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.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discutan con su pareja lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “Priya y Kiran lanzaron clips en un juego en el que usaban fracciones y una recta numérica. Los clips de Priya son azules. La imagen muestra el final del juego” // “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.”
• “¿Cuál podría ser una regla de este juego si la imagen muestra que Priya ganó?” // “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)
• “¿Cuál sería una regla si Kiran, quien usó los clips negros, hubiera ganado?” // “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.
• “En grupo, jueguen una versión del juego de lanzar clips. Para jugar el juego, van a lanzar algunos clips y a usar fracciones para marcar en dónde caen” // “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.”
• “Primero, haremos el tablero del juego. Doblen su tira de papel por la mitad y después otra vez por la mitad. Con cuidado, péguenla en su espacio de trabajo (como el pupitre o el piso) y marquen las fracciones de referencia” // “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

• “Mientras juegan, identifiquen el mejor lugar para pararse y cómo lanzar los clips” // “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.
• “¿Cuál fue el mejor lugar para pararse? ¿En qué pensaron mientras lanzaban los clips?” // “Where was the best spot to stand? What did you think about when you tossed?”
• “Examinen todas las fracciones que anotaron en su recta numérica. ¿Qué observan?” // “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.)
• “¿Qué hicieron cuando un clip cayó en un lugar donde era difícil escribir la fracción?” // “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
• “Revisen los resultados de sus lanzamientos. Con su pareja, invéntense un juego nuevo” // “Take a look at the results of your tosses. With your partner, invent a new game.”
• “Piensen en lo quieren que sea la meta del juego y decidan lo que alguien tendría que hacer para ganar” // “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
• “¿Qué cosas tienen que considerar en su grupo para decidir las reglas del juego?” // “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.

• “¿Qué les gusta de las reglas que inventaron?” // “What do you like about the rules you chose?”
• “¿Qué partes de sus reglas hacen que su juego sea justo para los dos jugadores?” // “What parts of the rules make your game fair to both players?”

The purpose of this activity is for students to share their games with their classmates. This provides pairs the opportunity to articulate their rules and check to see if they are clear to their audience. Based on feedback, they will be able to revise their directions.

• Groups of 4
• “Cuando los creadores de juegos producen un juego nuevo, una de las cosas que hacen es probar su producto nuevo para ver si funciona y si a la gente le gusta. Esto se llama hacer una ‘prueba de campo’” // “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’.”
• “Con mucha frecuencia, ellos descubren que tienen que hacer ajustes de mejora cuando le piden a personas que prueben lo que crearon. Eso es lo que vamos a hacer” // “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

• “¿Cómo ajustaron el juego para mejorarlo? ¿Sus cambios fueron sobre logística o sobre matemáticas o sobre las dos?” // “How did you revise your game to make it better? Did your changes have to do with logistics or mathematics or both?”