ACCELERATED MATHEMATICS CHAPTER 3 FRACTIONS TOPICS COVERED:

Transcription

1 ACCELERATED MATHEMATICS CHAPTER FRACTIONS TOPICS COVERED: Divisibility Rules Primes & Prime Factorization Greatest Common Factor Least Common Multiple Fraction sense Adding and subtracting fractions and mixed numbers Equations with adding and subtracting fractions Hands-on multiplying fractions Multiplying fractions and mixed numbers Applications of multiplying fractions Hands-on dividing fractions Dividing fractions and mixed numbers Applications of dividing fractions

2 Activity -: Divisibility Rules A number is divisible by if the ones digit is even. A number is divisible by if its last two digits are divisible by. 6 A number is divisible by 6 if it is divisible by both and. Divisibility Rules A number is divisible by if the sum of its digits is divisible by. A number is divisible by if it ends in 0 or. 9 A number is divisible by 9 if the sum of its digits is divisible by 9. 0 A number is divisible by 0 if its last digit is 0. Circle the numbers that are divisible by Circle the numbers that are divisible by Circle the numbers that are divisible by Circle the numbers that are divisible by Circle the numbers that are divisible by Circle the numbers that are divisible by Circle the numbers that are divisible by

3 Activity -: Divisibility Rules. The Southlake Carroll Marching Band is getting ready to perform at halftime of the football game. With 6 musicians, can the marching band form equal rows of? of? of? of 6? of 9?. True or false: All numbers divisible by are also divisible by 0.. True or false: All numbers divisible by 0 are also divisible by.. True or false: All numbers divisible by 9 are also divisible by.. A giant pizza is divided into 8 pieces. What are the different numbers of people you can divide it among so that there are no pieces left over? 6. What is the smallest number you find that is divisible by,,, 6, 9, and 0? Complete the table. Answer Y (yes) or N (no) for each box. Divisible Divisible Divisible Divisible Number by by by by 7. Divisible by 6 Divisible by 9 Divisible by ,.,97 6., Marty said to Doc, So we are going to travel back in time. What year did you set the Delorean for? Doc replied, I can t remember exactly, but I do remember the following: If you divide the year by, you ll get a remainder of. If you divide the year by,,, 6, 7, or 9, you ll also get a remainder of. What about 8? Do you also get a remainder of? No, said Doc. Marty then knew which year they were off to. Which year? The page numbers of a book are numbered to 00. How many page numbers meet these conditions: A. The page numbers have the digit and are also divisible by. B. The page numbers contain the digit but are not divisible by. C. The page numbers do not have a but are divisible by.

4 Activity -: Primes and Composites. Circle all the prime numbers in the first row.. Draw a line through the first column (except for ) and through the third and fifth columns.. Draw a line through the second column (except for ).. Draw a diagonal line from the in the top row (not including the ) to the in the side column. Repeat with diagonal lines between the pairs of s in the side columns.. Draw a diagonal line down and to the right between the first 7 in the left side column to the first 7 in the right side column. Do this again for the second 7 s. 6. Circle any number that does not have a line through it. 7. Explain why you are left with just the primes. Side Column First Second Third Fourth Fifth Sixth 6 7 7A Side Column A B 7B 0 7A A C B B C

5 Prime number with million digits is the biggest ever found By Jacob Aron It s time for a new prime to shine. The largest known prime number is now 7,07,8, smashing the previous record by nearly million digits. This mathematical monster was discovered by Curtis Cooper at the University of Central Missouri in Warrensburg as part of the Great Internet Mersenne Prime Search (GIMPS), a collaborative effort to find new primes by pooling computing power online. It has,8,68 digits in total. The GIMPS software automatically crunches through numbers, testing whether they are prime that is, only divisible by themselves and one. Cooper s computer actually found the prime on 7 September 0, but a bug meant the software failed to send an alert reporting the discovery, meaning it went unnoticed until some routine maintenance a few months later. Cooper also discovered the previous record holder in February 0, which was 7,88,6, a number with more than 7 million digits, along with two other older records. He has received a $000 prize from GIMPS for each discovery. All of the GIMPS primes are Mersenne primes, which take the special form p, where p is also a prime number. Only 9 are known, and the GIMPS project has discovered the last. The prime numbers are infinite, and there is little practical use in discovering one, but the search is a good way to put computing hardware through its paces.

