area perimeter volume 2

TIPS: Section 2 Developing Perimeter and Area Formulas Queens Printer for Ontario, 2003 Page 1

Developing Perimeter and Area Formulas

Context Stand in any room, in any building, or on any street and you can see the context for a measurement

problem. Students learn through posing their own measurement problems, drawing from their prior knowledge,

and making connections. When students learn how to develop measurement formulas for the basic shapes, they are building

understanding that can be extended when they encounter a new irregular shape. Optimization problems, such as maximizing the area enclosed by a length of fence, require the use

and development of formulas.

Context Connections

Signs/Logos/Symbols Designing Irregular Surfaces Sports Field

Packaging Frames Buildings Tracks

Fencing Sewing Fractions Other Connections

Manipulatives tiles cubes 3-D models nets geoboards

Technology The Geometers Sketchpad Fathom calculators/graphing calculators spreadsheet software

Other Resources http://standards.nctm.org/document/chapter6/meas.htm http://mmmproject.org/dp/mainframe.htm http://www.shodor.org/interactivate/activities/perm/index.html


Connections Across Grades Selected results of word search using the Ontario Curriculum Unit Planner Search Words: develop, understand, comparison, relate, investigation, pose, surface area, composite, irregular, formula

Grade 6 Grade 7 Grade 8 Grade 9 Grade 10 solve problems related to the

calculation and comparison of the perimeter and the area of regular polygons

relate dimensions of rectangles and area to factors and products, e.g., in a rectangle 2 cm by 3 cm the side lengths are factors and the area, 6 cm2, is the product of the factors

understand the relationship between the area of a parallelogram and the area of a rectangle, between the area of a triangle and the area of a rectangle, and between the area of a triangle and the area of a parallelogram

understand the relationship between area and lengths of sides and between perimeter and lengths of sides for squares, rectangles, triangles, and parallelograms

develop rules for calculating the volume of rectangular prisms, generalize rules, and develop formulas, e.g., Volume = surface area of the base height

pose problems by recognizing a pattern, e.g., comparing the perimeters of rectangles with equal area

solve problems related to the calculation and comparison of the perimeter and the area of irregular two-dimensional shapes

understand that irregular two-dimensional shapes can be decomposed into simple two-dimensional shapes to find the area and perimeter

develop the formula for finding the area of a trapezoid

develop the formulas for finding the area of a parallelogram and the area of a triangle

develop the formula for finding the surface area of a rectangular prism using nets

estimate and calculate the perimeter and area of an irregular two-dimensional shape, e.g., trapezoid, hexagon

ask questions to clarify and extend their knowledge of linear measurement, area, volume, capacity, and mass, using appropriate measurement vocabulary

develop the formula for finding the circumference and the formula for finding the area of a circle

develop the formula for finding the surface area of a triangular prism using nets

estimate and calculate the radius, diameter, circumference, and area of a circle, using a formula in a problem-solving context

Academic identify, through investigation, the

effect of varying the dimensions of a rectangular prism or cylinder on the volume or surface area of the object

identify, through investigation, the relationships between the volume and surface area of a given rectangular prism or cylinder

pose a problem involving the relationship between the perimeter and the area of a figure when one of the measures is fixed

solve simple problems, using the formulas for the surface area and the volume of prisms, pyramids, cylinders, cones, and spheres (Note: Students should develop these concepts.)

Applied solve problems involving the area

of composite plane figures, e.g., combinations of rectangles, triangles, parallelograms, trapezoids, and circles

solve problems involving perimeter, area, surface area, volume, and capacity in applications (Note: Students should develop these concepts.)

Academic and Applied solve simple problems, using the

formulas for the surface area of prisms and cylinders and for the volume of prisms, cylinders, cones, and spheres

Academic determine the

properties of similar triangles, e.g., the correspondence and equality of angles, the ratio of corresponding sides, the ratio of areas, through investigation, using dynamic geometry software

Applied determine some

properties of similar triangles through investigation, using dynamic geometry software

construct tables of values, sketch graphs, and write equations of the form y = ax2 + b to represent quadratic functions derived from descriptions of realistic situations, e.g., vary the side length of a cube and observe the effect on the surface area of the cube

