GHSGT-Math-Obj. 26-27-Perimeter, Area, Volume

1

GHSGT-Math-Obj. 26-27-Perimeter, Area, Volume

INTERVENTION

LESSONS

FOR GHSGT

MATH OBJECTIVES

26 & 27

Perimeter, Area, & Volume

Perimeter, Area, Volume - Lesson 1

GHSGT Objective :

26Finds the perimeter and area of plane figures (such as polygons, circles, composite figures) and surface area and volume of simple solids (such a rectangular prisms, pyramids, cylinders, cones and spheres)

27Calculates perimeter and area of plane figures; finds appropriate measures of objects and their models prior to such calculations for basic polygons and circles

Warm-up / Activator

Experiment with 1” square tiles or grid paper. Find two shapes with an area of 6 square units and a perimeter of 12 units.

Compare your shapes with your neighbor. Are they the same shape?

Now, draw at least two shapes with a perimeter of 18 units and different areas. How do you find the perimeter of any shape? How is the perimeter measure different from the area measure?

/ 2
1
3
/ 4
5

Mini Lesson

Find the area of each shape and explain how you found it.

Polygons are often described by giving their base and height. How can you determine where the base and height measures are located for each shape in the grid?

For each shape 1-5, identify the base and height and explain how to find the area using the appropriate formula.

To the right is a family of parallelograms. Find the area of each parallelogram. What patterns do you see? Why do you think these parallelograms are called a family?

Work Session

1. The Ace Sign Company makes traffic signs for the state road department. A model of the signs and their approximate measurements are given below.

a. One of the costs that Ace must consider is the cost of metal. If metal costs $1.00 for every 1000 square centimeters, what is the cost of the metal for each sign?

b. After the signs are cut, the edges must be sanded to prevent metal splinters. If the cost of sanding is 2 cents for every centimeter, what will it cost to sand each sign?

2. Lara is helping her family build a recreation room in their basement. The room will be 28 feet by 20 feet. They have already put up the walls.

a. The family wants to tile the floor. Lara decides to buy 1-foot-square tiles. How many tiles will she need?

b. The tiles Lara has selected cost $0.75 each. How much will the tile floor cost?

c. Lara needs to buy baseboard to put along the wall. How much baseboard does she need?

d. The baseboard comes in 10-foot and 16-foot lengths. How many boards of each length should Lara buy?

3. Is it possible to make a shape with the specified properties below? If so, sketch the shape.

a. a triangle with an area of 16 cm2 and a height of 2 cm

b. a rectangle with a perimeter of 20 cm

c. a parallelogram with a pair of opposite side lengths 10 cm and an area of 96 cm2

d. a rectangle with a perimeter of 24 cm and an area of 20 cm2

Closing

Students should explain their work to the class. Ask leading questions to guide students to understanding the concept.

Students should summarize what they learned about the difference between area and perimeter and how to find each for various types of polygons.

Perimeter, Area, Volume - Lesson 2

Warm-up / Activator

Mathematicians have found a relationship between the diameter and circumference of a circle. You can discover this relationship by measuring different circles and looking for patterns. Use a tape measure (or string and yardstick) to measure the diameter and circumference of each circular object. Record your results in a table with these column headings.

ObjectDiameterCircumference

Make a coordinate graph of your data. Use the horizontal axis for diameter and the vertical axis for the circumference.

Study your table and your graph, looking for patterns and relationships that will allow you to predict the circumference from the diameter. Test your ideas on some other circular objects. What is the relationship between the diameter and the circumference of any circle?

Mini Lesson

There are at least four measurements that are useful for describing the size of a circle: diameter, radius, area, and circumference.

How can you find the circumference of a circle if you know the diameter?

How can you find the diameter of a circle if you know the circumference?

How is the diameter related to the radius of a circle?

What are two possible formulas for the circumference of a circle?

Squaring a Circle activity: Three circles are drawn below. A portion of each circle is covered by a shaded square. The sides of each shaded square are the same length as the radius of the circle. We call such square a “radius square.”

For each circle, cut out several copies of the radius square from a sheet of grid paper. Find out how many radius squares it takes to cover the circle. You may cut the radius squares into parts if you need to. Record your data in a table with these column headings:

Circle Radius of circle Area of radius square Area of circle Number of radius squares needed

Describe any patterns you see in your data. If you were asked to estimate the area of any circle in “radius squares,” what would you report as the best estimate?

How can you find the area of a circle if you know the diameter or the radius?

How can you find the diameter or radius of a circle if you know the area?

Work Session

Use the given measurements and the formulas for circumference and area of circles to find the missing measurements.

