solar energy harvesting â€“ for the sows teacher annotations solar

Solar Energy Havesting Teacher Annotations of 25

Solar Energy Harvesting – For the Sows

Teacher Annotations

Solar Energy Harvesting Name __________________________ Preliminary Activity Part 0: Strings, Stamps & Dice – A check for understanding on measurement. 1. Measurement is counting how many standardized units it takes to determine

the magnitude/size of something. 2. Length is counting _the number of standard lengths of string.

a. The units for length tell you how long each standard piece of string is.

b. Give three examples of length measurements and include reasonable units.

i. ___Height of the room in feet____________________

ii. ___Perimeter of the room in feet_______

iii. ___How far you travel to get to school in miles_____________ 3. Area is counting the number of square stamps it takes to cover a surface.

a. What makes a square a square? _Equilateral-Quadrilateral_ and

_Equiangular_. b. The units for area tell you _the standard size of each square stamp._

How does measuring area depend on measuring length? It takes two

identical string measurements to build a standard sized square stamp, thus the stamp size depends on the string measurement.

c. Give three examples of area measurements and include reasonable units.

i. ___Area of this page in sq cm___

ii. ___Area of the room in sq yards____

iii. ___Surface area of the school in sq. ft.____


4. Volume is counting _the number of standard sized dice it takes to fill an object.

a. The units for volume tell you _the standard size of the dice being counted.

b. Give three examples of volume measurements and include reasonable units.

i. ___Amount of space in a refrigerator in cubic feet_____

ii. ___Amount of air in your lungs in cubic centimeters____

iii. ___Amount of concrete used to build a dam in cubic yards__ 5. Three versions of the perimeter formula for a rectangle are:

P = _L + L + W + W_ = _2L + 2W_ = _2(L + W)_, and all three versions are really just counting _the 4 sides of a rectangle: two Lengths, and two widths_. a. Sketch and label a picture to illustrate how to count the correct number of

one-inch “string-lengths” for the perimeter of a 3 inch by 5 inch rectangle.

3 + 3 + 5 + 5 = 16inches 6. The formula for area of a rectangle A = _L x W square units____________, which

counts the number of _square stamps__ it takes to cover the rectangle. It does this quickly by first counting _the stamps_ in a _row (or column)_, and then counting _rows(or columns) of stamps and multiplying the two values_... a

groups-of-groups counting method.

a. Show your understanding by sketching and illustrating the groups-of-groups method to count the number of “square-stamps” it takes to cover a 3” x 5” rectangle.

123

1 2 3 4 5

123

1 2 3 4 5


5 stamps in one row 3 rows So there are 3 rows of 5 stamps 3 x 5 = 15 square inches 7. The formula for volume of a rectangular-prism V = _L x W x H_.

a. Another way to find area is to use your understanding of area to count the number of “dice” it takes to cover the base(Area of the base), and height (H) to count the groups of “dice” in each layer:

V = _(Area of the base) x H_.

Show your understanding by labeling the pictures to show how to calculate volume by counting “dice” two different ways for a 3” x 4” x 5” rectangular

123

1 2 3 4 5

1 2 3 4 5 1 2

3 4

3

2

1

Base width

Height = Number of layers

Base length

Area of the base = W x L = 5 x 4 = 20 sq. in.

Also number of dice it takes to cover the base and in each layer.

Volume = (Area of Base) x Height = 20 x 3 (3 layers of 20) = 60 cubic inches

1 2 3 4 5

Base width = Group of 5

1 2

3

Base length = Group of 4

4

3

2

1 Height = Group of 3

Volume = W x L x H = 5 groups of 4 groups of 3 = 5 x 4 x 3

= 60 cubic inches


Typical thermal solar collectors

Solar Energy Overview – The sun produces solar radiation that travels about 8½ minutes to reach the earth. (TC-8) This solar power strikes the Earth’s upper atmosphere with enough energy, that if one square meter of it were converted to electricity it could power about fourteen 100watt light bulbs. (TC-9) Unfortunately, not all of this energy reaches the ground because some is reflected and absorbed by the atmosphere, clouds, dust and pollution. (TC-10) By the time sunlight reaches the ground in North America, there is about 150 to 375 watts per square meter available…in other words about enough for two to five 75watt bulbs (assuming all the energy is converted to electricity).

The main advantage of solar energy, is the source is free. The obvious disadvantage is the sun sets in most places taking away this free source of energy on a daily basis. To get through the night, most systems designed to “harvest” solar energy have two main components:

• Collectors that catch an area of sunlight (There are two main types): (TC-11) • Solar thermal collectors collect it as heat. • Photovoltaic cells collect and convert it directly to

electricity like on a solar-powered calculator. • An energy-storage device to hold extra energy

generated during the daytime that can be used up when solar energy is not available.

