MATHEMATICS 10 COMMON



MATHEMATICS 10C

MEASUREMENT

High School collaborative venture with

Harry Ainlay, Jasper Place, McNally, Queen Elizabeth,

Ross Sheppard and Victoria Schools

[pic]

Harry Ainlay: Mathias Stewart, Colin Veldkamp, Cindy Wilson

Jasper Place: Lisa Lei, Clayton Maleski

McNally: Neil Peterson

Queen Elizabeth: Michael Freed, Scott Nytchay

Ross Sheppard: Silvia Fernandes-Isidoro, Clarence Harker, Dean Walls

Victoria: ,Shannon Sookochoff

Facilitator: John Scammell (Consulting Services)

Editor: Rosalie Mazurok (Contracted)

2009 - 2010

TABLE OF CONTENTS

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|STAGE 1 DESIRED RESULTS |PAGE |

| | |

|Big Idea |4 |

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|Enduring Understandings |4 |

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|Essential Questions |4 |

|Knowledge |5 |

| | |

|Skills |6 |

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|Stage 2 ASSESSMENT EVIDENCE | |

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|Teacher Notes For Transfer Task |7 |

| | |

|Transfer Task | |

| Emergency Room Situation | |

|Teacher Notes for Emergency Room Situation and Rubric |8 - 9 |

|Transfer Task |10 - 14 |

|Rubric |15 |

|Possible Solution |16 -19 |

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|Stage 3 LEARNING PLANS | |

| | |

|Lesson #1 A World Without Standard |20 - 23 |

|Lesson #2 Explore Imperial |24 - 27 |

|Lesson #3 Explore Metric |28 - 31 |

|Lesson #4 Imperial to Metric…and Back |32 - 35 |

|Lesson #5 Surface Area |36 - 38 |

|Lesson #6 Volume |39 - 41 |

| | |

|APPENDIX - Handouts | |

| | |

|Measurement Unit Handouts |43 - 47 |

Mathematics 10 Common

Measurement

| |

|STAGE 1 Desired Results |

[pic] Big Idea:

Students will gain real and meaningful connections to the units of measure, providing them with the capacity to describe the world.

[pic] Enduring Understandings:

Students will understand that…

• Imperial and SI are related systems of measurement.

• different situations require different units of measure and different measurement tools.

• estimating is useful to approximate and validate measures.

• surface area and volume are related to the dimensions of 3-D objects.

[pic] Essential Questions:

• What would the world be like without measurement?

• How do we communicate measure?

o Who keeps the standards?

• In what circumstances is accuracy important?

o When is estimation good enough?

[pic] Knowledge:

|Enduring |Specific |Knowledge that applies to this Enduring Understanding |

|Understanding |Outcomes | |

| | | |

|Students will understand that… | |Students will know… |

| |*M1 | |

|Imperial and SI are related systems of | |the different units of measure for each system. |

|measurement. | | |

| |*M2 |that conversions within and between systems exist. |

| | | |

|Students will understand that… | |Students will know… |

| | | |

|different situations require different units |*M1 |the advantages and disadvantages of various measurement tools|

|of measure and different measurement tools. | |in diverse contexts. |

| | |appropriate units of measure for varying lengths. |

| | | |

|Students will understand that… | |Students will know… |

| |*M1 | |

|estimating is useful to approximate and | |referents for common units of |

|validate measures. |*M2 |measure. |

| | | |

| | | |

|Students will understand that… | |Students will know… |

| | | |

|surface area and volume are related to the |*M3 |the connections between objects, their dimensions and the |

|dimensions of 3-D objects. | |related formulas. |

| | | |

*M = Measurement

[pic] Skills:

|Enduring |Specific |Skills that apply to this |

|Understanding |Outcomes |Enduring Understanding |

| | | |

|Students will understand that… |*M1 |Students will be able to… |

| | | |

|Imperial and SI are related systems of | |measure using referents or either of the systems of standard |

|measurement. |*M2 |measure. |

| | | |

| | |compare and convert measurements. |

| | | |

|Students will understand that… | |Students will be able to… |

| | | |

|different situations require different units|*M1 |justify choice of tools and units. |

