Exploring Equivalent Fractions
Lesson 10 of 14
Objective: SWBAT explore making equivalent fractions.
For this number hook, I give students a riddle rather than a magic trick. I roll five dice and show the dice under the document camera. Then I tell students, "The name of the game is Polar Bears Around the Ice Hole. The name of the game is important! How many polar bears are there?"
The following is an example of how this looked during this riddle:
The first roll produced 4, 6, 1, 3, 2. "Six," said Billy. "No, two," Johnny replied. The next roll was 5, 1, 5, 2, 4. "Four?" said Billy. "No, eight," Johnny said. The next rolls were 3, 5, 3, 1, 2. There were 8 polar bears. The next rolls were 6, 2, 1, 2, 4. There were no polar bears. How does Johnny figure out the number of polar bears?
The answer to this riddle is quite simple, but one that my students have not figured out yet. My students LOVE this riddle and based on previous experience, I predict I will get many requests to play this again, often.
Dice all look the same. On a die, the 1, 3, and 5 all have a dot in the center. The 3 has 2 dots on either side of the center dot, and the 5 has 4 dots around the center dot. Johnny simply counted the number of dots around the outside. A "3" has 2 "petals around the rose, or polar bears around an ice hole." The "5" has 4 "petals" or "polar bears." Roll some dice and it will become clear!!
My students have worked at this number hook now for several days. They are not getting tired of it at all. Out of 80 students, I currently have about 20 that have figured this trick out. They are very good about not telling other students so others still have an opportunity to figure it out on their own. It has been interesting to observe which students have figured this trick out to now. Many of my students who I would say have above average math fact fluency and high accuracy levels with computation, are not figuring this trick out. While my students who may not have their facts memorized at this point, many of which have thoughts about being "bad" at math simply because they don't have facts memorized, have been figuring this trick out. I love watching their faces as they figure it out. It's helped with their confidence tremendously.
In previous lessons, students practiced comparing fractions with number lines, area models, and with the benchmark fraction 1/2. In this lesson students explore equivalent fractions to build towards a future lesson in which students will compare fractions by creating the same numerator or denominator. The common core stresses students have a strong conceptual understanding before applying the algorithm of finding equivalent fractions. When students find the same numerator or denominator, they are applying an algorithm for equivalent fractions.
I begin this lesson by asking students if they would rather have 2/6 or 4/12 of a pizza and to think about why?
Note: I do not expect students to know that these are equivalent fractions, but a few of my students do have this knowledge. I do not provide the answer but rather listen to their ideas. This is meant to be a conversation piece that will be clarified throughout the lesson. I also use this to facilitate a discussion at the end of the lesson during the student wrap up portion of the lesson. This will allow students to change their initial response to the question incorporating what they learn throughout the lesson.
Then I give each student a piece of paper and crayons. I instruct students to fold the paper in half. (I do not specify which way to fold in half and allow students to choose.) Then students use a crayon to shade in half of their paper. Next, I instruct students to fold the paper in half again. Then I have students open their paper.
I ask questions like: How many parts are there now in the whole? (Four, which is the denominator) How many of those four parts are shaded? (Two, which is the numerator). What fraction is shaded? (2/4) What is the relationship /between 1/2, and 2/4? (Looking for equivalence or the same value)
I record the fractions on the board in a number sentence. (1/2 = 2/4)
Then I instruct students to fold the paper back up so that 1/4 is showing. Just as I did before, I instruct students to fold the paper (1/4) in half again. I instruct students to open their paper. Again, I ask questions very similar to the ones above.
How many parts are now in the whole? (Eight, which is the denominator) How many of these eight equal parts are shaded? (Four, which is the numerator) What fraction is shaded? (4/8) Again, I ask what the relationship between the fractions is and I record the fractions on the board for students to see.
Next, I instruct my students to fold their paper up again so they can only see 1/8. I repeat the same process again, having students fold the paper in half and make observations about what is happening to the shaded part (numerator) and the whole piece of paper (denominator.)
Then I repeat the above procedures again, but have students start with another piece of paper and fold it into thirds. I direct students to color in two thirds. I then go through a series of questions about what students notice. I guide students to make connections between patterns they see, which ultimately leads students to discover how to make equivalent fractions by multiplying the numerator and denominator by the same number, which represents a whole or 1.
In this photo you can see a student working on thirds as well as see the paper on the right with the fold marks from coloring 1/2 and making equivalent fractions.
You can see in this student video below, a student who is really trying to make sense of equivalent fractions noticing the pattern that appears between numerators and denominators. This student is explaining how he knows that 1/3 is equivalent to 4/12. I had written on the board 1/3 = 2/6 = 4/12. You can hear in this very short clip, (and see this student's productive struggle) that he notices the pattern of multiplying both numerator and denominator by 2 each time.
You will hear me ask this student how he/she could tell that 1/3 is equivalent to 4/12. Notice the wait time I give this student. While it's not immense, it is definitely a strategy I think about often. Allowing students time to think is critical. I can be guilty of jumping in to quickly to give students answers or hints. Notice in this video a redirected question and then wait time to allow the student to think.