• Heat can be stored in any mass such as a tank of water or a slab of concrete. (TC-12)

• Electricity is generally stored in batteries. Block diagram of three main types of solar energy harvesting systems (TC-13)

Thermal Solar Collector

Thermal Solar Collector

Photovoltaic Collector

Thermal Storage (Water)

Electrical Storage (Batteries)

Power Plant

Thermal Load (House, heat, hot water, industrial process)

Electrical Load (House, power grid)


Cross-section of a typical active water-type solar heating system.

Summary • When collecting solar power the amount of energy available is directly

proportional to area of sunlight collected. So if you double the patch of sunlight collected, you would have twice as much energy available to harvest.

• The energy to be stored is directly related to the area of the sunlight collected, and it is usually stored in a mass or volume of substance like a tank of water or a slab of concrete. (TC-14)

Solar Heating System Design – (TC-15) We will now explore one type of solar heating system: The active water-type solar system which transfers heat from a roof collector to a well-insulated under-floor water storage tank. Warm water from the storage tank is pumped through pipes in the concrete floor of the building to heat the floor, which heats the building. Whenever the temperature difference between collector and storage is less than 10oF, the water supply to the collector shuts off and it drains back to storage because the water is no longer being heated by the sun at an efficient rate. The water circuit for heating the animal area is a separate piping arrangement, with the pump taking water directly from the storage tank and sending through pipes in the floor. A boiler is needed to provide supplemental heat when solar energy is insufficient. (TC-17) Collection Requirements The energy requirements can be found in the following table that gives the required area of sunlight to harvest:

Square Feet of collection per an animal unit

**Area of Wall Collector

**Area of Roof Collector

Live Stock Type Air Type Air Type Water Type

Sow & Litter 20 20 30 (TC-18)

Nursery Pig 2 2 2.5

Dairy Calf 10 10 15

**Values shown are based on farmer experience and empirical calculations, and should be considered approximate.


Storage Requirements – Storage size is directly related to collector surface Storage needed for each square foot of collector surface

Storage Medium Rock Concrete Water

1.5 Cubic feet 1.5 Cubic feet 4 Gallons (TC-19)

**The energy storage requirements for 10 sq. ft of collector would be 15cu. ft. of rock or concrete or 40gal. of water.


Solar Energy Harvesting Name ________________________ Overview Instructions: Read the Solar Energy Harvesting handout, and use the information to answer the questions below on solar Energy. 1. How long does it take for solar radiation to travel from the Sun to the Earth? (TC-20)

About 8.5 minutes 2. Label the diagram to indicate the amount of energy that reaches the upper

atmosphere.

14 3. How many watts of solar energy on average reaches the ground?

__150 to 375 __watts per ___1 square meter___.

a. Show how to determine the number of 15-watt compact-fluorescent light bulbs would this power.

bulbs1015150 =÷ bulbs312315350 =÷

b. Give three reasons for why less energy reaches the ground.

i. ____________________

ii. ____________________

iii. ____________________ (TC-21)

4. What is the main advantage of solar energy? The source is free during the day. 5. What is the main disadvantage of solar energy? The sun is not always available

_1m_x_1m_(__1__ m2) Patch of Sunlight

‘s 100watt


6. Describe the two main components of solar energy harvesting a. The __Collectors___, which ___absorb/catch solar energy_. There are

two primary types: i. __Thermal_ to collect ___heat energy_ ii. Photovotaic to collect & convert solar energy directly to

electricity_ iii. The energy storage device to be used later when the sun is

down or not available to generate solar energy.

There are two primary types:

iv. Heat stored _in a mass such as water or rock_. (TC-22)

v. Electricity stored _primarily in batteries_. (TC-23) 7. If 1 square meter of collected energy requires 150 liters of water for storage, then

show how to determine the energy storage requirements for 8 square meters?

Since there is a direct variation between collection and storage, scale the original system by a factor of 8. (TC-24)

liters12001508 =⋅ for 8 square meter of collection OR

Xlitersliters

Xlitersmsq

msqlitermsq

Xlitersmsq

liters

=

=⋅

=

1200.1

.8150.8.1

150

8. Sketch and label a diagram for a water based solar heating system. Include the four

main components: a. Collectors b. Energy storage c. Heating units for the floor d. Backup heating

Arrows indicate pipes and flow of water


Solar Energy Harvesting Architect: ____________________________ Heating System Design Objective: Design an active water-type solar heating system for the floor of a sow farrowing house (a building for female pigs to build their nests and birth their litters of piglets). Part 1: (TC-25) Energy Requirements (You will need to reference the Solar Energy Overview handout for Sow & Litter collector requirements, and water storage requirements.)

1. Read the Solar Energy Overview, and use the information to help with Part 1.

TC-26) 2. Complete the table for solar collector and energy storage requirements (TC-27)

Number

of Sows with

Litters

Solar Collector

requirements in square

feet

Energy Storage

requirements in gallons of

water 1 30sq ft 120 gal 2 60 sq ft 240 gal 3 90 sq ft 360 gal 4 120 sq ft 480 gal 5 150 sq ft 600 gal … … … 20 600 sq ft 2400 gal … … …

n 30n sq ft 4(30n) gal

OR 120n gal

3. Explain your method to determine solar collector requirements. Each sow & litter requires 30 sq ft of solar collector (a direct variation/proportion), so multiply 30 sq ft by the number of sows & liters to determine the number of sq ft of solar collector required. See table for algebraic answer.