|of measure and different measurement tools. | | |

| | |use different measuring tools. |

| | | |

|Students will understand that… | |Students will be able to… |

| |*M1 | |

| | |choose a referent to estimate a measure and explain the |

|estimating is useful to approximate and | |process used. |

|validate measures. |*M2 | |

| | |use estimation to validate other types of thinking. |

| | | |

| | | |

|Students will understand that… | |Students will be able to… |

| | | |

| | |solve problems using some of the following strategies: |

|surface area and volume are related to the |*M3 |a provided formula |

|dimensions of 3-D objects. | |deriving a formula |

| | |manipulating a given formula |

| | |drawing nets |

| | |building models |

*M = Measurement

| |

|STAGE 2 Assessment Evidence |

1 Desired ResultsDesired Results

[pic] Emergency Room Situation

Teacher Notes

There is one transfer task to evaluate student understanding of the concepts relating to measurement. A photocopy-ready version of the transfer task is included in this section.

Each student will:

• Estimate the surface area of his/her body using referents and/or geometric calculations.

Teacher Notes for Emergency Room Situation Transfer Task

The following website could be used to calculate and check the estimate of the surface area of the body:



Estimation tools

paper

hands

towels

cheese cloth

clothing

Suggestions for Differentiated Instruction

The students may choose to use a formula to actually calculate the surface area of their bodies.



Come up with a way to convert from Imperial square units to Metric square units.

Suggestions for Extensions

Volume

• Calculate the volume of your body.

o Immerse your body in a tub full of water, calculate your overflow. You may use composite shapes.

o Blood Volume Calculator



Teacher Notes for Rubric

• No score is awarded for the Insufficient/Blank column , because there is no evidence of student performance.

• Limited is considered a pass. The only failures come from Insufficient/Blank.

• When work is judged to be Limited or Insufficient/Blank, the teacher makes decisions about appropriate intervention to help the student improve.

[pic] Emergency Room Situation - Student Assessment Task

Scenario:

Sirens are blaring . . .

Nurse 1: “The ambulance is coming and there is a patient who has been burned from

head to toe.”

Nurse 2: “Quick! Call the skin bank and request 1000 square inches of skin”

The Doctor assesses the patient…

Doctor: (motions to you) “Hey, Intern! You are the exact same size as the patient.

Quick, calculate to see if they have ordered enough skin!!”

The following criteria must be considered:

Part A

• Choose a referent to make an estimation of the SA, surface area of the human body in square inches. Provide a full explanation of your results.

• Imagine the body as a composite figure composed of standard 3-Dimensional shapes that we will be examining in class. Provide a detailed calculation in square inches.

• Compare your initial estimation to your final answer.

• Summarize your findings.

Part B

The patient has lost 30% of his/her bodily fluids . . . calculate the volume of his/her body and find out how much fluid would be necessary to replenish this amount.

Initial Skin Graft Preparation Form

Patient’s Name:

Height: Weight: Gender:

Initial estimate of needed graft area:

Detailed Skin Graft Preparation Form

Detailed calculation of graft area:

[pic] Assessment Mathematics 10C

Emergency Room Transfer Task Rubric

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| |

Lesson 1

A World Without Standard

|[pic]STAGE 1 |

| |

|BIG IDEA: |

| |

|Students will gain real and meaningful connections to the units of measure, providing them with the capacity to describe the world. |

| |

| | |

|ENDURING UNDERSTANDINGS: |ESSENTIAL QUESTIONS: |

| | |

|Students will understand that… |What would the world be like without measurement? |

| | |

|different situations require different units of measure and different|How do we communicate measure? |

|measurement tools. |Who keeps the standards? |

| | |

|estimating is useful to approximate and validate measures. |In what circumstances is accuracy important? |

| |When is estimation good enough? |

| | |

|KNOWLEDGE: |SKILLS: |

| | |

|Students will know… |Students will be able to… |

| | |

|appropriate units of measure for varying |choose a referent to estimate a measure and explain the process used.|

|lengths. | |

| |justify choice of tools and units. |

|referents for common units of measure. | |

| |use estimation to validate different ways of measuring. |

[pic]Lesson Summary

• Students will see relevance of numbers in measurement by trying to describe objects without using numbers.

