This unit is a seventh grade Pre-Algebra introduction to three-dimensional measurement. The unit starts with an introduction to surface area and volume by building on their prior knowledge of perimeter and area. The students explore different shapes of boxes, beginning with cubes and working through hexagonal prisms. After working on prisms, they explore cylinders and their properties. The students then move on to cones and spheres using what they have learned already to investigate the properties of these shapes. They finally do some work with changing the dimensions of boxes and seeing how that affects the volume and surface area.

Overview:

You will need centimeter grid paper throughout this unit.

Unit 1: Building Boxes

Objectives: to develop the concept of surface area by counting the number of unit squares needed to wrap a rectangular box. To explore the relationship between the surface area of a box and the total area of the unit squares needed to wrap the box. To develop the concept of volume of a rectangular box, and strategies for finding it, by filling a box with unit cubes.

Day 1: Making Cubic Boxes

Teacher should begin by defining cube and flat pattern. They should explain what a unit cube is also.

The students should begin the lesson by drawing as many flat patterns as they can for a cube. They should sketch the patterns on centimeter grid paper.

For homework I gave them a few different flat patterns and asked them whether or not they can be folded into a cube.

Day 2: Making Rectangular Boxes

The teacher should begin by explaining that not all boxes are shaped like cubes. Most boxes are rectangular.

The students should begin by drawing flat patterns of rectangular boxes on their centimeter grid paper. They should find the total area of the pattern that they have drawn. They should then identify what the dimensions are of each face of the box. After finding the area, they should try to figure out how many unit cubes will fit into each box.

For homework, the students were given a rectangular box and it’s dimensions. They were told to sketch the flat pattern and find the area of each flat pattern. They were then told to find the total area and compare their answers.

Day3: Flattening a Box

The teacher should begin by defining rectangular prism. I chose to do this activity by dividing the classes into groups and giving each group a different sized box. The groups then had to find the dimensions of their box and draw a flat pattern of it. The activity is entitled "How Big is Your Box?" I collected this assignment at the end of class and graded it.

Day 4: Testing Flat Patterns

For this activity, the students are given the flat patterns of boxes drawn by a packaging company. They are not given the dimensions and are asked to figure them out. Each student should be given a Labsheet 1.4 for this activity. They are asked compare the dimensions of each box to the dimensions of the faces of each box. They should then find the total area of all of the faces of the box. Finally they need to figure out how many unit cubes it takes to fill the box. This assignment will take all of class and be assigned for homework if not completed.

Unit 2: Designing Packages

Objectives: to develop strategies for finding the surface area of a rectangular box. To determine which rectangular prism has the least surface area for a fixed volume. To reason about problems involving surface area.

Day 5: Packaging Blocks

This unit begins with a discussion about finding the right size box for a particular product. The teacher should explain that the volume of a box is the number of unit cubes that can fit into that box. They should also explain that the surface area of a box is the total area of all of its faces.

To begin the activity, the students are told that a toy company wants to arrange 24 blocks into a rectangular prism and they are asked to give the dimensions of some boxes that these blocks will fit into. After finding the different box dimensions, they are asked to determine which box has the smallest surface area.

For homework, the students are given 3 different size rectangular prisms made from inch cubes and asked to find the length, width, and height of each prism. They are then asked to find the surface area and to tell how many blocks will fit in each box.

Day 6: Saving Trees

This activity begins with a discussion about how boxes with a smaller surface area save money for the companies using them. The students are given 3 scenarios; 8 cubes, 27 cubes, and 12 cubes and asked to figure out what the least amount of packaging material is required. After figuring this out, they should be able to establish a pattern about surface area.

Unit 3: Finding Volumes of Boxes

Objectives: to develop a strategy for finding the volume of a rectangular prism by filling it with unit cubes, and to recognize that the number of cubes in the bottom layer is equal to the area of the base. To determine that the total number of unit cubes in a rectangular prism is equal to the area of the base times the height, and to discover that this strategy works for any prism. To learn that the surface area of a prism is the sum of the area of its faces, and to apply this strategy to any right prism. To reason about problems involving volume and surface area.