4. Explain your method to determine energy storage requirements (TC-28)

Each sq ft of collector needs 4 gallons of storage, so find the number of square feet of collector then multiply by 4. OR Each sow and litter needs 4x30 = 120 gallons of thermal storage, so multiply the number of sows & liters by 120 gallons to find the total storage. See table for algebraic answer.


5. There are approximately 7.5 gallons in a cubic foot. Show how to find the number of cubic feet for:

a. One gallon of water. ..331...5.7

..11

.. ftcuftXcugalftcu

galftXcu

=⇒=

b. For energy storage using water for one sow with a litter?

1 sow & liter 120 gal storage (from table). ..16..5.7

..1201

.. ftcuftXcugal

ftcugal

ftXcu=⇒=

c. For energy storage using water for 20 sows with litters? TC- 29) 1 sow & liter 16cu. ft. (from #4b). 20 x 16 = 320cu. ft.

6. How many half-Gallon milk cartons would it take to fill a cubic foot? Surprised? 7.5 ÷ .5 = 15 half-gallon milk cartons. (TC-30)

Part 2: Determine solar collector needs for a 18-sow farrowing house. (TC-31) Solar Collectors – Thermal solar collector panels come in several sizes, but there are some standard modular sizes. The prices can vary depending on the manufacturing methods employed but usually fall in the $20 - $60 per square-foot range. 1. Assume you are buying panels that cost $30 per square-foot, to complete the table.

Panel Style

Solar Panel Size

Number of Sq. Ft.

Cost in Dollars

A 2ft x 4ft 8 sq. ft. $240

B 4ft x 4 ft 16 sq. ft. $480

C 4ft x 8ft 32 sq. ft. $960

D 6ft x 12ft 72 sq. ft. $2160

E 6ft x 20in 10 sq. ft. $300

F 78in x 36in 19.5 sq. ft. $585

G 19.5ft x 36in 58.5 sq. ft. $1755 2. How many times more area than panel A is:

i. Panel C: __4 times_

ii. Panel D: __9 times_

(TC – 32)

TC-33)


3. Sketch and label pictures showing why panel C costs four times more than panel A and is a scaled version of A. (TC-34) (Hint: Think Area.) 2ft

4ft 4ft 8ft 4. Sketch and label pictures showing why panel D costs nine times more than panel A

and is a scaled version of A. (TC-35)

2ft 4ft 6ft 12ft 5. What would be the dimensions of a panel that was 4.5 times larger than panel A?

2 x 4.5 = 9ft by 4 x 4.5 = 18ft So 9ft by 18 ft

6. Panels C and D are scaled up versions of panel A, yet panel B is not. Explain why

panel B is not a scaled up version of panel A. (TC-37) Because Panel B is only doubles one of Panel A’s dimensions rather than doubling both dimensions. 7. Complete the table with help from the information from question #2. (TC-38)

Panel Style

Perimeter of Panel in Feet

Scale Factor

of Panel

A

Scale Factor

Squared

Area of Panel in Square

Feet

A 12 ft 1 12 = 1 8 sq. ft. C 24 ft 2 4 32 sq. ft. D 36 ft 3 9 72 sq. ft

X** 54 ft 4.5 20.25 162 sq. ftY** 108 ft 9 81 648 sq. ft

**Panels X and Y are new panels scaled up from Panel A

Panel A Panel A

Panel A

Panel A

Panel A

Panel C It takes 4 Panel A’s to have same area as Panel C. Panel C is scaled up by a factor of 2.

Panel A Panel A

Panel A

Panel A

Panel A

Panel A

Panel A

Panel A

Panel A

Panel A

It takes 9 Panel A’s to have same area as Panel D. Panel d is scaled up by a factor of 3.

String Scale Factor

N

Stamp Scale Factor

M

N2

M

(TC-36)


8. Find the areas and costs for the hypothetical panels X and Y if they are scaled versions of panel A. (TC-39)

a. X: _162sq ft_ _$4860_

b. Y: _648sq ft_ _$19,440_

Summarize your understanding 9. Find a relationship between the scale factor of the perimeter and the costs of the

panel compared to panel A. (Hint: How many times more area than panel A is each panel?) (TC40)

The linear scale factor for perimeter is squared to find the scale factor for area. Take the square root of the area scale factor to get the linear scale factor. 240$2 ⋅= scaleAPanelCost

10. Explain the relationship of perimeter to area when objects are scaled up.

The linear scale factor for perimeter is squared to find the scale factor for area. Take the square root of the area scale factor to get the linear scale factor. 2

scalescale PA = and scalescale AP =

a. How many square inches are there in one square foot? (Hint: It’s not 12…think about scaling up a square inch to a square foot.) (TC-41)

Think scaling 1sq in by a linear factor of 12, so area scale factor is 122 = 144. So, 1 x 144 = 144 sq in.