Summary of Prior Learning and Next Steps In earlier years, students: build understanding and familiarity with measurement attributes (length, height, width, perimeter, area, etc.); become familiar with both standard (metric) and non-standard units of measure (e.g., area of a page is four pencil cases); estimate regular and irregular areas using grid paper (e.g., surface area of a puddle); explain the difference between perimeter and area and understand when each measure should be used; develop rules and formulas for perimeter and area of rectangles, squares (Grade 5). Note: Students use area formulas for triangles and parallelograms in Grade 6 before the formulas are developed in Grade 7. In Grade 7, students: focus on developing area and perimeter formulas for triangles and some quadrilaterals (rectangle, square, trapezoid, parallelogram); determine perimeter and area of complex shapes by decomposing into simple 2-D shapes which leads to determining the surface area of a

rectangular prism by constructing the net; after demonstrating understanding of the grade-level formulas, should become proficient in the use of the formulas. In Grade 8, students: focus on developing formulas for the area and circumference of a circle; after demonstrating understanding of the circle, should become proficient in the use of formulas. In Grade 9 Applied, students: are introduced to cones, cylinders, pyramids, and spheres surface area and volume formulas are developed and used; use formulas developed in earlier grades as a good foundation for grade-level investigations; after demonstrating understanding, should become proficient in the use of the formulas for surface area of prisms, pyramids, cylinders, cones,

and spheres. In Grade 10, students: proficiency is assumed concepts become tools for investigations and building new knowledge.


Instruction Connections

Suggested Instructional Strategies Helping to Develop Understanding Grade 7 Use non-standard units for measuring length, area, and volume students

must think conceptually about the required measurement to measure a desk. Ask students which type of measurement they are going to make and then choose an appropriate non-standard unit.

Attribute Unit of Measurement Length length of a ruler, length of a piece of paper Area area of one face of a book, area of a sticky

note Cut a piece of paper into several pieces reassemble and compare the

perimeter and areas to help students understand that two different shapes can have the same area but different perimeters.

Use transparent grid paper to estimate areas of irregular shapes, e.g., picture of a puddle experiment with different types of grid paper to develop understanding of units of measure.

Discuss appropriate units of measure, reasons for standard measures, e.g., cm, and selection of appropriate units depending on precision requirements, e.g., would you measure the length of a pencil and the distance to the next city with the same standard measure? Explain.

Develop the formula for the area of a rectangle by using geoboards collect data in charts start with whole number dimensions then use grid paper to move to non-whole number dimensions.

Make connections between the two common formulas for the area of a rectangle. (Area = l w and Area = b h) the formula Area = b h (base height) connects to the formulas for the area of a triangle, a trapezoid, and a parallelogram.

Develop the formula for the area of a parallelogram by using paper models students can use their knowledge about rectangles to determine the area of a parallelogram, e.g., any parallelogram can become a rectangle therefore the area formula is the same (A = b h).

Use The Geometers Sketchpad sketches to facilitate the development of area formulas (the pictures above can become dynamic with GSP).

Develop the formula for the area of a triangle using the formula for the area of a parallelogram two identical paper triangles can be arranged to form a parallelogram since 2 triangles are used the area of one triangle is one-half of the area of a parallelogram A = (base height).

Grade 8 Ask questions to clarify understanding about measurement, attributes, units

of measure, and unit choices. Develop the formulas for the circumference and area of a circle through

investigation using paper circles, string/rope, and The Geometers Sketchpad use the formulas only after students have developed them.

Connect students knowledge about rectangles and circles to surface area of cylinders. Although the formula for surface area of cylinders is not part of the Grade 8 expectations, all necessary concepts are components of the Grade 8 curriculum.

Construct nets for triangular prisms, use prior knowledge about composite 2-D shapes to determine the net area then develop a formula.

Grade 9 Applied Develop the formulas for surface area of prisms, pyramids, cylinders, cones,

and spheres before using the formulas. Use concrete materials students pose problems. Investigate relationships between perimeter and area by graphing data with

graphing calculators students estimate and hypothesize. Pose what if questions. Support students in developing proficiency in using formulas.

When grid paper is used to estimate areas, the area is expressed in terms of grid squares. It may be necessary to apply a scale factor to approximate the actual area represented by the drawing or diagram.

To establish a conceptual basis for understanding perimeter and area, students can use physical models to measure the perimeter (units needed to go around) and area (units needed to cover).

Height, length, slant height, side, and base are confusing to students, e.g., any side of a parallelogram can be called a base. Demonstrate that if you cut out the shape and placed any base on the floor, the height would be the perpendicular distance from the floor.

Shapes can be decomposed (separated) into simple 2-D shapes to find area or perimeter. However, when finding the perimeter of the illustrated shape, students may need to be reminded not to add the interior side of the rectangle and/or the interior diameter of the semi-circle.

