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© 2013 by Brad Fulton and TTT Press Array
We Go!
An Engaging Visual Representation of Factors, Multiples, Primes, and Composites and More For grades 3, 4, and 5 By Brad Fulton
Educator of the Year, 2005
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© 2013 by Brad Fulton and TTT Press Brad Fulton
Educator of the Year
Brad Fulton
PO Box 233, Millville, CA 96062 (530) 547-­‐4687 b r a d @ t t t p r e s s . c o m ♦ Consultant
♦ Educator
♦ Author
♦ Keynote presenter
♦ Teacher trainer
♦ Conference speaker
Known throughout the country for motivating and engaging teachers and students, Brad has coauthored over a dozen books that provide easy-to-teach yet mathematically rich activities for busy
teachers while teaching full time for over 30 years. In addition, he has co-authored over 40 teacher
training manuals full of activities and ideas that help teachers who believe mathematics must be
both meaningful and powerful.
Seminar leader and trainer of mathematics teachers
♦ 2005 California League of Middle Schools Educator of the Year ♦ California Math Council and NCTM national featured presenter ♦ Lead trainer for summer teacher training institutes ♦ Trainer/consultant for district, county, regional, and national workshops Author and co-author of mathematics curriculum
♦ Simply Great Math Activities series: six books covering all major strands ♦ Angle On Geometry Program: over 400 pages of research-­‐based geometry instruction ♦ Math Discoveries series: bringing math alive for students in middle schools ♦ Teacher training seminar materials handbooks for elementary, middle, and secondary school Available for workshops, keynote addresses, and conferences
All workshops provide participants with complete, ready-­‐to-­‐use activities that require minimal preparation and give clear and specific directions. Participants also receive journal prompts, homework suggestions, and ideas for extensions and assessment. Brad's math activities are the best I've seen in 38 years of teaching! Wayne Dequer, 7th grade math teacher, Arcadia, CA
“I can't begin to tell you how much you have inspired me!” Sue Bonesteel, Math Dept. Chair, Phoenix, AZ
“Your entire audience was fully involved in math!! When they chatted, they chatted math. Real thinking!” Brenda McGaffigan, principal, Santa Ana, CA
“Absolutely engaging. I can teach algebra to second graders!” Lisa Fellers, teacher
© 2013 by Brad Fulton and TTT Press References available upon request
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© 2013 by Brad Fulton and TTT Press Array We Go!
An Engaging and Visual Representation of
Factors, Multiples, Primes, and Composites and More
Lesson Duration: 1 to 4 class periods
Required Materials:
Procedure:
ý Centimeter grid paper
(included)
1.
Divide the class into groups of four students. 2.
Pass out copies of the centimeter grid paper (included in this handout), scissors, glue, and construction paper. ý Scissors
ý Glue
ý Construction paper
3.
Explain that students will be assigned numbers to work with. Their task is to cut out as many rectangles as possible such that each one has an area in square centimeters equal to their number. For example, if their number is 15, they could cut out a rectangle that is three centimeters by five centimeters. They could also cut out one that is five by three centimeters. For the purpose of this activity, both of these should be accepted. They can also make a one by fifteen and a fifteen by one. Thus there are four possible rectangles that have an area of 15 square centimeters. 4.
Each group should make these four solutions and glue them to one sheet of construction paper. On the back of the paper, they should write “15” in large numerals. 5.
The task is to repeat this for all the numbers from one through 25 (excluding 15). You should now decide if you want the class to make one set, or if you would rather each group made a set of their own. If you choose the former option, you will need less materials but some students will have less work to do (prime numbers) while others have more (composites). They can help one another if you wish. Another option is to divide the numbers among the teams so that each team has approximately the same amount of work to do. Thus each team should have some primes and some composites. 6.
As students work, they will notice that some numbers have very few rectangle solutions (primes) while others have many solutions (composites). They will also notice that even among the composites, some have more solutions than others. 7.