Day 7: Filling Rectangular Boxes

In this activity, the students are given four different boxes and asked to figure out how many unit cubes can fit into each. After finding the number of cubes, they are asked to find out what the surface area of each of the boxes is. They are then asked questions like; how many cubes can fit into a single layer of each box? How many identical layers can be stacked into each box?

For homework, the students are given rectangular prisms made of unit cubes similar to the ones from the previous unit. They are asked to answer similar questions.

Day 8: Burying Garbage

This is a short activity about finding volume using a real world application. The students are told that there is a city that has set aside land on which to bury garbage. The city wants to dig a rectangular hole with a base measuring 500 feet by 200 feet and a depth of 75 feet. They are told that the population of the city is 100,000. It is estimated that on average a family of four throws away 0.4 cubic feet of garbage every day. With this information, the students are asked to find the volume of the hole and asked how long it will take for the hole to fill up.

For homework, the students are given the dimensions of a classroom and asked to sketch it, find the volume, and the total area of the walls, floor, and ceiling. They are also asked to sketch a prism with a base of 40 cm squared and a height of 5 cm. They are asked to find the volume and discuss whether or not everyone has the same prism.

Day 9: Filling Fancy Boxes

This activity is centered on prisms that are non-rectangular. The students are introduced to triangular prisms, square prisms, pentagonal prisms, and hexagonal prisms. They will try to find the volume of these by figuring out how many unit cubes will fit into each prism. I found that this worked best by having the students construct their own paper prisms. This took a whole class period to do and after they are done, they will need to save their prisms for the next activity. They should start with four identical sheets of paper. They should fold on piece into three congruent rectangles and tape it to form a triangular prism. They should fold the second sheet into four congruent rectangles and tape it to form a rectangular prism. The remaining two pieces of paper should be folded into 5 and 6 congruent prisms and taped to form a pentagonal and hexagonal prism.

Day 10: Filling Fancy Boxes Continued

After constructing all of their prisms, the students should find the volume and surface area of each of them. They should use their centimeter grid paper to figure out the area of the bases.

Unit 4: Cylinders

Objectives: to develop strategies for finding the volume and surface area of a cylinder. To compare the process of finding the volumes and surface areas of cylinders and rectangular prisms. To investigate interesting problems involving the volumes and surface areas of cylinders and prisms.
 
 
 
 

Day 11: Filling a Cylinder

The students are introduced to cylinders and their properties in this unit. They are told that the volume of a cylinder is the number of unit cubes that can fit into a single layer multiplied by the number of layers they have (or the height). They are asked to construct a paper cylinder and find the volume of it. The students are to figure out how many cubes will fit into the bottom layer of the cylinder and how many layers they will have.

For homework, the students are given three different sizes of soft drinks and the prices of each. They are asked to find the best value out of the three. They are also asked to determine what features of a cylinder would be measured in centimeters, centimeters squared, and centimeters cubed.

Day 12: Making a Cylinder from a Flat Pattern

Each student is given a Labsheet 4.2 with a flat pattern of a cylinder in it. They are asked to find the dimensions, the surface area, and the volume of the cylinder. They may cut out the cylinder and tape it together if they wish.

For homework, the students are given a problem about a storage tank. They are given the radius and the height and asked to make a sketch, find the volume, and find the surface area. They are given another problem where they are told that they cylinder has a radius of three and that sand is poured into it 1 centimeter deep. They are asked to find the volume of the sand. They are also given that the height of the cylinder is 20 cm and asked how many layers of sand are needed to fill the cylinder. Finally they are given the dimensions of a soft drink can and the dimensions of a classroom. They are asked to determine how many cans will fit into the classroom.

Day 13: Designing a New Juice Container

In this problem, students are told that a juice can has a height of 8 cm and a radius of 2 cm. They are asked to find the volume and design a rectangular juice box with the same volume. They should draw their rectangular box on centimeter grid paper.