6 Activity -: Primes and Composites A prime number is a number with only factors. Composite numbers have more than factors. Using your 00 Board, answer the following questions.. What is the smallest prime number that is greater than 0?. What is the smallest prime number that is greater than 0?. and 7 are called twin primes because they are both primes and they differ by two. List all twin primes between and 00.. Find composite numbers in a row.. On Activity -, why didn t we have to keep going and cross out all multiples of 9? 6. Which of the primes,,, and 7 divide into 8? Cross out the boxes containing composite numbers to discover the hidden message. D 7 P 6 I R 8 V 9 I M P 60 S K 9 S O 9 Z Q A R M E S D M I V H I N B U R T T F O A I C T R O N E U M A S F G O R K E Q F R C I M E T K N D L N I R E R T 7 E 9 S 7 A S 7 D 7 R 8

7 Activity -: Monomial or not? The following are all examples of monomials: x x y z x 7z x y x y The following are all examples of things that are NOT monomials: x + y x + x y x z + WHAT IS A MONOMIAL? A monomial is a.

8 Activity -6: Prime Factorization and Monomials Consider the expression x. Since this notation means x, it follows that and x are factors of x. Determine whether each expression is a monomial... m. y +. xyz. c 6. 6( xy ) 7. q + r 8. 66h 9. fgh Why isn t zero composite? For a number to be composite it has to be able to be written as a product of two factors, neither of which is itself. Zero has an infinite number of factors; however they are all multiplied by zero to equal zero. Create a factor tree for each number. Write your answer using exponents. Show all work/answers on separate paper Create a factor tree for each number. Write your answer without using exponents. Show all work/answers on separate paper xy 0. x z. pq e f. mn. 6 jk m p 6. 0ab 7. ac 8. 0c d xy 0. 0s t. 860x y. 000x y. 0a b. What is the maximum number of prime numbered dates in any two consecutive months? On a prime date both the month and the date are prime numbers. For example, / is a prime date. How many prime dates occur this year? I am a two-digit prime number. The number formed by reversing my digits is also prime. My ones digit is less than my tens digit. What number am I? Use the prime factorizations of these numbers to find the seventh number in the pattern: 8, 8,, 0, 7,

9 Activity -7: Triple Venn Diagrams You can use a triple Venn diagram to find the GCF and LCM of set of monomials. Below is an example of how to complete a triple Venn diagram. Find the GCF and LCM of 6, 90, and 0. The prime factorization of 6 = The prime factorization of 90 = The prime factorization of 0 = GCF = 6 because all three numbers have both a and a in common. times equals 6. LCM = = 60 because you multiply all terms in the diagram. Notice an easy way to do this is to take the 0 circle and just multiply it by the only number,, sitting outside this circle.

10 Activity -8: Greatest Common Factor (GCF) Find the GCF of each set of numbers or monomials.. 8,. 8, 6.,0. 7,8,. 0, 8, ,68, 0 7. ab, 0ac 8. xy, , xy x z 0. 0 x, 0xy. 80 ac, 0a c. pt, 9 p t, pt 6. am, 8a m 0 x y, 6x y 60 zw, 0 w m, 0 m, m 7. Find the GCF of each expression. 7x + 9x 7 0y 8n 8 x x a 8 + a mn n x 7 6x + 0 6x + 0x x + xy + 0y 80x + 0x 8. A lady wants to buy plants for her garden and wants to plant them in rows with the same number of plants in each row. Which of these would give her the most choices? 8 flats of 6 plants OR 7 flats of 8 plants OR flats of 0 plants 9. What numbers between 0 and 00 have the greatest number of factors? 0. What calendar date(s) has the most factors? have exactly factors?..... Rebecca s little sister Tina has 8 yellow blocks and 0 green blocks. Tina builds some number of towers using all 88 blocks. What is the greatest number of identical towers that Tina can build? How many green and yellow blocks are in each tower? Mr. Mangham has some cookies. They can be divided evenly among 9 students. They can also be divided evenly among 6 students. What are two possibilities for the number of cookies? A band of pirates divided 8 pieces of silver and 8 gold coins. Since pirates are known to be fair about sharing equally, how many pirates were there? Identify the number that satisfies all three conditions: a) It is a composite between 6 and 7. b) The sum of the digits is a prime number. c) It has more than factors. The number 6 has exactly four whole number divisors:,,, and 6. What is the smallest number with exactly five whole number divisors?