Once the students have completed a set of cards, display them in the front of the room for all to see as you discuss the following questions and activities. © 2013 by Brad Fulton and TTT Press Notice that the words rectangle and factor can be used interchangeably. Factor is more mathematically precise, but the term rectangle is more visual. Factors a. Which numbers have a rectangle with a width or height of two squares? (2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24) b. Turn these cards over so that the rectangles are on the back and the even numbers are showing to the class. c. What do you call these numbers? (Even numbers) d. What are the next three even numbers? (26, 28, 30) e. What are the other numbers called? (Odd numbers) f. What are the next three odd numbers? (27, 29, 31) g. Which rectangles have a dimension of three squares? (3, 6, 9, 12, 15, 18, 21, 24) h. Turn these cards over as you did for the even numbers so the numbers show. i.
If the class has made more than one set of cards, you may wish to leave the multiples of two up while working with the multiples of three. This will allow students to identify which numbers have common factors of both two and three. (6, 12, 18, 24) j.
Repeat these steps for other multiples such as four and five as desired. Prime Numbers a. Which numbers have only two rectangles? (2, 3, 5, 7, 11, 13, 17, 19, 23) b. Turn over the cards of the prime numbers so the rectangles are hidden and the numbers themselves show on the back. c. Are all of these numbers odd? (No, 2 is even, but the rest are odd.) d. Numbers with exactly two factors (rectangles) are called prime numbers. What are the next three numbers that would be prime? (29, 31, 37) e. Is there a pattern to the prime numbers? (No) © 2013 by Brad Fulton and TTT Press Composite numbers a. Which numbers have more than two rectangles? (4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24) b. Which number(s) has the most factors? (24 has eight factors.) c. Which number(s) have the next most factors? (12, 18, and 20 have six factors.) d. Numbers with more than two factors are called composite numbers. What are the next three composite numbers? (26, 27, 28) e. What are their factors? (26: 1, 2, 13, 26; 27: 1, 3, 9, 27; 28: 1, 2, 4, 7, 14, 28) f. Which number(s) have only one factor? (1 is neither prime nor composite for this reason.) Advanced questions a. Which numbers have an odd number of factors? (1, 4, 9, 16, 25) b. Which numbers have a square arrangement (as opposed to only rectangles)? (1, 4, 9, 16, 25) c. These are called square numbers for this reason. Do you see a pattern to the square numbers? (Some students may notice that you can find them by multiplying a number by itself: 1x1=1; 2x2=4; 3x3=9; 4x4=16; 5x5=25. Others may notice that the square numbers are separated by odd numbers: 1+3=4; 4+5=9; 9+7=16, 16+9=25) d. What are the next three square numbers? (36, 49, 64) e. How many square numbers are there between 1 and 100? (Ten: 1, 4, 9. 16, 25, 36, 49, 64, 81, 100) f. Numbers whose factors (besides themselves) add up to the number are called perfect numbers. The factors of 6 (other than 6) are 1, 2, and 3, which add up to six. What is the next perfect number? (28=1+2+4+7+14) g. Numbers whose factors (besides themselves) add up to a sum greater than the number are said to be factor rich. Which numbers are factor rich? (12: 1+2+3+4+6=16; 18: 1+2+3+6+9=21; 20: 1+2+4+5+10=22; 24: 1+2+3+4+6+8+12=36) © 2013 by Brad Fulton and TTT Press Answer key Number Solutions 1 1x1 2 1x2 2x1 3 1x3 3x1 4 1x4 2x2 5 1x5 5x1 6 1x6 2x3 7 1x7 7x1 8 1x8 9 1x9 10 4x1 3x2 6x1 2x4 4x2 8x1 3x3 9x1 1x10 2x5 5x2 10x1 11 1x11 11x1 12 1x12 2x6 3x4 4x3 13 1x13 13x1 14 1x14 2x7 7x2 14x1 15 1x15 3x5 5x3 15x1 16 1x16 2x8 4x4 8x2 16x1 17 1x17 17x1 18 1x198 2x9 3x6 6x3 9x2 19 1x19 19x1 20 1x20 2x10 4x5 5x4 10x2 20x1 21 1x21 3x7 21x1 22 1x22 2x11 11x2 22x1 23 1x23 23x1 24 1x24 2x12 3x8 25 1x25 5x5 7x3 4x6 25x1 © 2013 by Brad Fulton and TTT Press 6x2 6x4 12x1 18x1 8x3 12x2 24x1 If you also want to incorporate data collection and data displays into the activity, you can have the students graph the number of factors using the activity master provided. Procedure: 1.
