# To Float or Not to Float

 Author: Kevin Metzger Unit Title: To Float or Not to Float Grade: 10 Subject: Geometry/Algebra II Estimated Duration: 8-48minute class periods Unit Activities: Activity 1: Building a container Activity 2: Building a Boat EDP Activity 3: Building an Aluminum Boat Activity 4: Communicating Results Background Knowledge:  Students will already have an idea on volume and surface area.  However, we will go in depth with these concepts using 3-dimensional figures as well as figures that may not be necessarily a geometrical shape.  Students will need to understand how to solve basic equations as well.  Finally, students will need to be able to find the area of different geometrical shapes. Date: July 2014
##### The Big Idea (including global relevance)

Almost 71% of the earth’s surface is covered in water.  With that being said, we as humans have had to manage large bodies of water for centuries.  This would include traveling across large bodies of water.  The use of boats has become a regular method of transportation.  Boats have been made to hold large amounts of weight and be able to withstand the elements during transportation.  The physics behind boat making is precise in that it maximizes the surface area of the boat as well as the volume of which it can hold items.  A thorough analysis of volume and surface area is necessary to understand the physics behind the design and structure of boats.

##### The Essential Question
1. What surface area and volume provide the most effective means of boat design?
2. How can surface area and volume be used to create boats?
##### Justification for Selection of Content

Surface area and volume is widely used in mathematics.  Area of 2-dimensional objects is introduced in earlier grades prior to high school.  When students reach their freshmen or sophomore years in high school they are expected to find the surface area and volume of 3-dimensional objects.  These types of problems are commonly found on the Ohio Graduation Tests (OGT) as well as the ACT and SAT tests that students need prior to going to college.  Last year, 41% of the co-taught Math 2 classes failed the OGT test.

Surface area and volume is widely used in the construction of boats.  By having students build a scale model of a boat, they are able to use surface area and volume to reach a bigger picture of boat making.  How much weight can the boat hold?  In today’s society we are all about shipping as many goods as fast as possible.  So by maximizing the weight a boat can hold, we are able to ship a large amount of goods.  Students will consider these applications in their co-taught math 2 course.

##### The Challenge

Students will be required to make an aluminum foil boat that holds at least 1000gm of water.  They will use the following materials:

• Sheet of aluminum foil
• Hot glue
##### The Hook

Students will complete an engineering design activity that requires them to build a structure given a certain amount of card-stock and tape.  Their structure will have to hold the most water (without leaking) for at least 2 minutes.

##### Teacher's Guiding Questions
1. How does surface area relate to boat design?
2. How does volume relate to boat design?
3. What is buoyant force?
4. What other forces act on boats?
5. What is the concept of floating?
6. How can something as heavy as an aircraft carrier float?

##### ACS (Real world applications; career connections; societal impact)
1. Real World Applications:  Students will understand how objects move through certain medians such as water and air.  The concept of a floating boat is very similar to the concept of a “gliding” airplane.
2. Career Connections:  Boat engineer, structural engineer, aeronautical engineer, Boat Designers (recreational, commercial, and military).
3. Societal Impact:  Students will better understand how boats work.  They will understand how much weight a boat can hold as well as how much space the boat can actually take up with respect to how much weight it can hold.
##### Engineering Design Process (EDP)

Students will use the engineering design process when building their overall structure for their aluminum foil boats.  They will have to decide how to maintain the overall constraints on size as well as how much weight the structure can hold.  Students will work in engineering design teams to design the boat before actually making the boat.  They will record all of their calculations in their engineering design notebooks.  Students will also have a chance to refine their design after testing their first design.

A.SSE.1 Interpret expressions that represent a quantity in terms of its

context.★

a. Interpret parts of an expression, such as terms, factors, and coefficients.

b. Interpret complicated expressions by viewing one or more of

their parts as a single entity. For example, interpret P(1+r)n as the

product of P and a factor not depending on P.