11 Activity -9: Least Common Multiple (LCM) Multiple comes from multi meaning many and pli meaning fold. Fold a piece of paper in half, in half again, and again. The resulting number of pieces is eight times the number of original pieces. Remember: Every number has a multitude of multiples! List the first six multiples of each number or algebraic expression x. Find the LCM for each set of numbers or monomials..,, 6., 6, 8 7., 6, 0 8., 6 y 9., 0, 0 0. x,x. 6 x,8 x,0x. 8, 0, 0. 6, 8, 6., 6, 0. x, 0 xy,0xyz 6.,, , 80, ,, 6 9.,, 0., 6, 8. x, x. x, y., k k. h, 8. 6 p, 8 p, p 6. x, x, , 0, k k k 8. a b, 6b c 0. 8 g, 7h. 00 f g,0 f h. c, c, 7c 0 x y, y z 8 a bc, abc,6ab c w x, x y,6y z Advanced Venn Diagram Problem. Every red car at an auto show was a sports car. Half of all the blue cars were sports cars. Half of all sports cars were red. There were forty blue cars and thirty red cars. How many sports cars were neither red nor blue?

12 Activity -0: GCF and LCM How fast can you solve all the problems on this page? Can you solve them all mentally? GCF st monomial nd monomial LCM x x x x 00 x x x 0x xy 6xy x y 8x y z 7 6x y 6 x y x x x x y x 0x 8xy x 6x y 6xyz 9 7x y 6 60x y y... The GCF of two numbers is 80. Neither number is divisible by the other. What is the smallest that these two numbers could be? The GCF of two numbers is 79. One number is even and the other number is odd. Neither number is divisible by the other. What is the smallest that these two numbers could be? Ms. Wurst and Mr. Pop have donated a total of 90 hot dogs and 6 small cans of fruit juice for a math class picnic. Each student will receive the same amount of refreshments. What is the greatest number of students that can attend the picnic? How many cans of juice will each student receive? How many hot dogs will each student receive?

13 Activity -: Least Common Multiple Word Problems Solve.. At one store hot dogs come in packages of eight. Hot dog buns come in packages of twelve. What is the least number of packages of each type that you can buy and have no hot dogs or buns left over? Tongue Tickler Tooth Paste comes in two sizes: 9 oz. for $0.89 oz. for $.9 A. What is the LCM of 9 and? B. If you bought that much toothpaste in 9-oz. tubes, how much would it cost? C. If you bought that much toothpaste in -oz. tubes, how much would it cost? D. Which tube gives you more Tongue Tickler Toothpaste for the money? In the school kitchen during lunch, the timer for pizza buzzes every minutes; the timer for hamburger buns buzzes every 6 minutes. The two timers just buzzed together. In how many minutes will they buzz together again? Two ships sail steadily between New York and London. One ship takes days to make a round trip; the other takes days. If they are both in New York today, in how many days will they both be in New York again? The high school lunch menu repeats every 0 days; the elementary school menu repeats every days. Both schools are serving sloppy joes today. In how many days will they both serve sloppy joes again? Two neon signs are turned on at the same time. One blinks every seconds; the other blinks every 6 seconds. How many times per minute do they blink on together? Gear B has teeth. Gear C has 8 teeth. How many teeth should be on gear A if each turn of gear A is to produce a whole number of turns of the shafts attached to B and C? A company ships in two different sized boxes. One box is inches long and the other is 8 inches long. What is the shortest length crate the company can use to ship its product in either sized box without having extra space? A construction company uses -foot long concrete blocks for the width and -foot long blocks for the length of any rectangular building. What is the shortest length square building the company could construct? On Southlake Highway there are rest stops every 0 miles. Lauren s family stops at the rest stops every 0 miles. Kyle s family stops every 0 miles. The two families began their trips from the same place. What is the shortest distance the two families must drive before they stop at the same rest stop? Write your own word problem that involves finding the least common multiple. Provide the solution.

