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 still love it. Out of 80 students, I currently have about 26 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.
For this warm up, I ask students to complete a one minute quick write. For this quick write students write as many fractions equivalent to 1/2 as they can in the one minute. After the one minute, I ask students to stand up if they wrote more than 8,more than 10, then 15, and finally 20.
In this photograph you can see a student writing as many fraction as she can. You can see that she doesn't appear to be going in any pattern.
Students end this warm up by sharing with their partner the fractions they came up with. I ask students how they know they came up with fractions equivalent to 1/2 and listen to their responses. Students respond with statements like, "Well, if you have one whole that is the same size and you have 3 of 6 pieces or 4 of 8 pieces, you will get the same amount, they are just different sized pieces or parts." Some students referred back to the paper folding activity in the previous lesson.
This is the second exploratory lesson in which students will discover equivalent fractions.
The concept of equivalent fractions is needed in many applications involving fractions. Many students of all ages experience difficulties in their attempt to find equivalent fractions. Students either do not know how to find equivalent fractions or do not make the connection between equivalence and size. Other misconceptions might include students applying whole‐ number rules to their work with fractions. Some students also believe that the bigger the denominator, the bigger the piece.
In a previous lesson, students identified equivalent fractions with denominators that were multiples of two and three through a paper folding activity. My students are just beginning to understand the concept of equivalent fractions and I want to ensure, through this lesson, that this concept is solid before I introduce students to the algorithm for creating same numerator or same denominator.
Students have access to fraction bar manipulatives during this lesson. While the cards have fractions equivalent to 1/3 like 3/9 and 3/18 for 1/6, students use strategies they know to determine if these fractions are equivalent. The fraction bar models I have will only help students identify equivalent fractions for fractions up through twelfths. I did not teach my students the number algorithm of multiplying or dividing by one whole to determine equivalent fractions prior to this lesson, which is why this lesson is so powerful for students.
Students play the Fraction Go-Fish game in trios for this lesson. The trios are not grouped by ability. This is important for this lesson because I want students who are struggling or lacking strategies to compare and make equivalent fractions to benefit from hearing their peers thinking.
While students work, I observe and ask students questions like; How do you know these fractions are equivalent? How do you know these fractions aren't equivalent? Tell me more, Explain how you got that, and Can you prove it? These questions allow me to informally assess if students are where they need to be in understanding equivalent fractions. This can be done as I walk around and observe how the students are doing during the game.
In this video, you can listen in and watch a student as he is trying to figure out if he has an equivalent fraction that another student asked for.
Students play the game until there are about 20 minutes left of the class period. This 20 minutes is used for the wrap up section of the lesson and allows me to help students make connections with new learning.
Note: In the resource section, you will see another version of fraction cards that could be used to differentiate instruction.
In order to help students make meaningful connections between what they have experienced in the game with identifying and making equivalent fractions I lead a brief conversation about patterns students notice.
I lead students in a discussion about fractional parts taking up the same amount of space being equivalent fractions. 1/2 and 4/8 take up the same amount of space in a region or bar fraction model.
To end the lesson, I ask students to model 3/4 and 6/8 on their whiteboards. I ask students to discuss with their learning partner whether these two fractions are equivalent.