Using Area and Volume to Problem Solve
Using Area and Volume to Problem Solve
Objectives
In this unit, students will use radius, diameter, and perimeter to solve for area and volume. They will solve problems using attributes of triangles, quadrilaterals, and circles. They will find, confirm, and use relationships involving perimeter, circumference, area, and volume to solve real-world problems. Students will:
- calculate the perimeter and area of a variety of triangles and quadrilaterals.
- determine the minimum and maximum area of a rectangle given a fixed perimeter.
- discover the relationship between the diameter and circumference of a circle.
- discover the relationship between the radius and area of a circle.
- calculate the area and circumference of a circle.
- calculate the volume and surface area of right prisms and cylinders.
- apply perimeter, circumference, area, surface area, and volume calculations to solve real-world problems.
Essential Questions
- How can we use the relationship between area and volume to help us draw, construct, model, and represent real situations and/or solve problems of area and volume?
Related Unit and Lesson Plans
Related Materials & Resources
The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.
- Changing areas of rectangles and irregular figures given a specific perimeter
http://www.shodor.org/interactivate/activities/AreaExplorer/
- Calculate the perimeter and area of rectangles and irregular figures generated by computer
http://www.shodor.org/interactivate/activities/ShapeExplorer/
- Math Made Easy mini-lesson on using the formula for area of a trapezoid without a calculator, 4:22 min.
https://www.youtube.com/watch?v=2PhMRRBVvEg&feature=related
- Math Made Easy mini-lesson on finding the area of any regular polygon using apothem without a calculator, 9 min.
https://www.youtube.com/watch?v=eQhgozrRiYI
- Enter any value for radius to have circumference and area of a circle instantly calculated
http://www.calculatorsoup.com/calculators/geometry-plane/circle.php
- First 1000 digits of pi
http://www.factmonster.com/ipka/A0876705.html
- Math Made Easy mini-lesson on using the area formula to find the area of a circle without a calculator, 5 min.
https://www.youtube.com/watch?v=OBnKYoOpdsM&NR=1
- Song about parts of a circle and the formulas for area and circumference; use as introduction, review, or memory tool; 1:15 min.
https://www.youtube.com/watch?v=lWDha0wqbcI&feature=related
- Game in which counting cubes helps students visualize volume of irregular three-dimensional block figures
http://www.coolmath-games.com/0-countcubes/index.html
- Nutshell Math mini-lesson on surface area and volume of solids; prisms are used as examples, 4 min.
https://www.youtube.com/watch?v=hgj--7-afDM&feature=related
- Geometry videos of surface area and volume lessons including cylinders, cones, prisms, and pyramids
http://www.mathplayground.com/mathvideos.html
- Calculate the surface area and volumes of rectangular and triangular prisms by adjusting the dimensions; for use by students
http://www.shodor.org/interactive/activities/SurfaceAreaAndVolume/
Formative Assessment
-
View
Multiple Choice Items
1. What does the perimeter of a polygon represent?
A
the sum of the length and width
B
the size of the space enclosed by the figure
C
the length of the segment going across the figure and through its center
D
the sum of the lengths of all its sides
2. Select the statement below that correctly describes rectangle PQRS.
A
Perimeter = 13 cm, Area = 26 cm2
B
Perimeter = 16 cm, Area = 40 cm2
C
Perimeter = 26 cm, Area = 40 cm2
D
Perimeter = 40 cm, Area = 13 cm2
3. Jane is trying to determine the length of ribbon to buy to decorate the outer edge of her circle design. What is the length, to the nearest whole inch, of ribbon Jane should buy?
A
18 inches
B
28 inches
C
36 inches
D
38 inches
4. Charles drew the outline of a triangle on the pavement to play a game of marbles with friends. Based on the information given, what type of triangle did Charles draw?
A
equilateral
B
isosceles
C
right
D
scalene
5. Using the figure from number 4, what is the perimeter of the triangular game Charles drew?
A
P = 68 inches
B
P = 70 inches
C
P = 210 inches
D
P = 420 inches
6. An equilateral triangle can also be which of the following types of triangles?
A
acute
B
obtuse
C
right
D
scalene
7. Which is a true statement about isosceles triangles?
A
They always have a 90o angle.
B
They are always an acute triangle.
C
They always have three congruent sides.
D
They always have at least two congruent angles.
8. A chord is a line that is drawn from one side of a circle to the other as shown below.
Which of the following could also be considered a chord?
