Reflections on Numbers
E
Multiplication and Division
Section E: Reflections on Numbers 47 9. Three students used their calculators to generate random natural
numbers. Tim got 36 as his first number, Pam got 41, and Joyce got 50. Who is most likely to end up with a whole number after generating a second natural number and dividing? Justify your reasoning.
10. Describe the kinds of numbers you might get when you divide two whole numbers.
Press
i Using a calculator, create two random natural numbers ranging from 1 to 100.
Arrow over to PRB and select randInt(
TEST MATH
1 , 100 , 2 )
Press
ENTER
Press
iii Divide the first random number by the second.
Determine whether or not the result is a natural number.
Record your results.
and record the two random numbers.
ENTER
ii Repeat this process by continuing to press until you have 10 results.
iv Combine your results with those from the rest of the class.
Then use these numbers to explore the chance of getting a whole number quotient when you divide two randomly chosen natural numbers.
Based on your results, what is the chance of getting a natural number quotient when you divide two randomly chosen natural numbers?
MATH NUM CPXPRB :rand
2:nPr 3:nCr 4:!
5:randInt(
6:randNorm(
7:randBin(
randInt(1,100,2)
randInt(1,100,2) {86 73}
Random Number Activity
Reflections on Numbers
E
Consider the first quadrant of the coordinate system again.
11. a. On Student Activity Sheet 4, assign a new value to each of the grid points by dividing the first coordinate by the second.
For example, point Awith coordinates (3, 2) now has a value of 3
2(3 2 32). Leave your answers as fractions reduced to lowest terms. Label this grid the Division Grid.
b. Use the Division Grid to find the chance of getting a whole number quotient when you divide two randomly chosen whole numbers. Compare this with your answer to problem 7.
12. a. Circle the grid points that have whole number values.
b. There are more whole numbers circled in some columns than in others. Explain why this happens.
13. Use color pens to circle all of the points with the same value in one color. Describe the patterns that you notice.
14. a. Describe how to travel on the grid between two points, each having a value of 1
2. y
A
0 x 1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
3 2
Section E: Reflections on Numbers 49
Reflections on Numbers A
15. a. Compare the locations of the points labeled 2 with those of the points labeled 12. Describe what you find.
b. The points labeled 2
3 and 3
2 are reflections of one another over the line formed by connecting all points with a value of 1.
Name some other values that are reflections over this line.
16. Consider the Subtraction Grid and the Division Grid.
a. For each grid:
• Describe the location where points have the same value.
• Describe the location of any whole numbers.
b. Consider the line yx. Using each grid, decide what values lie on this line. Explain why this happens.
c. What happens on each grid when you use zero as the first or the second number?
d. Are there points that have the same value on both grids?
Explain.
Here are all four quadrants of the rectangular coordinate system, labeled counterclockwise from Quadrant I to Quadrant IV.
17. a. On Student Activity Sheet 5, recreate the Division Grid by assigning a value to each of the grid points by dividing the first coordinate by the second.
b. Do the observations you made in problems 12 and 13 still hold true? Describe any other patterns that you notice.
y
1
0 1 2 3 4 5 6 7
1 1 2 3 4 5 6 7
2 3 4 5 6 7 2 3 4 5 6 7
x Quadrant I
Quadrant II
Quadrant III Quadrant IV
19. a. If you square a whole number, will you always get a whole number as a result?
b. If you raise a whole number to some power other than two, will you always get a whole number as a result?
c. If you take the square root of a whole number, will you always get a whole number as a result?
d. How can you tell whether or not you can give an exact number for a square root?
The inverse operation of squaring is taking the square root. When you take the square root of some whole numbers, you cannot write them as integers or fractions. In these cases, you can use the square root sign ().
For example, you cannot write the square root of two as a whole number or a fraction; you must leave it as 2 . Numbers that you cannot express as rational numbers are called irrational numbers.
20. a. What is the square root of five? Is it irrational? Why or why not?
b. Which of the following numbers are irrational?
1.69, 36, 14, 10, π
When you divided two integers, you found that the quotient was not always a whole number but was often a fraction or decimal number.
