I tell students we have all solved different problems involving fractions and verbally explained our work. Today we will learn how to write explanations to support our answers.
To pull students deeper into the lesson, I began to talk about a situation that happened during my twelfth birthday party. I asked my mom for a pair of roller skates and she told me “no”. I remember asking my mom why… I go on to explain that justifying is a way that helps us understand why things are they way they are, especially when we are learning.
I ask a couple of volunteers to tell of a similar situation. Many students think that the explanation just is not enough. However, I explain that in math an explanation can help teachers determine if you know the skill or not.
We will be working on the following Mathematical Practices in this lesson:
MP.2. Reason abstractly and quantitatively.
MP.4. Model with mathematics.
MP.7. Look for and make use of structure.
MP.8. Look for and express regularity in repeated reasoning.
During this part of the lesson I really want to focus on how I will demonstrate the use of problem-solving. I want students to know why and how they solve problems and why it is necessary to ask questions in order to understand the content being taught.
To began, I focus more on moving students towards the intended outcome of this lesson. I draw an unequal fraction on the board. I ask students to explain if this picture represents the fraction 2/5.
Discussion: This particular model is suppose to represent 2/5, however, the sections are not divided equally. I ask students does this fraction represent 2/5. I ask them to think of why, or why not as they determine what their explanation is going to be.
Bring students deeper into the lesson: When students are comfortable with explaining their answers, I place more fractions on the board and ask them to work along with me explaining how and why the given fraction models are correct or incorrect.
For instance, I place a fraction on the board, and ask students to explain why this fraction model is not 2/4.
Students should respond by saying the fraction model is not divided into 4 equal parts, so the shaded part cannot be 2/4. If students are not able to respond I will repeat this step and continue to practice with them until a level of understanding is reached.
How can I make it better
After students respond to the given question, I ask students can this response be improved. If so, how can it be improved? Possible answers may be it is not detailed enough. As students respond I make notes of these problem areas and ask questions to bring out the correct problem solving techniques. I ask can the sentences be more detailed. Can you explain step by step what you did to come to this conclusion? Can you draw an illustration of the correct way 2/4 can be illustrated. Then we work together rewriting a better response.
Student response: No. The parts are not equal.
Revised response: The fraction model is not divided into four equal parts; therefore this fraction does not represent 2/4.
Some students may come up with other ways to explain their answers, and that is ok, as long as they can justify how they got their answers. Make note of the students who seem to be having difficulty, and work with them in a smaller group on how to problem-solving. Working in a small group allows me to understand where the students are in their learning. I use their own talk, to connect them to the information they need in order to grow.
I move students into groups of three. I give each group a word problem to solve. I explain that we will solve these problems in the exact same way we did during the modeling stage.
Make sure you illustrate your own visual model to explain your answers. As students are working, I circle the room to assist as needed.
Checking for Understanding:
What do you know so far? Can you support your solution? (Some students still have a hard time illustrating equal parts, so I give them strips of graph paper.) Again, I only assist when needed and I take anecdotal notes as students are working to help me better understand what they have learned so far. graph paper.pdf
I allow student about fifteen minutes or so to work on the given problem.
In this portion of the lesson I ask students to return to their assigned seats. I give each student a set of questions to solve. I encourage them to write out a summary of how they solve each answer.
In writing it is important for you to edit your writing. When you all complete your explanations, I want you to go back over your writing to see if can be stated better mathematically.
How did you get your answer?
Did anyone solve differently?
Can you demonstrate?
Does this make sense?
Most student rely on their writing to help them explain their method, however, their responses are a bit vague. We will continue working on word problems throughout the year.