• Measure objects using different referents. Discuss varied responses to show the importance of standardization.

[pic] Lesson Plan

A World Without Numbers

Discuss what the world would be like without numbers. How would you describe lengths? How would you compare? Consider sharing the story “Neil’s Numberless World” by Lucy Coats.

Activate prior knowledge through discussion of situations where students may have already measured without tools using a referent (e.g. football yards, golf estimates, etc.)

Using Referents

Prepare an activity where students would measure various objects using referents of their choice (e.g. body parts, common objects at hand, etc.). To differentiate you could suggest 10 objects and ask students to choose 5 (see Going Beyond for enrichment opportunities). Then compile results to investigate the need for standardized measure. Once students choose a referent they should record it on the My Referent Table handout (See Appendix).

The Need for Standards

A separate activity that could be included with the previous one is to have students measure the same object using the same referent (hands, for example) and investigate if this is standardized enough. If not, how could the system be improved?

Consider asking for the My Referent Table (See Appendix) as an exit slip.

[pic] Going Beyond

The choice of objects would be a great way to differentiate this lesson. To enrich you could choose objects that are very small, or ones that are larger and/or slightly beyond their reach. For example, the width of coins, the height of a wall or gym etc.

When completing the third column of the My Referent Table handout, encourage examples of items not used during the classroom activity.

[pic]Resources

Foundations and Pre-calculus Mathematics 10 (Pearson: sec 1.1)

Math 10 (McGraw Hill: sec 1.1)

Neil’s Numberless World by Lucy Coats

My Referent Table (See Appendix)

A variety of objects to measure

[pic]Assessment

An Exit Slip - students should show their own systems of measurement.

[pic] Glossary

exit slip – a task to be submitted to be able to leave at the end of class

referent – something that serves as a reference to which something else may be compared

standard system – an authorized model used to define a unit of measurement

system of measurement – a system of measurement units such as the Metric

System or the Imperial System

Lesson 2

Explore Imperial

|[pic]STAGE 1 |

| |

|BIG IDEA: |

| |

|Students will gain real and meaningful connections to the units of measure, providing them with the capacity to describe the world. |

| |

| | |

|ENDURING UNDERSTANDINGS: |ESSENTIAL QUESTIONS: |

| | |

|Students will understand that… | |

| | |

|Imperial and SI are related systems of measurement. |How do we communicate measure? |

| |Who keeps the standards? |

|different situations require different units of measure and different| |

|measurement tools. |In what circumstances is accuracy important? |

| |When is estimation good enough? |

|estimating is useful to approximate and validate measures. | |

| | |

|KNOWLEDGE: |SKILLS: |

| | |

|Students will know… |Students will be able to… |

| | |

|the different units of measure for this system. |measure using referents and this system of standard measure. |

| | |

|that conversions within these systems exist. |justify choice of tools and units. |

| | |

|the advantages and disadvantages of various measurement tools in |use different measuring tools. |

|diverse contexts. | |

| |choose a referent to estimate a measure and explain the process used.|

|appropriate units of measure for varying lengths. | |

| |use estimation to validate different ways of measuring. |

|referents for common units of measure. | |

[pic]Lesson Summary

• Students will become familiar with units of Imperial and relate these to their referents.

• Students will be able to determine appropriate units to measure varying lengths.

• Have students estimate and measure objects using Imperial measurement tools.

[pic] Lesson Plan

Imperial Standards

Measure a medium sized object (exactly 3 feet) using the student’s medium referent from the previous day. Have students compare their results. Point out that “Sarah” is queen and her referent is the correct one. After introducing this standard ask students to try and relate their measures to hers. This leads nicely into the video on body measures.

My Human Ruler

Introduce students to measuring tools that are in Imperial units focusing only on whole numbers. Using appropriate tools have students develop a human ruler, deciding what part of their body best compares with an inch, foot and yard. Record decisions on My Human Ruler (see Appendix).