12 x 12 to cover area with stamps = 144 sq in.

11. Find the cost in dollars per square inch for your solar collector panels.

Cost per square foot is $30, so 30 ÷ 144 ≈ $.21 per square inch.

OR 21$...144

..1..1

30$$ ≈⋅=insq

ftsqftsq

X

12. If the side lengths of panel E were scaled up by a factor of 3.5 a. How many times larger would its perimeter be?

3.5 times because it is a 1-dimensional scaling of a string measurement

b. How many times larger would its area be? (TC-42)

3.52 = 12.5 times more area because the string scale factor is squared to get the equivalent stamp scale factor.


13. This relationship between linear dimensions and area only works for scaled objects. Show this is true by demonstrating it doesn’t work for non-scaled objects. Sketch and label a picture of panel A and another panel with double the Area that doesn’t follow the perimeter - area scaling rule. (Hint: Don’t scale.) (TC-43)

P = 12ft & A = 8sq ft P = 36ft & A = 16sq ft Linear scale factor is 1:3 which would predict an area scale ratio of 1:9, but the actual area ratio is 1:2. So it doesn’t work.

14. This new relationship can also be used to go backwards. Practice this relationship

by completing the table. Assume each object listed is a scale version of the others:

Object #

Dimensions in feet L x W

Perimeter in feet

Area in sq. ft.

1 10 x _50_ 120 500

2 _40_x_200_ 480 8000

3 __1__x__5__ 12 5 4 __5__ x 25 60 125 5 _2.5_x_12.5_ 30 31.25

6 .2 x __1__ 2.4 .2

7 __.5_x_2.5_ 6 1.25 8 _20_x_100_ 240 2000 9 __.1_x_.5__ 1.2 .05

10 _200_x_1000 2400 200000 15. Determine the cost and the number of collector panels needed for the design of a

18-sow farrowing house: (TC-45)

a. Using panel style A. From Overview handout 18 sow & liters at 30sq ft per sow & liter need: 18 x 30 = 540sq ft of collector required. Each Panel A has 8sq ft per panel, so 540 ÷ 8 = 67.5 Panel A’s needed. A .5 panel cannot be used, so need 68 panels. Each panel costs $240, so 60 x $240 = $16,320

b. Using panel style E. Need 540sq ft (from #15a). Each Panel E has

10sq ft, so 540 ÷ 10 = 54 Panel E’s. Each Panel E costs $300, so Total cost is 54 x $300 = $16,200

Panel A Panel A Panel A

String Scale Factor

N

Stamp Scale Factor

M

N2

M

(TC-44)


Part 3: Determine solar energy storage needs for a 18-sow farrowing house. Energy Storage – The heated water from the solar collectors will be stored in a rectangular tank underneath the floor of the building. The water can be circulated through the floor later for heat when it is needed. The buildings floor plan calls for the building to be 32 feet wide by 40 feet long. 1. Show how to determine the cubic feet of water storage required for your systems

design of a 18-sow farrowing house. (TC-46)

18 sow & liters need 540sq ft of collector (from #15), and each square foot of collector needs 4 gallons of water. So 540 x 4gallons = 2160 gallons of storage needed. Since there are 7.5 gallons per 1cu ft, 2160 ÷ 7.5 = 288cu ft.

2. Complete the table for tank designs, and give three other possible tank dimensions

that would work for your storage requirements. (TC-47) Tank

Design Number

Length in feet

Width in feet

Height in feet

Area of Tank Floor in

square feet Volume in cubic feet

1 28.8 40 .25 1152 288cu ft.

2 2.5 18 6.4 45 288cu ft.

3 6.8 6.8 6.23** 46.24 288cu ft.

4 10 10 2.88 100 288cu ft.

5 6 6 8 36 288cu ft.

6 10 4 7.2 40 288cu ft. **Note: Calculate to the nearest hundredth of a foot. 3. The tank with the least amount of surface area will lose the least amount of heat.

Determine which tank design would be best for your project if you want to minimize heat loss due to surface area? (TC-48)

A cube would be the shape to minimize surface area (a sphere would be best but is hard to build), so Tank Design #3 is the best in the table( ..262)23.68.6(4)8.68.6(2 ftsq≈⋅+⋅ ). The ideal cubic tank would be have edge lengths of 60.62883 ≈ ft. (TC-49)

4. Which would be the worst? Explain your reasoning. (TC-50)

Tank Design #1 is the worst because it has the most surface area and thus will lose the most energy. ( ..2323)25.8.28(2)25.40(2)1152(2 ftsq≈⋅+⋅+ )


Part 4: Modifications to your design. (TC51) 1. Your client was impressed enough with your design that you have been asked to

come up with different versions of another design scaled up in different ways. You are given an original tank that is a cube: 4ft x 4ft x 4ft, and asked to make changes to it.

