Who in the heck ordered fractions? Comparing Who in the heck ordered fractions? Ordering and Fractions using The Least Common Multiple © Mike’s Math Mall

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Least Common Multiple The Least Common Multiple of two numbers is the smallest number that can be evenly divided by both numbers. Sweet! It looks like we have this concept mastered! For example, what’s the smallest number that can be evenly divided by both 2 and 3? If you said 6, you’re right! 6 is the “LCM” of 2 and 3! Umm…not so fast, Sparky!

Least Common Multiple 3: 3, 6, 9, 12, 15, 18… 4: 4, 8, 12, 16, 20,… Let’s find the LCM of 3 and 4. You can do this by creating a multiple list for both numbers: 3: 3, 6, 9, 12, 15, 18… 4: 4, 8, 12, 16, 20,… The LCM of 3 and 4 is 12. Were you able to do that one in your head?

Show Your Stuff! Let’s practice! 1) 6 and 9 2) 5 and 6 Find the LCM of each pair of numbers by listing their multiples. 1) 6 and 9 2) 5 and 6 3) 4 and 10 4) 12 and 15 LCM = 18 LCM = 30 6: 6, 12, 18, 24,… 9: 9, 18, 27,… 5: 5, 10, 15, 20, 25, 30, 35,… 6: 6, 12, 18, 24, 30, 36,… LCM = 20 LCM = 60 4: 4, 8, 12, 16, 20, 24,… 10: 10, 20, 30,… 12: 12, 24, 36, 48, 60, 72,… 15: 15, 30, 45, 60, 75,…

Least Common Multiple Great question! OK! But what if the numbers are bigger, like 18 and 30, and are harder to multiply out? You just used the word cool with math! Stop it! At least my mom thinks I’m adorable! Great question! I have a cool method that will help you find the LCM of any set of numbers. Fair enough, but if we’re talking about things that aren’t cool, let’s start with that spasmodic face.

Problem 1: Find the LCM of 18 and 30. Least Common Multiple Problem 1: Find the LCM of 18 and 30. Step 1: Put the two numbers side-by-side: 18 30 Step 2: Draw an “upside-down division box” around them: 18 30 Step 3: Divide both numbers by the smallest possible prime number. 18 30 2 9 15 →

Least Common Multiple Problem 1: Continued 18 30 Step 4: Draw a box around the new numbers and divide them by the smallest possible prime number: 18 30 2 15 9 3 3 5 Step 5: When you can no longer evenly divide both new numbers, multiply all numbers on the outside of the boxes. 2 × 3 × 3 × 5 = 90 90 is the LCM

face…painful…so…painful! Least Common Multiple I have to admit that was kinda cool, but there’s a lot of steps. Can we do another one? Sparky’s face…painful…so…painful! What’s that? Absolutely! But I have a question for you. When you make that face, does it actually hurt?

Problem 2: Find the LCM of 16 and 24. Least Common Multiple Problem 2: Find the LCM of 16 and 24. 16 24 2 2 8 12 2 4 6 2 3 2 × 2 × 2 × 2 × 3 = 48

Problem 3: Find the LCM of 12, 18, and 24. Least Common Multiple Problem 3: Find the LCM of 12, 18, and 24. Keep dividing until no pair of bottom numbers have a common factor. A number that’s not divided, like the 3, is brought down. 12 18 24 2 3 6 9 12 2 2 3 4 3 1 2 Am I dorky if I actually think this is fun? Sweet! 2 × 3 × 2 × 1 × 3 × 2 = 72 Well…not too dorky!

Show Your Stuff! Let’s practice! 1) 7 and 21 2) 12 and 40 Use the “upside-down division box” method to find each LCM. 1) 7 and 21 2) 12 and 40 7 21 7 1 3 2 12 40 6 20 3 10 7 × 1 × 3 = 21 2 × 2 × 3 × 10 = 120 LCM = 21 LCM = 120

Show Some More Stuff! 3) 24 and 32 4) 20, 30, and 50 LCM = 300 Use the “upside-down division box” method to find each LCM. 3) 24 and 32 4) 20, 30, and 50 24 32 2 12 16 6 8 3 4 20 30 50 2 10 15 25 5 3 2 × 5 × 2 × 3 × 5 = 300 2 × 2 × 2 × 3 × 4 = 96 LCM = 300 LCM = 96

Least Common Multiple You just used the word great in the same sentence with fractions! Seriously? So what’s the big deal with finding all of these least common multiples? Oh, that’s just lunch creeping back up on me! Knowing how to find the least common multiple will be great when we start working with fractions. I sense some fractional distress! Moving right along…

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the least common denominator do Comparing Fractions If you can find the Least Common Multiple of a pair of numbers, then you can find the Least Common Denominator of a pair of fractions. So what does the least common denominator do for us? Nice?... Easier?...Fractions? I seriously doubt that! They’re the same! Just check this out! Least common denominators are nice because they keep the numbers in our fractions smaller, making them easier to compare and order.

