Number Theory - Learning Mathematics Through Reading and Writing Activities - Pre-Algebra Out Loud

Pre-Algebra Out Loud: Learning Mathematics Through Reading and Writing Activities (2013)

Chapter 4. Number Theory

WHAT? Introduction

The Common Core State Standards recommend that sixth-grade students be able to write and interpret numerical expressions and find common factors and multiples. This chapter introduces these concepts, using reading and writing strategies and activities to reinforce understanding and may even allow students to appreciate the playfulness and aesthetics of numbers. Students and teachers often consider number theory to be the fun part of math! The stronger the base that students have with operations involving prime and composite numbers and factors, the easier it will be to apply this knowledge to algebra.

WHY? Objectives

In this chapter, pre-algebra students will:

· Develop and solve number riddles

· Learn to paraphrase content

· Construct magic squares

· Work with semantic feature analysis tables to display their knowledge of number theory concepts

Mini-Lesson 4.1 Divisibility Rules

CCSS Standard 6.NS: Number System

Compute fluently with multi-digit numbers and find common factors and multiples.

Divisibility rules offer shortcuts for determining if a number is divisible by another number rather than having to do long division. Knowledge of these rules gives pre-algebra students a useful tool for mental math and problem solving.

A prime number is a positive whole number that has only 1 and itself as factors (divisors). For example, the number 2 has only 1 and 2 as its factors, and 13 has only 1 and itself as its factors. Thus, 2 and 13 are prime numbers. A composite number has more than two factors. For example, the number 4 has 1, 2, and 4 as its factors, making 4 a composite number. The number 1 is a special case: it is considered to be neither prime nor composite, because 1 = 1 × 1 × 1 × 1 ... Note that 1 is the only factor of itself.

Divisibility is a useful property of whole numbers. For example, 2 divides 8 and 5 divides 15.

1 Teaching Tip

It's important not to confuse the divisibility symbol (|) with the fraction bar, as “/” in the fraction images or 25/102. In fact, we would say here that 25 does not divide 102, since 25 and 102 have no factors in common. Be careful to sketch the “divides” sign as a vertical line to reinforce the difference.

Consider the following:

· images is true, since images.

· images is false since there is no whole number that when multiplied by 7 gives 20. The numbers 7 and 20 are said to be “relatively prime,” meaning they have no factors in common except 1.

· images is true since images. Care must be taken when dealing with 0 and divisibility, because in general a number cannot be divided by 0.

· images is false since there is no whole number that when multiplied by 0 gives 6.

The divisibility rule is that, in general, a whole number a divides another whole number b, images, iff there exists a whole number n so that images.

1 Teaching Tip

Iff means “if and only if.” It implies you could rewrite the rule with the condition reversed. For example, images implies images, and images implies images. Therefore, we write images iff images.

Knowing many of the following divisibility rules is useful when mentally dividing large numbers by small numbers. For example, 52,266 is divisible by 2 since the unit's last digit is even. It is also divisible by 6. Let's see if we can determine why.

Divisibility by 2: A number is divisible by 2 iff its last digit is 0, 2, 4, 6, or 8.

Divisibility by 3: A number is divisible by 3 iff the sum of the digits is divisible by 3, since images and images. Also, 15 is divisible by 3 since images.

Divisibility by 4: A number is divisible by 4 iff the number formed by the last two digits is divisible by 4. Example: 13,564 is divisible by 4 because images. Note that 52,266 is not divisible by 4 since 4 does not divide 66.

Divisibility by 5: A number is divisible by 5 iff the last number is 5 or 0. Note that 52,266 is not divisible by 5, but a number like 6050 is divisible since the unit's last digit is 0.

Divisibility by 6: A number is divisible by 6 iff it is divisible by 2 and divisible by 3. Remember that “and” means it must happen in both. Recall the number 52,266 from above. We can now see 52,266 is divisible by 6 since images and images.

Divisibility by 8: A number is divisible by 8 iff the last three digits form a number divisible by 8. Note that 52,266 is not divisible by 8 since 266 is not divisible by 8. Also, 4 does not divide 52,266 so 8 could not divide it.

Divisibility by 9: A number is divisible by 9 iff the sum of the digits is divisible by 9. Note that 52,266 is not divisible by 9 since images and 9 does not divide 21. Note also that 3 may divide a number, but that does not mean 9 will.

Divisibility by 10: A number is divisible by 10 iff its unit's digit is a 0. In fact, a number is divisible by images iff the last n digits of the number are 0s. Note that 4800 is divisible by images since it ends with two 0s and 250,000 is divisible by images since it ends with four 0s.