Fraction Strip

Start a discussion around lengths that aren’t whole numbers? We can estimate these, but how do we get exact lengths? Show students how to measure within an inch. A suggestion would be to utilize fraction strips. Start with a strip of paper, define it has a unit length, and then fold it in half to show measuring accurately to a half of an inch. Then show folding it in half again to develop an accuracy of a quarter of an inch. Point out that [pic]is equal to [pic]. Go back to fold it in half again creating an accuracy of an eighth. Again help students see that [pic] is equal to [pic]and [pic]is equal to [pic]. You could carry it to [pic]or [pic]if necessary.

Practise Measuring

Have students measure a few objects; the objects from the previous day would be ideal. If you have students measure the same objects as you could have them compare the measurements between the referent and the Imperial System.

Build-a-box

A closing activity would have students demonstrate that they know how to measure with Imperial measuring tools. For example, build a box that would be able to store a certain object with little wasted space. Have a box that will store an object so that the object is unable to move. Students will have to take measurements of the box and then use their measurements to construct a replica of the box. Test the accuracy of measurements by testing the fit of the object. Have students journal the steps they took in creating the box to develop communication skills. Encourage students to discuss the need for accuracy in their measurements.

The use of this box could be carried forward to future lessons regarding Metric System, Surface Area and Volume.

[pic] Going Beyond

Enrichment would come from the objects that students measure, both in size and shape. With shapes, it would be interesting to see how students would measure the perimeter of non-rectangular objects.

Challenge students to be precise and to refine their measurements. Provide students with opportunities to compare measurements and share strategies to improve their measurement accuracy

[pic]Resources

Math 10 (McGraw Hill: sec 1.2)

Foundations and Pre-calculus Mathematics 10 (Pearson: sec 1.2)

Imperial rulers

Imperial tape measures

sewing tape measure

yard stick

Video on Body Measures



[pic]Assessment

Assessment will be combined with next lesson.

[pic] Glossary

foot – a unit of length equal to 30.48 centimetres or12 inches

Imperial System – in the United Kingdom, belonging or conforming to the non-metric system of weights and measures that includes the foot, pound, and gallon

inch – a unit of length equal to 2.54 centimetres or[pic]of a foot

yard – a unit of length equal to 0.9144 metres or 3 feet

Lesson 3

Explore Metric

|[pic]STAGE 1 |

| |

|BIG IDEA: |

| |

|Students will gain real and meaningful connections to the units of measure, providing them with the capacity to describe the world. |

| |

| | |

|ENDURING UNDERSTANDINGS: |ESSENTIAL QUESTIONS: |

| | |

|Students will understand that… | |

| | |

|Imperial and SI are related systems of measurement. |How do we communicate measure? |

| |Who keeps the standards? |

|different situations require different units of measure and different| |

|measurement tools. |In what circumstances is accuracy important? |

| |When is estimation good enough? |

|estimating is useful to approximate and validate measures. | |

| | |

|KNOWLEDGE: |SKILLS: |

| | |

|Students will know… |Students will be able to… |

| | |

|the different units of measure for this system. |measure using referents and this system of standard measure. |

| | |

|that conversions within these systems exist. |compare and convert measurements within this systems. |

| | |

|the advantages and disadvantages of various measurement tools in |justify choice of tools and units. |

|diverse contexts. | |

| |use different measuring tools. |

|appropriate units of measure for varying lengths. | |

| |choose a referent to estimate a measure and explain the process used.|

|referents for common units of measure. | |

| |use estimation to validate different ways of measuring. |

[pic]Lesson Summary

• Introduce the Metric System by showing some challenges of the Imperial System.

• Students will become familiar with the metric units and relate these to their referents.

• Students will be able to determine appropriate units to measure varying lengths.

• Have students estimate and measure objects using metric measurement tools.

[pic] Lesson Plan

Imperial vs. Metric

Begin by highlighting some of the difficulties within the Imperial system. For example, when dealing with the fractions of an inch, adding measurements together is difficult for some. Conversions within the Imperial System are not straightforward and the ratios would need to be memorized. See Conversion Chart in Appendix.

