# Hot dog

Team Members and grade level they teach:

Colleen Nevius Math7/Algebra 1, Patty Hetrick Math 6,7,8 /Algebra 1, Robin Stewart Math 8, Almarie Campbell Math 7

Standard(s):
The student will develop a wide range of skills and strategies for problem solving
8.3 The student will
a) solve practical problems involving rational numbers, percents, ratios, and proportions; and
b) determine the percent increase or decrease for a given situation.

Lesson Study Title:

The “Apple Problem” (also presented as The Hotdog Problem)

Overview of the lesson and how it connects to STEAM:

Given the task to sell an (apple), on day one the storekeeper increases the initial price (not given) by a given percent. On day 2, the storekeeper further increases the price by a different percent. On day 3, the item is returned to its original price.

1. Task 1 is to find the percent increase of the item at the end of the second increase.

2. Task 2 is to find the percent decrease from Day 2 to Day 3

Anticipated student strategies: (BOARD PLAN with Sequencing and connections)

1. Add and subtract percent of the ORIGINAL price (20% + 25% = 45%), then subtract 45%
2. Find percent increases properly, but assume that the percent decrease to the original price will be the sum of the two increases (subtract 45%.)
3. “Ratio box:
4. Plug in \$ amounts and test
5. Draw percents in the form of Circle Graphs
6. Model % increase with bar graphs
7. Identify 20% increase as 120% (rather than adding 20% to original), then identify subsequent 25% increase as 125% of 120%. Identify 150% as 1/3 of the total 150%.
8. Notice that the 2 increases are 3/2 of the original, and that the original is 2/3 of the 2 increases (reciprocals)

Formative Assessment strategies: (Rubric/checklist)

Student Team will produce notes and poster reflective of problem solving strategy and solution:

Requirement: Expectation Observation:
Involvement All students engaged and contributing
Strategy Team uses mathematic strategy to solve the problem
Correct Mathematics Students apply % increase properly
Students apply subsequent % increase properly
Students identify % decrease to the original state
Correct Analysis Students interpret math solution correctly

Extended learning * Students recognize fractional interpretation of the increases and ultimate decrease

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?
Students will:
1. Recognize % change as a relative (not absolute) operation
2. Rationalize the math (explain their reasoning)

Students should be familiar with computing a change in amount given the percent change. Students should discover that there is an iterative process when taking the percent of an amount that has been previously increased or decreased.

Students will likely apply previously learned techniques for finding percent of a number and/or percent change.

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?
Anticipated responses:
1. Ratio box
2. Percent/Actual box
3. Use representative dollar amounts
4. Draw circle graphs
5. Cognitively add and subtract percents in the problem without using solution strategy
Expectation:
1. Small group strategy session with presentation
2. Manipulatives available
3. Teams assigned 2-3 per group

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?

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)?

Teams will be established and materials available at the side table.
The problem will be presented as a projection. Ground rules and timeline will be explained. Students will have the opportunity to read through the problem and ask questions about expectations or the problem.

Every student will receive a copy of the problem, and may work individually or consult with a teammate. Poster paper made available when team has a strategy (right or wrong) that they can defend.

Timing (45 minute period) :
5 min intro
10 +/- min teamwork/problem solving
10 +/- min poster production
15 min presentations (remove markers from desks!)
5 minutes wrap up

Teaching strategies:
• Visit tables
• Ask questions if needed to facilitate exploration:
o Does it make sense?
o Can you name it as a fraction?
o Does it work for other numbers?
o Is there a pattern? Can you describe it?
o Can you see a relationship between the original and the final values?
• Encourage varying strategies:
o Draw a picture
o Set up a Proportion
o Choose a trial price
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?
Solution Presentations order:
1. Best answer with least math reasoning (just ‘sees’ the solution)
2. Pattern recognition with graphic representation (drawings or graphs to show scale up/down)
3. Use of algorithm (% change)

Students using an algorithm will benefit from seeing other representations of math solutions, but may “rest on their laurels” if they go first, and not pay attention.

Students that just “see” the solution may have an intuitive grasp but be unable to explain their reasoning.

Putting a graphic strategy in the middle benefits both traditional learners and ‘intuitive’ math students.
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 show that each increase leads to a new 100%.
Students will apply subsequent % increase to the new amount.
Students will see that percents, decimals or fractions will all work to find the right answer to this problem.

References: