In this Cyberchase video segment, the CyberSquad is trying to figure out the location of Icky, the guardian of the Electric Eel of Aquari-Yum. The clue to his location lies in eight bubbles filled with pictures and numbers. The kids are told to find two numbers whose product is 20. Realizing that the pair of 10 and 2 is not an option, they use seashells to create an array to test out possible factors.
Determining a number's factors is a very important part of number theory. A factor of a number is a positive integer that divides evenly into the number. For example, the factors of 8 are 1, 2, 4, and 8. Factoring is an important tool used to divide whole numbers greater than 1 into two groups, prime and composite. A prime number has exactly two factors, 1 and itself. Composite numbers are whole numbers that have more than two factors. For example, the number 12 is composite because it has six factors, namely 1, 2, 3, 4, 6, and 12. Factors are useful in situations such as simplifying fractions or finding the greatest common factor or least common multiple of two numbers.
In order to represent factors visually, numbers can be modeled using rectangular arrays. For example, the number 18 can be shown as one row of 18, two rows of nine, or three rows of six. The composition of the array expresses the factors of a number. Consequently, these three arrays show that 1, 2, 3, 6, 9, and 18 are all factors of 18. When there are leftovers, neither the number of columns nor the number of rows is a factor. For example, if you try to express 18 using four rows, you find that you can put four in each row, but then there will be two left over, which is not enough to add one to each row. Therefore, 4 is not a factor of 18. Prime numbers can only be expressed as one unique array, which will look like one single row or column.
The factors of a number play a role in everyday life. For example, to determine the packaging and shipping options for a large number of items, manufacturers have to consider the arrangement of the items inside the package. If you look at a package containing six rolls of toilet paper, they are usually stacked three rolls wide and two rolls high. This shows that 2 and 3 are factors of 6. A package of 12 soda cans arranged as two rows of six is possible because both 2 and 6 are factors of 12. Now suppose you are arranging 24 photos, all of the same size and orientation, on a large cardboard presentation board. How can you arrange them so that you have an equal number of photos in each row? Knowing the factors of 24 can help you quickly consider your options: 1 row of 24, 2 rows of 12, 3 rows of 8, or 4 rows of 6.
To learn more about arrays, check out Harry the Substitute Teacher.
To find a lesson plan involving factors, check out Factors, Arrays and Commutativity.
DIGIT: What’s with the bubbles?
MATT: Hey! Each of these bubbles has a picture inside it!
INEZ: And each one has a number!
MANNY: Number pairs...find the place…
DIGIT: Number pairs find the place? What’s that supposed to mean?
MATT: Well, we’ve got eight bubbles with pictures and numbers in them.
MANNY: Twenty...need twenty..ah-shh..
DIGIT: Twenty what??!
INEZ: Maybe a pair of these numbers add up to twenty?
MATT: But we only have the numbers one through eight. There aren’t two numbers big enough to add up to twenty!
INEZ: Maybe instead of adding pairs we have to multiply two numbers.
MATT: But which two numbers make twenty when you multiply them together?
DIGIT: I know! Two times ten!
MATT: Hey, you’re right, Didge, two times ten is twenty. But we don’t have the number ten- so it can’t be that pair.
INEZ: How about we make rows and columns of seashells - like we did with the cells?
INEZ: Here's twenty shells.
DIGIT: Okay, but what if two and ten are the only numbers that make twenty when you multiply ‘em?!
INEZ: That’s what we’re going to find out.
INEZ: We’re looking for other factors of twenty.
INEZ: Other numbers that when multiplied together give us twenty with no leftovers.
MATT: Well...it can’t be three times six you’ve got leftovers.
DIGIT: Try five in a row, try five!!
INEZ: Four times five is twenty - with no leftovers!
DIGIT: Whoa! There’s more than one pair of numbers that multiply to twenty. Who knew? Let’s go to the bubbles!
MATT: Pair the numbers...find the place, that’s what Manny said.
INEZ: The pictures must be places!
INEZ: Okay, our pair of numbers is four and five. Bubble four is...a hot dog! And bubble five is...a hill.
DIGIT: Hot Dog Hill!
DIGIT: Icky must be at Hot Dog Hill!
MATT: Are we sure?
MATT: We still have bubbles seven and eight left.
INEZ: If we multiply three times seven it’s twenty-one and that’s more than twenty. So it can’t be any of those bubbles!
MATT: You’re right! Hot Dog Hill it is.
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