For this warm up I begin by reading the story Pizza Peril from Perfectly Perilous Math.
In summary, the story is about equivalent fractions. Students must figure out if the character in the story has enough money to buy pizzas. Students must also determine how many pizzas the character in the story should buy. Within the story, characters order whole pizzas and pizza slices. Students must add wholes and parts in order to determine the total amount of whole pizzas needed. Once they determine this they can then calculate to determine if the character has enough money to buy the pizzas.
Listen and watch below as a student displays a misconception about adding the wholes and the parts.
In this video you can see a student explaining a correct answer and one strategy used to determine the amount of pizza needed.
I like this problem because students can find the answer using many different strategies. It also clued me into which students are still struggling with the basic premise of fractions as being part of a whole. I was able to determine which students would benefit from a re-teach session to review such important concepts.
In this lesson, students continue to refine their strategies for adding mixed numbers. They work with a partner to solve two problems in this lesson.
I start by giving each pair of students 2 pieces of paper. Students will use one piece of paper for each problem. Besides solidifying strategies for adding mixed numbers, another goal for this lesson is that students justify their solutions and reasons. Students need to clearly explain the process they used and provide a high-quality justification for their response using appropriate mathematical language and representations. In order to justify their thinking, I ask students to ask themselves these questions:
Students answer the questions on their paper as well.
Next, I hand each partner group the following two problems. You can also access the problems by clicking this link - adding mixed numbers.docx.
Next, students work for about 20 minutes to solve both problems, concentrating on justifying their solutions. I actually set the timer for students to see. I have found this keeps them more on track and focused.
Listen in to this student explaining her thinking about the cider problem.
After the 20 minutes, I ask partners to return to their tables to share and discuss. For this share and discuss time students first exchange papers with their table mates. All students look at the solutions and strategies silently for 1 minute. After the first minute, students may respond and ask questions about the work and strategies of the other group.
In this video you can see and hear the questions a student has about the solution and strategies he's viewing.
When students share and discuss solutions I ask them to think about these things: How are all the strategies the same? How are they different? How do they all make sense? How could you model it if you didn't know how to get started. What are some different ways you decomposed and recomposed the fractions to make computation easier?
It is important that during this wrap up I help students make connections between their strategies and how to add mixed numbers.
Students should be able to demonstrate the concept of adding and subtracting mixed numbers with a variety of visual models and representations as well as strategies that build on and extend whole number addition and subtraction.
Students need to be able to support their reasoning about fractions. They need to focus on using precise mathematical language and constructing viable arguments when writing and speaking (SMP. 3: Construct viable arguments and critique the reasoning of others, SMP. 6: Attend to precision)
Students should be encouraged to use strategies from whole number computation to explore fraction computation. Examples: decomposing and recomposing fractions to make the computation easier or using addition to solve fraction subtraction problems.
During this debrief I list students answers, but then draw out and emphasize the strategies students used to add the numbers. For example, when one student stated that 6- 1/2 quarts is equivalent to 6- 2/4 I ask that student to explain his/her thinking so all students can be informed.