An important skill that you will develop throughout this course is making sense of a problem or situation. You will be asked to think and talk through challenging problems until they make sense to you. You will know that an idea makes sense when you understand it so well that you can explain it to others and answer their questions about it. In this lesson, you will make sense of a challenging logic problem and work with your team to explain your ideas.

1-33. TRAIL MIX

Rowena and Polly were making trail mix. Rowena had 4 cups of raisins, and Polly had 4 cups of peanuts. Polly poured exactly one cup of her peanuts into Rowena’s raisins and stirred them up, as shown in the diagram at right. Then Rowena poured exactly one cup of her new peanut-and-raisin mix back into Polly’s peanuts.

Did Rowena get more of Polly’s peanuts, or did Polly get more of Rowena’s raisins?

Your task: First decide by yourself what you think the answer to this question is. Then share your ideas with your team.Together make a guess (also called a conjecture) about which girl got more of the other’s snack item. Explain your conjecture with words, numbers and symbols, diagrams, models, or anything else you think will convince another student.

1-34. Rowena and Polly still cannot agree about who has more of the other’s item. Rowena is still sure that Polly got more of her raisins, and Polly is sure that Rowena got more of her peanuts. In order to make sense of what happened, they decided to try a simpler experiment.

Rowena got a cup of 10 red beans, and Polly got a cup of 10 white beans. Polly gave 3 white beans to Rowena, and Rowena stirred them into her red ones. Then she closed her eyes and chose 3 beans from her mixture at random to give back to Polly. The girls then examined each cup.

  1. Try their experiment a few times with a partner. What happens each time? Work with your team to find a way to explain why your results make sense.
  2. Would you have gotten similar results if you had exchanged 5 beans? 6 beans? 20 beans? Be ready to explain your thinking.
  3. With your team, consider whether your ideas about Rowena’s raisins and Polly’s peanuts have changed. If so, write and explain your new conjecture. If not, explain why you still agree with your original conjecture. Be sure to include anything you think will be convincing as you write down your ideas. Be prepared to share your ideas with the class.

1-35. LEARNING LOG

In this course, you will often be asked to reflect about your learning in a Learning Log. Writing about your understanding will help you pull together ideas, develop new ways to describe mathematical ideas, and recognize gaps in your understanding. Your teacher will tell you where your Learning Log entries should go.For your first entry, you will consider the process by which you worked with your team and your class to make sense of “Trail Mix” (problem133). Write a reflection in your Learning Log that addresses the following questions:

Title thisentry “Making Senseof a Challenging Problem” and label it with today’s date.

1) What did people say or what questions did they ask that helped you to make sense of this problem?

2) What did you say or what questions did you ask that helped you to make sense of this problem?

3) What would you advise another student to do to make sense of this problem?

Conjecture and Justify

A conjecture is a statement that appears to be true. It is an educated guess.

To justify a conjecture is to give reasons why your conjecture makes sense. In this course you will justify conjectures by using observations of a pattern, an algebraic validation, or some other logical method.

1-36.Bob and Mark decided to try the peanut and raisin investigation at home. Bob started with 10 peanuts on his tray, and Mark started with 10 raisins on his tray, as shown in the diagram below.

  1. Mark gave 3 raisins to Bob, as shown in the diagram below. How does the number of peanuts and raisins on Bob’s tray compare to the number of peanuts and raisins on Mark’s tray?
  1. Copy the diagram below. Circle a group of any 3 peanuts and raisins on Bob’s tray. Besure to circle some of each for a total of 3. Then give them to Mark. Does Bob now have more of Mark’s raisins, or does Mark have more of Bob’s peanuts?
  2. Now start from the beginning and repeat parts (a) and (b) assuming that 8 (instead of 3) snack items are handed from one student to the other. Explain your results.

1-37. Eli walked 12 feet down the hall of his house to get to the door. He continued in a straight line out the door and across the yard to the mailbox, a distance of 32feet. He came straight back across the yard 14 feet and stopped to pet his dog.

  1. Draw a diagram of Eli’s walking pattern.
  2. How far has he walked?
  3. How far from the house is he now?

1-38.Nadine and Diondra were working together to divide a circle into three equal parts. They came up with the diagrams shown below.Tanisha said, “One of these pictures is wrong.”

What do you think? Is one picture incorrect?If so, which one? Why?

1-39.Find apattern in each number sequence below. Then use your pattern to generate the next five numbers in the sequence. Explain the pattern.

  1. 2, 5, 3, 6, 4, ____, ____, ____, ____, ____
  2. 100, 99, 97, 94, 90, ____, ____, ____, ____, ____

1-40.Roundeach number to the specified place.

  1. 33.54296 (ten thousandth)
  2. 307,407 (thousand)
  3. 285.39154 (hundredth)
  4. 6811.09 (ten)