Every number that you can write as a quotient of two integers is called a rational number. The word rational comes from the word ratio.
18. a. Is every natural number also a rational number? Show two examples.
b. Is zero a rational number? Explain.
Reflections on Numbers
E
Powers and Roots
Section E: Reflections on Numbers 51
Reflections on Numbers E
21. a. Use Student Activity Sheet 6to locate 9 and –9 on the number line. Use arrows to point to the right location of the numbers as accurately as possible.
b. How can you locate 99 on the number line without the use of a calculator?
c. Place each of the following numbers on the number line. Use an arrow to point to the location of the numbers as accurately as possible.
–41
4, 0.01, 27, –0.99, π, 225, 10.5, 10.05, 3.52
22. a. Use the second number line on Student Activity Sheet 6to locate 2 and 3 .
b. 2 and 3 are both irrational numbers. Find a third irrational number that is located somewhere between 2 and 3 . Give this number.
c. Find a fourth irrational number that is located somewhere between 2 and the third number. Give this number.
23. a. Draw a number line that shows 5.1 on the left and 5.2 on the right.
b. 5.1 and 5.2 are both rational numbers. Find a third rational number that is located somewhere between 5.1 and 5.2. Give this number.
c. Find a fourth number that is located somewhere between 5.1 and the third number. Give this number.
The rational and irrational numbers combined together make up real numbers.
In the last two problems, you have investigated a property of real numbers: Between any two real numbers on the number line, you can always find another real number!
–10 0 10
Reflections on Numbers
In this section, you investigated the properties of real numbers. Here are the names of all the different real numbers.
Natural Numbers
Natural numbers are numbers 1, 2, 3, 4, and so on. The counting numbersis another name for natural numbers.
Examples: 1, 36, 65, 234, 2.57 106
Whole Numbers
All natural numbers and zero make up the whole numbers.
Examples: 0, 1, 49, 54, 2.54 104
Integers
All natural numbers, zero, and the opposite of the natural numbers, make up integers.
Examples: –34, –5, 16, 0, 100, 32, 99, 2.67 105
Rational Numbers
All numbers that you can write as a ratio of two integers are rational numbers.
Examples: –22.7, – 914, –6, –4, –13, – 3.7 10-4, 0, 3.7 10-4,
12, 14, –225, 22.7, 3.7 104
Irrational Numbers
Numbers that cannot be expressed as rational numbers are called irrational numbers.
Examples: The square root of 10 (10) cannot be written as an integer or as a fraction; it is approximately equal to 3.16.
Pi (π) is approximately equal to 3.14 or 31
7.
Real Numbers
The rational and irrational numbers combined make up real numbers.
Examples: –22.7, – 914, –6, –10, –4, –13, – 3.7 10-4, 0, 1 π
E
Section E: Reflections on Numbers 53 1. Is it possible to end up with a smaller number when you multiply?
Explain and give an example.
2. a. Is2
3 a rational or irrational number?
b. Is 112 a rational or irrational number?
c. Is 2 a rational or irrational number?
d. Would the numbers in 2a and 2b be rational if they were negative?
3. Consider a circle with a radius of 5 centimeters. On his calculator, Pablo computes the area of this circle as follows.
π25 Josie uses her calculator to compute:
3.14 25
a. Will Pablo and Josie get the same answer? Why or why not?
b. Will either of them get an exact answer? Why or why not?
4. a. Is 23a rational or irrational number?
b. Is 49a rational or irrational number?
5. Do negative irrational numbers exist? Explain your answer and give an example if appropriate.
How are ratios and rational numbers the same? How are they different?
Speed of Sound
Lena saw a flash of lightening and heard the thunder three seconds later. She knows a rule to estimate the storm’s distance—one mile for every five seconds.
1. What estimate will Lena get for the storm’s distance?
To understand why this works, you have to know that there is a big difference between the speed of light and the speed of sound. Sound travels through air with a speed of about 760 mi/h. Light travels almost 106times faster than sound.
2. a. What is the speed of light in mi/h? Write your answer as a numeral.
b. Use the speed of sound to calculate the distance in miles that sound can travel in three seconds.
c. Is Lena’s rule reasonable? Explain.