My Human Ruler

Introduce students to measuring tools (ruler, measuring tape, metre stick) that are in metric units. Using appropriate measuring tools have students develop a human ruler, deciding what part of their body best compares with a centimetre and metre. The intent here is to assist students to be better able to estimate lengths. Have students record their decisions on the My Human Ruler handout. (See Appendix)

Practise Measuring

Have students measure a few objects. The objects from the previous day would be ideal. If you have students measure the same objects as they did using their referents you could have them compare the measures between the referent, Imperial and the Metric System.

Measure-a-box

Prepare a closing activity that would have students demonstrate their ability to measure. For example measure the dimensions of the box that was built in the last lesson.

Why we need metric video (American Chopper) is a humorous look at the difficulties of the Imperial System.

[pic] Going Beyond

Enrichment would come from the objects that students measure, both in size and shape. With shapes it would be interesting to see how students would measure the perimeter of non-rectangular objects.

Challenge students to be precise and to refine their measurements. Provide students with opportunities to compare measurements and share strategies to improve their measurement accuracy.

Students could explore the use of callipers and micrometers.

[pic]Resources

Foundations and Pre-calculus Mathematics 10 (Pearson: sec 1.3)

Math 10 (McGraw Hill: sec 1.3)

Conversions Chart (See Appendix)

meter stick

metric rulers

metric tape measures

sewing tape measure

Why we need metric video (American Chopper)



Metric & Standard Measures Video Clip



[pic]Assessment

Assess Metric and Imperial measuring skills through the “Build a Box” Activity (see Lesson 2).

[pic] Glossary

centimetre – a unit of measurement equalling [pic]

metre – the basic unit of length in the International System of Units (SI)

metric system – a decimal system of weights and measures based on units of 10

millimetre – a unit of measurement equalling [pic]

SI – International System of Units (Metric System)

Lesson 4

Imperial to Metric…and Back

|[pic]STAGE 1 |

| |

|BIG IDEA: |

| |

|Students will gain real and meaningful connections to the units of measure, providing them with the capacity to describe the world. |

| |

| | |

|ENDURING UNDERSTANDINGS: |ESSENTIAL QUESTIONS: |

| | |

|Students will understand that… | |

| | |

|Imperial and SI are related systems of measurement. |How do we communicate measure? |

| |Who keeps the standards? |

|different situations require different units of measure and different| |

|measurement tools. |In what circumstances is accuracy important? |

| |When is estimation good enough? |

|estimating is useful to approximate and validate measures. | |

| | |

|KNOWLEDGE: |SKILLS: |

| | |

|Students will know… |Students will be able to… |

| | |

|the different units of measure for this system. |compare and convert measurements within and between systems. |

| | |

|that conversions within this systems exist. |justify choice of tools and units. |

| | |

|the advantages and disadvantages of various measurement tools in |choose a referent to estimate a measure and explain the process used.|

|diverse contexts. | |

| |use estimation to validate different ways of measuring. |

|appropriate units of measure for varying lengths. | |

| | |

|referents for common units of measure. | |

[pic]Lesson Summary

• Explain the importance of equivalent measurements within and between different systems. For example, 12 inches = 30 cm, 1000 m = 1 km.

• Have students convert within each system and between the Metric and Imperial Systems.

[pic] Lesson Plan

Warm Up

Review base 10 multiplications / divisions with a sequence of warm up questions.

• 32 x 10

• 32 x 100

• 32 x 1000

• 32 / 10

• 32 / 100

Importance of Conversions

Discuss conversions within the systems to help students understand the process. To help students understand the purpose behind conversions you can engage in a discussion around 2-D and 3-D real life situations. For example, after purchasing a shelving unit from “IKEA”, you see the box dimensions are 4 inches by 72 inches by 114 inches. If you are unfamiliar with the inch measurement, it is hard to judge whether the carton will fit in the back of your car. If we could convert the inches to feet it would make it easier to estimate.

Proportionality Activity – “How many … in a …”

To help students develop the comparison between measurements using their referents, have each student determine how many “human inches” will fit in a “human foot.” (Basically, compare how many thumb widths to their foot). In comparing results, ideally students will come to a consensus that 12 inches will fit in a foot. Once this has been established have students extend this knowledge to how many inches fit in 2 feet or in 3 feet. What if it was 2.5 feet? After students have developed a strategy for these conversions you could provide them with conversion factors (see conversion chart in Appendix) for other lengths to practice with. The hope is that students will discover the idea of proportionality.

