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1 NAME: OPTION GROUP: WJEC MATHS FOR AS BIOLOGY CELL CALCULATIONS Maths Covered in this booklet are: 1. Revision of Scientific Notation. 2. Significant figures, number of decimal points and rounding numbers. 3. Units of length and equivalent units. 4. The size of cells and organelles. 5. Surface area, volume and Surface area to volume ratio.

2 It s not the smartest people who achieve success. It s the people who procrastinate less and make fewer excuses as they take actions everyday towards the goals they want to achieve. Unknown 2

3 REVISION OF SCIENTIFIC NOTATION Also introducing rounding numbers, number of decimal places & significant figures. Worked examples on scientific notation 1. Express the following numbers in scientific notation. 450, 8750, , 0.875, , Express the following number as decimals. 8.9x10 1, 7.234x10-6, x10 3, 100x10-1, 0.5x

4 Exercises on scientific notation Q1. Express the answers to the following calculations in scientific notation. 567x45, 900x2, , 0.64x0.098, , , , Q2. State the exponent that would cause the following movement of the decimal point. (a) The decimal point has been moved 3 places to the right. (b) (c) (d) The decimal point has been moved 1 place to the left. The decimal point has been moved 4 places to the left. The decimal point has been moved 6 places to the right. Q3. What would be the exponent in the following? (a) A tenth. (b) (c) (d) A thousandth. A billion fold. A hundred fold. 4

Rounding numbers Place Values Here is the general rule for rounding numbers: 1. If the number you are rounding is followed by: 5, 6, 7,8 or 9 round the number up. 2.

5 Rounding numbers Place Values Here is the general rule for rounding numbers: 1. If the number you are rounding is followed by: 5, 6, 7,8 or 9 round the number up. 2. If the number you a rounding is followed by 0, 1, 2, 3 or 4 round the number down. What are you rounding to? When rounding a number, you first need to ask: what are you rounding it to? Numbers can be rounded to the nearest ten, hundred or thousand, and so on. Consider the number rounded to the nearest ten is rounded to the nearest hundred is rounded to the nearest thousand is 9000 Rounding and fractions Rounding fractions works exactly the same way as rounding whole numbers. The only difference is you round to tenths, hundredths and thousandths, and so on. For example: rounded to the nearest tenth is rounded to the nearest hundredth is rounded to the nearest thousandth is

6 Correcting to a specified number of decimal places Giving a number to the nearest tenth, hundredth etc. For example, µm is 1.4µm to the nearest tenth. As tenths are represented by the first decimal point, we write: µm = 1.4µm to 1 decimal place (1d.p.) A number given to the nearest hundredth, thousandth etc are said to be correct to 2 d.p and correct to 3 d.p respectively. So, µm = 1.38µm to 2d.p µm = 1.384µm to 3d.p. Are you sure that you understand how I have/have not rounded the numbers in the above examples? Number of significant figures A mature female egg cell has a size of approximately 123µm or nm. In each number, the first number (number 1) has a different place value although the number 1 is the first figure in each number. It is called the first significant figure. The figure 2 is the second significant figure. Here is the general rule for quoting a number to a certain number of significant figures. Reading any number from left to right, regardless of the decimal point, the first significant figure is the first non-zero figure. The second significant figure is the next figure, which can be a zero or any other number. You continue this for further significant figures. 6

7 SECTION ONE Calculating the size of cells in scientific notation and converting between different units of length. We will use the following equation in this section: As well as scale bars. Magnification = Size of image Actual size Below is a table listing the different units of length. 7

8 Notes on cell sizes 8

Worked examples on cell sizes using the magnification equation 1. The images below and on the next page are electron micrographs of a bacterium and a mitochondrion.

9 Worked examples on cell sizes using the magnification equation 1. The images below and on the next page are electron micrographs of a bacterium and a mitochondrion. The actual length of the bacterium is 2.7µm and the actual length of the mitochondrion is 1.175µm. Calculate: (i) The magnification of each micrograph. (ii) The actual width of the bacterium and the mitochondrion. 9

10 81mm 235mm Scale bars Scale bars are lines that are drawn next to images of cells and microorganism as shown below: The scale bars allow a quick calculation of the actual size of the specimen. 10

11 Notes on Scale Bars 11

12 Worked examples on scale bars 1. Using the scale bar calculate the (a) (b) Actual length and actual width of the organisms below. Using both the actual length and actual width calculate the magnification of the image. 2. Using the scale bar calculate the (a) (b) (c) Actual width of the nucleus in the cell below. Using actual width calculate the magnification of the image. Convert the answer to part b into mm and nm. 12

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14 Exercises on cell sizes Q1 A cell image has a magnification of X The width of the cell is 5.2cm. Calculate the actual width of this cell. Q2. The image below is of a chloroplast as seen by an electron microscope. The image magnification is X Structure B is a starch grain and structure L is a granum. (a) (b) Calculate the actual length of the starch grain and granum. The lines have been drawn in for you to measure. State your answer in mm, µm and nm. Calculate the percentage decrease in width between the starch grain and granum. 14

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16 Q3 Using the image below calculate the magnification of cell A. Q4 Using the image below, which has a magnification of X27500, draw a suitable scale bar next to the image. 16

Section Two Volume, surface area, surface area to volume ratio of a cube/rectangle, a cylinder and a sphere. Also, the circumference and area of a circle.