14 Activity -: Least Common Multiple Word Problems Solve.. Jim is planning a party for his class at school. He is going to buy hot dogs and hot dog buns to serve. The hot dogs come 0 to a package. The buns come 8 to a package. He doesn t want any hot dogs without buns and doesn t want any buns or hot dogs left over. How many packages of each does he need to buy?... Hannah is in charge of organizing school supplies to sell in the school store. She has pencils that come 0 to a package. She has pencil top erasers that come to a package. Each pencil needs a pencil top eraser. How many packages of each does she need so that every pencil has a pencil top eraser and every eraser has a pencil? Stephanie has a problem. Her English teacher gives a test every school days. Her mathematics teacher gives a test every school days and her social studies teacher gives a test every school days. She knows that she could have all tests on the same day. How often will she have three tests on one day? Lance is making cookies for the bake sale. He wants to use up the ingredients he is going to buy to make cookies. The recipe calls for cup of chocolate chips, cup of pecans, and cup of coconut. The chocolate chip package contains cups of chips. The pecan package contains cups of pecans. The coconut package contains 7 cups of coconut. How many packages of each will he need to buy to use up all of the ingredients? How many batches of cookies will that make?. How many dates per year are multiples of? 6. How many dates per year are multiples of? Danielle thinks of two numbers that are multiples of 9. The product of the two digits (neither a 9) of either of her numbers is also a multiple of 9. What are her numbers? The after-school program at DIS includes a craft session. Danny is planning on having children make ladybugs for their craft this week. Each ladybug needs Styrofoam hemisphere, two fuzzy stems for antenna, and six regular stems for legs. The Styrofoam hemispheres are in a package, the fuzzy stems are 6 in a package, and the regular stems are in a package. How many packages of each are needed to make ladybugs with no left over parts? How many ladybugs can be made with these packages?

15 Activity -: Juniper Green Juniper Green Round Rules of the game:. Two players play at a time. The first player selects an even number.. On each turn, a player selects any remaining number that is a factor or a multiple of the number just selected by his or her opponent.. The first player who cannot select a number loses Juniper Green Round Rules of the game: The rules this time are very similar to the first game, except you and your partner are now working together. Try to stay alive as long as possible by crossing out as many numbers as possible. The game is over when a player cannot select another number

16 Activity -: The Bubble Gum Factories The Mangham and Underwood Bubble Gum Factories (Taken from Lessons for Algebraic Thinking: Grades 6-8). Show all work on a separate sheet of paper to receive full credit. At the Mangham Bubble Gum Factory, lengths of gum are stretched to larger lengths by putting them through stretching machines. There are 00 stretching machines, numbered through 00. Machine does nothing to a piece of gum; machine stretches pieces of gum to twice their original length; machine triples the length of the gum, and so forth. So, machine, for example, will stretch a piece of gum to times its original length. Gum can go through as many machines as necessary and may travel through the same machine more than once. An order has just come in for a piece of bubble gum 6 inches in length. The factory has pieces of gum that are only inch in length, and machine number 6 is broken. Mr. Mangham, a factory worker, believes there is another way to create a piece of bubble gum 6 inches in length by using other machines. Show or explain how this could be accomplished It appears some of the machines in the factory are unnecessary because combinations of other machines could be used instead. Figure out which machines are actually unnecessary. List all unnecessary numbers. The Underwood Bubble Gum Factory is going to offer gum in the following lengths:, 8, 6, 6, 8. The Underwoods are about to order machines for their company and since they are expensive, they want to order the fewest possible. What is the fewest number of necessary machines needed to create all of these gum lengths? What is the number of each machine? Show all work. For each of the lengths at the Underwood Bubble Gum Factory, what other machines could have been used that were unnecessary? Show all work. Back at the Mangham Bubble Gum Factory, which lengths between and 00 would come out if the bubble gum went through five machines and all five machines were necessary ones? It can travel through the same number machine more than once. Show all work. You know the gum can go through five machines and stay less than 00. Can it go through six and stay less than 00? Seven and less than 00? Which lengths between and 00 require the greatest number of runs through necessary machines? How did you figure out your answer? It can travel through the same number machine more than once. Show all work. Suppose you now have lengths of gum up to 00. Are you going to need to buy more necessary machines? If so, do not list them all, but what do the machines have in common? Show all work. How many necessary machines are required for a 60-inch length? Which lengths between 00 and 00 inches require going through six necessary machines? Which lengths between 00 and 00 inches require going through seven necessary machines? Show all work. SHOW ALL YOUR WORK AND MAKE SURE I CAN UNDERSTAND WHERE ALL YOUR NUMBERS CAME FROM.