Sample Problem

Another problem to present to students would be comparing two runners, one running a mile in 15 minutes and the other running 1000 yards in 12 minutes. Which runner ran further? Faster?

Importance of Conversions Between

For conversions between imperial and metric you could begin by discussing the need for this, and the need to communicate in a common language. Examples of this include: Speed limits in different countries, lengths of material, volumes of quantities, etc.

Ask students to explore the following proportions:

• How many cm in an inch?

• How many inches in a cm?

• How many feet in a metre?

• How many metres in a foot?

Follow up by illustrating the number of centimetres in an inch using the “Inch vs. Centimetre Image” (see Appendix) and highlight known proportions on the “Conversion Chart” (see Appendix).

Practise Conversions

Provide students with opportunities to practise conversions (refer to textbook).

Encourage the continued use of referents and estimation to verify the reasonableness of all answers.

Tip: Check the reasonableness of answer by recognizing that a greater number of smaller units is required to express an equivalent measure in larger units.

[pic] Going Beyond

Have students look up other units. The focus has been on linear units. Students could look into 2-D and 3-D units (the names, the conversions).

Challenge students to do conversions of large numbers in scientific notation.

[pic]Resources

Math 10 (McGraw Hill: sec 1.3)

Foundations and Pre-calculus Mathematics 10 (Pearson: sec 1.3)

Conversion Chart (See Appendix)

Objects to Measure

Inches to cm Image (see Appendix)

[pic]Assessment

Provide students with a list of measurements already converted. Make some with mistakes so that they have to determine which are done correctly and which need to be corrected.

Students should attempt some conversions and then check their answers using referents and measurement tools.

Use the box that students created (See Lesson 2) and check the accuracy of their Imperial and Metric measurements by converting both ways.

[pic] Glossary

conversion – a change from one measuring or calculating system to another,

or a calculation done to bring about the change

metric prefixes – relating to or using the metric system of measurement

proportional – possessing a constant ratio

Lesson 5

Surface Area

|[pic]STAGE 1 |

| |

|BIG IDEA: |

| |

|Students will gain real and meaningful connections to the units of measure, providing them with the capacity to describe the world. |

| |

| | |

|ENDURING UNDERSTANDINGS: |ESSENTIAL QUESTIONS: |

| | |

| |What are meaningful applications of surface area and volume? |

|Students will understand that… | |

| |What are the advantages and disadvantages of each shape? |

|surface area and volume are related to the dimensions of 3-D objects.|What fascinates artists, mathematicians and designers about the |

|. |beauty of shapes? |

| |What is the importance of basic figures? |

| |How do degree, exponents and dimensions relate? |

| | |

|KNOWLEDGE: |SKILLS: |

| | |

| |Students will be able to… |

| | |

|Students will know… |solve problems using some of the following strategies: |

| |a provided formula |

|the connections between objects, their dimensions and the related |deriving a formula |

|formulas. |manipulating a given formula |

| |drawing nets |

| |building models |

[pic]Lesson Summary

• Review formulas from Junior High to determine surface area of prisms and cylinders.

• Apply this understanding to the surface area of pyramids, cones and spheres through the use of nets and real world connections.

• Given appropriate formulas, solve for an unknown dimension.

[pic] Lesson Plan

Review Surface Area

Students should have an understanding of surface area of prisms and cylinders from Junior High.

To begin this lesson it would be worthwhile to check the students’ understanding through discussion, practice questions or providing real objects that students would measure and then calculate the surface area. It may be worthwhile to investigate student understanding of surface area. For example, some students may know it as the application of a formula and others may understand that surface area can be determined by combining the areas of all individual faces.

Review Idea: To review surface area of rectangular prism, have students calculate the surface area of the box they created (See Lesson 2).

Importance of Surface Area

It may be worthwhile to facilitate a discussion around the meaningful applications of surface area. Discussion topics may include:

• What fascinates artists, mathematicians and designers about the beauty of shapes?

• What is the importance of basic figures?

