The Value of Guess and Check

1. The Value of Guess and Check • The Value of Guess and Check, Mathematics Teaching in the Middle School, March 2010, p 392-398 • Providing students a structure for employing the guess-and-check method for solving word problems leads to an understanding of quantitative relationships and builds a bridge to algebra. P 393

2. Connection • NCTM Algebra Standard says that middle grades should enable students to “represent and analyze mathematical situations and structures using algebraic symbols. • Guess and check is a powerful problem-solving strategy that can connect a conceptual understanding of word problems with a symbolic representation. P 398

3. Let’s Begin • Coby is building a rectangular playpen for his dog, Max. He has 48 feet of fencing and plans to make the length of the pen 6 feet longer than the width. What will be the dimensions of Max’s playpen? • Guess and check

4. Structuring Guess and Check • Arrange “guesses” and accompanying work in a table. • Begin • What information is given? • What am I trying to find? • What do I need to know before I can do anything? • What am I going to guess to start with? • Once a guess is made other columns are built.

5. The Table

6. New Questions • What is the relationship between the quantities in the word problem? • Now that I have made a guess, what do I know? • How can I use the results from my previous guess to make a better guess? • How do I know if my next guess should be greater or less than my previous guess?

7. The Table-Another Guess

8. Guess and check builds a bridge between finding a concrete solution to the problem and creating a more abstract equation to represent the problem.

9. The Table-Another Guess and a General Case

10. Getting Started • What information is given? • What am I trying to find? • What do I need to know before I can do anything else? • Create a table • Check the solution • Adjust the next guess by the final outcome.

11. Try this one using a table • A farmer has 12 animals, all chickens and horses. If the animals have a total of 34 legs, how many chickens are there?

12. A Possible Table

13. Looking Back • Emphasis is on sense making over merely applying rote computational strategies. • Guess and check gets students over the hurdle of not knowing what the problem says, what it means or how to get started. • Using a “key word” approach promotes a “plug and chug” solution with limited understanding of underlying relations within the problem. • Bob has three times as many books as Mary. • Is it 3B = M or 3M = B

14. Remember • Arrange guesses • Keep track of guess and other numbers in a table • Use the results from previous guess to make a better next guess • Guess and check is as much about developing mathematical reasoning and problem solving as it is about finding a correct solution.

15. Guess and Check • Builds a bridge between finding a concrete solution to the problem and creating a more abstract equation to represent the problem. • Is a powerful problem-solving strategy that connects a conceptual understanding of world problems with a symbolic representation. • Is simply one technique in the toolbox • Is one strategy, the most important part is to know when to employ the tool and when not to employ the tool.

16. Take Time To • Engage in a frequent analysis of quantitative relationships so as to describe and represent embedded relationships. • Emphasize sense making over merely applying rote computation strategies

18. Another Problem • Katie is thinking of three numbers. The greatest is three times as large as the least, or smallest number. The middle number is 5 more than the smallest number. The sum of the three numbers is 65. What are the three numbers?

20. Some birds are on two branches of a tree. If a bird goes down a branch the numbers are the same, if the bird goes up a level there are twice as many on the top as bottom.