The Comparing Game

 

 

• Students use a die and make decisions to create a fraction that will end up larger than their opponent. Students use strategies to place numbers in the numerator, denominator, or the reject boxes.

 

• Students also have an opportunity to engage in conversation with their partner to decide whose fraction is greater and use multiple representations to compare the two fractions.

 

 

The Comparing Game gives students an opportunity to compare fraction sizes while playing a strategic game with a partner. This involves the teacher anticipating a number of things that might occur during the lesson, such as thinking ahead to what students might do correctly as well as areas of struggle. The goal is not to provide students with answers or procedures, but enough support for students to engage in the mathematics inherent in the task. Thinking ahead to how you will navigate through the lesson, such as supporting students who struggle with concepts, is an important consideration in this phase.

 

 

Part 1

 

 

During anticipating, you want to plan for possible solutions students may come up with in relation to a task. The Thinking Through a Lesson Protocol (TTLP) will help guide you in this activity with questions to think more deeply about when implementing a task. You will want to think about possible solutions students might have, methods they may use, misconceptions they may have, and also errors they might make. You can do this by brainstorming possible student approaches and devise strategies for misconceptions ahead of time.

 

Considerations for this CDT:

• When first playing the game, students may not have any strategies for placing numbers at their disposal. It’s okay to let them play a few rounds before expecting them to make intentional decisions.

 

• Students may assume that the “1” should always be discarded because it is a small number. They may not see that “1” in the denominator is a great way to create a large fractional number.

 

• In addition, students may not recognize the magnitude of improper fractions or know how to compare these with other fractions.

 

• Students should be encouraged to use multiple forms to represent and compare fractions (number line, fraction bars, etc.)

 

Resources: Comparing Game Lesson Plan; Comparing Game Rules; Comparing Game Worksheet

 

 

 

Part 2

 

 

The second part of anticipating involves engaging students in the investigation. You will plan ahead of time how you might help students move forward who are having a hard time getting started with the problem or challenging students who seem to find the investigation easy.

 

Considerations for this CDT:

• Before intervening in this investigation, it is best to allow students the opportunity to play the game. Yet, it would also be appropriate to have students brainstorm their strategic thinking before beginning as a comparison in their own reflection.

 

• You will want to keep in mind that students may not be able to easily determine the winner of the game and may not utilize manipulates to help them determine the larger fraction.

 

• Some students might find this game easy. You will want to prepare extensions for the game ahead of time, such as using multiple dice, having variations of the game available for students, and even adapting the comparing game worksheet to have less reject boxes and multiple boxes in the numerator and denominator.

 

 

The goal of monitoring is to learn more about students’ ways and means of operating. You will likely have a variety of student responses and your goal is to challenge each student appropriately. Questions are used to help you address this goal. Monitoring is not about giving correct answers, procedures, or solutions. In order to push students thinking, you need to ask students questions, listen to their thinking, and find ways to extend their thinking.

 

Considerations for this CDT:

• Be sure to utilize questions in the question guide for this investigation. You will find many questions to help you understand students’ reasoning, such as: How do you know you won, what strategies have you used, what types of numbers should the denominator have, the Numerator, what strategies didn’t your opponent use?

 

• For students who are having a hard time determining which fractions are larger, you may need to suggest representations for them to try (such as fraction bars)

 

• For students who find this game easy, you could change the goal of the game (closest to ½, have them omit the reject boxes, or even use multiple dice to create larger fractions)

 

• Keep your eye out for non-traditional and unique strategies to highlight when students are sharing in the connecting phase.

 

Resource: Comparing Game Question Ideas

 

 

 

The goal of connecting is to have students present their work to others to foster mathematical connections among students. You use the information gained from the monitoring phase to help guide this part of the lesson. You will want to know the various ways students are thinking before calling on them to share. Students can use the sharing time to explain struggles they overcame and their transition from concrete to more abstract ways of representing problems. The main outcome from this sharing should be focused on exposing students to various ways the investigation can be addressed, regardless of their current level of operating.

 

Considerations for this CDT:

• You will want to allow students who struggled with various aspects of the game to share first, highlighting the ways they overcame these obstacles.

 

• Have students share strategies their partners used that they thought were effective/ineffective.

 

• Have advanced students share some of the various ways they ended up playing the game and how this changed the strategies they used.

 

• Have students share the multiple ways they determined whose fraction was greater (of smaller).