For composite shapes, have students write word statements then replace the words with appropriate variables or formulas, e.g., in the previous diagram if the circle had a diameter of 4 m and the base of the rectangle was 10 m then:Perimeter = (perimeter of a semi-circle) + 2 sides + 1 side

= d 2 + 2b + d = (4) 2 + 2(10) + (4) etc.

When students cant remember a formula encourage them to recall how they developed the formula.

Use manipulatives, e.g., interlocking cubes, to help students understand that the volume of a rectangular prism is the area of its base times its height students need to understand that this basic formula is true for any prism.

For a 2-D polygon the base can be any side; however, for a 3-D prism the base is the face that stacks to create the prism. This face determines the name of the prism. Discuss.

Give students opportunities to progress through different representations (concretediagramssymbolic) use formulas only after students have personally developed them.


Connections Across Strands

Grade 7 Number Sense

and Numeration Measurement Geometry and Spatial Sense Patterning and Algebra Data Management

and Probability understand fractions by

connecting them to area use order of operations

in measurement formulas represent perfect squares

and their square roots in a variety of ways, e.g., by using blocks, grids

See Connections Across Grades, p. 2

identify, describe, compare, and classify geometric figures

identify, draw, and construct three-dimensional geometric figures from nets

recognize and sketch three-dimensional figures

build three-dimensional figures and objects from nets

explain why two shapes are congruent, e.g., bases of prisms

use patterns interpret a variable as a

symbol that may be replaced by a given set of numbers

write statements to interpret simple formulas

translate simple statements into formulas

systematically collect, organize, and analyse data

use conventional symbols, titles, and labels when displaying data

connect area to bar and circle graphs

Grade 8 Number Sense

and Numeration Measurement Geometry and Spatial Sense Patterning and Algebra Data Management

and Probability use order of operations

in measurement formulas use exponents in

measurement formulas develop understanding of

squares and square roots through formula for area of circle

develop estimation skills in working with


identify, describe, compare, and classify geometric figures

identify, draw, and construct nets for 3-D figures

investigate the Pythagorean relationship using area models and diagrams

apply the Pythagorean relationship to area problems

construct heights (line segments) of parallelograms, trapezoids and triangles using a variety of methods including paper folding

use patterns in algebraic terms

identify, create, and solve simple algebraic equations [formulas]

use the concept of variable to write equations [formulas] and algebraic expressions

write statements to interpret simple equations [formulas]

evaluate formulas translate complex

statements into formulas

systematically collect, organize, and analyse primary data

connect area to bar and circle graphs

Grade 9 Applied Number Sense and Algebra Measurement and Geometry Analytic Geometry Relationships

manipulate first-degree polynomial expressions to solve first-degree equations

demonstrate facility in operations with percent, ratio and rate, and rational numbers, as necessary to support other topics of the course, e.g., analytic geometry, measurement

use a scientific calculator to evaluate formulas

use the Pythagorean theorem substitute into measurement

formulas and solve for one variable, with and without the help of technology

use algebraic modelling

determine relationships between two variables [in measurement formulas] by collecting and analysing data

compare the graphs of linear and non-linear [measurement] relations

describe the connections between various representations of relations

demonstrate an understanding that straight lines represent linear relations and curves represent non-linear relations

construct tables of values, graphs and formulas of linear and non-linear relations

determine values for relations by using formulas


determine the relationship between variables in measurement formulas, e.g., the radius and area of a circle form a relationships that is non-linear

graph relationships that are determined by measurement formulas, e.g., investigate the relationship between the height and radius of a cone if the volume remains constant

select the equations of straight lines from a given set of equations of linear and non-linear relations, e.g., C = 2r and A = r2

identify the geometric significance of m and b in the equation y = mx + b through investigation, e.g., investigate the geometric significance of 2 in C=2r

create tables of values, plot points, graph lines, by hand and with technology

Summary or synthesis of curriculum expectations is in plain font

Verbatim curriculum expectations are in italics


Developing Mathematical Processes Grade 7 Short-answer Questions Addressing the Different Mathematical Processes

Name: Date:

Knowing Facts and Procedures Calculate the area of the given trapezoid. Show your work.

Hint: A = (a + b)h2

Reasoning and Proving Ryan baked a rectangular chocolate cake. His sister ate part of it. Now Ryan has to cut the rest of the cake to share equally with his brother David. Show where he should make the cut(s). Explain your answer.

Communicating The T-Square Tiling Company makes ceramic floor tiles. Note: The square tiles that are shown are the same size. AB and CD have the same length. Explain how Janet could use the formula for the area of a trapezoid to convince Meredith that the inside dark areas are the same size.

