# Mango Problem

Team Members and grade level they teach:

Standard(s):
• 7th : Rational numbers, conversions with fractions and decimals
• Proportional reasoning
• Understanding part to whole relationships

Lesson Study Title: The Mangoes Problem

Overview of the the lesson and how it connects to STEAM: (Please attach the task & blackline masters)
Uses 21st Century Skills of Critical Thinking, Creativity, Collaboration and Communication
• “Can You See It” Warm-Up Worksheet
• Mangoes Problem Worksheet

Anticipated student strategies: (BOARD PLAN with Sequencing and connections)
• Work backwards from a picture starting with 4
• Represent each fraction separately
• Find a common denominator and add them
• Multiply denominators
• Pictures
Formative Assessment strategies: (Rubric/checklist)

Understanding- What did you notice about patterns and how to approach the problem? Were you able to relate the problem to something in your background or any prior knowledge that you may have?

Strategy- Was a mathematical strategy chosen and applied correctly? Discuss strategies prior to working on the problem, so students have a starting point. Strategies: guess and check, draw a picture, create a table, chart or graph, work backwards, solve a simpler problem, etc.

Communication-Is reasoning explained using words, pictures, tables etc.?

Goals
What are your mathematical goals for the lesson:
1) What do you want students to understand as a result of this lesson? …as a result of this unit?
2) What mathematical processes are you working to develop?
3) How does this lesson contribute to their continuing development as learners?

Prior Knowledge
In what ways does the task build on students’ previous knowledge? What definitions, concepts, or ideas do students need to know in order to begin to work on the task? What questions will you ask to help students access their prior knowledge? Mathematical Goals:

1) Fractions as not just part of a whole, they can be part of a set. The definition of a whole can change when you take a part away.
2) Recognize the pattern in the number of mangoes taken
3) Apply and adapt variety of strategies to solve problems
4) Self-monitor and reflect on problem solving process
5) Communicate their mathematical thinking clearly to the class
6) Representing parts of the whole as rational numbers in different forms

Prior Knowledge:
• Students should know how to represent fractions. They should recognize the fractions as unit fractions and the relationship they have to the whole.
• Students should know how to represent 1 whole in multiple ways.
Anticipating Student Responses
Identify the ways in which the task can be solved.
• Which of these methods do you think your students will use?
• What misconceptions might students encounter?
• What errors might a student make?

Expectations
What are your expectations for students as they work on and complete this task?
• What resources or tools will students have to use in their work?
• How will the students work — independently, in small groups, or in pairs — to explore this task? How long will they work individually or in small groups/pairs? Will students be partnered in a specific way? If so in what way?
• How will students record and report their work? Anticipating Student Strategies:
• Students start backwards and reason by using inverse operations- for example if 4 is ½ of a number they could double it to work backwards
• Students can draw pictures beginning with the 4 mangoes and working backwards to represent how many mangoes there were before each person ate a fraction of them
• Students may not know what to do with the fractions. They may try one operation such as addition or multiplication
• Students could view the mangoes as 1 unit and break it into pieces

Expectations:
• Students will have unifix cubes , markers, blank paper, and counters available and chart paper for the group
• Teacher will read the problem, with students following along
Students will solve the problem independently for 5-10 minutes
Students will work in groups of 3-4 to collaborate on a strategy to use to solve (20 min)
Students will share ( 15 min)
• Students will be grouped heterogeneously
• Each student records work on their own sheet with the problem
• As a group, they will record one strategy on chart paper that will be shared with the class
Launch
How will you introduce students to the task so as not to reduce the problem solving aspects of the task(s)?
What will you hear that lets you know students understand the task(s)?

Explore/Monitoring
As students are working independently or in small groups:
• What questions will you ask to focus their thinking?
• What will you see and hear that lets you know how students are thinking about the mathematical ideas?
• What questions will you ask to assess students’ understanding of key mathematical ideas, problem solving strategies, or their representations?
• What questions will you ask to advance students’ understanding of the mathematical ideas?
• What questions will you ask to encourage students to share their thinking with others or to assess their understanding of their peers’ ideas?

Engagement:
How will you ensure that students remain engaged in the task?
• What will you do if a student does not know how to begin to solve the task?
• What will you do if a student finishes the task almost immediately and becomes bored or disruptive?
• What will you do if students focus on non-mathematical aspects of the activity (e.g., spend most of their time making a beautiful poster of their work)?

Whole Class Discussion/Selecting, Sequencing, Connecting
Which solution paths do you want to have shared during the class discussion in order to accomplish the goals for the lesson?
• Which will be shared first, second, etc.? Why?
• In what ways will the order of the solution paths helps students make connections between the strategies and mathematical ideas? Warm up and Launch:
Provide students with pictures and play “Can you See”. Ask them how they can see ¼, ½, etc. of the stars.

Teacher and students will have a discussion about 21st Century Skills and Problem-Solving Strategies – record the responses

Introduction Question:
I have 8 bananas. My sister ate half of them. My brother ate the remaining ½. How many bananas do I have now?

Write the ideas students have for this problem and post them to use with the mango problem.

We will read the problem together, and teacher will work with students to ensure they understand the context of a problem.

Possible prompting questions:
• What is a mango? What does it look like? Why would someone want one?
• If you take mangoes away from the basket, what would you expect to happen?
• Make a list of what you know

Exploring/Monitoring:
• What is the question asking you to do?
• Can you state what the question is asking for in your own words?
• What strategies have you used in other problems that might be helpful for this?
• Make a list of the given information
• Is there a way you could represent the mangos? How might you use a manipulative?
• Is there another way you could show this problem?

To Assess Student Understanding:
• How do you know if your answer is reasonable? (Not too big or small?)

When students are sharing their thinking in groups:
• Use your own words to explain what strategy your classmate used
• How did you start solving this problem?
• Have students compare strategies. How do you understand the problem? Example: if someone did it in a table and doesn’t understand the problem, how can they understand it from their peer who drew a picture
• How did your classmate represent the mangoes in the problem?

Engagement:
For students who finish early, have them represent it in another way or prove their work by justifying the answer/ possible debate

Student who does not know where to begin:
• Where do you think you should start?
• How does this problem relate to our warm up?
• How can you represent the mangos?
• someone in your group a clarifying question

BIG POINT: We’re looking for your thinking and not the end result

Sequencing:
Begin with concrete strategies such as students who represented and grouped each mango. Move to more abstract representations such as breaking a whole into unit fractions.

Students can make connections between grouping a fraction of mangoes to seeing the mangos as a whole unit that can be broken into unit fractions. As a unit fraction is taken off such as 1/6, the whole now changes into fifths and so on. Students will hopefully be able to see the part to whole relationship and relate it to more abstract strategies.

Assessment
What will you see and hear that lets you know that students in the class understand the mathematical ideas addressed in the lesson?
Students will…
• Notice patterns in the problem
• Explain and justify their strategy
Know and explain if their strategy is reasonable

References: http://home.avvanta.com/~math/mango

http://illuminations.nctm.org/Lesson.aspx?id=1037