17 Section Two Volume, surface area, surface area to volume ratio of a cube/rectangle, a cylinder and a sphere. Also, the circumference and area of a circle. In this section, we will cover: 1 Calculations on the circumference and area of a circle. 2 Calculations on cell volume, surface area and surface to volume ratio using cubes/rectangles, cylinders and spheres. The equations used in this section are: Cube/Rectangle Volume = Length x width x breath V = L x W x B Surface area = 6 x length x width SA = 6 x L x W Cylinder Volume = pi x radius squared x height V = π x r 2 x h Surface Area = 2 x pi x radius squared + 2 x pi x radius x height. SA = 2 x π x r x π x r x h Pi = 3.14 Sphere Volume = 4/3 x pi x radius cubed V = 4/3 x r 3 Surface area = 4 x pi x radius squared SA = 4 x π x r 2 Circle Area = pi x radius squared A = π x r 2 Circumference = 2 x pi x radius C = 2 x π x r You can watch videos by clicking the link below to help you with this section. 17

18 1. The circumference and area of a circle. Notes from video 18

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20 Worked Examples on the circumference and area of a circle. 1. (a) Calculate the circumference and the area of a circle with a radius of: (i) 23mm, (ii) 4m, (iii) 23cm, (iv) 4.128µm, (v) 5.3nm (b) Express your answers to part a as (i) correct to 2 decimal places and (ii) to 3 significant figures. 20

21 Exercises on the circumference and area of a circle. 1. Consider the electron micrograph below and calculate the area and circumference of organelle A (assume it s a perfect circle). A 21

22 2. (a) Consider the electron micrograph of the nucleus below. The circumference of the nucleus is 8.6µm. Calculate the area of the nucleus and the magnification of the image. (b) Confirm, using calculations, that the scale bar in the above image is of the correct value. 22

23 (c) How many times smaller is area of the nucleolus compared to the nucleus? 23

24 2. Cell volume, surface area and surface to volume ratio of cubes/rectangles, cylinders and spheres. Notes from videos 24

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The cells of this tissue are rectangular. The dimensions of one columnar cell is 6.2µm in length and 2.

27 Worked examples on cell volume, surface area and surface area to volume ratio of cubes/rectangles and spheres. Q1. The light microscope image below is of columnar epithelium from the human small intestine. The cells of this tissue are rectangular. The dimensions of one columnar cell is 6.2µm in length and 2.3µm in width and depth. Calculate the volume and surface area of this cell. Express your answer to 3d.p. 27

Q2. Cuboidal epithelium is a tissue composed of cells shaped like a cube. Cuboidal epithelium can form tube structure like the one from the kidney in the image the below.

28 Q2. Cuboidal epithelium is a tissue composed of cells shaped like a cube. Cuboidal epithelium can form tube structure like the one from the kidney in the image the below. The 5 cuboidal cells next to the black arrow are arranged in a monolayer and the dimensions of each cell are 2.1µm by 2.1µm by 2.1µm. Each cell is attached to the next cell. Calculate: (i) (ii) (iii) The total volume of the 5 cells. The surface area of the 5 cells. The surface area to volume ratio. Express your answers to 3 significant figures. 28

29 Q3. Centrioles are organelles that have a role in cell division. Their structure is shown below. The length of the centriole is approximately 500nm and the diameter is approximately 250nm. Calculate the volume of the centriole and quote your answer rounded to the nearest tenth. 29

30 Exercises on cell volume, surface area and surface area to volume ratio of cubes/rectangles, cylinders and spheres. Q1. A typical diameter of a nucleus is 8µm. (i) (ii) Calculate the volume of this nucleus. Calculate the circumference of this nuclei. Express each answer to the 4 decimal places. 30

31 Q2. Below is an electron micrograph of yeast cells. A Assume cell A is a perfect sphere, calculate (i) The volume (ii) The Surface area (iii) The area (iv) The circumference Express your answer to 2 d.p. 31

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Decimal Addition: Remember to line up the decimals before adding. Bring the decimal straight down in your answer.

Summer Packet th into 6 th grade Name Addition Find the sum of the two numbers in each problem. Show all work.. 62 2. 20. 726 + + 2 + 26 + 6 6 Decimal Addition: Remember to line up the decimals before