Do you ever hear people talk about the factors they consider when they make a decision? Ifyou wanted to decide what movie to see, factors that could affect your decision might include who is in the movie, what the movie is about, and which movies are playing at the nearby theater. Factors that affect decisions are like the numbers that affect a product.
In mathematics, factors are numbers that create new numbers when they are multiplied. A number resulting from multiplication is called a product. In other words, since 2(3) = 6, 2 and 3 are factors of 6, while 6 is the product of 2 and 3. Also, 1(6) = 6, so 1 and 6 are two more factors of 6. Thus, the number 6 has four factors; 1, 2, 3, and 6. In this lesson, you will use an extended multiplication table to discover some interesting patterns of numbers and their factors.
1-73. Have you ever noticed how many patterns exist in a simple multiplication table? Such a table is a great tool for exploring products and their factors.
Fill in the missing products to complete the table.
With your team, describe at least three ways (besides simply multiplying the row and column numbers) that you could use the table to figure out the missing numbers.
1-74. Gloria was looking at the multiplication table and noticed an interesting pattern.
“Look,” she said to her team.“All of the prime numbers show up only two times as products in the table, and they are always on the edges.”
Discuss Gloria’s observation with your team. Then choose one color to mark all of the prime numbers. Why does the placement of the prime numbers make sense?
1-75. See how many patterns you can discover in the multiplication table. Use the suggestions in parts (a) through (c) below to help you get started.
Gloria’s observation in problem 1-74 related to prime numbers. What other kinds of numbers do you know about? Begin by brainstorming a list of kinds of numbers you have discussed. (You may want to look back at Lesson 1.2.3 to refresh your memory.) Where do these different kinds of numbers appear on the table?
What patterns can you find in the locations of the numbers of each type? Be ready to explain your observations.
Notice how often different types of numbers appear. Record your observations. Do you find any patterns that makesense? Explain.
1-76. Consider the number 36, which could have been Ann’s number in part (b) of problem1-62.
Choose a color or design (circle or x the number) and mark every 36 that appears in the table.
Imagine that more rows and columns are added to the multiplication table until it is as big as your classroom floor. Would 36 appear more times in this larger table?If so, how many more times and where? If not, how can you be sure?
List all of the factor pairs of 36. (A factor pair is a pair of numbers that multiply to give a particular product. For example, 2 and 10 make up a factor pair of 20, because 2 · 10 = 20.) How do the factor pairs of 36 relate to where it is found in the table? What does each factor pair tell you about the possible rectangular arrays for 36? How many factors does 36 have?
1-77.Frequencyis the number of times an item appears in a set of data. What does the frequency of a number in the multiplication table tell you about the rectangular arrays that are possible for that number?
Gloria noticed that the number 12 appears, as a product, 6 times in the table. She wonders, “Shouldn’t there be 6 different rectangular arrays for12?” What do you think? Work with your team to draw all of the different rectangular arrays for 12 and explain how they relate to the table.
How many different rectangular arrays can be drawn to represent the number 48? How many times would 48 appear as a product in a table as big as the classroom? Is there a relationship between these answers?
In problem 1-76 how many different rectangular arrays could be drawn to represent the number 36? How many times did it appear as a product in a table as big as the room?
Does the pattern you noticed for 12 and 48 apply to 36? If so, why does this make sense? If not, why is 36 different?
1-78. PRIME FACTORIZATION
What are all the factors of 200?
A prime factor is a factor that is also a prime number. What are the prime factors of 200?
Sometimes it is useful to represent a number as the product of prime factors. How could you write 200 as a product using only prime factors? Writing a number as a product of only prime numbers is called prime factorization.
Tatiana was writing 200 as a product of prime numbers. She shared with her team the beginning of her work, which is shown at right.
200
= 4 · 50
= 2 · 2 · 10 · 5
= 2 · 2 · 5 · 2 · 5
Notice that Tatiana uses a “dot” ( · ) to represent multiplication. This is a way to show multiplication without using an “x”. Try to use this method now so that when you learn algebra, you are not confused about the use of the letter x as a variable.
What process do you think was going through Tatiana’s mind when she wrote 200 as a product of prime factors?
Do you think it matters what products Tatiana wrote in her second step? What if she wrote 10 · 20instead? Finish this prime factorization using Tatiana’s process.
1-79. Write the prime factorization of each of the numbers below.
100 36 54 600
1-80.When you write a number as a product of prime factors, often you have many factors to record. Explore this idea further in parts (a) through (c) below.
How many prime factors did you need to represent part (d) above?
To make it easier to record prime factors we use exponents.
Do you remember how repeated addition can be written in shorter form using multiplication? For example,10 + 10 + 10 + 10 + 10 can be written as 5 · 10. Similarly, repeated multiplication can be written in shorter form using exponents:10 · 10 · 10 · 10 · 10 = 105.
