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-<chapters><chapter id="ch1" title=""><section id="sect1" title=""><clip id="gcf" startSegId="s0" title="Greatest Common Factors" url="gcf.swf"><seg id="s0" templateId="" url=""><args title="Introduction"/><next branch="false" value="s1"/></seg><seg id="s1" templateId="" url=""><args text="If you want to figure out how a gadget works, you can read a manual ... or you can take it apart. That's more fun, and you often learn a lot more. Just don't break something that belongs to someone else. "/><next branch="false" value="s2"/></seg><seg id="s2" templateId="" url=""><args text="Just as you can take apart a watch to see how it works, you can take numbers apart by breaking them down into prime factors. If you're familiar with prime factors, let's find out how to use them to solve other kinds of problems."/><next branch="false" value="s3"/></seg><seg id="s3" templateId="" url=""><args text="Breaking numbers down into their prime factors will help us solve Greatest Common Factor problems. What's a greatest common factor? Lest you think it's some highfalutin abstract concept, let's look at a silly example: it's a little game called Cow Stacking."/><next branch="false" value="s6"/></seg><seg id="s6" templateId="" url=""><args text="Cow Stacking is easy in principle. You take one cow, and stack her on top of another one. You have to stack two herds: 12 black and white Holsteins, and 8 Norwegian Reds. Go ahead, try it out. "/><next branch="false" value="s7"/></seg><seg id="s7" templateId="" url=""><args text="But it gets trickier. There are rules. Rule Number One: You can't mix herds in a stack. You can only stack cows of the same herd. That means you'll have to make at least two stacks. Also, there isn't room in this pasture to make more than six stacks. You get to choose how many stacks to make."/><next branch="false" value="s8"/></seg><seg id="s8" templateId="" url=""><args text="Rule Number Two: All of your stacks must contain the same number of cows."/><next branch="false" value="s9"/></seg><seg id="s9" templateId="" url=""><args text="Rule Number Three: You have to use all of the cows."/><next branch="false" value="s10"/></seg><seg id="s10" templateId="" url=""><args text="Got the rules? So, what's the highest stack you can build if you stick to those rules? We'll come back to Cow Stacking and answer this question in a bit, but first, let's learn about greatest common factors."/><next branch="false" value="s29"/></seg><seg id="s29" templateId="" url=""><args title="Greatest Common Factor"/><next branch="false" value="s30"/></seg><seg id="s30" templateId="" url=""><args text="What is a greatest common factor? Let's look at an example. The factors of 10 are 1, 2, 5, and 10. The factors of 15 are 1, 3, 5, and 15. In other words, 10 and 15 share the factors 1 and 5. These are the common factors of 10 and 15. "/><next branch="false" value="s31"/></seg><seg id="s31" templateId="" url=""><args text="Among these common factors of 10 and 15, the largest is 5. That's the greatest common factor, also known as the GCF. Easy, right?"/><next branch="false" value="s32"/></seg><seg id="s32" templateId="" url=""><args aPop="That's not correct. 1 is a common factor of 9 and 12, but it's not the greatest common factor." aText="1" bPop="That's correct. The common factors of 9 and 12 are 1 and 3, so 3 is the GCF." bText="3" cPop="That's not correct. Is 6 a factor of 9?" cText="6" correct="b" dPop="That's not correct. Is 9 a factor of 12?" dText="9" text="So, what's the GCF of 9 and 12?"/><next branch="false" value="s33"/></seg><seg id="s33" templateId="" url=""><args text="Finding a GCF of two numbers is easy in principle. First, find the factors of each number. Then, find the factors they share. Finally, take the greatest of those common factors. That's the GCF! "/><next branch="false" value="s34"/></seg><seg id="s34" templateId="" url=""><args text="Usually, the tricky part is finding all the factors. For larger numbers, that can be quite a bit of work. Let's look at how we can use prime factors to simplify that process."/><next branch="false" value="s35"/></seg><seg id="s35" templateId="" url=""><args text="For example, let's find the greatest common factor of 27 and 99. We'll start with their prime factors."/><next branch="false" value="s36"/></seg><seg id="s36" templateId="" url=""><args text="How can you find the GCF from the prime factors? First, let's group the prime factors and arrange them from least to greatest. "/><next branch="false" value="s37"/></seg><seg id="s37" templateId="" url=""><args Answer="9" correct="That’s correct! Excellent work." incorrect="That's not the GCF. Check whether there's another pair of common factors." text="Once you have all the prime factors in front of you, finding the GCF is just a matching game. Try clicking on a 3 in the first column, then a 3 in the second column. 3 is a factor of 27 and a factor of 99, so 3 is a factor of the GCF. "/><next branch="false" value="s37a"/></seg><seg id="s37a" templateId="s37" url=""><args Answer="9" correct="That’s correct! Excellent work." incorrect="That's not the GCF. Check whether there's another pair of common factors." text="27 and 99 have another prime factor in common. When you've found both pairs of common factors, click 'Submit'. The product of the common factors is the GCF."/><next branch="false" value="s38"/></seg><seg id="s38" templateId="" url=""><args text="Let's continue practicing! Keep in mind that this program automatically finds the product of the common factors for you. When you solve GCF problems on paper, you'll have to find the product yourself."/><next branch="false" value="s39"/></seg><seg id="s39" templateId="" url=""><args Answer="4" correct="That’s correct!" incorrect="Try again. Check that you’ve found all of the common factors." text="Find the GCF of 28 and 36."/><next branch="false" value="s40"/></seg><seg id="s40" templateId="" url=""><args Answer="5" correct="Excellent work!" incorrect="That is not the correct answer. Try again." text="Find the GCF of 45 and 25."/><next branch="false" value="s41"/></seg><seg id="s41" templateId="" url=""><args Answer="6" correct="Yes, that’s the correct answer!" incorrect="That is not correct. Be sure that you’ve found all of the common factors." text="Find the GCF of 54 and 24."/><next branch="false" value="s42"/></seg><seg id="s42" templateId="" url=""><args Answer="14" correct="Great work! That's correct." incorrect="That is not correct. Did you find all of the common factors?" text="Find the GCF of 14, 70, and 42. You'll need to select the primes that all three numbers have in common. "/><next branch="false" value="s43"/></seg><seg id="s43" templateId="" url=""><args text="So how about that Cow Stacking game?"/><next branch="false" value="s44"/></seg><seg id="s44" templateId="" url=""><args answer="4" correct="That’s right! Excellent work!" incorrect="Try again! Check that you’ve followed all the rules." text="Click and drag the cows to stack them! Follow the rules and see how high you can make the stacks. Remember, you get to choose how many stacks to make. To enlarge the rules, just click on them."/><next branch="false" value="s45"/></seg><seg id="s45" templateId="" url=""><args text="Could you have figured out how many cows would be in each stack before you even started stacking?"/><next branch="false" value="s46"/></seg><seg id="s46" templateId="" url=""><args text="You have to divide both herds of cows into equal stacks. So the height of the stacks has to be a factor of 12 and a factor of 8."/><next branch="false" value="s47"/></seg><seg id="s47" templateId="" url=""><args text="Then, you wanted to make the stacks as high as possible. So the height of the stacks is the GCF of 12 and 8."/><next branch="false" value="s48"/></seg><seg id="s48" templateId="" url=""><args answer="4" correct="Fantastic! That’s correct." incorrect="Check that you’ve found all the common factors, and try again!" text="Find the GCF of 12 and 8."/><next branch="false" value="s62"/></seg><seg id="s62" templateId="" url=""><args text="Excellent work! You can use these tools on pencil and paper whenever you need to find greatest common factors. Here's how you could find the GCF of 16 and 28 on paper."/><next branch="false" value="s71"/></seg><seg id="s71" templateId="" url=""><args title="Review"/><next branch="false" value="s72"/></seg><seg id="s72" templateId="" url=""><args text="In an earlier module, you learned about finding prime factors."/><next branch="false" value="s73"/></seg><seg id="s73" templateId="" url=""><args text="Here's how to find the greatest common factor of two or three numbers."