A.SSE.3 Choose and produce an equivalent form of an expression

to reveal and explain properties of the quantity represented by the

expression.★

a. Factor a quadratic expression to reveal the zeros of the function

it defines.

b. Complete the square in a quadratic expression to reveal the maximum

or minimum value of the function it defines.

c. Use the properties of exponents to transform expressions for

exponential functions. For example the expression 1.15t can be rewritten

as (1.151/12)12t 1.01212t to reveal the approximate equivalent

monthly interest rate if the annual rate is 15%.

A.CED.1 Create equations and inequalities in one variable and use them

to solve problems. Include equations arising from linear and quadratic

functions, and simple rational and exponential functions.

A.CED.2 Create equations in two or more variables to represent

relationships between quantities; graph equations on coordinate axes

with labels and scales.

A.CED.4 Rearrange formulas to highlight a quantity of interest, using the

same reasoning as in solving equations. For example, rearrange Ohm’s

law V = IR to highlight resistance R.

##### Unit Activities

Lesson 1:  Understanding Surface Area and Volume:

Day 1:  Review of Surface Area of various 3D figures as well as pre-assessment: Activity 1 Worksheet A.

Day 2:  Review of Volume of various 3D figures. Activity 1 Worksheet B

Day 3:  Activity 1 Worksheet C – The Hook.  Students will be given an 8.5inx11in piece of card-stock.  They will be instructed to work in their engineering design teams to develop a structure that holds the most amount of water for a total of 2 minutes.  This structure must be able to close and open in order to measure the amount of water.

Day 4:  Activity 2 Worksheet D.  Students will be given the opportunity to discuss the big idea and essential questions based on the “hook” from the previous day.  This is where I will attempt to have students develop the idea of floating on their own.  Students will be given the constraints of their overall challenge.  They will be expected to work on the engineering design process with their engineering design teams.  Students will record all steps to the engineering design process in their engineering notebooks.

Lesson 2:  Surface Area and Volume in Building Boats:

Day 5:  Activity 3 Worksheet E.  Students will build their first structure for an aluminum foil boat that can hold 1000grams of water.

Day 6:  Students will test their design Activity 3 Worksheet F.

Day 7.  Students will refine their design to make it hold more weight or meet the first 1000gm requirement.

Day 8. Activity 4 Worksheet H.  Students will present their overall experiment with the first and second designs.  They will be expected to describe, in detail, any calculations to determine the maximum weight that their structures can hold.

##### Where the CBL and EDP appear in the Unit

CBL:  Worksheet C, Worksheet D, Worksheet E

EDP:  Worksheet D, Worksheet E, Worksheet H

##### Misconceptions

Students will confuse how to find the surface area and volume of their own boats.

##### How to Make This a Hierarchical Unit

Surface Area and Volume is used in Math 1, Math 2, Math 3, and pre-calculus at Milford High School.  I focused this unit on the 3 dimensional surface area and volume.  We could also increase the level of rigor by having students study the forces on the boat including buoyant force.  Most students will learn this in their physics class.  However, to increase to another level of mathematics, buoyant force would be a great topic to discuss.

##### Reflection
1. The students were able to find the solution that resulted in concrete meaningful action.  During the phase of students testing their designs, they were able to successfully conclude the concept of surface area (more than volume) played a huge role in the outcome and result of their designs.  Students were able to make conclusions, using surface area, as to why their boats may not have held as much as they anticipated.
2. The content that I chose for this lesson was chosen based on the history of my students.  The class that I teach is a co-taught class where students tend to struggle in mathematics.  Almost 80% of my students will move on to a vocational school setting.  Creating a lesson where they can use their hands allows them to better understand this type of mathematics.

##### Reflection

The chart above shows the amount of questions students correctly answered on the pre-assessment as well as the amount of questions answered on the post assessment.  The red bars clearly show a significant growth from pre to post assessment.

1. I believe the purpose for selecting the unit was met.  Students were able to learn the EDP and also obtain mastery of the concept.  They stayed engaged the entire time we completed the lesson.
2. Next time I implement this lesson I will try to incorporate more physics and science applications.  These applications are what keeps students engaged in the lesson!