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!!
Note: My students have worked at this number hook now for five days. They still love it. Out of 80 students, I currently have about half 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. I love watching their faces as they figure it out. It's helped with their confidence tremendously.
Once students know the answer and can explain to me how they know the answer, they are members of the Polar Bear Club. By the end of the year, all fourth graders are members of the Polar Bear Club. I have never had a class where all students did not eventually figure out the riddle. It gives us a good reason to have Klondike Bars in the middle of May when the weather gets warm.
For this warm up I ask students to use a rectangular area model, or fraction bar, to show that 2/3 is equivalent to 8/12.
I am looking for my students to divide the rectangle along the length into three parts. Two parts are shaded, so 2/3 of the rectangle is shaded. To show that 2/3 is equivalent to 8/12, I want my students to show that when the rectangle has 12 equal parts, the shading will cover 8.
They then should be able to divide the rectangle into 12 equal parts. Students should have 3 parts. (Some will also note that 12 = 3 • 4). Students should conclude that since there were three parts that are divided into 4 more parts that the answer is 8/12.
Every fractional amount has many names. The equivalent fraction names for a given amount may make fractions seem a little slippery and difficult to work with, but they are also a great part of the power and versatility of fraction notation. In this lesson, I start by reviewing some of our previous work with fractions by reviewing the meaning of numerator and denominator as well as what equivalent fractions are.
Then I ask students several questions that are key to this lesson's success. I ask students what happens to any number when we multiply it by one? What happens to any number when divided by one? I write some examples on the board like;
456 x 1 = 456
64,256 x 1 = 64,256
Any number x 1 = That number
Are fractions numbers? Yes!
I emphasize these questions and answers throughout the lesson, especially the idea that fractions are numbers and any number multiplied by one results in the same number. This is important for students to remember as they find common denominators in this lesson.
Next, I ask students to name the strategies for comparing fractions that they know thus far. Students respond with using a number line, using an area model, and using the benchmark of 1/2. I tell students that they will learn another strategy for comparing fractions today called, Finding Common Denominators.
I have found in the past, my students need background information about the word "common." I ask them if they can think of something we all have in common. Students respond with things like, "We're all fourth graders! We're all students. We all go to the same school!" I tell them that yes, those are all things we have in common, something that is the SAME about all of us. I tell students that when we find common denominators, we are finding a denominator that two or more fractions have that are the same. This seems to help with their understanding. This allows me to refer back to the word common, or the same, through out the lesson as students make sense of this strategy.
Then I play this learnzillion video that talks about comparing fractions by finding common denominators.
One of the strengths of this video is the review in the beginning of the video. This is one simple way I can reach students that at still struggling with the concepts of numerator and denominator. Students who are struggling benefit from the review and students who are solid with understanding also benefit from hearing the academic language in order to use fraction specific vocabulary.
As the video is playing, I stop it several times to help my students make sense of what the video is expressing. The first place I stop the video is when the narrator says that the first step in finding common denominator is to find the least common multiple of all three denominators. This is very confusing language for students. I break the phrase down into smaller more manageable chunks by first asking students what the narrator means when she uses the word multiple. I briefly review with students what a multiple is. Then I break it down even further by asking what they think "least common multiple" means. Many students have no ideas about what it could mean, while others have vague ideas. I then show students a list of multiples for the fraction in the video. I tell them that the "least common multiple" is the multiple that appears for all of the denominators FIRST. I tell students that the denominators may have other common multiples, but least common multiple refers the the common multiple that is the lowest. Finding least and greatest common multiples is a sixth grade common core standards, and I do not stress the language in fourth grade for this reason. I want my students to understand that finding a common denominator or numerator is one way to compare fractions and know and be able to do that. I know their understanding of this will deepen over time in fifth grade when students multiply fractions and in sixth grade when finding the greatest common factor and least common multiple.
After the video I have my students write 5/8 and 6/9 in their math notebooks. Together, I lead the students through this example of how to find a common denominator. We do the steps together very similar to the video. This example helps me re-stress the steps and procedures as well as discuss the importance of why multiplying by one whole works to create an equivalent fraction.
After this example I display the birthday cake problem - birthday-fractions-4nf2.
Students work with their learning partner to solve the birthday cake problem, however, each students is responsible for completing and turning in a solution. As students work, I circulate around the room, guiding and questions students thinking. This problem definitely provides some productive struggle as students wrestle with the two different kinds of cake, the two different fractions, and comparing those fractions with a newly learned skill. I use the birthday cake problem to group students for re-teaching opportunities. Students who may be able to model the problem, but not solve using numbers or common numerators/denominators, will be given opportunities to work more intensely with this skill in order to be proficient.
In this video, you can hear a student explaining how she knows she has found a common denominator.
For this lesson wrap up, I ask student groups to share their findings. I only build in 10 minutes for this share, but it is a very important 10 minutes. As you can see in this student video, I invite student groups to come up to the document camera and ask the pair to report their ideas to the class. This 10 minute time period only allows for about 3 groups to share, but as groups are working, I am circulating around the classroom. As I move around the classroom, I look for student groups that show different methods and strategies and ask them to share during the share time. This is an important step in students sharing work. In order to help progress all students' learning, I need to ensure that I help facilitate this sharing by strategically picking which groups I want to share. When my students see a variety of strategies and solutions, they have more opportunities to be successful. For example, some students could see that 4/8 was the same as 1/2, so when they compared fractions and made a common denominator of 10 while other students used 4/8 and 3/5 and made a common denominator of 40. These different strategies help my students make connections and strengthens their flexible thinking about numbers.