A B C D Hint: A = (a + b)h

2

Making Connections Andreas backyard is rectangular. Its dimensions are 15.0 m by 10.0 m. Andreas family is making a garden from the patio doors to the corners at the back of the yard. The patio doors are 2.0 m wide. Determine the area of the garden. Show your work.

Expectation Measurement, 7m37: Estimate and calculate the perimeter and area of an irregular two dimensional shape, e.g., trapezoid, hexagon.


Developing Mathematical Processes Grade 8 Short-answer Questions Addressing the Different Mathematical Processes

Name: Date:

Knowing Facts and Procedures Calculate the area of the given circle. Show your work.

r = 2.5 cm

Hint: A = r2

Reasoning and Proving Westview School has a track in the playground.

You want to run 2 km every day. Determine how many times you have to go around the track. Show your work.

Communicating Janice works at Buttonique a company that makes buttons. To determine the cost of producing a new button, she needs to know the surface area of one side of the button. Explain how Janice would determine this area.

Making Connections Pax Man figure has a radius of r units.

r

Which of the following formulas could be used to determine the perimeter of Pax Man?

a) 2r 14+ r + r b) 0.75r 2

c) 34

(2r) + r + r d) 2r 14

Give reasons for your answer.

Expectation Measurement, 8m47: Estimate and calculate the radius, diameter, circumference, and area of a circle, using a formula in a problem solving context.


Developing Mathematical Processes Grade 9 Applied Short-answer Questions Addressing the Different Mathematical Processes

Name: Date:

Knowing Facts and Procedures Determine the surface area of an open-topped cylinder that has a diameter of 8.0 cm and a height of 10.0 cm. Show your work.

Hint: There is only one circular face. A= r2 + 2rh

Reasoning and Proving A company produces labels for cans of food. A label goes completely around the can. Each can is 12.0 cm high and has a diameter of 8.0 cm. The labels are printed side by side on a long roll of paper that is the width of 1 label. label 1 label 2 label 3 label 4 etc.

Determine the number of labels that would fit on a roll of paper that is 20.0 m long. Show your work.

Communicating Jeremy says there are three formulas for the surface area of a cylinder:

1) A= r2 + 2rh 2) A= 2r2 + 2rh 3) A= 2rh

Explain his reasoning.

Making Connections How much cardboard would it take to make a box with dimensions 40 cm x 50 cm x 30 cm? Show your work.

Expectation Measurement and Geometry MG2.02: Solve simple problems using the formulas for the surface area of prisms and cylinders and for the volume of prisms cylinders cones and spheres.

12 cm

8 cm


Developing Proficiency Grade 7

Name: Date:

Proficiency Target Met. [Practise occasionally to maintain or improve your proficiency level.] Still Developing [Search out extra help and practise until you are ready for another opportunity to demonstrate proficiency.]

1. Determine the area of each shape. Show your work.

a)

Answer: ________________ b)

Answer: ________________ 2. Determine the perimeter of each shape. Show your work.

a)

Answer: ________________ b)

Answer: ________________

Expectation: Estimate and calculate the perimeter and area of an irregular two-dimensional shape, e.g., trapezoid, hexagon

4 cm

5 cm3 cm

8 cm

6 cm2 cm

2 cm

3 cm

5 cm6 cm

6 cm

12 cm

10 cm

3 cm

2 cm


Developing Proficiency Grade 8

Name: Date:


1. Determine the area of each shape. Show your work.

a)

Answer: ________________ b)

Answer: ________________ 2. Determine the perimeter of each shape. Show your work.

a)

Answer: ________________ b)

Answer: ________________

Expectation: Estimate and calculate the radius, diameter, circumference, and area of a circle, using a formula in a problem solving context

diameter = 6.5 cm

radius = 3.3 cm

radius = 2.7 cm

diameter = 7 cm


Developing Proficiency Grade 9 Applied

Name: Date:


1. Determine the surface area of each shape. Show your work.

a)

Base: 2.0 cm 5.0 cm Height = 3.0 cm

Answer: ________________ b)

Closed on both ends h = 12 m, r = 3

Answer: ________________

2. Determine the volume of each shape. Show your work.

a) h = 4.2 cm, r = 2.2 cm

Answer: ________________

b)

h = 23 cm, r = 10 cm Answer: ________________

Expectation: Solve simple problems using the formulas for the surface area of prisms and cylinders and for the volume of prisms cylinders cones and spheres


Extend Your Thinking Grade 7

Name: Date:

Find 3 different ways to determine the area of:

1

2

3

Expectation: Estimate and calculate the perimeter and area of an irregular two dimensional shape, e.g., trapezoid, hexagon


Extend Your Thinking Grade 7 (Answers) Possible ways students could respond.

