Illuminations: Seeing Double

# Seeing Double

 In this lesson, students encounter the magical effect of reflection as they use a mirror to visually experience the concept of doubling quantities. This experience focuses student’s attention to using doubles as a strategy to make number operations easier, specifically addition. Literature is used as an introduction to provide a connection and motivation, a game is used for discovery and practice, and writing is used as closure to this lesson.

### Learning Objectives

 Students will: write equations to represent doubles create a pictorial representation of an equation with doubles verbally define the word “double” as being made up of two identical parts or groups

### Materials

 Two of Everything, by Lily Toy Hong Handheld mirrors Dice Plastic counters Seeing Double Activity Sheet Doubling Pot Activity Sheet Our Class Doubling Book Cover Sheet

### Instructional Plan

Begin this lesson by reading the book, Two of Everything. This is a Chinese folktale about a hardworking, modest couple, Mr. and Mrs. Haktak, who discover a magical pot. When something is put into the pot, it is automatically doubled. This book creates a common experience about the concept of “doubling” to which the entire class can relate. After the book is complete initiate a conversation about the book with the class by asking some of the following questions:

• What was special about the pot that Mr.Haktak found?

[Everything he put into the pot was doubled.]

• What happened when Mr. Haktak fell into the pot?

[Two Mr. Haktaks came out of the pot.]

• What happened when Mr. and Mrs. Haktak put 5 coins in the pot?

[Ten coins came out.]

After the students have confirmed an understanding of the book, write equations to show what happened to Mr. Haktak and the coins Write 1+1=2 to represent Mr. Haktak being doubled, and write 5+5=10 to represent the coins being doubled.

To begin the main activity, sit in a circle with the students. Explain that you have a very special “doubling machine” (mirror) that is going to show them how different things are doubled. Hold up the “doubling machine” next to a student and ask the class to explain what happened. [The student was doubled and now you can see two students. Now ask how they could write that as an equation. [1+1=2.] Repeat this with different objects around the room.

Have the students remain in a circle, and ask them if they would like to use their own “doubling machine.” Explain that they are going to play a game and use the “doubling machines” to help them. Create a larger version of the Seeing Double activity sheet and model how to play the game as you sit at the head of the circle and the students watch.

1. roll one number cube
2. place the number rolled in the square on the Enlarged Seeing Double Activity Sheet
3. select that number of plastic chips and place them on the desk
4. have your partner hold up the doubling machine behind the chips so that you can see the reflection
5. count all of the chips (the real ones, plus the reflected ones)
6. write an equation

For instance, let’s say you roll a 4. Then place 4 chips on the desk. With students watching, place the mirror behind the chips so that the reflection of all 4 chips is visible. Then count all 8 chips. Finally, write an equation that represents this situation: 4 + 4 = 8.

Have students follow these same six steps, and then switch roles with their partner.
Divide the students into pairs and give each pair a number cube, plastic chips, and a handheld mirror. In addition, give each student in the pair a Seeing Double activity sheet. Let the students play this game until they have written four equations. If students finish early they may turn their paper over and continue rolling the number cube and recording equations.

Closure
When the class is finished, have them clean up their game and again sit in a circle. In the circle, ask students to imagine that they have a doubling pot. What would they put inside? Record any ideas that they may have on chart paper. Explain that they are going to be authors of their own book called Our Class Doubling Pot. Instruct the students to choose something they would like to put in their pot--it can be from the class generated list or one of their own ideas. Then have each student select one of the four equations from their Seeing Double activity sheet and circle it. They will use that equation as a basis for what they are going to write on their page of the book. Give each student a Doubling Pot activity sheet to create their page for the class book. They will write their equation and create an illustration to represent it. Print a copy of the Our Class Doubling Book cover sheet for the class book, and write all students’ names on the cover. To compile the class book, place all of the student’s pages underneath the cover and bind the book by stapling along the left side, or by punching three holes down the left side and securing it with ribbon or yarn.

Student examples:

### Questions for Students

 What does it mean to be a double? [Two equal sets or groups.] When playing the game, how did you get each addend and the sum? [The first addend was the number of chips outside of the doubling machine (the real chips), the second was the number of chips inside the doubling machine (the reflections), and the sum was both sets added together.] Is 6+7 a double? Why? [No, because 6 and 7 are not the same, so they can not be a double.] If you put 10 coins in the doubling pot, how many would come out of the doubling pot? [Twenty, because 10+10=20.] If 8 coins came out of the pot, how many did you put into the pot? [Four, because 4 + 4 = 8.] Could you give me an example of other doubles? [Answers will vary.]

### Assessment Options

 The students’ Seeing Double Activity Sheet and page in the book can be used to assess the student’s understanding and ability to write equations to represent doubles and create a pictorial representation of an equation with doubles. Throughout the lesson, circulate, observe and question the students as they play the Seeing Double game. Use the questions from Questions for Students to assess students’ understanding. An individual assessment can also be done to assess a student’s understanding of the concept of doubles. Place a pile of 4 chips and a pile of 3 chips on an overhead projector so that all students can see them. Then ask the students to write on a piece of paper if those chips show a double. If not, have the student draw or write what would have to be changed in order for it to become a double.

### Extensions

 As an extension to this lesson, patterns and algebra can be explored by changing the rules of the “magic pot”. Instead of doubling, everything that is put into the pot could increase by 3. Have students play the game again, but this time they must write the equation for a number increasing by 3 instead of a number being doubled. Challenge students to create their own rules for the magic pot, and have other students play the game based on their rules. Make a list of all of the equations with doubles that were generated from playing the Seeing Double game. Then ask the students to find patterns that emerge during doubling. [They are all even; The sum of consecutive doubles increases by 2 each time] Then create the corresponding subtraction equations represented by these doubles: 2-1=1 4-2=2 6-3=3 As an introduction to this lesson students can practice basic addition facts and number recognition by using the Concentration and How Many under the Shell tools. The Ten Frame and The Abacus tools could be used by the teacher or the student to represent doubles in a different way.

### Teacher Reflection

 Are the students able to relate the concept of doubles to events, experiences or people in their own life? What additional activities could be done to help the struggling learner master the concept of doubles? Did the students have the proper background knowledge to be successful at this lesson? What concepts could have been introduced or reviewed prior to this lesson? What activities could be given as an extension of this lesson for a student to do with their parents?

### NCTM Standards and Expectations

 Algebra Pre-K-2Use concrete, pictorial, and verbal representations to develop an understanding of invented and conventional symbolic notations. Model situations that involve the addition and subtraction of whole numbers, using objects, pictures, and symbols. Number & Operations Pre-K-2Connect number words and numerals to the quantities they represent, using various physical models and representations. Develop fluency with basic number combinations for addition and subtraction. Use a variety of methods and tools to compute, including objects, mental computation, estimation, paper and pencil, and calculators.

### References

 Bachman, Vicki. First Grade Math. Sausalito, CA: Math Solutions, 2003.Brown, Clara Lee, Jo Ann Cady and Thomas E. Hodges. “Supporting Language Learners.” Teaching Children Mathematics, April 2010.
 This lesson was prepared by Jamie Piecora as part of the Illuminations Summer Institute.

1 period

### NCTM Resources

 Principles and Standards for School Mathematics (Book and E-Standards CD)

 More and Better Mathematics for All Students
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