For homework, the students are asked to look around their homes and find three different size cylindrical objects. They need to find the dimensions, volume, and surface area of each.

Unit 5: Cones and Spheres

Objectives: to develop strategies for finding the volumes of cones and spheres. To find the relationships among the volumes of cylinders, cones, and spheres. To reason about problems involving cylinders, cones, and spheres.

Day 14: Comparing Spheres and Cylinders

To start this unit, the teacher should explain what cones and spheres are and show examples of each. In this problem, you will need Play-Doh, transparency film, transparent tape, and a ruler. The teacher should divide the students into groups and have each group make a sphere with their play-doh. Using their transparency film, they should wrap it around the sphere and trim the height so that it is as tall as the sphere. They now have a cylinder with a height equal to the diameter of the sphere. After this, they should flatten the sphere so that it fits snugly into the bottom of the cylinder. They should measure the new height of the play-doh. They should then be able to determine the relationship between the volume of a sphere and the volume of a cylinder.

Day 15: Comparing Cones and Cylinders

This activity is similar to the previous one. Each group should have a cylinder made out of the transparency film and a piece of stiff paper. The groups should construct a cone so that the tip touches the bottom of the cylinder. They should trim around the lip of the cylinder so that the height of the cone is equal to the height of the cylinder. They should then fill the cone with rice or sand and empty the contents into cylinder. They should repeat until the cylinder is filled so that they can see the relationship between the volume of a cone and a cylinder.

For homework, the students are told that a city has a water storage tank in three different shapes: a cylinder, a cone, and a sphere. Each tank has a radius of 20 ft and a height of 40 ft. They are asked to sketch each tank and find the volume of each. They are then given three diagrams, one of a cylinder, one a cone, and one a sphere, all with a height of 4 and a radius of 2. They are asked to find the volume and compare what they have found. The last problem they are given tells them that they have an ice cream cone with a radius of 1 inch and a height of 5 inches. If they put a scoop of ice cream in it with a one-inch radius, how many scoops will fit in the cone?

Day 16: Melting Ice Cream

The students are posed with this problem: Olga and Serge buy ice cream from the ice cream parlor. They think about buying one for Olga’s little sister but decide that the ice cream will melt by the time they get home. Serge wonders, "If the ice cream all melts into the cone, will it fill the cone?" Olga buys her ice cream in a cone and Serge gets a scoop in a cylindrical cup. Each container has a height of 8 and a radius of 4. Will the melted ice cream fill Olga’s cone? Will it fill Serge’s cup?

For homework the students will do more review with finding the volume of cones and spheres.

Unit 6: Scaling Boxes

Objectives: to apply strategies for finding the volume of rectangular prisms to designing boxes with given specifications. To investigate the effects of varying the dimensions of rectangular prisms on volume and surface area and vice versa.

Day 17: Building a Bigger Box

The introduction to this unit discusses problems with waste in our society. They talk about making compost out of the waste and give students a recipe for a compost box. The recipe is as follows:

• Start with an open rectangular wood box that is 1 foot high, 2 feet wide and 3 feet long. This is called a 1-2-3 box.
• Mix 10 pounds of shredded newspaper with 15 quarts of water, and put the mixture in the 1-2-3 box.
• Add a few handfuls of soil.
• Add about 1000 red worms (about 1 pound).

With this information, the students are told to design a box with doubled edges. They should draw their new 2-4-6 box on centimeter grid paper. The boxes should have open tops. The students should then figure out how much shredded paper and water they will need for their new box. How many worms will she need? How much plywood will she need to build the box? How many pounds of garbage will the box be able to decompose in one day?

This is an excellent group class work assignment. For homework the students should do more review with finding volume of rectangular prisms and figuring out how much of something will fit into it.