The prime factorization of 200 from problem 1-78 was 2 · 2 · 2 · 5 · 5. How could you write this with exponents?
Write your answers from problem 1-79 in exponent form.
1-81. WHY DOES IT WORK?
Work with your team to analyze an interesting pattern in the multiplication table.
Choose any four numbers from the multiplication table that form four corners of a rectangle. For example, you could choose 6, 15, 14, and 35, as shown in bold in the table below.
Multiply each pair of numbers at opposite corners of the rectangle. In this example, you would multiply 6times35 and 14times15.
What is the pattern? Work with your team to test enough examples to be convinced about whether there is a consistent pattern.
Additional Challenge: Why does this pattern work? Work with your team to explain why it makes sense that the products of opposite corners of any rectangle in the multiplication table are equal.
1-82. With your team, list everything you know about the following products. For each product, assume that the table has many, many rows and columns and extends long distances in each direction.
A number that appears twice in the multiplication table.
A number that appears seven times in the multiplication table.
A number that appears eight times in the multiplication table.
Additional Challenge: In general, what do you know about any number that appears in the multiplication table an even number of times?An odd number of times? Be ready to explain your thinking.
1-83. Additional Challenge: There is a special (and very rare) kind of number called a perfect number. Here is how to tell whether a number is perfect:
Make a list of all the factors of the number, except itself. For example, the factors of 6 are 1, 2, 3, and 6, so list the numbers 1, 2, and 3.
Add all the numbers on the list and if the sum is equal to the original number, the number is perfect! For example, 1 + 2 + 3 = 6, so 6 is a perfect number.
Can you find another perfect number?
1-84. LEARNING LOG
Reflect about the number characteristics and categories that you have investigated in this lesson and the previous lesson. Then, in your Learning Log, summarize the characteristics of numbers (such as prime and composite). Also, describe how you can use these properties to write the prime factorization of a number. Title this entry “Characteristics of Numbers and Prime Factorization” and label it with today’s date.
1-85.Use your multiplication table to figure out the missing number in each of the following number sentences. Each missing number is represented by n.
15n = 225
11n = 143
1-86. Jack has four tiles and wants to find out how many different shapes he can make withthem.
Sketch all of the arrangements that Jack could make with his tiles so that all of the tiles touch at least one other tile completely along a side. Assume that no tiles can overlap. How many arrangements are there?
For each diagram that you drew in part (a), find the area (the “tiles”) and the perimeter (the “toothpicks”). What do you notice?
1-87. Write four different fractions that are equal to 1.
1-88. Copy and complete the table of multiples below (count by 2’s and count by3’s).
Two / 2 / 4 / 6Three / 3 / 6
Write down all the numbers that appear in both rows. Describe any pattern(s) that you notice.
What is the smallest number that appears in both rows? This number is said to be the least common multiple of 2 and 3.
Find three more common multiples of 2 and 3.
Can you find the largest number that is a common multiple of both 2 and 3? If so, what is it? If not, explain why not.
1-89. How many “hands” long is your desk?
Using your hands as units of measure, first estimate (without actually counting) the number of “hands” that you think will fit across the length of your desk.
Now measure and record the length of your desk using your hands.
1-90. Write a whole number in the box in the fraction that makes it:
Equal to 1.
Greater than 1.
Less than 1.
Equal to 0.
Greater than 100.
1-91. Study the dot pattern below.
Sketch the 4th and 5th figures.
How many dots will the 50th figure have?
1-92.Write the prime factorization of each of the numbers below.
24 52 105
1-93. Throughout this book, key problems have been selected as “checkpoints.” Each checkpoint problem is marked with an icon like the one at left. These checkpoint problems are provided so that you can check to be sure you are building skills at the expected level.
This problem is a checkpoint for using place value to round and compare decimal numbers. It will be referred to as Checkpoint 1.
Use your knowledge of place value to round the decimals to the specified place in parts (a) through (c). Place the correct inequality sign (< or>) in parts (d) through (f).
a) 17.1936 (hundredths)
b) 0.2302 (thousandths)
c) 8.256 (tenths)
d) 47.2__47.197
e) 1.0032__1.00032
f) 0.0089__0.03
If you needed help solving these problems correctly, then you need more practice.Consider getting help outside of class time. From this point on, you will be expected to solve problems like this one quickly and easily.
1-94. Simplify the expressions in parts (a) through (f). Then answer the questions in part (g) using complete sentences.
a) 13 · 1
b) 1 · 5.5
c) 6 ·
d) 12 · 2
e) 4 ·
f) 14 ·
g) Use these examples to answer the following questions:
What happens when you multiply a number by one?
What happens when you multiply a positive number by a positive number less than one?
What happens when you multiply a positive number by a number greater than one?