Some examples of real world connections are: sod, wrapping paper, paint, wall paper, flooring, sculpting, etc.

Surface Area of Pyramids, Cones and Spheres

Once all students seem to have arrived at similar understandings, have them apply this knowledge to pyramids, cones and spheres. Using a net approach will be easy for a pyramid. For a cone you would need to show [pic] as the formula for calculating the area of the sector. With a sphere, you can use an orange. Take the peelings and piece it together making 4 circles that have the same radius as the radius of the orange. For teacher reference see video: Surface Area of a Sphere

Missing Dimensions

After students have practiced determining surface area you would then need to give them a surface area and ask them to determine a given dimension.

[pic] Going Beyond

Have students discover/prove why [pic] can be used to determine the area of the sector which is the side of the cone.

[pic]Resources

Foundations and Pre-calculus Mathematics 10 (Pearson Publishers: sec 1.4, 1.7)

Math 10 (McGraw Hill: sec 2.1, 2.2)

Surface Area of a Sphere



[pic] Glossary

dimension – a measurement of something in one or more directions such as length, width, or height

net – a set of polygons in a plane, all connected by certain edges such that when "folded up" form a polyhedron

square centimetre – any area equivalent to the area formed by a 1 cm by 1 cm square

square metre – any area equivalent to the area formed by a 1 m by 1 m square

Lesson 6

Volume

|[pic]STAGE 1 |

| |

|BIG IDEA: |

| |

|Students will gain real and meaningful connections to the units of measure, providing them with the capacity to describe the world. |

| |

| | |

|ENDURING UNDERSTANDINGS: |ESSENTIAL QUESTIONS: |

| | |

| |What are meaningful applications of surface area and volume? |

|Students will understand that… | |

| |What are the advantages and disadvantages of each shape? |

|surface area and volume are related to the dimensions of 3-D objects.|What fascinates artists, mathematicians and designers about the |

|. |beauty of shapes? |

| |What is the importance of basic figures? |

| |How do degree, exponents and dimensions relate? |

| | |

|KNOWLEDGE: |SKILLS: |

| | |

| |Students will be able to… |

| | |

|Students will know… |solve problems using some of the following strategies: |

| | |

|the connections between objects, their dimensions and the related |a provided formula |

|formulas. |deriving a formula |

| |manipulating a given formula |

| |drawing nets |

| |building models |

[pic]Lesson Summary

Students will develop formulas for and solve problems relating to the volumes of prisms, pyramids, right cones, right cylinders, and spheres. An exploration of the relationship between pyramids and prisms as well as that of cylinders and cones will be done.

[pic] Lesson Plan

Stacking Activity

Develop the volume of a prism by “stacking pancakes”.

• By stacking various shapes students can see that the volume of a prism is the “area of the base multiplied by the height”.

• Discuss the relationship between degree, exponents and dimensions.

The “[pic]” Relationship

Explore the relationship between prisms and pyramids, cones and cylinders, with the same base and height through the following two activities:

• 3-D Mystery Model Activity (See Appendix).

• Prism Capacity Activity

o Students will try to fill, using rice, various prisms by using the corresponding pyramid of that shape to discover the [pic] relationship.

o See YouTube Clips: Deriving the Formula – Volume of Cone and Deriving the Formula – Volume of Pyramid .

.

Volume in the World

Discover Real World Connections of the Volumes of Shapes.

• Some possible activities include a home project exploring the volume of concrete, aquarium, dirt, mulch, etc.

• Possible problems include a pirate ship that has a hole and will sink when 50% of the ship is filled with water. If the ship fills at a certain rate, how long until the ship sinks?

[pic] Going Beyond

Explore building composite structures and the volume of material needed to construct them.

Students may solve problems in Imperial and Metric Systems.