A
center
B
circumference
C
diameter
D
radius
9. The height of the cylinder shown below is labeled h.
The height, h, is used to find which of the following?
A
circumference
B
diameter
C
radius
D
volume
Multiple-Choice Answer Key
1. D
2. C
3. B
4. D
5. B
6. A
7. D
8. C
9. D
Short-Answer Items:
10. Janelle has 16 feet of fencing to make a pen for her rabbit, Fluffy. She would like Fluffy to have as much room as possible to move around in the pen. Find the maximum possible area for a rectangular pen with a fixed perimeter of 16 feet. Use a table to show all possibilities and explain your choice.
11. A circle is shown. Label and draw the different parts of a circle.
A. diameter
B. radius
C. center
D. Another word for circumference is ___________________. Label the circumference.
12. What is one set of possible dimensions for the rectangle and triangle such that their perimeters are equal to one another? What is the perimeter? Remember that there are many correct solutions for this situation.
Note: Figures are not drawn to scale.
Short-Answer Key and Scoring Rubrics:
- Janelle has 16 feet of fencing to make a pen for her rabbit, Fluffy. She would like her Fluffy to have as much area as possible to move around in the pen. Find the maximum possible area for a rectangular pen with a fixed perimeter of 16 feet. Use a table to show all possibilities and explain your choice.
Width
Length
Perimeter
Area
1 ft
7 ft
16 ft
7 ft.2
2 ft
6 ft
16 ft
12 ft.2
3 ft
5 ft
16 ft
15 ft.2
4 ft
4 ft
16 ft
16 ft.2
5 ft
3 ft
16 ft
15 ft.2
6 ft
2 ft
16 ft
12 ft.2
7 ft
1 ft
16 ft
7 ft.2
Since the amount of fencing for the perimeter is 16 feet, the sum of the length and width needs to be half of 16, or 8 ft. Each combination of sums in order, from least to greatest, is listed in the table. The perimeters all added up to 16 feet using P = (l + w) × 2. The areas were found using A= l × w. The greatest area was found in the middle of the table, when the rectangle dimensions form a square 4 ft × 4 ft.
To create the maximum area for Fluffy, Janelle should make the pen 4 ft × 4 ft.
Points
Description
2
- Written explanation and table are complete, correct, and detailed.
- Pen size answer is correctly stated and explained (4 ft × 4 ft pen with area 16 sq ft).
- Student’s answer demonstrates thorough understanding of the fixed perimeter concept.
1
- Written explanation and table are partially incorrect or partially missing.
- Pen size answer is incorrect, possibly due to a miscalculation in the table, or misunderstanding of the meanings of table values.
- Student’s answer demonstrates limited understanding of the fixed perimeter concept.
0
- Written explanation and table are mostly incorrect or missing.
- Answer for pen size is not stated, and correct answer is not displayed in the table or work.
- Student’s answer demonstrates no understanding of the fixed perimeter concept.
11. A circle is shown. Label and draw the different parts of a circle.
A. diameter
B. radius
C. center
D. Another word for circumference is perimeter. Label the circumference.
Points
Description
2
- All four parts drawn and labeled on the circle.
- Understands that circumference is perimeter (and/or the distance around the circle).
- Figure order is correctly stated (A, C, B) with clear reasoning in the work and explanation.
- Student demonstrates thorough understanding of circles and the concept of how perimeter relates to circles.
1
- Three of the four parts of the circle are labeled.
- Error in what the circumference means.
- Student understands the parts of the circle.
0
- Most of the parts are labeled incorrectly.
- Student doesn’t show understanding of what circumference is.
- Student demonstrates no understanding of the parts of a circle.
12. What is one set of possible dimensions for the rectangle and triangle such that their perimeters are equal to one another? Remember that there are many correct solutions to this situation, and note that these figures are not drawn to scale.
Possible answers (but not limited to):
6×4 and (5-7-8) p=20
10×3 and (10-10-10) p=30
5×8 and (6-8-12) p=26
2×4 and (3-4-5) p= 12
*Remember that for a triangle the longest side must be between (not equal to) the sum of the two other sides and their difference (a − b < c < a + b).
Points
Description
2
- Written work for finding equal perimeters of the figures is complete and detailed.
- The perimeters and dimensions are stated and labeled correctly.
- Student’s answer demonstrates thorough understanding of the concept of perimeter.
1
- Written work for finding equal perimeters is partially incorrect or has some parts missing.
- Only one of the dimensions is stated incorrectly. Labels are incorrect or missing.