17 Bubble Gum FAQs * How do I know if a machine is necessary or unnecessary? Let s say you need a piece of 6 in. gum. Can you make it without machine #6? Yes, you can use machines # and #. So machine #6 is unnecessary. Let s say you need a piece of 7 in. gum. Can you make it without machine #7? No, the only way to get to 7 is with machine 7. So machine #7 is necessary. * Can I put gum through the same machine more than once? Yes, you can put it through as many times as you wish. * Can I put gum through different machines? Yes. For example, you can put it through machine # and then machine #. * What happens if I put gum through machine # three times? Machine # multiplies the length of the gum times. Therefore the first time you put it through it will become inches. The second time it will become inches and the third time 8 inches.

18 Activity -: Basic Fractions Find three fractions equivalent to each of the following Write each fraction in simplest form Write each improper fraction as a mixed number Write each whole or mixed number as an improper fraction Order each set of fractions from greatest to least. 9.,, 8...,,,, 8 8,,, ,, 8 6 7,, 7,, 8 6 7,,, 6 8 0

19 Activity -6: Basic Fractions/Addition & Subtraction Use your fraction sense to solve each problem.. Name a fraction between and.. Name a fraction between and.. Name a fraction between and whose denominator is 6.. Name a fraction between and whose denominator is 0.. Name a fraction that is halfway between and Name a fraction that is halfway between and. 7. How many fractions are there between one-fourth and one-half? Use the clues to discover the identity of the mystery fraction. 8. My numerator is 6 less than my denominator. I am equivalent to. 9. The GCF of my numerator and denominator is. I am equivalent to My numerator and denominator are prime numbers. My numerator is one less than my denominator. My numerator and denominator are prime numbers. The sum of my numerator and denominator is. My numerator is divisible by. My denominator is divisible by. My denominator is less than twice my numerator. My numerator is divisible by. My denominator is divisible by. My denominator is more than twice my numerator. My numerator is a prime number. The GCF of my numerator and denominator is. I am equivalent to one-fifth. Add or Subtract. Write each answer in simplest form.. + = = = = = = = 7 8 =

20 The Basics of Negative Fractions The number above represents negative two and one-half. If you owed someone $.0 you could write this number as dollars. On a number line, the mixed number would be placed here: It is located between the negative and the negative. Other ways to think of this number is as or + because the negative sign applies to both the and the. When converting a negative mixed number to a negative improper fraction, ignore the negative sign, convert like normal, and then place the negative sign back in at the end. Ex. Three times five equal fifteen plus one equals 6, so 6.

21 Activity -7: Improper Fractions/Mixed Numbers To convert from an improper fraction to a mixed number:. Divide the denominator into the numerator.. Place the remainder over the divisor. To convert from a mixed number to an improper fraction:. Multiply the whole number times the denominator.. Add the original numerator to your answer and this is the new numerator.. Place the numerator over the same denominator. Example: Example: r = = 9 + Write each improper fraction as a mixed number. For the first three problems you complete draw pictures that show how the improper fraction and mixed number are equivalent Write each whole or mixed number as an improper fraction Describe in writing and with a picture how 7 compares with. 7. Which is larger, or? Explain it! Show it! Prove it! 6. Find five fractions between one-fourth and one-half.

22 Activity -8: Fractions/Decimals Converting fractions to decimals Fraction to a decimal Divide! Example : = = 0.6 If the decimal keeps repeating use bar notation. Example : = = 0.6 Write each repeating decimal using bar notation Express each fraction or mixed number as a decimal. Use bar notation, if necessary Order each set of rational numbers from least to greatest..,.,.0,..,,., ,,. 6.,0., ,, 0., ,.,, 9 7,,, 0 8,, 0.8, , 0.7, 0.7, 7%, 7.%

