Problem-Solving Perseverance - Pre Algebra

1. What should a problem solver do when a chosen strategy stops making progress?

   Switch to a different strategy, such as drawing a diagram, making a table or trying a simpler case.

2. Using an organized list, how many handshakes occur when each of 5 people shakes hands once with every other person?

   10 handshakes: $4 + 3 + 2 + 1 = 10$.

3. A number is doubled, then 5 is added, and the result is 29. Working backward, what is the number?

   12. Undo the steps in reverse: $29 - 5 = 24$, then $24 \div 2 = 12$.

4. Why should intermediate results be checked during a multi-step problem?

   An error in an early step carries into every later step, so catching it early saves the whole solution.

5. Keychains sell for 3 dollars and bookmarks for 2 dollars. Exactly 100 items are sold for at least 240 dollars. What is the fewest keychains?

   40. With $k$ keychains, $3k + 2(100 - k) \ge 240$ gives $k + 200 \ge 240$.

6. Sam is 3 times as old as Ana. In 5 years, their ages will add to 38. How old is Ana now?

   7. With Ana at $a$, $(a + 5) + (3a + 5) = 38$, so $4a = 28$.

7. What does the strategy "solve a simpler problem" mean?

   Try smaller numbers or fewer cases to find a method, then apply that method to the original problem.

8. In the four-step process (understand, plan, solve, look back), what happens in the look back step?

   You check that the answer is reasonable and actually answers the question asked.

9. How should each wrong guess be used in the guess and check strategy?

   To adjust the next guess: if a guess gives too large a result, try a smaller one, and the reverse.

10. Two consecutive integers add to 47. What are they?

    23 and 24. Solve $n + (n + 1) = 47$ to get $n = 23$.

11. A snail climbs 3 ft up a 10 ft well each day and slides back 2 ft each night. On which day does it reach the top?

    Day 8. After 7 days and nights it is 7 ft up, and on day 8 it climbs 3 ft to reach 10 ft.

12. When is the strategy "work backward" most useful?

    When the final result is known and you need to find the starting value.

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