Unit title*

 Equivalent Fractions – “The Black Whole”

Subject area*

 Mathematics

Secondary subject

Grade*

 3^rd or 4th

Duration/time*

 Lesson 1- 40 minutes
 Lesson 2 – 20 minutes
 Lesson 3 – 25 minutes
 Lesson 4 – 30 minutes

Overview*

Third graders often have problems understanding equivalent fractions.  Using the TI-15 calculator and the game “The Black Whole," students experience both hands on and visual practice.  I tried this on summer school students and found they quickly figured it out and were independently discussing equivalent fractions with each other.  They also quickly noticed what happened when different fractions were added together on the calculators.  Directions for the game "The Black Whole" are below. Be sure and look for the variation in Lesson 4)

Technology overview*

click on level for description

Guiding questions*

What happens when we cut strips into different but equal sized pieces?  Which are the biggest?  Which are the smallest?  What happens if we put different sized pieces together? Can we make equal amounts using different sized pieces?  What happens if we add different sized pieces together? (Ask this one only if you are using the TI-15.)<![endif]>

Content standards and benchmarks*

(Mastery)

MA.IV.I.1  Develop an understanding of whole numbers; read, write, and count using whole numbers; investigate basic concepts of fractions and decimals.

MA.IV.2.1  Represent whole numbers, fractions, and decimals using concrete, pictorial and symbolic representations.

MA.IV.2.2  Explore and recognize different representations for the same number and explain why they are the same.

Content standards and benchmarks*

(Secondary)

Technology standards and benchmarks*

(Mastery)

3-5.1.1   Use keyboards and other common input and output devices efficiently and effectively.

3-5.4.3   Use general purpose productivity tools and peripherals to support person productivity, remediate skill deficits, and facilitate learning throughout the curriculum.

3-5.6.8  Use technology resources (e.g. calculators, data collection probes, videos, educational software) for problem solving, self-directed learning, and extended learning activities.

Technology standards and benchmarks*

(Secondary)

Resources

*1 set of paper strips for each student  (Strips are the same size and in the colors, black, orange, blue, green, and yellow. 12 X 2 inch strips work well.)

*1 die for each student (Die should be marked with ½, ¼, 1/8, 1/16, 2/8, 2/16.)  If you don’t have blank dice, put sticky circles on regular die and write the fractions on each side.

*1 marker or crayon per child

*1 scissors for each 2 students

*1 TI –15 calculator for each student  

Check out these two websites:

 http://wneo.org/hotlists/msandelemmath.htm

       (for other math/technology activities)

  http://www.ti.com/calc/docs/guides.htm
        (for information about TI-15 calculators)

Title of Lessons/Activities

Lesson 1 - 40 minutes

Take the black strip and tell the students it is 1 black whole.

Take the orange strip and tell students to fold it in half.  Students should mark each piece ½, then cut on the fold line.

Repeat for the other strips, marking and cutting the blue into fourths, green into eighths, and yellow into sixteenths.

Have each pair of students take a die.  The first student rolls the die, then picks a fraction strip that matches the fraction on the die and places it on his/her black strip.

The next student takes his/her turn.

Students continue alternating turns, trying to cover their black strip exactly.  The first one to do this, to make a black whole, wins the game.

Lesson 2 - 20 minutes

Give each student a TI-15 calculator.

Have students turn on the calculator and practice putting in and adding mixed fractions.  Once they are comfortable with that, have them add mixed fractions until they show a number over 1.  Practice subtracting the fraction that put them over 1.  For example:  If the calculator shows 7/8 and they add ½, the calculator will show 1 3/8.  Be sure they are able to then subtract the ½ so they can get back to where they were. This is a good time to discuss what is happening so you are sure students understand.

Lesson 3 - 25 minutes

Have students get out the strips and die from lesson 1 and the calculators from lesson 2.

Have them play the game but this time also add the fractions on the calculators.  Each time a student adds a fraction on the calculator, she shows it to her partner.  Later on, when they had free math time, students played the game with the calculators and die and did not even use the colored strips.

Lesson 4 - 30 minutes

This is my variation of the game and I like it a lot more than the basic game. 

Once the students are familiar with the game, I had them take the black whole, one ½ strip, two ¼ pieces, four 1/8 pieces, and eight 1/16 pieces.  The rest of the pieces were put away.  As the students played the game this time, they soon ran out of pieces and had to start “trading” pieces in order to cover the black whole.  It was interesting to listen to them discuss what to do.  For example, I overheard one twosome discuss what to do when one of them needed ¼ and had already used both of his fourth pieces.  His opponent suggested he use four of his 1/16 pieces.  He responded that he’d rather use two of his 1/8 pieces because it was easier.  The first boy agreed and they continued to play.

 Equivalent Fractions –“The Black Whole”  a game from Catherine Carlson at othercinsf@aol.com  
Variation from Linda Curran   lcurran@dsisd.k12.mi.us

Example student materials

Assessment*

 Using their fraction pieces, ask students to show you as many ways of making ½ as they can. Repeat for ¼ and 1/8 if needed. 

Reflection

    When summer school students beg to play the game and are able to verbalize what equivalent fractions are without teacher assistance, I think the lessons were successful.
    The great thing about TI-15s is the fraction function.  My students knew how to use TI-15 calculators before doing this activity.  To teach them, I spent about 20 minutes with them, doing some activities from the manual.  This website will tell you more about TI-15s:  http://www.ti.com/calc/docs/guides.htm

Home activities

 Students can easily make the game themselves if the teacher gives them sticky circles to put on their die as well as a list of the fractions to write on the paper strips and die.

Credits

Catherine Carlson taught me the game “The Black Whole," a game adapted from an activity in Family Math from the Lawrence Hall of Science, University of California, Berkeley,California.
   Submitted by Linda Curran, third grade, Lakeside Elementary School, Manistique, Michigan         lcurran@dsisd.k12.mi.us