The number 142,460 is divisible by 2, 3, 4, 5, 6, and 10 but not 9:

· It is divisible by 2 since the unit's digit is 0.

· It is divisible by 3 since the sum of the digits, images, and 3 divides 15.

· It is divisible by 4 since the last two digits give the number 60, and 4 divides 60.

· It is divisible by 5 since 142,460 ends with zero.

· It is divisible by 6 since 2 and 3 are relatively prime, and both divide 142,460.

· It is not divisible by 9 since the sum of its digits is not divisible by 9.

· It is divisible by 10 since the unit's digit is a 0.

When you need to find and use the quotient, divisibility rules help only up to a point. But if you are interested in finding the factors of a multidigit number, then knowing all of the basic divisibility rules makes the process go much faster.

Activity 4.1: Number Riddles

WHAT? Description

This activity may be used with many different mathematical processes but works especially well with the divisibility rules. Students work in pairs to create riddles based on the model described below.

WHY? Objectives

During this activity, pre-algebra students:

· Work in pairs to create number riddles

· Review and discuss the divisibility rules

· Compete and cooperate to find more complex riddles to solve

· Use or learn to use numbers in a creative manner

HOW? Example

Here is a sample number riddle: “I am thinking of a number whose digits add up to a number divisible by 9. What other numbers must divide my number?”Answer: 1 and 3.

Pairs of students can share and solve each other's riddles, discussing how they found the answers.

Worksheet 4.1: Number Riddles Using the Divisibility Rules

Name _____ Date _____

Directions: Try to answer the following number riddles.

1. I am thinking of a number that is divisible by 6. What other whole numbers will divide my number?

2. My number is also divisible by 4 and less than 20. Can you guess my number?

3. I am thinking of a number that is divisible by 12. What other whole numbers will divide my number?

4. My number is also divisible by 11, but NOT divisible by 8. Can you find my number?

5. Is there more than one possibility for riddle 4? _____ If so, what is that possibility?

6. I am thinking of a number that is divisible by 9 and between the numbers 30 and 50. Must the number be divisible by 3? _____ By 2? _____

7. If the number in riddle 6 is NOT divisible by 4, what is my number?

Now it's your turn. With a partner, brainstorm and write your riddle. Trade riddles with another pair, and try to solve each other's. When you are finished, discuss the problem and create another one.

Solutions for Worksheet 4.1

Name _____ Date _____

1. I am thinking of a number that is divisible by 6. What other whole numbers will divide my number? 2 and 3

2. My number is also divisible by 4 and less than 20. Can you guess my number? 12

3. I am thinking of a number that is divisible by 12. What other whole numbers will divide my number? 1, 2, 3, 4, 6

4. My number is also divisible by 11, but NOT divisible by 8. Can you find my number? 132

5. Is there more than one possibility for riddle 4? Yes. If so, what is that possibility? 924

6. I am thinking of a number that is divisible by 9 and between the numbers 30 and 50. Must the number be divisible by 3? Yes. By 2? Not necessarily

7. If the number in riddle 6 is NOT divisible by 4, what is my number? 45

Activity 4.2: In Your Own Words: A Paraphrasing Activity

WHAT? Description

One of the most common excuses that students give for not reading material in their textbook is that they do not understand the language. This activity helps students target and interpret key concepts. By rewriting mathematical passages, students demystify and make personal meaning of mathematical content.

Assign students small portions of the pre-algebra text to read and rewrite in their own words. This activity works equally well with concept definitions, theorems, and examples. Having students read their own versions to each other allows student writers to consider different interpretations and pinpoint misconceptions. If writing is handed in, you can assess your students' understanding of the material.

Rewording or rewriting the divisibility rules is a good way to remember them.

WHY? Objectives

During this activity, pre-algebra students:

· Paraphrase the content on assigned portions of the text

· Focus on the important ideas in the content or lesson

· Share their paraphrasing activity with peers

HOW? Example

Text

Paraphrase

There are three types of positive integers: prime numbers, composite numbers, and “1.” Prime numbers only have two factor, 1 and itself. Composite numbers have more than two factors. The number 1 is unique since it does not fit in either the primes or the composites.

Prime numbers are like 2: it has only 1 and 2 as its factors; 13, for example, has only 1 and 13 as its factors. Composite numbers are not prime like the number 4, which has three factors: 1, 2, and 4. The number 1 is different: it has only one factor, and that is 1!

Worksheet 4.2: In Your Own Words: A Paraphrasing Activity

Name _____ Date _____

Directions: In your pre-algebra book, reread the lesson on divisibility rules. Then follow the prompts below, paraphrasing (putting into your own words) the content.