- Student’s answer demonstrates limited understanding of the concept of perimeter.
0
- Written work for finding perimeters and dimensions is completely incorrect or missing.
- Dimensions answers are incorrect, incomplete, or missing.
- Student’s answer demonstrates no understanding of the concept of perimeter.
Performance Assessment:
This assessment may be completed individually or in student pairs. Provide students with the Landscape Your Lot Summary sheet (M-6-2_Landscape Your Lot.docx), centimeter grid paper, 24 × 36 inch (or larger) chart paper, ruler, and markers or colored pencils.
Landscape Your Lot
You will design a yard plan that must include the specific requirements below. Each centimeter you draw will represent a foot (1 cm:1 ft).
- Begin by drawing the back wall of a house 30–40 feet wide near the edge of your paper. The back yard is adjacent to this.
- The yard must have an overall area between 4,000 and 5,000 square feet. Show the dimensions you chose on your plan, and show your area calculations.
- The perimeter of the yard must have a fence around it that must go up to the sides or back corners of the house to enclose the entire back yard area. Draw the fence, label the lengths of each segment, and calculate the number of feet of fencing needed to enclose the yard.
- You are adding a rabbit kennel using 24 feet of special fencing. Design a rabbit kennel with the maximum possible area using this fencing. Show how you determined the dimensions. Draw and label your kennel on the plan.
- Add in two to four different elements to your back yard. One of the elements needs to be a three-dimensional figure. You must calculate the dimensions (as appropriate: length, width, height, perimeter, area, surface area, and volume) of each element. Be sure to label each element.
Draw your design on the chart paper using rulers and grid paper. Show your calculations and answers on the Landscape Your Lot sheet (M-6-2_Landscape Your Lot.docx). Label your answers. Remember to use the scale 1 cm: 1 ft, and show this scale on your design. The final design must be neatly drawn (using rulers and compasses where appropriate) and colored. Be sure to include a summary table that is neat and detailed.
Performance Assessment Scoring Rubric:
Points
Description
4
- Four to five elements are incorporated in the design and labeled.
- Back yard design is neat, accurate, creative, detailed, and well presented.
- Student’s work and explanations are clear and logical, possibly with multiple methods or incorporates unique use of space and resources.
- Perimeters, areas, and volumes are all correct and labeled accurately.
- Summary table is neat, thorough, and detailed.
- Student demonstrates advanced understanding of the mathematical ideas and processes related to perimeter, area, and volume.
- Student worked beyond the problem requirements, possibly by including extra design elements, combined shapes, and/or incorporating technology.
3
- Three of the elements are incorporated in the design and most are labeled, and one of them is three dimensional.
- Back yard design is lacking some detail and presentation but neat.
- Student’s work and explanations are present, but may lack detail, be missing, or illogical in some places.
- Perimeters, areas, and volumes are mostly calculated correctly (one to three mathematical errors) and most are correctly labeled.
- Summary table is mostly complete.
- Student demonstrates basic understanding of the mathematical ideas and processes related to perimeter, area, and volume with some misconceptions.
- Student meets most of the problem requirements.
2
- Three of the required elements are incorporated in the design, but none of them are three dimensional.
- Back yard design contains some inaccuracies, is lacking detail, and is not clearly presented.
- Student’s work and explanations are unclear, illogical, or missing several elements.
- Perimeters, areas, and volumes contain several calculation errors (four to six) and some missing or incorrect labeling.
- Summary table is missing several required segments.
- Student demonstrates limited understanding of the mathematical ideas and processes related to perimeter, area, and volume.
- Student partially meets the problem requirements.
1
- One to two of the required elements are incorporated in the design.
- Back yard design contains several inaccuracies, is incomplete, and/or poorly presented.
- Most of the student’s work and explanations are illogical or missing.
- Perimeters, areas, and volumes are mostly incorrect (six or more errors) and most labels are missing or incorrect.
- Summary table has more than half of the required entries missing.
- Student demonstrates substantial gaps in understanding of the mathematical ideas and processes related to perimeter, area, and volume.
- Student does not meet most of the problem requirements.
0
- Has none of the required elements in the design.
- Student’s work and explanations are missing.
- Perimeters, areas, and volumes are all calculated incorrectly or are missing.
- Summary table is nearly blank or missing.
- Student demonstrates no understanding of mathematical ideas and processes related to perimeter, area, and volume.
- Student does not meet the problem requirements.
DRAFT 10/07/2011
