Lesson 17

The purpose of this warm-up is to draw students’ attention to the number line, which goes from 0–2, and the locations of tossed paper clips relative to the fractions on the number line.  

While students may notice and wonder many things, highlight observations about the locations of the paper clips on the number line—how they are clustered or spread out and how their locations connect to the labeled benchmark fractions.

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “Priya and Kiran were playing a paper clip tossing game that uses fractions and a number line. Priya’s paper clips are blue. The image shows the result of one game.”
• “What could be a rule for this game if this picture represents a win for Priya?” (Sample responses: The winner is the player with:
  • more paper clips greater than 1 
  • the paper clip closest to 2 
  • the most paper clips on the benchmark fractions 0, \(\frac{1}{2}\), 1, \(\frac{3}{2}\), and 2)
• “What if it was a win for Kiran, whose paper clips are black?” (Sample response: The winner is the player with:
  • more paper clips between 0 and 1
  • the paper clip closest to 1
  • the fewest paper clips on the benchmark fractions 0, \(\frac{1}{2}\), 1, \(\frac{3}{2}\), and 2)

In this activity, students use their understanding of benchmark fractions and equivalent fractions to play a game that involves fractions on the number line. They toss paper clips on a game board that is a number line, and then write fractions to label the locations where the paper clips land. In previous lessons, students labeled fractions on a number line with premade tick marks. This activity gives students an opportunity to label them independently. Before playing the game, students prepare the game board and materials.

• Each group of 2 needs 1-inch paper strips and 10–12 paper clips.

• Groups of 2
• Give each group strips of paper, markers, paper clips.
• “Work with your group to play a version of the paper-clip tossing game. To play the game, you will toss some paper clips and use fractions to label where they land.”
• “First, we’ll make the game board. Fold your paper strip in half and then in half again. Carefully tape it down to your workspace (desk or floor can work) and label the benchmark fractions.”
• 5 minutes: partner work time

• “As you’re playing, determine the best spot to stand and how to toss the clips.”
• Monitor for students who:
  • use benchmark fraction strategies and numerical strategies for finding equivalent fractions to label the fractions
  • consider what to do with clips that do not land on the strip
• 5 minutes: game playing time

• Invite previously selected students to share their strategies for tossing and labeling the locations of the paper clips.
• “Where was the best spot to stand? What did you think about when you tossed?”
• “Take a look at all the fractions you recorded on your number line. What do you notice?” (Sample responses:
  • They are ordered from smallest to greatest.
  • We used lots of sixths and eighths.
  • There are lots of different denominators.)
• “What did you do if a paper clip landed in a location where it was difficult to name the fraction?” (I found the closest fraction that was easy to find and used that to help. I used equivalent fractions.) 

• Groups of 2
• “Take a look at the results of your tosses. With your partner, invent a new game.”
• “Think about the goal of the game and decide what someone would need to do to win in a fair game.”
• 1 minute: partner think time
• “What are some things your group needs to think about when you’re making up the rules for the game?”
• 1 minute: whole-class discussion. List should include elements such as:
  • How does someone win the game?
  • Is it possible to have a tie game?
  • How many tosses does a player make?  
  • Does a player toss all their clips at once, one after another, or do players take turns?
  • What happens if a clip lands off the adding machine tape?
  • Do you measure the front, middle, or back of the paperclip?
  • Do you want to limit the possible fractions that can be on your number line (for example, only \(\frac{1}{2}\)s, \(\frac{1}{4}\)s, and \(\frac{1}{8}\)s)?
  • Is there a penalty if a player miscalculates their location?
  • How far are players away from the game board?  
  • Are they standing, crouching, or sitting?
  • Are they in line with or facing the number line when they toss?  
• Distribute new paper strips and masking tape. 

• 10 minutes: partner work time
• Monitor for groups that consider multiple aspects of the game or come up with a unique rule.

• “What do you like about the rules you chose?”
• “What parts of the rules make your game fair to both players?”

• Groups of 4
• “When game makers produce a new game, one of the things they do is test their new product to see if it works and if people like it. This is called doing a ‘field test’.”
• “Very often, they discover the first time they ask people to try what they made, they need to revise it to make it better. That’s what we’re going to do.”

• 10 minutes: play time
• Monitor for groups who:
  • discuss the fairness of a rule
  • suggest strategies for winning or tossing the clips a certain way

• “How did you revise your game to make it better? Did your changes have to do with logistics or mathematics or both?”