# Math. So we would say that the volume of this cube is: cubic units.

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1 Math Volume and Surface Area Two numbers that are useful when we are dealing with 3 dimensional objects are the amount that the object can hold and the amount of material it would take to cover it. For instance, if you worked for Quaker Oats you would want to design a box that could hold 12 oz of oats; if you needed to wrap a birthday gift, you would want to know how much paper you would need. The amount it takes to fill a container is called the Volume of the container. The amount it takes to cover a container is called the Surface Area of the container. Volume Volume is measured in Cubic units: cubic inches, cubic cm, etc. What are cubic units? Cubic units are cubes that measure 1unit high x 1 unit wide x 1 unit deep. The volume of a cubic unit is 1 unit cubed. To find volume you need to know how many of these cubic units can fit into the container. I. You have been given a foam cube and some cubic units. Find the dimensions of the foam cube. Measure only the solid part not the crags. Round your answer to the nearest unit. The dimensions are: x x Now fill the foam cube with the cubic units. How many cubic units did it take to do this? So we would say that the volume of this cube is: cubic units. Do you see a relationship between the dimensions of the cube and its volume? Explain II. Now stack 2 cubes, one on top of another. Find the dimensions of this rectangle. Again do not include the crags in your measurement. Round your answers to the nearest unit. The dimensions are x x 1

2 How many cubic units do you think it take to fill this rectangle? So we would say that the volume of this rectangle is: cubic units Do you see a relationship between the dimensions of the rectangle and its volume? Explain It s easy to see how finding the volume by filling a container with cubic blocks may not be very practical. We need a quicker way to find the volume. Based on the findings above, state a formula in words for finding the volume of a Cube and the volume of a Rectangle: Volume of a Cube = Volume of a rectangle = Another way to find Volume Stack the two foam cubes again. What is the area of one of the smaller sides of the rectangle (the base)?. Measure the height of the rectangle and then multiply this by the area of the base. What do you get? Compare this number to the volume that you found using the cubic units. What do you notice? Explain this method of finding the volume. In words, write out the formula for finding the volume using this method. Volume = Volume of a Cylinder It is difficult to find the volume of a cylinder using the cubic blocks. But the second method we used to find the volume of a rectangle works great for a cylinder, too. To use it, you need to recall how to find the area of a circle. The formula is: A= Use this second method to find the volume of a cylinder whose base has a radius of 3cm and height of 5 cm. Volume = cubic cm or cm 3 More practice: Table 1 2

3 Radius of the base Area of the base Height of the Cylinder 4 in 6 in 7 cm 12 cm 2.5 cm 8.25 cm Volume Surface Area Area is always measured in square units : square inches, square cm, etc. What are square units? Square units are squares that measure 1 unit on a side. They have an area of 1 square unit. To find the surface area of a container we need to know how many square units it takes to cover the container. I. You have been given some square units. Begin by opening up the foam cube so that it is flat. Now cover it using the square units. How many square units does it take to do this? So we would say that the surface area is: square units. What is the relationship between the number of square units and the dimensions of the figure? Explain how you could find the surface area of this figure using the dimensions of the pieces. State a formula for finding the Surface Area of a Cube: Surface Area = Could this same formula be applied to finding the Surface Area of other shapes? Explain Surface Area of a Cylinder 3

4 You have been given an open cylinder. Measure the height of the cylinder in inches. Find the circumference of the base of your cylinder. Recall the Circumference = the distance around a circle. The formula to find the circumference is: C = 2*π*radius or π*diameter Now flatten out your cylinder. This flattened cylinder is a rectangle. The measure of two sides of the rectangle are equal to the height of the cylinder. The measure of the other two sides is the same as the of the base of the cylinder. What is the area of the rectangle? Area = square inches or in 2. Can you see that the area of this rectangle is the surface area of the cylinder? Stat a formula in words to find the Surface Area of a Cylinder Surface Area of a Cylinder = Now suppose you had a closed cylinder, one that has a top and bottom, like the Quaker Oats box. Would this make a difference in the surface area? Why? Explain how you could find the surface area of the closed cylinder. In words, write out a formula to find the surface area of a closed cylinder. Surface Area of a closed cylinder = 4