2. Modifications A: You are limited to a tank height of 4 feet due to rock underneath

the building. So you can only change the length OR the width of the tank, OR both. Show your understanding by completing the table.

Design Version

Linear Scale factor used

Tank Floor dimensions

In feet

Tank Floor Area in

square feet

Tank Volume In cubic

feet

Area Scale factor

Volume Scale factor

Original Tank 1 4ft x 4ft 16sq.ft. 64cu.ft. 1 1

Tank Width

doubled 2 4ft x 8ft 32sq.ft. 128 cu.ft. 2 2

Tank Length

and Width doubled

2 8ft x 8ft 64sq.ft. 256cu.ft 4 4

Tank Length Tripled

3 4ft x 12ft 48sq.ft. 192cu.ft. 3 3

Tank Length

and Width Tripled

3 12ft x 12ft 144sq.ft. 576cu.ft. 9 9

Tank Length 7

times longer

7 4ft x 28ft 112sq.ft. 448cu.ft. 7 7

Tank Length

and Width 7 times more

7 28ft x 28ft 784sq.ft. 3,136cu.ft. 49 49

Tank Length Halved

.5 4ft x 2ft 8sq.ft. 32cu.ft. =21 .5 =

21 .5

Tank Length

and Width halved

.5 2ft x 2ft 4sq.ft. 16cu.ft =⎟⎠⎞

⎜⎝⎛

2

21 .25 =⎟

⎠⎞

⎜⎝⎛

2

21 .25


3. What is the pattern for a 1-dimensional linear change on the tank’s Width, Floor-area and volume? Make sure to indicate how the scale factor(s) help predict results. A linear 1-dimensional change Changes only one linear dimension, and thus only changes the area and the volume by the same linear factor. (TC-52)

change by linear factor of tank floor by factor of N W = old W x N, Area = old Area x N, V = old Vol. x N 4. What is the pattern for a 2-dimensional change? Make sure to indicate how the

scale factor(s) help predict the results. Two linear 1-dimensional changes Affect both linear floor dimensions, and changes the area and the volume by the square of the linear factor.

change by linear factor of tank floor by factor of N W = old W x N, L= old l x N, Area = old Area x N2,

V = old Vol. x N3

5. Optional Modifications B: (TC-53)Now assume there is not a 4 foot height

restriction…In other words it can be any height you need it to be. So, the design version has you change all three dimensions by the same factor. Show your understanding by completing the table built from the same original 4ft x 4ft x 4ft tank.

Design Version (TC-54)

Tank dimensions

in feet (Best Tank)

Tank Volume in cubic feet

Tank Surface

Area nearest square

foot

Scale factor for

tank dimensions

Scale factor for

Tank volume

Scale factor for

tank surface

area

Double the tank

dimensions

L = 8 W = 8 H = 8

512 384 2 8

4

Triple the tank

dimensions

L = 12 W = 12 H = 12

1,728 864 3 27

9

Four times the tank

dimensions

L = 16 W = 16 H = 16

4,096 1,536 4 64 16

Five times the tank

dimensions

L = 20 W = 20 H = 20

8,000 2,400 5 125

25

Six times the tank

dimensions

L = 24 W = 24 H = 24

13,824 3,456 6

216

36


6. (Optional) What is the pattern that scaling all linear dimensions (length, width and height) have on the tank’s dimensions, Surface area and tank volume? Make sure to indicate how the three different scale factors help predict results. (TC-57) Original object

Scaled in three Linear dimensions One of the reasons the client was asking about different scaled designs is because they heard of a liquid mixture better than water for storing the thermal energy. The mixture is called a glauber salt slurry. It uses a special kind of salt mixed in with water, and it effectively cuts the solar collecting requirements in half. (TC-58 7. (Optional) Scale a design that uses a cube-shaped storage tank of 1024 cubic feet

of water down to what the glauber salt slurry will allow. Be sure to indicate your methods. Your answer should include: (TC-59) • The amount of solar collector area required. • The amount of thermal storage required. • The number of solar panel needed specified by type. • The dimensions of the storage tank • The surface area of the storage tank • Scale factors linking all the dimensional changes between the old and new

designs. Original Tank volume 1024 New 1024 ÷ 2 = 512cu ft 512cu ft 512 x 7.5gal/1cu ft = 3840gal 3840gal ÷ 8gal/1sq ft of panel = 480sq ft of collector panel Panel A @ 8sq ft 480 ÷ 8 = 60 Panel A’s The volume scale factor is .5, so the linear scale factor is 7937.5.3 ≈ The area scale factor is the square of the linear scale factor which is 6300.7937. 2 ≈ The original tank dimensions were L = W = H = 3 1024 ≈ 10.079 so scale each one with the linear scale factor:

L = W = H = feet87937.079.10 =⋅

If a linear scale factor of N is applied to all three linear dimensions then: • Edges scaled by the linear factor

• L = old Lengths x N • Area scaled by square of linear factor

• A = old Area x N2 • Volume Scaled by the cube of the linear

factor • V = old Vol x N3

• To find Linear factor from Area factor square root the area factor

• To find linear factor from Volume factor f


The original tank’s surface area was 610sq ft(to the nearest square foot), so scale this by the area scale factor: ..3846300.610 ftsq=⋅ of surface area for the new tank (to the nearest foot). Preliminary Activity TC-1This Preliminary Activity is an optional activity for students that need extra help with the concepts of measurement for length, area and volume using the context that many typical measurements can be done by counting a standardized unit.