Peter is more familiar with metric units. He uses this rule — one kilometer for every three seconds.
3. Which rule is more accurate, Lena’s rule or Peter’s rule? Explain.
4. Calculate the speed of sound in feet per second.
Additional Practice
Section A Speed
When you stand in front of a rock wall and you clap your hands, you may hear an echo. The sound waves move from your hand to the wall, then reflect off the wall and travel the same distance back to you.
Kim heard an echo a half a second after she clapped her hands.
5. How far was Kim standing from the wall? You may want to
You can hear an echo when the time the sound travels from you to the wall and back again is more than 0.05 seconds.
6. What is the minimum distance you need to stand from the wall to hear an echo?
Additional Practice 55
Section B Notations
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 1 10-12Watts per square meter (W/m2).
Here are some sounds and an estimation of their intensity level.
1. 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?
Sound Source Intensity (in W/m2) Times Greater Than Threshold of Hearing (TOH)
TOH 1 10–12 1
Rustling leaves 1 10–11 10
Whisper 1 10–10
Library conversation
Normal conversation
1 10–9
1 10–6
Additional Practice
A new scale, the decibel scale, makes it easier to work with these small numbers. A sound intensity of 0 decibel (dB) represents the threshold of hearing.
You can use the table on Student Activity Sheet 7to 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 1 10–2W/m2. b. How many decibels does a MP3 player produce?
The threshold of pain is 1014TOH. This level causes pain and permanent hearing damage.
c. What is the decibel level for threshold of pain?
Sound Source Intensity (in W/m2)
Times Greater Than TOH Threshold of Hearing (TOH) 1 10–12 1
Rustling leaves 1 10–11 10
Whisper 1 10–10
Library conversation
Normal conversation
1 10–9
1 10–6
Decibel Level
0 dB 10 dB 20 dB 30 dB
Additional Practice 57
Additional Practice
Section C Investigating Algorithms
Throughout history, people have used different algorithms to do the same calculations. The gelosia or lattice method is an algorithm for multiplication used in Italy around 1500 A.D. Today, some people use a shortened version of this algorithm. Here is an example of the
problem Harvey worked (24 49) using the lattice method. You might recall the product is 1,176.
1. a. Explain how the lattice algorithm works.
b. Check your explanation by using the lattice method to multiply 52 63 and 254 647. Use a calculator to check your results.
2. a. Compare the lattice algorithm with the algorithm Hattie used on page 26. What are the advantages of using the lattice method? What are the disadvantages?
b. Compare the lattice method with the method used by Clarence shown on page 27.
Michael creates the following story to solve the division problem:
212 18.
“I walked 21
2 miles. I know that a city block is 1
8 of a mile. Each mile has eight blocks, so instead of 21
2 18, I can calculate 21
2 ____.”
3. a. Complete Michael’s story to rewrite the division problem 212 18as a multiplication problem.
b. Solve the division problem: 21
2 18.
1 2
1
4 9
7 6
4
8 1 8
1 6 3 6
Harvey has a piece of tape 212 feet long. He wants to cut the tape into pieces that are14 foot long.
4. a. Write a division problem that matches this story.
b. Solve the problem using Michael’s multiplication.
c. How can you calculate 21
2 15?
Additional Practice
Section D Operations
1. a. Use the area model to illustrate the distributive property for finding the product of 5 24.
b. Use the area model to calculate 6 12.5.
2. Show how you can use the distributive property to make some calculations easier. Don’t forget to find each product.
a. 2 27 c. 3 99
b. 5 49 d. 6 28
Here are two multiplication problems illustrating the area model.
3. Copy and complete each model. Include the multiplication problem associated with each model along with the correct answer.
1 2
1 2 4
6
20
3 15
600
4. Make special number sentences for all whole numbers from 1 through 10.
Here are the rules for making the special number sentence.
• You must use the number 2 exactly five times.
1. Use Student Activity Sheet 8to assign a value to each grid-line intersection by multiplying the first coordinate by the second. For example, if point Ahad the coordinates (3, 2), the value of point A would be 6 because 3 2 6.