5 Practice finding the surface area for the cylinders in Table 1. Assume that each cylinder is closed. Final Question: Is the volume the same for an open cylinder or a closed cylinder of the same size? Explain and give an example to support your claim. Science Does Cell Size Count In this experiment you will be using agar blocks of to represent cells. The blocks contain phenolphthalein, a ph indicator dye. You will soak the blocks in NaOH for ten minutes; this will turn the phenolphthalein pink. You will then determine which block shows the greatest diffusion of the NaOH. Objectives: Materials: 1. Determine the effect of cell size on diffusion 2. Discover the relationship between surface-area and volume 3. Discuss why cells divide o plastic spoon o scalpel o 3 agar blocks of different sizes (3.0 cm, 2.0 cm, 1.0 cm) o small metric ruler o paper towels o 500 ml beaker filled with 250 ml of 0.4% NaOH o gloves and goggles Procedure: Table 1 1. Using the spoon, carefully place all three cubes in the beaker of NaOH 2. Swirl the beaker gently and begin timing immediately. 3. Let the experiment run for ten minutes 4. While the experiment is running complete Table 1. 5

6 Cube Size 1 cm Surface Area (cm 2 ) Volume (cm 3 ) Surface Area to Volume Ratio 2 cm 3 cm surface area= number of surfaces X length X width volume = length X width X height surface area to volume ratio = surface area volume 5. At the end of ten minutes use the spoon to remove the cubes from the beaker and blot each gently using a paper towel. 6. Cut each cube in half using the scalpel 7. For each cube calculate the distance in mm from the edge of each cube to the center of each cube. Do this by recording the length of one side by 2. Record this in Table Using the ruler, measure the distance in mm that the NaOH has traveled into the cube and record this in Table 2. 6

7 9. Calculate and record in Table 2 the percent of the total distance the NaOH has traveled. % = distance traveled X 100 total distance Table 2 Cube Size 1cm Total Distance to Center of Cube (mm) Distance NaOH has Diffused (mm) % Distance the NaOH has Travelled 2cm 3cm Conclusions: answer in complete sentences. 1. Which block has the greatest surface area? 2. Which block has the greatest surface area to volume ratio? 3. Into which block did the NaOH diffuse the most? 4. If the blocks were actual cells, which would be the most efficient in terms of permitting materials to diffuse across the cell membranes. 5. What happens to the surface area to volume ratio of a cello as it grows? 6. Based on this experiment why are what is an advantage of being multicelluar as opposed to single cellular? Carol Hay, Jessie Klein, Jane Wiggins Middlesex Community College 7

8 Education Lesson Plan I. Name of Activity: Making Patterns Main Curriculum Area: Math Teacher to Child Ratio: 1-15 Age Group: Older Pre-K Length of Activity: minutes for each activity II. Curriculum Frameworks for Preschool Experiences Guideline: Math Category: Patterns and Relations Learning Guidelines: (# 7) (# 8) (# 9) Guideline: Math Category: Shapes, Spatial and Sense Learning Guidelines: (# 10) Guideline: Math Category: Number Sense Learning Guidelines: (# 1) (#2) III. Describe the Activity in Detail: IV. Goals of the Lesson: At the completion of this lesson, the learner should be able to: 1. Recognize numbers 2. Identify shapes V. Objectives of Activity: At the completion of this lesson, the student should be able to: Physical: Move and match patterns Emotional: Cognitive: Apply number sense/spatial and shapes Social: Work with their partner and one on one with the teacher Objectives for Diverse Learning Styles: At the completion of this lesson, the learner should be able to: Visual: To recognize colors, shapes, patterns and spatial Auditory: To hear and match the shapes/colors/patterns Kinesthetic: Touches and places the pattern on the pattern 8

9 VI. Materials, Preparation, and Set-up Teachers will provide shapes, patterns, and dowels that the children will use in the lesson. 5 Children at each table and the material will be ready for them to use. The four tables will have the different shapes, sizes, patterns, sequencing. Teacher will demonstrate the use of the material to the children before they begin the lesson. Teaching of the Lesson There are three activities: 1. In a group each child will be given a shape (different one to each child) Teacher will explain the patterns and how they work and fit together. Teacher provides a pattern on the flannel board. The child with the corresponding shape will show the teacher and then place it on the flannel board where it belongs. Several different shapes will be used to provide children with the opportunities to recognize patters, shapes and colors. All children will have a turn. 2. One/One Teacher/Student Teacher provides a student with a pattern on a sheet of paper with shapes. The teacher instructs the child to create the pattern themselves underneath the pattern on the flannel board. 3. Self-Producing Teacher provides different color shapes and a strip of paper. The child works independently to create its own pattern. Observe each child s ability to create different patterns. (i.e. A,B A,B,B,A ABC, ABCA etc.. The teacher observes each child s ability and documents success and or difficulty the child has with the patterns. VIII. Reference Source: MAC with Carol Hay 9

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