I like to help my students make connections between mathematical concepts and their own lives. Today, I'm using a math prompt to encourage multiple different student responses.
"I encountered fractions this weekend when..."
Students spend 5 quiet minutes working in their math journals, while I circulate to check in on their progress. After 2-3 minutes if I see students who are struggling with getting ideas on paper, I ask them a few prompting questions about their weekend, rephrasing what they say into fraction examples to help them generate some ideas.
At the end of 5 minutes, I ask students to move to the carpet, where we have a short meeting to share the purpose and goals of the lesson.
To open the math meeting, students share their journal entries about a weekend encounter with fractions.
Then, I address the students about the purpose of this lesson. The lesson is designed to help you improve your communication about fractions. I have collected evidence from conferences and your work (a variety of informal assessments) that demonstrate why we need a "back to basics" day.
In an open conversation, I share with the students that today will be a back to the basics of fractions day. I share with them that through conferences and listening and recording their small group work, I learned that when communicating with one another, their language is vague and often inaccurate. Before we are able to move forward to learn more about fractions, some basic ideas need to be reviewed.
Today, the focus of the lesson will be around the first points.
I use an example of the students' dialogue about fractions to start the guided practice of the lesson. It is important for students to understand that the data we collect as teachers means more than a report card grade. I tell them, "statements like these tell me that you are not understanding the roles different parts of a fraction play, you are not considering the numerator and denominator when you read fractions, or you are - and you are not communicating it clearly. Either way, I need to spend more time working with you to help strengthen your basic fraction number sense and how you communicate that knowledge precisely (MP6).
This sentence resembles what I hear when you are talking to one another about fractions. 2/3 and 5/6 are equivalent because they are both 1 away from the whole.
I ask students:
Do you agree with this statement? Why? Why not?
Students talk in their groups and then share their thinking. I draw two number lines to support their ideas. It is my goal to help students state that 2/3 is one larger piece away from 3/3 and 5/6 is one piece that is smaller than the third from the 2/3 (1/2 that size) away from 6/6.
The students have been looking at the numerator without considering the denominator. I spend time reviewing these terms and discussing their role. Then compare 2 more fractions using the think-aloud model to make my expectations about precision clear. This time I choose two fractions with denominators are more dramatically different (4ths and 10ths) and add fraction titles to help those who were struggling with the first comparison.
Based on the student responses and limited participation I choose to focus on just one fraction for my 3rd example.
I ask students to place 6/8 on a number line, while they were working, I circulate around the room. At first, only one student made 8 spaces on his paper and labeled it 6/8. The other students "estimated" where it would be.
I encourage students to be more precise and prove that they know where 6/8 is on the number line. They use this prompts, as well as the model one student puts on the board to make 8 sections and mark the 6th.
I ask the students to talk in their groups about 6/8. Is it closer to 1/2 or a whole and how to you know?
Students debate about this estimate because "6/8 is directly between 4/8 and 8/8" As I circulate this time, I check student dialogue to see if they are considering the size of the pieces (denominator) in their thinking.
Correct example: "4/8 is 2 eights less than 6/8. And 6/8 is 2 eights from a whole"
Incorrect example: "4/8 is 2 away from both".
Students whose dialogue has not progressed meet with me for more gradual release during the independent practice. Those who are ready are provided with an opportunity to practice.
Students work in heterogenous groups (shape groups) to practice placing more fractions on a number line.
They work with 5/6, 3/7, 8/15 and 9/10.
At this time, students are expected to create a number line, place the fraction, then determine if it is closest to 0, 1/2. or 1 and explain their reasoning, verbally and then in writing.
As you can see in the video, students learn so much from talking about their math with peers. This student demonstrates how talking through an idea can be revised.
I choose to add in 3/7 and 8/15 as fractions to discuss, because the challenge of considering 3.5/7 and then talking about it allows for students to have a rich discussion; complete with questions and math arguments.
Students are provided with 10 minutes at the end of class to revisit the game fraction war. Today, they are to determine the larger fraction based on number sense, and explain their thinking using the precise language we practiced today.
Students are reminded that the numerator and denominator must both be considered when thinking about a fractions position compared to 0, 1/2, 1 or another fraction.
I circulate while students play to listen for a more precise dialogue between groups of students.