Additional Practice 59
Additional Practice
Section E Ref lections on Numbers
x y
–7 –6 –5 –4 –3 –2 –1 0 1
–1 –2 –3 –4 –5 –6 –7 2 3 4 5 6 7
1 2 3 4 5 6 7
Quadrant II Quadrant I
Quadrant III Quadrant IV
A6
a. Color the grid points that have a perfect square number. Describe the location of all the perfect square numbers.
b. Describe the location of all the negative numbers.
c. Color the grid points that have a prime number. Describe the location of all the prime numbers.
How would you classify numbers that have a repeating decimal? Are they rational or irrational numbers? Can every repeating decimal be written as a fraction?
Here is a sophisticated way to find the fraction that is associated with 0.7. The notation 0.7means that 7 repeats indefinitely.
2. a. Use 9 0.77777… 7 and what you know about the
relationship between multiplication and division to complete ________0.7777…
b. What fraction equals 0.7777…?
c. Apply this technique to find the fraction for 1.5. Use a calculator to check your result.
10 1 9
0.77777… 7.77777…
0.77777… 0.77777…
0.77777… 7 (Subtracting)
Section A Speed
1. Helen’s average speed was 4 km/h.
You may have used a ratio table to find your answer.
2. a. The six means 6,000,000,000, which is six billion.
b. 6.4 109
3. The speed of the earth at the equator is about 1,042 mi/h. Note that it doesn’t make sense to write this answer with decimals.
You may have used the following strategy.
The earth completes one revolution in one day, or 24 hours.
At the equator, the distance around the earth is about 2.5 104 miles, which is 25,000 miles.
To find the speed in mi/h you can use a ratio table.
4. The average speed of the earth around the sun is 6.6 104mi/h, or 66,000 mi/h.
You may have used the following strategy.
The earth travels 5.8 108miles in 365 days.
One day 24 hours, so 365 days 8,760 hours.
Distance (in km) 18 36 4
4 9 1
Time (in hr) 1 2
2 9
2 9
Distance (in mi) 25,000
24 1
Time (in hr)
24
24 1,042
1,000 8.76
5. Mercury’s average orbital speed is faster than Earth’s average orbital speed.
Your strategy may be different from this strategy:
Mercury travels 48 km/s.
One hour 3,600 seconds, so that is my target.
Mercury’s average speed is about 17 104km/h.
Earth’s average speed is about 6.6 104mi/h.
Comparing 6.6 miles and 17 km, 17 km is more than 6.6 miles.
Mercury travels faster around the sun.
6. Here are the answers written in scientific notation along with one sample solution strategy.
a.
b.
c.
d.
Answers to Check Your Work 61
Answers to Check Your Work
Distance (in km) 48
1 60
Time (in sec)
60 60
60 60 2,880
3,600 172,800
6 106
(3 104) (2 102) 3 104 2 102 (3 2) (104 102) 6 106
2 104
(2 1010) 106 2 (1010 106) 2 104
2.45 103
Writing 2,450 in scientific notation is 2.45 103. (2 103) (4.5 102) 2,000 450
2,450
1.4 103
Writing 1,400 in scientific notation is 1.4 103. (2 103) (6 102) 2,000 600
1,400
7. a. Your opinion might vary from these.
• I do not think it is fair because it is unlikely that all of Larson’s timekeepers had a faster reaction time than all of Devitt’s timekeepers.
• It could be fair since the race was timed manually, and the timekeepers would have had different reaction times.
• I think it is fair because the swimmers were too close to distinguish actual finishing times. More than likely, the judges decided that John Devitt finished first, so they had to award him the race in spite of the recorded times.
b. Larson was about 18 cm behind, which is more than the length of a hand. This distance is visible, but it occurs in 0.1 second.
Here is one strategy.
When Devitt finished, Larson had to swim 0.1 second.
Larson swam 100 meters in 55.2 seconds.
Using 100 m 10,000 cm to set up this ratio table:
Larson was about 18 cm behind, which is more than the length of a hand. In less than 0.1 seconds, I can understand why the manual timers had difficulty reacting to this visible distance.