/><next branch="false" value="e0"/></seg><seg id="e0" templateId="" url=""><args title="Exercises"/><next branch="false" value="e0a"/></seg><seg id="e0a" templateId="" url=""><args text="Grab a pencil and paper! In the following five exercises, you'll practice finding greatest common factors. After each question, the answer will be explained."/><next branch="false" value="e1"/></seg><seg id="e1" templateId="" url=""><args aPop="Good try — 2 is a factor of all three numbers, but it isn’t the greatest common factor." aText="2" bPop="You’re right. The number 4 goes into all three numbers, and it's the largest number that does so." bText="4" cPop="That's not the correct answer. Think about it. 14 is far bigger than 4, so it can’t be a factor of 4, so it can’t be the greatest common factor of the three numbers." cText="14" correct="b" dPop="That's not the correct answer. Sorry, 28 is neither a factor of 4 nor of 52. Try again." dText="28" text="1) Find the GCF of 4, 28, and 52."/><next branch="false" value="e2"/></seg><seg id="e2" templateId="" url=""><args text="In order to find the GCF, first find the prime factors of each number. Then, multiply together the common factors. The greatest common factor is 4."/><next branch="false" value="e5"/></seg><seg id="e5" templateId="" url=""><args answer="9" correct="Brilliant! 9 is correct." incorrect="That's not the correct answer. Be sure you've broken down the numbers into their prime factors correctly." text="2) Find the Greatest Common Factor of 99, 72, and 63."/><next branch="false" value="e6"/></seg><seg id="e6" templateId="" url=""><args text="In order to find the GCF of 99, 72, and 63, find the prime factors of each and multiply together the common factors. So the greatest common factor is 9."/><next branch="false" value="e9"/></seg><seg id="e9" templateId="" url=""><args answer="6" correct="Good work! The answer is 6." incorrect="That is not the correct answer. Be sure you've broken down the numbers into their prime factors correctly." text="3) Find the GCF of 108 and 42."/><next branch="false" value="e10"/></seg><seg id="e10" templateId="" url=""><args answer="6" correct="Good work! The answer is 6." incorrect="That is not the correct answer. Be sure you've broken down the numbers into their prime factors correctly." text="Begin by finding the prime factors of 108 and 42. Multiply together all of the factors that they have in common to find the GCF."/><next branch="false" value="e11"/></seg><seg id="e11" templateId="" url=""><args answer="5" correct="You’re right! 5 students per team would work." incorrect="Check your work and try again. You’re looking for a number that divides evenly into both 25 and 20." text="4) Mrs. Scanlan's homeroom has 25 students and Ms. Ayres's class has 20 students. The two classes are dividing into many equal-sized teams to play lazer tag. No team can have students from both classes in it, and every student must be on a team. What's the largest number of students per team?"/><next branch="false" value="e12"/></seg><seg id="e12" templateId="" url=""><args answer="5" correct="You’re right! 5 students per team would work." incorrect="Check your work and try again. You’re looking for a number that divides evenly into both 25 and 20." text="This problem is asking you to find the largest factor of both 25 and 20. That's the GCF! 5 is the only prime factor that they share. So the GCF is 5."/><next branch="false" value="e17"/></seg><seg id="e17" templateId="" url=""><args answer="22" correct="22 is exactly right! Good work." incorrect="That’s not the correct answer. Check whether you’ve found the prime factors correctly." text="5) Find the GCF of 66 and 110."/><next branch="false" value="e18"/></seg><seg id="e18" templateId="" url=""><args answer="22" correct="22 is exactly right! Good work." incorrect="That’s not the correct answer. Check whether you’ve found the prime factors correctly." text="What should you do first when finding the GCF? Find the prime factors! Then multiply the common prime factors. The GCF is 22."/><next branch="false" value="e21"/></seg><seg id="e21" templateId="" url=""><args text="Congratulations! You've completed the Greatest Common Factors module!"/><next branch="false" value=""/></seg></clip><clip id="clip2" title="title2"/><clip id="clip3" title="title3"/><clip id="clip4" title="title4"/><clip id="clip5" title="title5"/></section></chapter></chapters></app>
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