Factoring

This page contains the same material as in the PDF booklet Factoring Enrichment, which you can download for free by clicking here Factoring Enrichment - PDF. It is meant as an enrichment material to accompany the Factoring book.

How to use this book

This book is an enrichment book designed to help teachers and parents provide their students/children with more activities about the lessons in the Factoring book.Those activities were selected to enhance the Factoring book lessons and to enrich students with more general knowledge and skills.

There is no specific order that you have to follow, although the activities are structured in the same order of the lessons in the Factoring book.

Activities were selected to provide the appropriate enrichment at each stage and for each lesson. Usually, using activity of lesson 3 while the student had only studied lesson 1 would not work well since the student would be lacking some knowledge and skills required to make Lesson 3’s activity fun and enriching.

Some of the activities are more difficult. Do not worry if your students/children don’t do well on any specific activity. As long as they can do the practice questions in the Factoring book they are well prepared..

Section 1: What is Factoring?

The purpose of this lesson is to introduce the idea of factoring. The activities are around this idea but do not require real knowledge of factoring, only of squares. It is used to tie other topics with factoring.

Activity #1

Ask students to think of problems in which factoring can help.

Example: you need to make plans for your birthday. There are going to be 3 girls and 4 boys and you plan to make them compete against each other (the girls vs. the boys). You decided to give pieces of candy as a prize. However, you need to find a number, which regardless of which group wins, will divide evenly between the members of the winning group, without a remainder. You don’t want a remainder to avoid squabbling over it.

Note: do not expect them to come up with correct or really original questions. Some of the questions will be variants or a copy (with different numbers) of the example, which you should show them, or really a division question and not factoring. Use the division type of questions to show that the answer, if a whole number, is also a factor.

Activity #2

Find the factors of the numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 900, 1600, 2500, 3600, 4900, 6400, 8100, 10,000.

Ideas

1. Students practice their number sense by seeing a long list of squares and finding their root.
2. The idea of a square and root can be introduced and discussed.
3. A comparison between the smaller squares and their larger counterparts can be drawn. 1 is compared to 100, and the we see that the roots are 1 vs. 10; 4 is compared to 400 and we see that the roots are 2 vs. 20; 9 is compared to 900 and we see that the roots are 3 vs. 30, etc.
   1. It can be used as a pattern recognition exercise.
   2. It can be used for discussing the place value system (PVS) again, the structure of numbers, and the multiplication procedure.
      1. The correspondence between the squares and roots brings about the issue of 0 (place holder) in the PVS.
      2. Breaking 20 to 2x10 and so on helps in discussing the structure of numbers.
      3. Seeing that 20x20 is like doing 2x2x10x10 (and for all other examples) can bring about a better understanding of the multiplication algorithm and its connection to the structure of numbers and the PVS.
4. The notion of exponential series/growth can be introduced (informally, implicitly). See Activity 2 for how to do it.

Connected Activities

1. Class settings: Divide the class to small groups and do a competition of “who knows the root fastest” without showing them all the numbers.

• To help students who are not as quick with arithmetic, you might want to give them the numbers in sequence. While the quicker students will catch faster, after a few (and discussing the sequence) the slower students will catch up and it will be an even race.

• To further make it more fun for the slower students, introduce the first four numbers (4 to 25) as “warm up” and let the competition start only with 36.

2. Class work or homework: drawing the size of the squares and roots in the following way

• This activity, which is fun for students to do since they like seeing the visual aspect of the growth, illustrates the power of exponential growth. The roots are y=x series while the squares are y=x2 series.

• This activity can also be used to introduce charting. Due to the highly ordered nature of the data students find it easy to draw.

• It can also be used to show a spreadsheet and how to use the graphing function. Different graph types can be experimented with and students can vote for which one they think illustrates the point [different growth rates] best.
  • It is critical that students first draw by hand for them to really grasp the growth aspect of it. Especially for those who are “hands on” learners

Section 2: What are Prime Numbers?

The purpose of this lesson is to introduce the idea of prime numbers and to understand the difference between primes and divisible numbers.

Activity #1: Finding Primes

Students receive a list of the numbers 1 to 100 in a line or a column. They are asked to figure out ways to find the prime numbers on that list.

A hint (to give to your students): think of the number 2. How can you eliminate all the numbers between 2 to 100 that are divisible by 2? Given that, what is the next step?

• The process outlined in the hint is the Sieve of Eratosthenses, who was a Greek mathematician, and the first to figure this method.