Comparing Fractions It’s a close call! Which fraction is greater, 𝟕 𝟏𝟎 or 𝟓 𝟖 ? Thanks for painting that picture for us, Sparkington! Unfortunately, I had a close call the morning after eating two grande chili burritos at Ricky’s Rocko Taco! It’s a close call! Oh, yeah! TMI? If we can rewrite these fractions with a Least Common Denominator (LCD), then all we have to do is compare the numerators.

Step 1: Find the LCM of 10 and 8. Comparing Fractions Which fraction is greater, 𝟕 𝟏𝟎 or 𝟓 𝟖 ? Step 1: Find the LCM of 10 and 8. Remember! LCM = LCD 10 8 2 5 4 2 × 5 × 4 = 40 The Least Common Denominator is 40. →

Comparing Fractions 𝟕 𝟏𝟎 = 𝟐𝟖 𝟒𝟎 𝟓 𝟖 = 𝟐𝟓 𝟒𝟎 Which fraction is greater, 𝟕 𝟏𝟎 or 𝟓 𝟖 ? Step 2: Create equivalent fractions with LCD’s of 40. 𝟕 𝟏𝟎 ×𝟒 = 𝟐𝟖 𝟒𝟎 𝟓 𝟖 ×𝟓 = 𝟐𝟓 𝟒𝟎 ×𝟒 ×𝟓 →

Comparing Fractions 𝟐𝟖 𝟒𝟎 > 𝟐𝟓 𝟒𝟎 , so 7 10 > 5 8 Which fraction is greater, 𝟕 𝟏𝟎 or 𝟓 𝟖 ? Step 3: Compare numerators to decide which fraction is greater. 𝟕 𝟏𝟎 = 𝟐𝟖 𝟒𝟎 𝟓 𝟖 = 𝟐𝟓 𝟒𝟎 𝟐𝟖 𝟒𝟎 > 𝟐𝟓 𝟒𝟎 , so 7 10 > 5 8

possibly get you to show me another example? Comparing Fractions Could I possibly get you to show me another example? I promise! Promise to keep your burrito eating mishaps to yourself? Then it’s a deal!

Comparing Fractions Compare the fractions: 𝟓 𝟔 and 𝟕 𝟖 2 × 3 × 4 = 24 1) Find the LCM of 6 and 8. 6 8 2 3 4 2 × 3 × 4 = 24 The Least Common Denominator is 24. →

Comparing Fractions 𝟓 𝟔 = 𝟐𝟎 𝟐𝟒 𝟕 𝟖 = 𝟐𝟏 𝟐𝟒 𝟐𝟎 𝟐𝟒 < 𝟐𝟏 𝟐𝟒 , Compare: 𝟓 𝟔 and 𝟕 𝟖 2) Create equivalent fractions. 𝟓 𝟔 ×𝟒 = 𝟐𝟎 𝟐𝟒 𝟕 𝟖 ×𝟑 = 𝟐𝟏 𝟐𝟒 ×𝟒 ×𝟑 𝟐𝟎 𝟐𝟒 < 𝟐𝟏 𝟐𝟒 , so 5 6 < 7 8 3) Compare.

Since 7 and 5 can’t be divided evenly, just multiply to find the LCD. Show Your Stuff! Let’s practice! Find the LCD for each pair of fractions. 4 12 2 6 1 3 1) 𝟐 𝟕 and 𝟏 𝟓 2) 𝟑 𝟒 and 𝟗 𝟏𝟐 7 5 Since 7 and 5 can’t be divided evenly, just multiply to find the LCD. 2 × 2 × 1 × 3 = 12 7 × 5 = 35 3) 𝟐 𝟏𝟓 and 𝟏 𝟏𝟎 4) 𝟓 𝟏𝟐 and 𝟗 𝟐𝟎 2 12 20 6 10 3 5 5 15 10 3 2 5 × 3 × 2 = 30 2 × 2 × 3 × 5 = 60 →

Show Your Stuff! 2 7 > 1 5 3 4 = 9 12 2 15 > 1 10 5 12 < 9 20 Using the same problems from the previous slide, create equivalent fractions and compare them using <, =, >. 1) 𝟐 𝟕 and 𝟏 𝟓 LCD = 35 2) 𝟑 𝟒 and 𝟗 𝟏𝟐 LCD = 12 𝟐 ∙ 𝟓 𝟕 ∙ 𝟓 𝟏 ∙ 𝟕 𝟓 ∙ 𝟕 = 𝟏𝟎 𝟑𝟓 = 𝟕 𝟑𝟓 𝟑 ∙ 𝟑 𝟒 ∙ 𝟑 𝟗 ∙ 𝟏 𝟏𝟐 ∙ 𝟏 = 𝟗 𝟏𝟐 2 7 > 1 5 3 4 = 9 12 3) 𝟐 𝟏𝟓 and 𝟏 𝟏𝟎 LCD = 30 4) 𝟓 𝟏𝟐 and 𝟗 𝟐𝟎 LCD = 60 𝟐 ∙ 𝟐 𝟏𝟓 ∙ 𝟐 𝟏 ∙ 𝟑 𝟏𝟎 ∙ 𝟑 = 𝟒 𝟑𝟎 = 𝟑 𝟑𝟎 𝟓 ∙ 𝟓 𝟏𝟐 ∙ 𝟓 𝟗 ∙ 𝟑 𝟐𝟎 ∙ 𝟑 = 𝟐𝟓 𝟔𝟎 = 𝟐𝟕 𝟔𝟎 2 15 > 1 10 5 12 < 9 20