The worksheet is intended to be a set of notes for a class conversation guided by the teacher. Have the students try to answer a question with their previous individual knowledge, then discuss it with their group, and then as a class. Work through it in natural chunks of information creating a flowing conversation about measurement.

For length this can be modeled with a string (1” for example). Area is a count of square stamps completely covering a surface without overlap. And, volume can be modeled by counting the number of dice it takes to fill up an object.

You can get students to quickly grab on to this by always asking “Are you trying to count strings, stamps or dice?”

Students should understand that the standard formulas are really just “nice” ways to quickly count certain patterns.

Make sure students understand that it makes no sense to say 10 inches = 10 sq. inches = 10 cubic inches because they are counting different things…strings, stamps and dice as it were.

Ask students to label their answers with units, and in particular make them use SQ. IN and CU. IN. instead of superscript notations IN2 and IN3 to emphasize what it is they counted. TC-2 In question #2 be sure to emphasize COUNTING TC-3 In question #5 ask students what they are counting for perimeter and circumference. Ask for other examples for perimeter. TC-4 This is also a good opportunity to have students discuss why all of these are really counting the same thing by giving examples and sketching pictures of their explanations. TC-5 In question #5a if students have problems, have them utilize square-tile manipulatives to count edges. Start with one tile and gradually build up to the 3x5 figure so students understand that they are counting the outside edges of the final figure.

Be sure students understand the difference between perimeter and area.


Solar Energy Overview TC-8 The class objective for this part of the lesson is to gain context for studying the fundamental theorem of geometry (10MEO1). There is a 2-page worksheet that students can use as they read this overview and discuss the main ideas with their group. Be sure to circulate amongst the groups and verify they are getting the main ideas (See the answer key for the Overview Worksheet). This is a great opportunity to ask probing questions that set up the students to discoveries later linking 1-, 2- and 3-dimensioanl changes to each other. Eg. Is doubling area a 1-, 2- or 3-dimensional change? What about doubling volume? Tripling both? (Answer for all: 1-dimensional or linear) Note: Make students mark their area units with sq. ft. rather than a superscripted 2 to emphasize what is being counted. Continue with this idea for volume by using cu. ft. TC-9 Link this idea to the electric meters on the outside of houses, and bring in an old electric bill that indicates how many kilowatt hours were used. Good discussion questions could include:

• How many 100watt light bulbs you could light up for 1 hr with that energy used on the bill?

• How long fourteen 100watt bulbs could burn? • How long could fourteen 15watt compact fluorescent bulbs burn for the same

amount of energy. The Key is to get them comfortable with direct variation questions for later in the lesson TC-10 This is a good time to integrate some science into the mathematics classroom, and possibly current events for social studies. Encourage students to engage and share their knowledge to help take ownership of this lesson. TC-11 Students must link collection of solar energy to area in order to be successful. Use probing questions with the groups to determine if they understand this. E.g. • What do you need to change with the number of collectors if you need 5 times more

energy than I currently have? • What if you doubled the number of rows and columns of collectors? Tripled both? Manipulative: 50 per group: Paper models of solar collectors. (See teaching materials) TC-12 Students must link energy storage to volume in order to be successful. See water and battery examples. Manipulatives: 27 – 64 per group: Counting Cubes or equivalent. A cubic amount that can also form several different square values (1, 4, 9, 16 etc.) Use questions similar to the area probing question here, except link them to volume. Be sure to ask questions that get students to look at 1-, 2- and 3-dimensional scaling or arrangements of their cubes…e.g. Start with 2x3 cube arrangement then

• Double it in 1-dimension. • Double it in 2-dimensions.


• Double it in 3-dimensions. Ask how much more energy can be stored in the new arrangement than in the original. TC-13 This is just to illustrate the main components of a solar collection system and their connections:

• Collection • Storage • Usage load

Note: (Power plant coverts thermal energy to electric.) TC-14 This is an important link that students must have in order to utilize this context for success in the lesson. Teachers are strongly encouraged to have students explore this connection with their manipulatives in their groups, and then in closure for the entire class.

