzle grid. Each cell contains a symbol that tells you something about the digit in that cell. These clue grids may be put in the puzzle grid in any way they’ll fit without turning or flipping. Some clues are fixed — i.e., the clue grids can be fit onto the puzzle in only one way. Other clue grids can be fit onto the puzzle in a few different ways.

The Clues
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The letter E stands for an even number (2, 4, 6, or 8). An E with a slash through it means the number is not even.
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A square stands for a square number (1, 4, or 9). A square with a slash through it means the number is not square.
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A triangle stands for a triangular number (1, 3, or 6). A triangle with a slash through it means the number is not triangular.
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A cube stands for a cubic number (1 or 8). A cube with a slash through it means the number is not cubic.
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The letter P stands for a prime number (2, 3, 5, or 7). A P with a slash through it means the number is not prime.
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A number stands for itself. A number with a slash through it means any number but that number.

Before watching the video segment, you might like to try this sample puzzle and its solution. When viewing the video segment, keep the following questions in mind:
See Note 2 below.

 a. What fundamental ideas (content) about number and operations is the teacher trying to teach? b. What mathematical processes does the teacher expect students to demonstrate? c. How do students demonstrate their knowledge of the intended content? What does the teacher do to elicit student thinking?

Video Segment
In this segment, Ms. Donnell explains the That’s Logical! puzzles and has students try to solve a puzzle with two fixed clues.

You can find this segment on the session video approximately 6 minutes and 56 seconds after the Annenberg Media logo.

Problem A1
Answer questions (a), (b), and (c) above.

Problem A2
At what point(s) in the lesson are the students learning new content?

Problem A3
Discuss the role of manipulatives in Ms. Donnell’s lesson. How do they help deepen the students’ knowledge of the content area?

Problem A4
Ms. Donnell’s lesson was based on Session 6 of this course. Discuss the ways in which her lesson was similar to and different from the one in this course. What adaptations did she make, and why?

### Notes

Note 1
This session uses classroom case studies to examine how students in grades 3-5 think about and work with number and operations. If possible, work on this session with another teacher or a group of teachers. Using your own classroom and the classrooms of fellow teachers as case studies will allow you to make additional observations.

### Solutions

Problem A1
a.
This lesson deals with basic ideas of number theory — for example, understanding specific characteristics that numbers may have, such as being prime or even. The students also need to know factors in order to be able to decide which numbers are square, as well as addition of consecutive numbers in order to find triangular numbers. All this in combination makes them think about the multiple characteristics that numbers may have as well as the relationships between those numbers.
b.
The students are using spatial reasoning, number theory, and logical reasoning. They use spatial reasoning to place the clue in the grid, number theory to identify the characteristics of the clue for a particular square in the grid, and logical reasoning when clues fall on top of one another. Since the puzzles are beginner-level, students can use these processes one at a time as they solve each puzzle. In later puzzles, as the level of difficulty increases, students have to use number theory clues and spatial clues simultaneously to place the clues in the solution grid.
c.
After reviewing the categories (clues) with students and modeling the activity, the teacher has students work in groups to demonstrate their knowledge and understanding of the given material. The nice thing about this activity is that students can often detect on their own when their thinking is off and the clues don’t match, as is the case in this video segment. Then the teacher’s role is to help students review and correct their work.

Problem A2
The students first review some terms and clues that they are relatively familiar with (except perhaps such terms as triangular or square numbers, etc.). Then, as the activity gets more challenging, they need to consider multiple characteristics of each number and compare the numbers with one another. Again, that may be new for some students. Finally, the lesson reinforces spatial reasoning, which is rarely taught in classrooms but is extremely important.

Problem A3
Manipulatives play a key role in this type of lesson. By manipulating and playing with the physical clues, students are able to make visual connections that help enhance their understanding. Also, the grid greatly helps students organize and keep track of their data and, as a result, solve the problem correctly. Doing the lesson strictly in abstract terms without the aid of manipulatives would pose a much greater challenge. Notice, however, that the lesson helps guide the students to gain more familiarity with abstract reasoning (for example, figuring out that if a number is prime and even, it can only be equal to 2).

Problem A4
Many topics in this course were based on number theory. This lesson adapts some of those ideas to a level suitable to Ms. Donnell’s class and presents them in an introductory manner. The use of manipulatives, the teacher’s modeling of the activity, and the presentation of activities that increase in challenge level are all examples of techniques that Ms. Donnell used to adapt the lesson to her classroom.