So how’d you do on the practice problems, Sparky? Comparing Fractions So how’d you do on the practice problems, Sparky? Compared to 0%, I would have to say I did awesomely! I only missed 3 out of 4. Is that good? I guess it is if you’re OK with a 25%. That’s one way of looking at it. Hopefully, you’ll do better with ordering fractions!

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Ordering Fractions 𝟓 𝟗 , 𝟐 𝟑 , and 𝟕 𝟏𝟐 → Ordering a group of fractions from least to greatest requires finding the LCD of all the fractions. The “upside-down division box” works great for this. So, let’s say that you wanted to order the following fractions from least to greatest: 𝟓 𝟗 , 𝟐 𝟑 , and 𝟕 𝟏𝟐 →

Ordering Fractions 𝟓 𝟗 , 𝟕 𝟏𝟐 , 𝟐 𝟑 𝟓 ∙ 𝟒 𝟗 ∙ 𝟒 = 𝟐𝟎 𝟑𝟔 𝟐 ∙ 𝟏𝟐 𝟑 ∙ 𝟏𝟐 Next, create equivalent fractions with LCD’s of 36. 𝟓 𝟗 , 𝟐 𝟑 , and 𝟕 𝟏𝟐 First, find the LCD of all fractions. 𝟓 𝟗 , 𝟐 𝟑 , and 𝟕 𝟏𝟐 9 3 12 3 1 4 𝟓 ∙ 𝟒 𝟗 ∙ 𝟒 = 𝟐𝟎 𝟑𝟔 3 × 3 × 1 × 4 = 36 Last, compare numerators and list the original fractions from least to greatest. 𝟐 ∙ 𝟏𝟐 𝟑 ∙ 𝟏𝟐 = 𝟐𝟒 𝟑𝟔 𝟓 𝟗 , 𝟕 𝟏𝟐 , 𝟐 𝟑 𝟕 ∙ 𝟑 𝟏𝟐 ∙ 𝟑 = 𝟐𝟏 𝟑𝟔

Ordering Fractions My bad! 25% positive? Fractions used to freak me out, but I think I’m getting the hang of this comparing and ordering stuff. Positively sure, sir! That hurts! 25% positive? My bad! Excellent! Are you sure you’re ready for some ordering fractions’ practice?

You can Do it! Order the following fractions from least to greatest. 1) 5 16 , 7 24 , 3 8 2 × 2 × 2 × 2 × 3 × 1 = 48 𝟓 ∙ 𝟑 𝟏𝟔 ∙ 𝟑 = 𝟏𝟓 𝟒𝟖 2 16 24 8 8 12 4 6 3 1 𝟕 ∙ 𝟐 𝟐𝟒 ∙ 𝟐 = 𝟏𝟒 𝟒𝟖 7 24 , 5 16 , 3 8 𝟑 ∙ 𝟔 𝟖 ∙ 𝟔 = 𝟏𝟖 𝟒𝟖

Let’s get Crazy! Order the following fractions from least to greatest. 𝟑 ∙ 𝟑𝟎 𝟓 ∙ 𝟑𝟎 = 𝟗𝟎 𝟏𝟓𝟎 2) 3 5 , 8 15 , 11 25 , 7 10 𝟖 ∙ 𝟏𝟎 𝟏𝟓 ∙ 𝟏𝟎 = 𝟖𝟎 𝟏𝟓𝟎 5 15 25 10 5 1 3 2 𝟏𝟏 ∙ 𝟔 𝟐𝟓 ∙ 𝟔 = 𝟔𝟔 𝟏𝟓𝟎 5 × 1 × 3 × 5 × 2 = 150 𝟏𝟏 𝟐𝟓 , 𝟖 𝟏𝟓 , 𝟑 𝟓 , 𝟕 𝟏𝟎 𝟕 ∙ 𝟏𝟓 𝟏𝟎 ∙ 𝟏𝟓 = 𝟏𝟎𝟓 𝟏𝟓𝟎

So how’d you do this time, Spartacus? Ordering Fractions So how’d you do this time, Spartacus? The coincidence is eerie! I got one out of two right this time. If I’m not mistakin’, that makes 50%! Hey! Wanna see a belly wave? If I’m not mistaking, that’s the same percentage of belly you have covered! I’m gonna pass! © Mike’s Math Mall