Suggest that 1-Solar collector requires 2-cubes of storage, and revisit the previous probing questions for solar collectors and energy storage. The difference is that the number of collectors is linked to the number of cubes and vice verse. Make sure that they maintain the 1:2 ratio throughout. TC-15 Be sure to make references to the Cross-section diagram with the students. They will need to sketch their own system later in the worksheet. TC-16 There is not enough solar energy to generate heat for the system. TC-17 This might be a good time to let students share what special considerations might be needed for their region of the country, and how the unique problems might be solved. TC-18 This is the area of collection needed for each animal unit for this lesson. Use the other numbers in the table for probing questions to check their understanding of the information and it’s link to their solar collectors. E.g. Tell them each solar collector is 10 sq. ft. TC-19 This is the volume of storage needed for each square foot of collector for lesson.

The other numbers can be used for probing questions to check their understanding. To get a sense of volume, bring in half-gallon milk cartons. This will be helpful later when trying to get students to understand what a cubic foot of water is. (1cu. ft. = 7.5 gallons)

This might be a good time to look at different arrangements of milk cartons and to probe students understanding of conservation of volume and how to start minimizing surface area to avoid heat loss (more surface area means more heat loss.)


Solar Energy Harvesting – Overview TC-20 About 8.5 minutes. We see the visible light portion of this electromagnetic energy, but there is quite a bit more that is invisible ie. Infrared and ultraviolet which are lower and higher energy respectively than visible light. TC-21 Three of these:

• Reflected by the atmosphere • Absorbed by the atmosphere • Clouds • Dust • Pollution

Any reasonable similar idea

TC-22 Remind students how a sidewalk or a rock face holds its heat for a while after the sun is no longer shining on it. TC-23 You can demonstrate this is still a volume by using batteries as manipulatievs. They take up space (car batteries are nice rectangles), and to double your storage, you need twice as many batteries. TC-24 Students should be reminded that they are applying proportional reasoning here. Solar Energy harvesting – Heating System Design TC-25 Form heterogeneous groups of three or four students. Think-Pair-Share (Work as individual, compare and settle on answers with a partner, reach consensus as a group) is a good group function to work through each question.

Students have trouble seeing the difference between 1- dimensional, 2- dimensional and 3-dimensional changes. Part 1 emphasizes 1-dimensional changes even though the objects are 2- dimensional and 3-dimensional. Make sure they are getting this as you circulate around to the groups and probe their understanding. Keep building off of the ideas developed with the manipulatives from the Overview of Solar Energy handout. TC-26 This can be done as a teacher led activity to save time if needed. TC-27 This is a good time to circulate to groups and probe their understanding of linear changes even though these are 2- and 3-dimensional objects. Have groups demonstrate linear layouts of each row of the table and ask the class to explain why this is only a 1-dimensional change.

Use the Solar Collector manipulatives and blocks to help make connections between collector and storage requirements.


TC-28 Get students to answer these questions verbally as well as algebraically. Connect the three representations of the same information: Numeric, Algebraic and Verbal TC-29 These calculations can be worked out as an entire class to ensure the idea of gallons to cubic feet is flushed out conceptually TC-30 This is a great opportunity to start planting seeds for 1-, 2- and 3-dimensional changes to objects. Ask some “what if” questions such as:

• How long would the line of milk cartons be for 4.4 gallons. • If the milk cartons were arranged in a 3x5 grid, and the number of rows and

columns were scaled up by 2 or 3, how many milk cartons would you need to build the new version?

Start using questions to link changes in volume to surface area and vice verse, and nudge students to start thinking about how to minimize surface area for storage as this will minimize heat loss. This is a great segue to science examples. TC-31 Part 2’s main objective is to get students to understand the difference between 1- and 2-dimensional changes, and to make the connection how 2-dimensional changes affect linear changes, and vice verse.

Use the Solar Collector manipulatives here to demonstrate the difference between 1-dimensional and 2-dimensional changes. Students that don’t get scaling will miss how many dimensions must change for area to scale (2-dim change by same scale)

TC-32 This should be easy for students. If they struggle, be sure to check for understanding on calculating area of rectangles. TC-33 Make sure students catch the change in units.

TC-34 Students need to illustrate the area of Panel C is 4 times more than Panel A, and show that Panel A has been scaled up by a factor of 2 to get Panel C.

Probe students as to why the scale factor is 2 not 4. Get them to see it depends on the context of the question asked. This is the crux of student understanding.

TC-35 Must be complete answer. See question #3 above.

If students are not getting the idea of scaling up Panel A, have them use the Solar Panel manipulatives to build different scale versions by making sure they increase the vertical and horizontal number of panels by the same amount.

TC-36 This is a good question to use for probing students understanding of scaling. If they struggle, make up more questions like this with friendlier numbers and have them build models with the Solar Panel manipulatives.


TC-37 This is a key question that students must be able to answer correctly. It is worth stopping the entire class, and making sure everyone is on board by have students share other examples of scaled and non-scaled objects and firm up what it means” to scale”. Link this idea to dilations, and to multiplication. TC-38 As you circulate around to the groups, let students know they are looking for two patterns here, and a connection between them. Panels X and Y are just new panels.