Answers to Check Your Work
Section B Notation
Distance (in cm) 10,000
55.2 1 0.1
Time (in sec)
55.2 10
55.2 10 181.2 18
1. a. 1,000x means the hair is 1,000 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 0.0038 cm and written
2. a. 1010105105 b. 10310510-2
c. 100 100,000 102105 or 10-3 d. 10 1,000,000,000 101109or 10-8 e. 10-410 10-4101 or 10-5
f. 10-6100 10-6102 or 10-4 g. 10-710310-10
3. a. 34,200 b. 0.0342 4. a. 1.6 10-8
b. (3.5 103) (1.2 10-2) 3.5 1.2 10310-2 (3.5 1.2) (10310-2) 4.2 101 42
5. a. 0.00267 2.67 10-3 b. 0.00000678 6.78 10-6
c. 15 20,000,000,000 7.5 10-10
Answers to Check Your Work 63
Answers to Check Your Work
Section C Investigating Algorithms
1. The answer is 1,464, but you must show two different strategies.
Here are three sample strategies to calculate 24 61.
• Using Clarence’s algorithm: • Using a ratio table:
20 60 1,200
24 61 1,464 20 1 20 4 60 240 4 1 4
1 61
2 122
4 244
20 1,220
24 1,464
• Using the area model:
Adding up all the parts:
1,200 20 240 4 1,464,so 24 61 1,464
2. a. Everybody in your class probably made up a different context.
Compare and discuss your answer with another student.
Here are some contexts that fit with 3,000 28.
• Three thousand marbles shared equally among 28 students.
How many marbles will each student get?
• There are three thousand cubes to pack in boxes. Each box holds 28 cubes.
How many boxes do you need to pack all the cubes?
b. Your strategy might match one of these.
• Using a ratio table:
3,000 28 107 with 4 left over.
• Using mental computation:
100 28 2,800 7 30 210, so
7 28 210 14 196.
So 3,000 divided by 28 is 107.
c. Answers will vary, depending on your context in part a.
Sample responses based on sample contexts in part a:
• In the first story, there are four marbles left over. (In reality, four people will probably receive an extra marble each, but that cannot work for this problem because then the marbles
Answers to Check Your Work
20 1,200 20
60 1
4 240 4
1 28
100 2,800
7 196
107 2,996
3. a.
b. Here are two possible multiplication problems.
(50 2) (30 7) 1,924 or 52 37 1,924 4. Here are the answers along with some explanations.
a. 6 15 30
Changing both to fifths:
6 15 305 15 15 fits 30 times in 30
5, so the answer is 30.
b. 23
4 18 22
Changing both to eighths:
23
4 18 228 18
22 1 22 c. 5 12
3 3
Changing both to thirds:
5 12
3 153 53
If you want to know how many times 53 fits into 153, you can calculate 15 5 3.
5. a. Different stories are possible, for example:
• I have 1012 kilograms of peanuts, and I want to divide them in portions of 3
4kilogram.
b. 101
2 is the same as 102
4 , which is 42
4 . c. 1012 34 = 424 34
42 3 14
Answers to Check Your Work 65
Answers to Check Your Work
30 1,500 60
50 2
7 350 14
6. The average speed is 334 km/h.
Answers to Check Your Work
Distance (in km)
2 1
Time (in hr) 23
21 2 71 2 334 3 2
3 2
Section D Angles
1. Many explanations are possible. Here are two. If neither explanation is like yours, discuss your explanation with another student.
• Because the calculator gives me an error message when I enter 0 0.
• Because there is no one answer that works all the time. Some people might think that 0 0 = 1 because usually when you divide a number by itself, you get an answer of 1. Another person might think 0 0 0 because you have nothing to share and no one to share with. A third person might think 0 0 7 because rewriting 0 0 7 works as a multiplication problem (7 0 0).
You cannot have multiple answers for the same division problem.
2. Maybe you started to use your calculator to find the answer. But if you take a closer look at the last parenthesis, you see 9 9,
which is 0. And multiplying by 0 always gives 0 as an answer.
3.
–12
3 –2
–4
3 2
2
–