Day 18: Scaling Up the Compost Box

The students will revisit the compost box in this problem. They need to figure out how big of a box they should build if they want to get rid of 1 pound of organic waste per day. They should figure out the dimensions and then draw a scale model on their centimeter graph paper. They should then figure out how much plywood they would need to build the box and how much of each material to use for their new box.

For homework, students are given the dimensions of different boxes and asked to determine the volume.

Day 19: Looking at Similar Prisms

In previous lessons, the students studied similar two-dimensional figures. The teacher should explain that this idea could be applied to three-dimensional figures also. The teacher should explain to them what a scale factor is and how it applies to the compost box and the new boxes the classes created. They should find some other boxes that are similar to the 1-2-3 box and give their dimensions. They should then figure out the scale factor for the new boxes. They should now calculate the surface area of the new boxes and compare it to the original surface area. They should do the same with the volume.

For homework they should continue to review finding volume of all shapes of three-dimensional figures.
 
 

 2^nd period working hard on their prisms…

4^th period working hard as well…except

these guys who are hardly working !!!!

Standards this Unit Meets:

INSTASC:

1. Content Pedagogy: this is a very investigative unit. It works on several aspects of learning including individual, group, and whole class. It can be as structured or as unstructured as a teacher feels is appropriate.
2. Student Development: this unit involves several different types of learning that can benefit all students. It is not a discriminating lesson.
3. Diverse Learners: this unit uses individual, group, and whole class types of investigation.
4. Multiple Instructional Strategies: this unit is very hands on. It also works with more traditional styles of math.
5. Motivation and Management: as mentioned before, this unit can be as structured or unstructured the teacher feels comfortable with. I chose a very unstructured environment making class socialization easier.
6. Communication and Technology: I used as much technology as was available to me. A teacher could add more by using Geometry SketchPad.
7. Planning: this unit was planned around the required standards and what I thought was an appropriate way to approach them.
8. Assessment: I am not a big fan of tests so I chose to base my assessments around the in class activities and the homework assigned.
9. Reflective Practice: Professional Growth: I would constantly change the structure of the activities based on how things went in the previous class or on the previous days.
10. School and Community Involvement: I feel that I did a good job fostering relationships with the other teachers and administrators at my school. I used what I learned about the area I taught in to deal with the students more effectively.

7.8 The student, given appropriate dimensions, will estimate and find the area of polygons by subdividing them into rectangles and right triangles.

7.9 The student will investigate and solve problems involving the volume and surface area of rectangular prisms and cylinders, using concrete materials and practical situations to develop formulas.

7.10 The student will compare and contrast the following quadrilaterals: a parallelogram, rectangle, square, rhombus, and trapezoid. Deductive reasoning and inference will be used to classify quadrilaterals.

7.11 The student will identify and draw the following polygons: pentagon, hexagon, heptagon, octagon, nonagon, and decagon.

7.12 The student will determine if geometric figures (quadrilaterals and triangles) are similar and write proportions to express the relationships between corresponding parts of similar figures.

7.13 The student will construct a three-dimensional model using cubes, given the top, side, and/or bottom views, and determine the volume and surface area of the model.

7.14 The student will inscribe equilateral triangles, squares, and

hexagons in circles, using a compass and straightedge.
 
 
 
 

NCTM Standards Met:

Process:

Communication – this unit helped students learn to communicate with their peers in group work and with the teachers.

Connections – this unit showed connections with math they have done in previous units. They had to use what they knew prior to the unit to solve the problems involved.

Reasoning and Proof – this was a very investigative unit. It did not give the students the necessary formulas. Rather it made them figure them out for themselves.

Problem Solving – through problem solving, the students were able to learn the formulas for the volume and surface area of different prisms.

Representation – in this unit, the students created different representations of the prisms including three dimensional drawings, two-dimensional sketches, and scale models.

Content: Geometry

Analyze Characters: this unit helps students discover the properties of two and three-dimensional objects. It does investigative work with finding the perimeter and area of different shapes.

Use Visualization: given the dimensions of a shape or prism, the students will draw the object and be able to find the perimeter, area, or volume of the prism.