A B C

Solutions for Session 10, Part B

See solutions for Problems: B1 | B2 | B3 | B4 | B5 | B6 | B7

 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.