1. A dinner plate has a diameter of 9 inches. Find its circumference and area.

2. Suppose you tie together the ends of a 60-centimeter piece of string to form a circle. What is the diameter of the circle? What is the area of the space enclosed by the string?

3. Lydia’s mother decided to paint the semicircular patio in their back yard. Here is Lydia’s sketch of the patio, drawn on a grid. Each grid square represents 1 square foot.

a. What is the area of the patio? Explain how you found the area.

b. Each quart of nonskid paint covers 32 square feet. How much paint should Lydia’s mother buy if she plans to put one coat of paint on the patio? Show your work.

c. To keep grass from growing onto the patio, Lydia wants to plant a border around the patio. Since the patio is against the house, she only needs a border around the curved edge. How long will the border be? Show how you found your answer.

4. A circular cherry pie has a diameter of 11 inches and is cut into 8 equal-size pieces. What is the area of each piece of pie? Explain how you found your answer.

5. Use the diagram at the right to answer the following questions.

a. What is the perimeter of the figure? Explain your reasoning.

b. What is the area of the figure? Explain how you know.

6. Below is a drawing of a jogging track. Show your work for each question below.

a. What is the total distance around the jogging track?

b. How much area does the jogging track enclose?

c. If Tony wants to jog 4 miles, how many times will he have to jog around the track? (Remember that 5280 feet is 1 mile.)

Closing

Students should explain their work to the class. Ask leading questions to guide students to understanding the concept.

Students should summarize what they learned about the difference between area and circumference and how to find each measure for various circles and applications with circular objects.

Perimeter, Area, Volume - Lesson 3

Warm-up / Activator

On grid paper, draw a flat pattern for a rectangular box (prism) that is not a cube. Each side length of your pattern should be a whole number of units. Then, make a different pattern for the same box. Test each pattern by cutting it out and folding it into a box.

Mini Lesson

Use the box you constructed in the warm up activity and find the total area of each flat pattern. Describe the faces of the box formed from each flat pattern you made. What are the dimensions of each face? When the areas of all faces are added together, this is the surface area of the prism.

(The teacher or students should bring in assorted boxes with lids.) Take a box and find the dimensions in centimeters. Cut your box along the edges so that you can lay is out flat. Find the area of each face of the box. Do you notice any shortcuts for finding the surface area of a prism?

Draw a flat pattern for a rectangular prism with dimensions 2 inches by 3 inches by 5 inches. Find the surface area.

Explain how you find the surface area for a rectangular prism.

Work Session

1. Suppose you wanted to make a box to hold exactly 30 1-inch cubes.

a. Describe all the possible boxes you could make.

b. Which box has the least surface area? Which has the greatest surface area?

c. Why might you want to know the dimensions of the box with the least surface area?

2. The dimensions of a recreation center floor are 150 ft by 45 ft, and the walls are 10 ft high. A gallon of paint will cover 400 ft2. About how much paint is needed to paint the walls of the recreation center?

3. The bottom of a rectangular prism has an area of 50 cm2. If the box is 8 cm high, give at least three possibilities for the dimensions of the prism.

Closing

Students should explain their work to the class. Ask leading questions to guide students to understanding the concept. Students should summarize what they learned about the surface area of a rectangular prism.

Perimeter, Area, Volume - Lesson 4

Warm-up / Activator

(Bring in soup cans or other cans with labels.

Have students design flat patterns like the one

shown using the label and by drawing around

the circular bases. Using string and rulers

or measuring tape, students should measure

the circumference of the can and the

length of the label to observe that these

measurements are the same.)

Find the area of each part of the cylinder pattern.

How are the dimensions of the circles and the rectangle

in the flat pattern related to the dimensions of the

cylinder?

What dimensions of the cylinder would be used to calculate its surface area?

Mini Lesson

Shown is a scale model of a flat pattern for a cylinder. Each = 1 cm2

The surface area of the cylinder is made up of the two congruent circular bases and the rectangle that is the lateral surface of the cylinder. The length of the rectangle is equal to the circumference of the circular base. What measures do you need to find the circumference of the circle?

Once you have found the area of each circular base and area of the rectangle, this total area is the surface area of the cylinder.

Surface area = 2 (pi) radius2 + 2 (pi) radius x height

Work Session

1. Take two identical sheets of paper. Tape the long sides of one sheet together to

form a cylinder. Form a cylinder from the second sheet by taping the short sides

together. Imagine each cylinder has a top and a bottom. Which cylinder has

the greater surface area? Explain your reasoning.

2. A rectangular prism and a cylinder are shown.

Which of the shapes has the greater surface area?

