Contents. Notation. Additional Practice. Answers to Check Your Work. Section B. Base Ten 16 Dilution 17 Small Numbers 20 Summary 22 Check Your Work 23

Transcription

1 Contents Section ase Ten 6 Dilution 7 Small Numbers 20 Summary 22 Check Your Work 23 Additional Practice Answers to Check Your Work Student Activity Sheets Contents v

2 ase Ten In the previous section, you worked with small numbers with one or more decimals. You will now further investigate decimal numbers, such as a. What is the value of each digit in the number 9.78? b. What is the value in each digit in the number 97.8? c. How does the number 9.78 compare to the number 97.8? Describe how they are the same and how they are different. In the unit Facts and Factors, you learned that when you multiply a number by 0, you multiply the value of each digit by 0. Consider and When you divide a number by 0, you divide the value of each digit by Fill in each blank on Student Activity Sheet 2. 6 Revisiting Numbers

3 hundreds tens ones tenths hundredths hundreds tens ones tenths hundredths thousandths Here are two schemas for multiplying by ten and dividing by ten. 3. On Student Activity Sheet 2, complete the schema that shows dividing by ten. 4. Calculate without using a calculator. a d b e. 5,000 c f..25,000 Dilution You have to dilute many common household products before you use them. You dilute dish soap with water before you wash dishes. Pool water dilutes chemicals, such as chlorine, so the water can be clean. You dilute condensed soup with milk or water before you cook it. To investigate the dilution process, Shelia dilutes food coloring with water. ounce 9 ounces 0 ounces Stir it up She adds one ounce of food coloring into the container that contains nine ounces of water. Shelia has a 0-ounce solution. 5. What portion of Shelia s solution is food coloring? What portion is water? Next, Shelia pours one ounce of the solution into an empty measuring container. She adds enough water to make 0 ounces and stirs the solution. 6. What portion of Shelia s second solution is food coloring? Section : 7

4 Shelia loves the new color. She decides to do it again and pours one ounce of this second solution into another empty measuring container. She adds enough water to make a 0-ounce solution. 7. What portion of Shelia s third solution is food coloring? 8. Suppose Shelia repeats this process again. What portion of Shelia s fourth 0-ounce solution would be food coloring? 9. Copy and complete the table for six dilutions. Suppose the table extended up to 20 dilutions. 0. After 2 dilutions, what portion of the 0-ounce solution is food coloring? What about after 20 dilutions? Number of Dilutions Portion of the 0-oz Solution That Is Food Coloring 0. or or 00 You have probably found that it is just as tedious to write very small numbers as it is to write very large numbers. An abbreviation system like the one that you use for large numbers can be developed and used for very small numbers as well. When you were diluting food coloring, you divided the portion of food coloring in each solution by 0. Here is a way of describing this dilution process using arrow language Copy and complete the arrow string until you reach the sixth dilution. Here is an arrow string starting at 0,000 and repeatedly dividing the result by 0. 0,000 0, Copy and continue the pattern until you have eight arrows. 8 Revisiting Numbers

5 0, , You can display this pattern of dividing by 0 vertically. 3. a. Copy the vertical display on the left in your notebook. Write each result also as a power of 0. Do this only with the numbers for which you know the power of 0. b. Describe any patterns you notice in the powers of 0. How would you continue this pattern? c. How would you explain to someone that 0 0? 4. a. Look back at your chart of Shelia s food coloring dilutions. Suppose you have a solution that has a 0 4 portion of food coloring. How many dilutions have taken place? How do you know? b. How many dilutions have taken place if the solution is 0 6 food coloring? c. What portion of the solution is food coloring if you have done five dilutions? Write this number as a power of 0. d. Describe the relationship between the number of dilutions and the portion of the solution that is food coloring. Think about diluting the seventh solution to make the eighth solution. Remember that each dilution involves dividing by a. What is written as a power of 0? b. What is ? (Hint: Dividing by 00 is two dilutions.) 6. Show that you can write the result of 00,000 as a. Calculate b. Calculate Section : 9

6 Near the end of Section A, you created a general rule for multiplying powers of ten. If m and n are natural numbers, then 0 m 0 n 0 m+n. 8. a. Give examples to show that this rule is also valid for negative values for m and n. b. Create a general rule for dividing powers of ten. 9. Find each product or quotient. Write your answers both as a power of ten and as a numeral. a e b f c g d h Small Numbers Here is a photo of bacteria enlarged 0,000 times. 20. a. Measure the diameter of the image of one of the bacterium in the photo in millimeters (mm). b. Find the actual size of this bacterium. 0,000x Mycoplasmal pneumoniae is a type of bacteria known to cause a sore throat, bronchitis, and pneumonia. Mycoplasma bacteria range between mm and mm in size. 2. Can the bacteria from problem 20 be an example of the mycoplasmal pneumoniae bacterium? Justify your reasoning. 20 Revisiting Numbers

