A B C

Solutions for Session 10, Part A

See solutions for Problems: A1 | A2 | A3 | A4

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.