Rewrite the divisibility rule for 2 in your own words.

How are the divisibility rules for 3 and 9 alike and different?

If a number is divisible by 6, with what other numbers is it necessarily divisible?

The rules for divisibility for 4 and 8 are very similar. Explain.

Can you make a rule for divisibility by 12 or by 24? What makes these rules similar to the rule for 6?

Activity 4.3: Magic Square

WHAT? Description

The magic square activity combines a matching activity with the intrigue and mathematics of a magic square (Vacca & Vacca, 1999). The format of the matching activity consists of two columns, one for concepts and one for definitions, facts, examples, or descriptions. As the student solves the matching activity, he or she places the numbers in the proper square inside the magic square.

UnFigure

To check their answers, students add the numbers in each row, in each column, and on each diagonal. These sums should be equal. This sum is referred to as the square's “magic number.” Sometimes only the numbers in the rows will sum up to the magic number and one or more diagonal contains values that do not add up to the magic number.

Any mathematics topic that has a list of related features or rules, such as the divisibility rules, lends itself nicely to a matching exercise. In turn, a magic square gives a superb format for finding and posting the correct matches from the matching exercise.

WHY? Objectives

During this activity, pre-algebra students:

· Reinforce the meanings of concepts or words

· Learn about the features of a magic square

HOW? Example

The magic squares below are 3-by-3 unit squares. Notice that the cells in this particular magic square contain all whole numbers from 1 to 9 with no repeats. When you add the numbers in each column, in each row, and in each diagonal, your sum is 15. This number is called the magic number.

UnFigure

Worksheet 4.3: Magic Square: Divisibility Rules

Name _____ Date _____

Directions: Select the best answer for each of the concepts on the left from the numbered rules on the right. Put the number in its proper place in the magic square. Add each row and add each column. If these sums are the same number, you have found the magic number and correctly matched concepts with their rules.

Concepts

Rules

A. Divisibility by 2

1. Sum of even-numbered digits minus sum of odd-numbered digits is divisible by 11.

B. Divisibility by 8

2. Unit's digit is 0 or 5.

C. Divisibility by 3

3. The number formed by the last three digits is divisible by 8.

D. Divisibility by 11

4. Sum of digits is divisible by 3.

E. Divisibility by 6

5. Number is divisible by 2 and 3.

F. Divisibility by 10

6. Number formed by the last two digits is divisible by 4.

G. Divisibility by 4

7. Sum of all digits is divisible by 9.

H. Divisibility by 9

8. Unit's digit is 0, 2, 4, 6, or 8.

I. Divisibility by 5

9. Unit's digit is 0.

UnFigure

Mini-Lesson 4.2 Greatest Common Denominator and Least Common Multiple

CCSS Standard 6.NS: Number System

Compute fluently with multi-digit numbers and find common factors and multiples.

The greatest common factor and the least common multiple have many applications in mathematics. First, we discuss the greatest common divisor (or factor).

A common factor is a multiplier or divisor shared by two or more numbers. The greatest common factor (often called the greatest common divisor, GCD) is the largest factor that two or more numbers or terms have in common. The GCD is used to simplify fractions. Example: images.

There are two methods for finding the GCD of two or more numbers: the all-factors method and the prime factorization method.

All-Factors Method for Finding the GCD

1. Write out (or list) all the whole number factors for each of the numbers.

2. Circle or underline all common factors.

3. Select the greatest of the common factors.

Example: Find the GCD for the numbers 60, 96, and 156, using the all-factors method:

60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96

156: 1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156

Therefore, the GCD of 60, 96, and images.

1 Teaching Tip

Some students want to stop writing down factors when they find one factor that the numbers have in common; for example, 4 is a common factor of 60, 96, and 156. However, to find the greatest common factor, they must list all factors of each number.

Prime Factorization Method for Finding the GCD

1. Find all the prime factors for each of the numbers. Write each number as a product of its prime factors.

2. Express all factors that repeat in exponential form.

3. Find all the common prime factors to their least power.

Check by multiplying the factors they have in common to arrive at the GCD.

1 Teaching Tip

Every positive integer can be expressed as a unique product of prime numbers.

Example: images. This product of primes is equal only to 24.

Example: Find the GCD for 60, 96, and 156, using the prime factorization method. The last row gives the prime factorization for each of the numbers in the first row.

60

96

156

images

images

images

images

images

images

images

images

images

images

images

images

images

images

The numbers 60, 96, and 156 have images and 3 in common. Therefore, the GCD is images.

Next, we consider the least common multiple (LCM), also called the least common denominator (LCD), when adding or subtracting fractions.