Show how you know.

3. The circumference of a base of a cylinder is 16 (pi) cm. The height of the cylinder is 10 cm. Find the surface area of the cylinder.

Closing

Students should explain their work to the class. Ask leading questions to guide students to understanding the concept. Students should summarize what they learned about the surface area of a cylinder.

Perimeter, Area, Volume - Lesson 5

Warm-up / Activator

Find all the ways 36 cubes can be arranged into a rectangular prism. Make a sketch of each arrangement you find, and give its dimensions and surface area. You can organize your data in a table like the one below.

Rectangular Prisms with 36 cubes

Length / Width / Height / Volume / Surface Area / Sketch

What do you notice about the relationship between the dimensions and the volume of the prism?

Which of your arrangements has the smallest surface area?

Mini Lesson

Examine the rectangular prism that has dimensions 2 by 6 by 3 that you made from the 36 cubes. How many cubes would fill the bottom layer? How is this related to the area of the base? How many cubes are in each layer? How many identical layers will be stacked to have the volume of 36 cubic units? What connections do you see among the dimensions of the bottom layer, the number of layers, and the volume of the prism?

To find the volume of any prism (or cylinder), you can think of the process as a layering strategy. Once you know the area of the base of the prism (or cylinder), you then need to multiply the height of the prism to find the volume. The area is a two-dimensional measure (in square units or u2) and the height is a linear or one-dimensional measure. When these are multiplied (u2 x u), the units become cubic units (u3).

Find the volume of the prism shown.

Suppose the prism was laid on its side so its

base was 4 inches by 2 inches and its height

was 10 inches. Would this affect the

volume of the prism?

Explain your reasoning.

Cylinders often contain liquids and the volume or capacity of containers that hold liquids are often given in units like quarts, gallons, liters, and milliliters. They don’t tell you the number of cubic units, but there are cubic equivalences to these capacity measures. For example, a gallon equals 231 cubic inches and a milliliter equals 1 cubic centimeter.

When finding volumes of cylinders, you still consider the area of the base (circle area is approximately equal to 3.14 x radius square) and the height measures.

Work through the following problem: A cylinder has a radius of 3 cm. Sand is poured into the cylinder to form a layer that is 1 cm deep.

What is the volume of the sand in the cylinder?

If the height of the cylinder is 20 cm, how many layers of sand - each 1 cm deep - are needed to fill the cylinder?

What is the volume of the cylinder?

A cylindrical storage tank has a diameter of 15 feet and a height of 30 feet. Make a sketch of the tank and label its dimensions. What dimensions are needed to find the volume of the tank?

Show how you use the formula to find the volume of the tank.

Work Session

1. a. Sketch a rectangular prism with a base area of 40 cm2 and a height of 5 cm.

b. What is the volume of the prism you drew?

c. Do you think everyone in your class drew the same prism? Explain.

2. The city of Rubberville plans to dig a rectangular landfill. The landfill will have a base with dimensions 700 ft by 200 ft and a depth of 85 ft.

a. How many cubic feet of garbage will the landfill hold?

b. What information would you need to determine how long the landfill can be used until it is full?

c. An excavator was hired to dig the hole for the landfill. How many cubic yards of dirt will he have to haul away?

3. Look for an object in your classroom or neighborhood with a volume of about 60 cm3. Explain how you estimated the volume of the object.

4. Peter Dowdeswell holds the world record for eating pancakes in the shortest amount of time. He ate 62 pancakes with butter and syrup in 6 minutes 58.5 seconds. Each pancake was 3/8 inch thick and had a 6 inch diameter. If the pancakes were stacked, what shape would they resemble? What was the approximate height of the stack of pancakes? Show with your hands how high your estimate is. Find the actual height of the stack of 62 pancakes. How would you find the volume (or number of cubic inches) of the stack?

5. You are the manager of a new movie theater. You need to order popcorn boxes, and you must decide between a cylindrical box and a rectangular box. The cylindrical box has a height of 20 cm and a radius of 7 cm, and the rectangular box has a height of 20 cm and square base with 12 cm sides. The price the theater has to pay for each box is based on the amount of material needed to make the box. The theater plans to charge $2.75 for popcorn, regardless of the shape of the box. a. Find the volume of each box.

b. Find the surface area of each box.

c. Which box would you choose? Give reasons for your choice.

6. What features of a cylinder could be measured in the given units?

a. cm

b. cm2

c. cm3

Closing

Students should explain their work to the class. Ask leading questions to guide students to understanding the concept. Students should summarize what they learned about the volume of rectangular prisms and cylinders.

Assessment for Objectives # 26 & #27