2.
3.
4.
5.
Pass out the activity master after the students have created a set of the factor cards. Ask them how many factors 1 has (1). Have them color one square above the 1 on the horizontal axis. How many factors does 2 have? (2) Have them color two squares above the 2 on the horizontal axis. Repeat this for 3, which has two factors, and four, which has three. This will create a bar graph allowing them to compare numbers. Ask them these questions: a.
Do you see any patterns? (The bars tend to get higher toward the right.) b.
Which numbers tend to have more factors? (Numbers that are further down the horizontal axis.) c.
Is this always true? (No, some prime numbers are toward the right also.) d.
Could you go far enough down the number line that no primes ever showed up again? (Though they may not know the answer to this, there are always more primes.) © 2013 by Brad Fulton and TTT Press © 2013 by Brad Fulton and TTT Press 1
2
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Class_______________________________
Name______________________________
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Above each number color one square for each
rectangle you found.
Factor Graph
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
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Answer key © 2013 by Brad Fulton and TTT Press The Common Core Connection This activity addresses these Common Core Math standards:
3rd grade
Operations and Algebraic Thinking
•
CCSS.Math.Content.3.OA.A.1 Interpret products of whole numbers, e.g., interpret 5
× 7 as the total number of objects in 5 groups of 7 objects each. For example,
describe a context in which a total number of objects can be expressed as 5 × 7.
•
CCSS.Math.Content.3.OA.A.2 Interpret whole-number quotients of whole numbers,
e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are
partitioned equally into 8 shares, or as a number of shares when 56 objects are
partitioned into equal shares of 8 objects each. For example, describe a context in
which a number of shares or a number of groups can be expressed as 56 ÷ 8.
•
CCSS.Math.Content.3.OA.A.3 Use multiplication and division within 100 to solve
word problems in situations involving equal groups, arrays, and measurement
quantities, e.g., by using drawings and equations with a symbol for the unknown
number to represent the problem.
•
CCSS.Math.Content.3.OA.A.4 Determine the unknown whole number in a
multiplication or division equation relating three whole numbers. For example,
determine the unknown number that makes the equation true in each of the equations
8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ?
•
CCSS.Math.Content.3.OA.B.5 Apply properties of operations as strategies to
multiply and divide.2Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also
known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 =
15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of
multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 ×
(5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)
•
CCSS.Math.Content.3.OA.B.6 Understand division as an unknown-factor
problem. For example, find 32 ÷ 8 by finding the number that makes 32 when
multiplied by 8.
•
CCSS.Math.Content.3.OA.C.7 Fluently multiply and divide within 100, using
strategies such as the relationship between multiplication and division (e.g., knowing
that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of
Grade 3, know from memory all products of two one-digit numbers.
Measurement and Data
•
CCSS.Math.Content.3.MD.B.3 Draw a scaled picture graph and a scaled bar graph to
represent a data set with several categories. Solve one- and two-step “how many
more” and “how many less” problems using information presented in scaled bar
graphs.
© 2013 by Brad Fulton and TTT Press 4th Grade
Operations and Algebraic Thinking
•
CCSS.Math.Content.4.OA.A.1 Interpret a multiplication equation as a comparison,
e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as
many as 5. Represent verbal statements of multiplicative comparisons as
multiplication equations.
•
CCSS.Math.Content.4.OA.B.4 Find all factor pairs for a whole number in the range
1–100. Recognize that a whole number is a multiple of each of its factors. Determine
whether a given whole number in the range 1–100 is a multiple of a given one-digit
number. Determine whether a given whole number in the range 1–100 is prime or
composite.
In addition, this activity can be used beyond 4th grade with students who need
foundational work with multiplication. The advanced questions also provide extensions
that can be utilized in grades 5, 6, and 7.
© 2013 by Brad Fulton and TTT Press © 2013 by Brad Fulton and TTT Press