7 22. Copy these lines in your notebook and fill in the missing numbers. Size of Mycoplasmal acteria The last line shows how to write in scientific notation written in scientific notation is To write in scientific notation, you can start a pattern a. Continue the pattern until the number is in scientific notation. b. Write in scientific notation. c. Write what your calculator displays after you enter: 20 30,000,000,000,000 How do you write this answer in scientific notation? d. Write as a numeral. 24. Reflect Why is not written in scientific notation? How would you write this number in scientific notation? Section : 2

8 al Systems The number system we use today is a positional notation system based upon powers of ten. Each digit occupies a position connected to a power of ten. The position of each digit in a number determines its value. You can read the number 4.75 as: fourteen and seventy-five hundredths. You can write the number 4.75 as: You can also write the number 4.75 using powers of Dividing by Ten When you divide a number by 0, you divide the value of each digit by 0. hundreds tens ones tenths hundredths thousandths Scientific Scientific notation is a way of writing very large or very small numbers. A number written in scientific notation is a product of a number between and 0, and a power of ten. 45,000, ,000,000, so 45,000,000 written in scientific notation is Revisiting Numbers

9 , so written in scientific notation is Calculators display very small or large numbers using scientific notation. A calculator may display as: or as 2.5 E 05 Operating with Powers of 0 If p and q are whole numbers, then 0 p 0 q 0 p+ q, and 0 p 0 q 0 p q. Here is an enlarged photo of a human hair.. a. What does the,000x on the picture mean? b. Measure the width of the hair shown shown in the photo in centimeters. c. What is the actual thickness of this hair? Write your answer as a numeral and in scientific notation.,000x 2. Write each answer as a power of ten. a e b f c ,000 g d. 0,000,000, Write as a numeral. a b Section : 23

10 4. a. Use a calculator to find the answer of 4 250,000,000. Write your answer in scientific notation. b. Calculate the product of and Write in scientific notation. a b c. 5 20,000,000,000 How would you explain to someone that or 0.00? 24 Revisiting Numbers

11 Additional Practice Section s Sound is actually a form of energy. The intensity of sound is measured in Watts per square meter. The softest sound that a human ear can detect has an intensity of 0-2 Watts per square meter (W/m 2 ). Here are some sounds and an estimation of their intensity level. Sound Source Intensity (in W/m 2 ) Times Greater Than Threshold of Hearing (TOH) TOH 0 2 Rustling leaves 0 0 Whisper 0 0 Library conversation 0 9 Normal conversation 0 6. a. Consider the relationship between the threshold of hearing (TOH) and the sound of rustling leaves. Explain that the sound intensity of rustling leaves is ten times greater than the TOH. b. How many times greater is the intensity of a library conversation than the TOH? c. How many times greater is the intensity of a normal conversation than the intensity of a library conversation? Additional Practice 55

12 Additional Practice A new scale, the decibel scale, makes it easier to work with these small numbers. A sound intensity of 0 decibel (d) represents the threshold of hearing. Sound Source Intensity (in W/m 2 ) Times Greater Than TOH Decibel Level Threshold of Hearing (TOH) 0 2 Rustling leaves 0 0 Whisper d 0 d 20 d Library conversation d Normal conversation 0 6 You can use the table on Student Activity Sheet 7 to solve the following problems. 2. a. How many decibels is the sound intensity of a normal conversation? Music using an MP3 player has an intensity of 0 2 W/m 2. b. How many decibels does a MP3 player produce? The threshold of pain is 0 4 TOH. This level causes pain and permanent hearing damage. c. What is the decibel level for threshold of pain? 56 Revisiting Numbers

13 Section. a.,000x means the hair is,000 times larger than its actual size. b. The thickness of the hair in the picture is about 3.8 cm. c. The actual thickness is about cm and written cm. 2. a b c , or 0-3 d. 0,000,000, or 0-8 e or 0-5 f or 0-4 g a. 34,200 b a b. ( ) ( ) a b c. 5 20,000,000, (3.5.2) ( ) Revisiting Numbers

14 Student Activity Sheet 2 Use with Revisiting Numbers, pages 6 and 7. Name 2. Fill in each blank When you divide a number by 0, you divide the value of each digit by The following schemas show multiplying by ten and dividing by ten: hundreds tens ones tenths hundredths hundreds tens ones tenths hundredths thousandths Complete the schema that shows dividing by ten. 80 Revisiting Numbers Student Activity Sheet 2 Encyclopædia ritannica, Inc. This page may be reproduced for classroom use.

15 Name Student Activity Sheet 7 Use with Revisiting Numbers, page 56. You can use the table below to solve the following problems. 2. a. How many decibels is the sound intensity of a normal conversation? b. Music using an MP3 player has an intensity of 0 2 W/m 2. How many decibels does an MP3 player produce? c. The threshold of pain is 0 4 TOH. This level causes pain and permanent hearing damage. What is the decibel level for the threshold of pain? Sound Source Intensity (in W/m 2 ) Times Greater Than TOH Decibel Level Threshold of Hearing (TOH) 0 2 Rustling leaves 0 0 Whisper d 0 d 20 d Library conversation d Encyclopædia ritannica, Inc. This page may be reproduced for classroom use. Normal conversation 0 6 Student Activity Sheet 7 Revisiting Numbers 85