1

A1 = 3 4 = 12 cm2 A2 = 3 4 = 12 cm2 A3 = 3 14 = 42 cm2 Total area = 12 + 12 + 42 = 66 cm2

2

A1 = 3 7 = 21 cm2 A2 = 3 7 = 21 cm2 A3 = 3 8 = 24 cm2 Total area = 21 + 21 + 24 = 66 cm2

3

Area of entire rectangle = 7 14 = 98 cm2 Area A1 = 4 8 = 32 cm2 Required area = 98 32 = 66 cm2


Extend Your Thinking Grade 8

Name: Date:

Find 2 different ways to determine the area of the rectangle ABCD:

1 2

Expectation: Apply the Pythagorean relationship to numerical problems involving area and right triangles.


Extend Your Thinking Grade 8 (Answers) Possible ways students could respond. 1. A student still operating at the concrete stage may choose to use a scale drawing to approximate the

area.

The area is approximately 10 4.6 = 46 units2 using a scale drawing. (Note: Although the teacher may expect a student to apply a particular piece of mathematical knowledge, in a problem-solving context, the student may find some unexpected way to solve the problem.)

2.

Using Pythagoras relationship.

10

100

3664

68 222

==

+=+=

x

x

524h

h524

h1021ABEareaand68

21

ABEArea

==

==

2units4852410ABCDArea

==

3.

2units 48242 ABCD rectangle

246821ABE

ABCD rectangle21ABE

====

=


Extend Your Thinking Grade 9 Applied

Name: Date:

Find 2 different ways to fine the area of this patch of pavement.

1 2

Expectation: Calculate sides in right triangles, using the Pythagorean theorem, as required in topics throughout the course, e.g., measurement.


Extend Your Thinking Grade 9 Applied (Answers) Possible ways students could respond.

1.

Use Pythagoras relationship to determine that h = 4 m.

2

2

m12432

m643211

====

A

A

Total area = 6 + 12 = 18 m2

2.


2m64321321 ==== AAA

Total area = A1 + A2 + A3 + = 3 6 = 18 m2

3.

Use Pythagoras relationship to determine that h = 4 m. A (rectangle) = 6 4 = 24 m2

A (triangle) = 2m64321 =

A = 24 6 = 18 m2

4.


( )( )( )

2m18

4921

4632121)(trapezoid

==

+=

+= hbaA


Is This Always True? Grade 7

Name: Date:

Trevor travels from A to B by walking only south or east on the streets shown. The first time he follows the route indicated by the solid lines and determines that his walk was 16 blocks long. The second time Trevor walks from A to B by following streets south or east only, he follows the route indicated by the broken lines and determines that his walk was again16 blocks long. Will his trip always be 16 blocks long?

Answer Yes. Trevor will have to travel 8 blocks east and 8 blocks south, or a total of 16 blocks, no matter in what order he chooses to do the east and south parts of the trip. This concept can be connected to finding perimeter of a step shape, where sizes of the steps are not known. Perimeter is 2 12 + 2 22 = 68 cm since the 4 vertical steps on the right add to 12 and the 4 upper horizontal lengths add to 22.

A

B

N


Is This Always True? Grade 8

Name: Date:

1. Shelley says that the circle with diameter 4 has a smaller area than the square with side 4

and a larger area than the square with a diagonal 4. Is this true for any number that Shelley chooses?

2. Is it always true that the circumference of a circle is more than 6 times the radius of the

circle?

Answer 1. Yes.

2. Yes. C = 2r and 2 =& 2 3.1415 =& 6.283, which is more than 6. C > 6r


Is This Always True? Grade 9 Applied

Name: Date:

The largest rectangle with a perimeter 16 cm is a square. Is it always true that the largest rectangle with a given perimeter is a square?

Answer Yes. The largest number for area can be determined in a table of values. If perimeter is 16 cm, length plus width is 8 cm.

Length

(cm) Width (cm)

Area (cm2)

1 7 7 2 6 12 3 5 15 4 4 16 5 3 15 6 2 12 7 1 7

In a graph of area vs. length, the highest point on the graph identifies the area.

The highest point on my graph occurs when x = 4. This means that the length is 4. So, the width must also be 4 and it makes a square.

The largest area is 16 2cm . This happens when the length = width = 4 cm. This is a square.

area perimeter volume 2

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