Have students build a flowchart for the scaling relationship between 1- and 2-dimensional scaling factors. TC-39 If students have problems here, have them work out smaller problems with the Solar Panel manipulatives by using the area and costs per Panel A to build from and literally counting the amounts until they can generalize the relationship to solve this problem. TC-40 C Again, emphasize Verbal and Algebraic versions of these ideas. Encourage students to draw and label arrows with rules on the table to show connections to Panel A. TC-41 This is a classic, and a chance for them to see it in context. Manipulatives can help here by laying out 1x1” tiles to represent the left and bottom edges of a square foot and asking how to quickly count the entire thing. TC-42 If students are not sure of these questions, it is a good time to use graph paper or 3x5 cards to build it and other scaled versions until they “get it”. Note, let students cut the 3x5 cards if they need to. Another great manipulative is a geoboard. Part 3: Determine solar energy storage needs … TC-43 Make sure the entire class is on board with the counter-examples. It might be worth the time to build two collections on the chalkboard of scaled and non-scaled along with their perimeter and area relationships. TC-44 Let students struggle with this table for a bit, and then ask them to share successes and probe for what is blocking them.

Suggest students use the flow chart that connects the 1-dimensional scaling to the 2-dimensional scaling:

1-d scalar squared 2d scalar 2-d scalar square root 1d scalar Then get them to draw and label arrow on the chart labeled with scaling rules and

identify 1- and 2-dimensional scalars.


TC-45 This returns to the main design problem. Some extensions for students that get done quickly: What are different arrangements they could use the panels in? What would happen to the design if it were scaled up by a factor of 2? 3? 2.5? Ask them to choose one panel style over another and defend their choice. Ask them to mix 2 panel styles, and defend their choice. TC-46 They have already done problems like this in Part 1, so if they seem lost, have them refer back. Again, emphasize the linear connections to this problem even though this is a 3-dimensional answer.

This answer is used in the following table for the last column.

TC-47 Students need the correct answer of 288 cu ft to complete this table. This is a good opportunity to refresh the idea of prism Vol = Base-Area*Height. Note: Tank design 3 is a near-cube, and rounding is needed to keep the numbers nice TC-48 Suggest to students that building a formula might help speed things up here. Be sure to emphasize counting in terms of “Strings, Stamps and Dice”, and use manipulatives to help visualize what they need to count. Then have them play with the cube manipulatives and count the exposed faces of different designs. TC-49 Note: Cube roots are not tested on the WASL exam at this time. TC-50 This is an opportunity to introduce and explore surface to area ratios. It explains why ants don’t get hot in a microwave but a cup of coffee does, why babies get cold in a swimming pool before the adults etc. Part 4: Modifications of your design TC-51 This is the entire objective of the lesson in numeric form. You might want to walk them through the first one if they get stuck quickly, let students know they are looking for patterns, and rules for their patterns. Note that area has been worked in to get them to see both scaling factors at the same time.

Help students by getting them to expand their flow chart to include the rule to move from 1-dimensional to1-dimensional scalars. Have them show the rules with arrows on the Scale factor columns of the chart labeling 1-, 2- and 3- dimensional parts. Make sure students maintain their use of units on their answers where appropriate, and discuss why the scalars are unit-less if they are ready for that idea.

This is a good time to break out the manipulatives as needed to help reinforce the previous explorations, and link the patterns together for the Fundamental Theorem of Geometry.


TC-52 Make sure they connect that a single linear change results in: • Linear changes of length in that direction but not in the other two directions • Linear change to area • Linear change to volume Encourage them to sketch and label pictures, and to try and come up with an

algebraic rule that says the same thing as their verbal answer. If they have troubles finding the pattern, have them expand the table, and to model

the results with their groups using the manipulatives.

TC-53 This table is an extension to changes in three dimensions and goes beyond the requirements of the WASL exam. This is a great extension for students that find success throughout this activity. Have them look for the next link in the pattern which is cubing the linear scale factor. Also have students add one more element to their flowchart linking linear scale factors to cubic scale factors. This of course creates an opportunity to introduce cube roots. TC-54 Make sure students give a verbal description of their tank modification. TC-55 This is an optional pattern for the students to find but fits nicely for a complete picture of the fundamental theorem of geometry. TC-56 The basis for this modification is factual! It works like this due to the heat density, and because the operating temperature can be much lower (glauber salts melt at 90 degrees Fahrenheit which is the temperature they liberate their most heat during the transition from liquid to solid.) TC-57 This is a chance for the students to go through the whole process, and document their methods along the way. Emphasize leaving footprints of their methods, using correct units, and writing verbal justifications for their results. The volume is halved, so the linear dimensions will be the cube root of ½.”

Ask probing “why” question every chance you can here and don’t let students take shortcuts without reasons they can justify.

solar energy harvesting â€“ for the sows teacher annotations solar

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