A common multiple is a composite number that two or more numbers (not counting 1) divide. The least common multiple is the smallest number that two or more numbers can divide into.

When adding fractions, we need to find a common denominator and change the fractions into 1s with the common denominator in order to add the numerators.

As with the GCD, there are two methods for finding the LCM: the all-multiples method and the prime factorization method.

All-Multiples Method for Finding the LCM

1. Find several (at least five to start) of the multiples for two or more numbers.

2. Select the least of the multiples that the numbers have in common and you will discover the LCM.

Example: Find the LCM for 8, 12, and 30 using the all-multiples method:

8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120

12: 12, 24, 36, 48, 60, 72, 84, 96,108, 120

30: 30, 60, 90, 120, 150, 180, 210

By the all-multiples method, the LCM is equal to 120. It is usually written LCM images.

1 Teaching Tip

It is impossible to list all multiples of any number. So have students list all multiples up to the one multiple that the first two numbers have in common. Then they should list multiples of the third number until they find one that is in common with all three numbers. Stress that they may have to repeat this process more than once to find the LCM.

Prime Factorization Method for Finding the LCM

1. Find all the prime factors for each of the numbers. Write each number as a product of its prime factors.

2. Express all factors that repeat in exponential form.

3. Find all the different factors to their greatest powers.

4. Multiply, and you will have the LCM.

Example: Find the LCM using the prime factorization method. Recall the prime factorization for the numbers 60, 96, and 156:

60

96

156

images

images

images

images

images

images

images

images

images

images

images

images

images

images

The different factors are 2, 3, 5, and 13. The different factors to their greatest powers are 25, 3, 5, and 13. The LCM is the product of all the different factors (to their greatest power), which is images.

Example: Find the LCM (12, 30, 42) using the prime factorization method:

12

30

42

images

images

images

images

images

images

images

images

images

Therefore, the LCM images.

Activity 4.4: Semantic Feature Analysis

WHAT? Description

Semantic feature analysis (Baldwin, Ford, & Readance, 1981) is a reading strategy that asks students to complete a matrix showing how various terms and concepts are alike or different. The terms or concepts are related or fall under a particular category. The matrix itself consists of several columns. The first column contains a listing of the terms. The remaining columns contain headings spelling out or asking about features that the terms or concepts might have in common.

The GCF and LCM have comparable features, and the method for finding either one is similar. To use the prime factorization method to find either the GCF or LCM, first factor each of the numbers into a product of primes. In the all-factors and all-multiples methods, list all factors for the GCF and as many multiples needed to find the LCM. Using the semantic feature analysis is a good forum for displaying the similarities and the differences of these processes.

WHY? Objectives

During this activity, pre-algebra students:

· Compare and contrast features of related mathematical concepts

· Summarize this information

· Refer back to the completed matrix when reviewing for exams

HOW? Example

The following table displays an analysis of certain divisibility rules. Write yes or no (the answers are given in this example) in the cell according to the features along the top with the types of numbers along the left-most column.

images

Worksheet 4.4: Semantic Feature Analysis

Name _____ Date _____

Directions: Answer yes or no to the question asked in each cell. In the right-hand column, write out another name for each process.

images

Worksheet 4.5: Semantic Feature Analysis

The number 5 can be expressed in one rectangle model, meaning a rectangle divided into five connected and equal-sized units or squares. If the rectangle is a horizontal strip, its row by column sizes are written as 5 by 1. If the rectangle is a vertical strip, then it is expressed as 1 (row) by 5 (columns, or 1 by 5). Since the only factors for 5 are 5 and 1, it has only one rectangle model:

UnFigure

This rectangle model for 5 is 1 row by 5 columns. We will consider 5 columns by 1 row to have the same rectangular array.

The following two arrays, images and images, represent the number 4:

UnFigure

UnFigure

Directions: Fill in the table on the next page with the factors and the number of rectangles (factors) that can be formed by the indicated number. Then use your results to answer the questions following the table.

Note that each rectangular model represents a pair of factors.

The dimensions column gives each number's pair of factors, which are also the dimensions of each rectangular array. For example, 2 has only two factors: 1 and 2. Thus, 2 has one rectangular array with the dimensions 1 by 2—one row and two columns.

Number

Dimensions

Number of Rectangle = Factors

2

images

1

3

images

1

4

images, images

2

5

1

6

7

8

9

10

11

12

images

3

13

14

15

16

17

18

19

20

Questions:

· Which types of numbers—primes, composites, squares, and soon—have only one rectangle model?

· Which types of numbers have more than one rectangle model?

Extend the table to the number 36 and see if you can draw some other conclusions about the different types of numbers and their rectangular models or factors.