Using Factoring to Demystify Numbers - Strategies for Teaching

Using factoring to demystify numbers - strategies for teaching by Ann-Marie Hunter

Fun with Factoring

Factoring numbers in Math is a tool that can be used to diminish the mystery of numbers, building student confidence. I have found that when students are acquainted with how numbers are related, they feel much more empowered to explore new tasks using numbers.

With this in mind, one idea to use is to offer an inquiry challenge for students to identify all the factor pairs of numbers. It’s important to make the discoveries real for students by using tiles, beans, or (if no tactile materials are available) graph paper, to look for rectangles that can be made using a specific numbers of tokens.

For example, for the number 24, there are 3 different rectangles that can be made using 24 tokens. We can make 1 X 24, 2 X12, 3 X 8, and 4 X 6 rectangles. (The 24 X 1, 12 X 2, 8 X 3, and 6 X 4 rectangles are considered the same as the other four, with just different orientations.) These four rectangles, therefore, represent the factor pairs of the numbers, giving all the factors of the numbers as: 1, 2, 3 4, 6, 8, 12 and 24. I like to have the students write this out in a rainbow fashion (see slide below).

Then, following up on the inquiry, ask students to complete the pages that list all the numbers from 1 to 100, with places for the number pairs to be inserted. The quantity of factor pairs (number of lines beside each number) is given as a clue to this exploration – helpful for younger students! You might decide to assign particular numbers to individual students or pairs of students that can then be posted in the classroom master poster (enlarged). Students can then be asked to transport the factor pairs to their individual copies of the sheets. Here is a sample page from that series (posted on the PITA Wiki – MATH, with answer keys for each page).

Fantastic Factors – find the factor pairs! Page 1 1 = ______11 = ______21 = ______= ______2 = ______12 = ______= ______22 = ______3 = ______= ______= ______

4 = ______13 = ______23 = ______= ______14 = ______24 = ______5 = ______= ______= ______= ______6 = ______15 = ______= ______= ______= ______25 = ______7 = ______16 = ______= ______= ______8 = ______= ______26 = ______= ______= ______17 = ______9 = ______27 = ______= ______18 = ______= ______= ______10 = ______= ______28 = ______= ______= ______19 = ______= ______

20 = ______29 = ______= ______= ______30 = ______= ______= ______= ______

It is important to include the word “multiple” while talking about “factors” of numbers. A number is called a multiple of any of its factors. i.e. – 6 is a multiple of 2 and 6 is a multiple of 3 because 2 X 3 = 6.

As students are identifying factors, make a point of saying, for example, “Yes, we can see that both 9 and 12 are multiples of 3, as they both have 3 as a factor. Can you identify other multiples of 3 from your lists of factors?”

All the multiples of 3, for example, can be found by making a list of the answers to 1 X 3, 2 X 3, 3 X 3, 4 X 3, . . . giving multiples of 3 as: 3, 6, 9, 12, . . . (the list is endless! Whereas, the list of factors is finite.

I use the pneumonic: Multiples are many, Factors are few, to help students remember which is which!)

I liked to give students a visual method to identify multiples, such as:

1 X 3 2 X 3 3 X 3 4 X 3 5 X 3 3 , 6, 9, 12, 15, . . .

Continuing with that topic, after all the Fantastic Factors pages have been completed, a game called Factor Frenzy can be used to review and practise the skills. Here is a one-page description of the game, to be played using the ‘Fantastic Factors’ sheets for reference (at first) and then developing into games played using students’ memories only (referring to the sheets only as a check for each move in the game).

(P.S. The Hundreds Chart included here is an upside down Hundreds Chart, to build on students intrinsic sense of higher being bigger – rather than the usual Hundreds Chart that has 1 at the top and 100 at the bottom.)

Play FACTOR FRENZY!

Have fun helping your students become Factoring Wizards!

These factoring skills are beneficial in many areas of Math, including finding multiples of numbers, identifying common factors, facilitating calculations in geometry, using measurement tools, knowing timestables, … and simply feeling more connected to numbers.

Once the skills and knowledge of factoring have been developed, the talk in your Math classroom becomes much more fluent as you can freely talk about numbers using the factoring/multiples vocabulary, “Can you check your factor sheets to find a common factor for both the numerator and denominator of this fraction, so we can simplify it?” or “Ah yes, those two denominators are both multiples of 6 , aren’t they? Would you like to confirm that by checking your factor sheets?” As students refer to their factor sheets more and more, they build confidence with numbers. It’s great! FACTOR FRENZY 91 92 93 94 95 96 97 98 99 100

81 82 83 84 85 86 87 88 89 90

71 72 73 74 75 76 77 78 79 80

61 62 63 64 65 66 67 68 69 70

51 52 53 54 55 56 57 58 59 60

41 42 43 44 45 46 47 48 49 50

31 32 33 34 35 36 37 38 39 40

21 22 23 24 25 26 27 28 29 30

11 12 13 14 15 16 17 18 19 20

1 2 3 4 5 6 7 8 9 10

 A game for 2 or more players.  Equipment: different coloured markers or tiles. Reference sheets of factor pairs of numbers between 1 and 100 can be used at the beginning.)

– Player 1 chooses a number and circles it (or covers it with a tile), and gets that number for a score. – Player 2 circles (with a different colour) all the factors of that number (that are not already circled) and earns the sum of those numbers for a score. Player 2 then chooses a new number, circles (or covers) it and adds that number to his/her score. – Play goes to the next player to circle all factors of the previous player’s number and then to circle a new number, adding to his/her score the sum of the factors as well as the new number. (If there are no factors of their opponent’s chosen number, then he/she earns a zero – part of the strategy!) – The game continues until all numbers have been circled. – The player with the highest score wins.