Finding the Greatest Common Factor and Least Common Multiple

The most common method to find the greatest common factor (GCF) is to list all of the factors of each number, then list the common factors and choose the largest one.  Example: Find the GCF of 36 and 54.

1) The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36.

2) The factors of 54 are: 1, 2, 3, 6, 9, 18, 27, and 54.

Therefore, the common factor(s) of 36 and 54 are: 1, 2, 3, 6, 9, 18.  Although the numbers in bold are all common factors of 36 and 54, 18 is the greatest common factor.

To find the lowest common multiple (LCM), students are asked to list all of the factors of the given numbers. Let's say the numbers are 9 and 12.  

1) The multiples of 9 are: 9, 18, 27, 36, 45, 54.

2) The multiples of 12 are: 12, 24,  36, 48, 60.

As seen above, the least common multiple for these two numbers is 36.  

We often instruct our students to first list the prime factors, then multiply the common prime factors to find the GCF. Often times, if just this rule is given, students become lost in the process. Utilizing a visual can achieve an understanding of any concept better than just a rule. A two circle Venn Diagram is such a visual and will allow students to follow the process as well as to understand the connection between each step. For example: Let’s suppose we have the numbers 18 and 12.

1) Using factor trees, the students list all the factors of each number.

2) Now they place all the common factors in the intersection of the two circles. In this case, it would be the numbers 2 and 3.

3) Now the students place the remaining factors in the correct big circle(s).

4) That leaves the 18 with a 3 all by itself in the big circle. The 12 has just a 2 in the big circle.

5) The intersection is the GCF; therefore, multiply 2 × 3 to find the GCF of  6.

6) To find the LCM, multiply the number(s) in the first big circle by the GCF (numbers in the intersection) times the number (s) in the second big circle.

3 × GCF × 2 = 3 × 6 × 2 = 36. The LCM is 36.

This is an effective method to use when teaching how to reduce fractions,

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I have turned this method into a resource for Teachers Pay Teachers. It is 16 pages and begins with defining the words factor, greatest common factor and least common multiple. What a factor tree is and how to construct and use a Venn Diagram as a graphic organizer is shown. Step-by-step examples are given as well as student practice pages. How to use a three circle Venn Diagram when given three different numbers is explained. Two pages of blank pages Venn Diagrams are included for classroom practice. To learn more, just click on the price under the resource cover on your right. A free version is also available.

The ROOT of the Problem - Finding Digital Root to Reduce Fractions

When students skip count, they can easily say the 2's, 5's, and 10's which translates into easy memorization of those particular multiplication facts.  Think what would happen if every primary teacher had their students practice skip counting by 3's, 4's, 6's, 7's, 8's and 9's!  We would eradicate the drill and kill of memorizing multiplication and division facts.

Since many of my college students do not know their facts, I gravitate to the Divisibility Rules.  Sadly, most have never seen or heard of them.  I always begin with dividing by 2 since even numbers are understood by almost everyone.  (Never assume a student knows what an even number is as I once had a college student who thought that every digit of a number must be even for the entire number to be even.) We then proceed to the rules for 5 and 10 as most students can skip count by those two numbers.

Finally, we learn about the digital root for 3, 6, and 9. This is a new concept but quickly learned and understood by the majority of my students. (See the definition below which is from A Simple Math Dictionary available on TPT).

Here are several examples of finding Digital Root:

a) 123 = 1 + 2 + 3 = 6. Six is the digital root for the number 123. Since 123 is an odd number, it is not divisible by 6. However, it is still divisible by 3.

b) 132 = 1 + 3 + 2 = 6. Six is the digital root for the number 132. Since 132 is an even number, it is divisible by 6 and by 3.

c) 198 = 1+ 9 + 8 = 18 = 1 + 8 = 9. Nine is the digital root for the number 198; so, 198 is divisible by 9 as well as by 3.

4d201 = 2 + 0 + 1 = 3. Three is the digital root for the number 201; so, 201 is divisible by 3.

The first time I learned about Digital Root was about eight years ago at a workshop. I was beside myself to think I had never learned Digital Root. Oh, the math classes I sat through, and the numbers I tried to divide by are too numerous to mention! It actually gives me a mathematical headache. And to think, not knowing Digital Root was the ROOT of my problem!

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A teacher resource on Using the Divisibility Rules and Digital Root is available at Teachers Pay Teachers. If you are interested, just click under the resource cover on your right.

Using Math Riddles to Reduce Fractions, Recognize Equivalent Fractions, Identify Basic Percents

Do you need something besides a “drill and kill” activity to practice fractions and/or percents? This fraction riddle bundle is my newest bundled resource. It is a 33 page resource that is a fun and engaging way to utilize math concepts while keeping the students actively involved. 

Specific words are provided. The students are instructed to figure out the correct fractional part of each particular word. (Example: The first ½ of WENT would be WE. Notice that WE is also 2/4 or 50% of WENT.) If each fractional part is correctly identified, when the students write the fractional parts on the lines provided, a new word is created. Each group of new words becomes a riddle or the answer to a riddle.

It is important that students understand that a fraction and a percent represent the same thing; so, in the Snow Riddles handout, 25%, 50%, 75% and 100% are introduced.

 In March Riddles, specific questions are asked to acquaint the students with fun facts about the month of March. April Riddles introduces the students to several interesting historical facts that occurred during this month.

For each month, there are between 7-11 word fraction riddles; so, there are numerous ways to practice recognizing fractional parts, understanding equivalent fractions, identifying basic percents (25%, 50%, 75% 100%), and reducing fractions to lowest terms.

Instead of completing all of the monthly riddles in one day, the puzzles may be divided up and used as a focus activity, when a student finishes early, or when there is a short amount of time left before the next class or activity. An individual puzzle may be given each day, or the riddles can be interspersed throughout the week or month. Answers are included at the end of each month’s activities. The complete resource features six months (January, February, March, April, October, December) and contains a total of 49 fraction riddles.

If you prefer, each month of fraction riddles may be purchased separately; however, this resource bundles all six months for a discounted price. Just click the title under the cover page shown above, and download the preview to take a quick look at this new item.

  

Reducing Fractions with Pattern Sticks!

When working with fractions, many of my students seem confident in performing the different operations, but a few are still unsure of how to reduce fractions. 

Although I have stressed learning the Divisibility Rules for 2, 5, 10, and the digital root for 3, 6, 9,  some still have difficulty since they do not know their multiplication tables. As a mathematics tool, I have the students make Pattern Sticks, a visual and kinesthetic aid, similar to a multiplication chart like the one on the left. Notice that an extra column (blue) has been added to the chart. (In this space, a hole is punched so that a 1" ring can be inserted to store all of the sticks in one place.)

On the right are the directions for making the Pattern Sticks using a multiplication chart. 

(Side note: My students cut out individual Pattern Sticks which I prefer over cutting a multiplication chart apart.)

I then give the students fractions such as 9/36 to reduce. Using the Pattern Sticks, they search for a column where a 9 and a 36 are lined up in the same column. They easily find it on the 1 strip and the 4 strip. They then take the two strips and line them up so that the 9 is over the 36. (see illustration above) By moving to the left, they discover that 9/36 is the same as 1/4. This is 9/36 in its lowest terms. Also notice that all the fractions in the illustration are equivalent fractions - fractions that have the same value. The Pattern Sticks can also be used to determine what number to divide by and to change improper fractions to mixed numbers.

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If you are interested in learning more about Pattern Sticks and how to use them in your classroom, check out the resource entitled Pattern Sticks: A Math Tool for Skip Counting & Reducing Fractions at Teachers Pay Teachers.