Primes are important because mathemticians don't know how to easily tell if a number if a prime or not. They are also important because if mathematicians would figure certain riddles about primes they would be able to solve many other things in mathematics; if those things get solved scientists could use them for computers and physics.

Primes are also important because the main usage of primes is for encryption. Encryption is a way to transfer secret messages such that only the sender and the recipient could know what the message says, but no one else would be able to know it.

The explanation of how to do it can simplified to explain it to kids, but is not appropriate for this lesson. Students first have to go through the whole book to understand it. Therefore, the explanation is at the end.

Section 3:The Rules of Factors

Most of the section is simple and straightforward. Only sections H&I pose any difficulty. Therefore, practice consists of games that enhance number sense, memorization, and recall of the rules.

Activity #1: Factors Matching Game

Divide the class to teams. Each team receives cards with 3 numbers and 3 factors on them. They have to match the numbers with the factors.

Solution: 1-A, 2-C, 3-B. The cards are upside down and there are rounds with points. If the number of teams is 10, the fastest gets 10 points, the second 9, etc. Incorrect answer gets a penalty of 5 points and last place (so other teams do not suffer speed wise).

The cards should go from the easy to the difficult one. The card above is an easy one.

• Numbers that seem to have more than one factor make it more difficult. If instead of 363 we would have 374 it would be more difficult since 374 should be checked for 4 as well.
• Another level of difficulty is to have a possibility for multiple factors of the same number.
  • Example: Instead of 42 it can be 60 (both 4 and 5 are factors), and 352 (both 4 and 11 are factors)

Activity #2: For The Rule of Kids.

Note: the rule of kids is an important one because it allows the finding of many factors if we know one to start with.

A similar game to the one in activity #1. The cards should contain the following: trees of factors with some of the factors missing.

• Difficulty level depends on the number of factors that are missing and which ones.
• Another variation in the cards can be to have the factored number missing, as well as some of the factors. A simple one is when two factors that are co-factors (such as 3 &4 for 12) are shown, and therefore make it a multiplication problem. If no two co-factors are shown the difficulty is higher.

Note: similar game can be made for section I about the rule of sum and parts”.

Section 3: The Rules of Factors

As explained in the book, prime factorization is difficult in the sense that it can take long time. In fact, very large (with hundreds of digits) numbers might take so long that we cannot actually factor them with current technology. Activities should be based on easy numbers and on ensuring students understand the idea and can factor simple numbers. Students should not be required to (or drilled in) factor difficult and large numbers since it can only lead to frustration.

Activity #l: Building Numbers

Instead of factoring number a good activity is to build numbers from prime numbers. Give students a set of primes (example: 2,3, and 5) and ask them to build numbers. Then explore patterns in their sets. Explore and discuss with them ways to build numbers with those primes.

• This activity can serve to introduce powers or practice it if they know it already. The idea of power allows easy representation of many numbers using a limited set of factors.

Section 4: Prime Factorization

• Use the activities above with primes as the only factors that count.
• Advanced: visit the following page with the students and read about primes and “riddles” (unsolved problems) about primes:http://primes.utm.edu/
  

Internet Resources

Factorization related Games
24 game: factors version: an explanation of how to use the Factors/multiples edition of the 24 game. National Council of Teachers of Mathematics (NCTM) resources: The Factor Game: a nice, challenging game that has 3 levels. While the game seems simple, it is not really so, since what you do affects what your opponent can do and the other way around. You can play with a friend or the computer. Factorize The Product Game

Tests for Factors or Primes Factors Finder: It finds all the factors of a number and also does prime factorization (at the same time). Factor Pairs: You give a number and the finder shows you all the pairs of factors of that number. Factors Plot: it finds the factors of a number you provide and plots then on a grid. Primality check: you can check whether a number is a prime or not, as long as the number is smaller than 9007199254740991 = 253 - 1

Advanced Readings The Prime Pages: lots of material about primes, including lists of primes (even the first 15 million primes), unsolved problems about primes, and history of primes and prime factorization. Some of it is easy to read, and some is difficult. RSA security: factoring algorithms: RSA is a “security” company dealing with keeping data safe and private. The letters in the name are the first letters of the three scientists (Rivest, Shamir, and Adelman) who started the company based on their invention of how to encrypt data based on prime factorization.

• They promise large prizes to anyone who can factor “challenge numbers” (a list of big numbers). Each challenge number is a product of exactly two prime numbers.