[pic]Resources

Math 10 (McGraw Hill: sec 2.3)

Foundations and Pre-calculus Mathematics 10 (Pearson: sec 1.5-1.7)

Deriving the Formula – Volume of Cone

()

Deriving the Formula – Volume of Pyramid

()

[pic] Glossary

cubic centimetre – any volume equivalent to the volume formed by a 1 cm by 1 cm by 1 cm cube

cubic metre – any volume equivalent to the volume formed by a 1 m by 1 m by 1 m cube

exponent – the number in a power that represents how many times a base is used as a factor

Appendix

Handouts

My Referent Table

|When things are _______ |I would use ______ as a referent |to measure things like ________ |

|Small | | |

|Medium | | |

|Large | | |

My Human Ruler

|When I want to estimate using the units of ______|I compare to my _____________ |

|Inches | |

|Feet | |

|Yards | |

|Centimetres | |

|Meters | |

CONVERSION CHART

|Relationships between common Imperial Units |Relationships between Common Imperial Units and Metric Units |

|Length | 1 inch = 2.54 cm 1 cm = 0.3937 inches |

|1 mile = 1760 yards = 5280 feet | 1 mile = 1.609 km 1 km = 0.6214 miles |

|1 yard = 3 feet = 36 inches | 1 yard = 0.9144 m 1 m = 1.0936 yards |

|1 foot = 12 inches | 1 foot = 0.3048 m 1 m = 3.2808 feet |

Inch vs Centimetre Ruler

Here we can see that 1 inch = approximately 2.54 centimetres.

3-D MYSTERY MODEL ACTIVITY

Revised and used with permission from Bryan Quinn.

You will need one Pyramid A and four Pyramid Bs to complete this activity. Use the nets given to cut out and make the five pyramids from cardstock (or paper). Cut on solid lines and fold on the dotted lines and put glue on the tabs (or tape edges).

1. Without taping the pyramids together, arrange your 5 pyramids to make three congruent pyramids in a row, as shown below.

2. Start with your answer to #1 and tape together the five pyramids in such a way that the taped edges act as hinges, so that the pieces of the 3-D model above can be “folded up” to form a square prism

3. State the geometric property relating the volume of pyramids and prisms demonstrated by this model. Can we extend this property to other volume of prisms formulas?

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Implementation note:

Post the BIG IDEA in a prominent

place in your classroom and refer to it often.

Implementation note:

Ask students to consider one of the essential questions every lesson or two.

Has their thinking changed or evolved?

Implementation note:

Teachers need to continually ask

themselves, if their students are acquiring the knowledge and skills needed for the unit.

Implementation note:

Students must be given the transfer task & rubric at the beginning of the unit. They need to know how they will be assessed and what they are working toward.

Choose a referent to make an estimation of the SA of the patient in square inches. Provide a full explanation of your results.

Lab Work

Convert your dimensions from Imperial to Metric

|Body Part |Dimensions (inches) |Conversions (centimetres) |

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• Imagine the body as a composite figure composed of standard 3-dimensional shapes.

• Provide a drawing/model.

• Provide a detailed calculation.

•

Compare the estimate to calculation:

Names: ________________________

________________________

[pic]

[pic]

154 in2

/14

/15

Choose a referent to make

an estimation of the SA of

the patient in square inches.

Provide a full explanation of your results

Compare the estimate to calculation:

Other Research (if done):

Intern’s recommendation to the doctor:

Part B

Volume calculations:

This could be the completion of My Referent Table included in the Appendix.

Try to ensure students are measuring common objects with a variety of lengths. This will lead to the need for a standard.

The goal is to discuss the need for referents as an estimation tool.

The intent here is to assist students to better estimate lengths with Imperial units.

2902 in2.

2902 in2

Implementation note:

Each lesson is a conceptual unit and is not intended to be taught on a one lesson per block basis. Each represents a concept to be covered and can take anywhere from part of a class to several classes to complete.

Pyramid A

3

Glue Here

2 Pyramid Bs

Pyramid A

2 Pyramid Bs

Glue Here

Pyramid B

Pyramid B

Pyramid B

Pyramid B

Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (). Some terms can be found in more than one division. Some terms have animations to illustrate meanings.

Implementation note:

Teachers need to consider what performances and products will reveal evidence of understanding?

What other evidence will be collected to reflect

the desired results?

4

Emergency Room Situation

Sample Student Exemplar

[pic]

Intern’s recommendation to the doctor:

Compare the estimate to calculation:

Intern’s recommendation to the doctor:

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