23 Activity -9: Fraction Bar Notation Now that we have learned about bar notation with decimals here is a serious problem for you: How do you write That would be 0.9, right? Well, repeated forever equals what? Would you say that number is equal to or that it is less than??? Think about it. In ordinary math, this number equals one. Does your head hurt yet? So how can.9999 =? There are many different proofs of the fact that does indeed equal. So why does this question keep coming up? Do you agree that 0. is equal to? Remember doesn't mean "0.9" or "0.99" or "0.9999" or "0.999 followed by some large but finite (limited) number of 9's" never ends. There will always be another "9" to tack onto the end of So don't object to = on the basis of "however far you go out, you still won't be equal to ", because there is no "however far" to "go out" to; you can always go further. "But", some say, "there will always be a difference between and." Well, sort of. Yes, at any given stop, at any given stage of the expansion, for any given finite number of 9s, there will be a difference between and. That is, if you do the subtraction, will not equal zero. But the point of the "dot, dot, dot" is that there is no end; is inifinte. There is no "last" digit. So the "there's always a difference" argument betrays a lack of understanding of the infinite. We have learned that = 0... in decimal form. So + + = ( ) =. Reasonably then, = (0...) should also equal. But (0...) = Then must equal. If two numbers are different, then you can fit another number between them, such as their average. But what number could you possibly fit between and ?

24 Activity -0: Comparing and Ordering Fractions To compare and order fractions use the least common denominator (LCD). The LCD is the least common multiple (LCM) of the original denominators. Example Order from least to greatest:,, 8. Find the LCD of the denominators:, 8, and. The LCD is.. Write equivalent fractions. 6 8 = = = 8. Order the fractions. 6 8 < < Compare the following fractions using <, >, or = Order each set of fractions from greatest to least... 7.,, 8.,, 6. 8,,, ,, 8 6 7,,, ,, 8 6 Order each set of fractions from least to greatest ,,, 0. 0,,, 0 7.,,, ,,, 6 8

25 Activity -: Addition and Subtraction Example: 6 ( 8 ) First, change the problem to add the opposite. Then ask, Are there more positives or negatives? There more positives, so the answer will be positive. How many more positives? 8 6 = 6 So, the answer is + 6 Solve each equation. Write the solution in simplest form.. k = = n w = r = x = d = 0.. = w 6.. d = 8 6. = m 6 7 a = 8 8 b = 8 7 d = 6 8 q = + v = b = 8 p = 0 9 Evaluate each expression if 7 a =, b =, and c =. Write the solution in simplest form a + c 8. b a 9. c + b 0. c a + b. a + b + c. c a b Evaluate each expression if 7 x =, y =, and z =. Write the solution in simplest form x + y + z. x z. y ( z) 6. x + z 7. y x 8. z + ( y)

26 Activity: Multiplying Fraction Levels For each level, draw pictures to represent the problem. Look for the rule that works based on the answers in the pictures. Multiplication Level 0 Multiplication Level Multiplication Level 6 = 6 = 6 = 6 = 6 = = 6 = groups of groups of 6 Sample of how to read: Five groups of one-sixth Sample of how to read: Three groups of two-fifths Multiplication Level Multiplication Level Multiplication Multiplication Level Mixed Numbers Multiplication Level 6 Negatives Multiplication Level 7 Simplify before you multiply 8 8 = (or ) Sample of how to read: One-fifth of a group of ten 0 = (or 0 ) = = 0 = 6 = = 0 = Sample of how to read: One-half of a group of four-fifths You can not simply multiply the whole numbers and then the fractions! Show distributive property = drawing with pictures! Include negative values The commutative property allows simplifying before multiplying.

27 Activity -: Multiplying Fractions. Multiplication is the same as repeated addition when you add the same number again and again.. Times means groups of.. A multiplication problem can be shown as a rectangle.. You can reverse the order of the factors and the product stays the same.. When you multiply two whole numbers, the product is larger than the factors unless one of the factors is zero or one. 6. When you multiply two positive numbers, the product is larger than the factors unless one of the factors is zero or a fraction smaller than one.. Four times three means four groups of three. Three times four means three groups of four.. The problem three times four can be shown by a three by four rectangle.. The problem one-half times one-half can be shown by a two-by-two rectangle representing one whole. Draw a picture for. Estimate the answer. Use the distributive property.. Estimate the answer. = 0. Why not? 8 Example #: You can model of. Show. Example #: Model of. Show. Divide into thirds. Shade of the. of = = 6 Find of each. Combine the halves. of = = =

28 Activity -: Multiplying Fractions What multiplication expression is represented by each model?... Draw a model to represent each product.. of 6. of Find each product. You do not have to simplify improper fractions into mixed numbers of 9 of 7 7 of 8 0 of of e = of. 7 d = n = of of of 8 9 of = t 6. = k = t. 9 = c 9 h = 7 f = Write true or false.. <.. > > < 7 7

29 Activity -: Multiplying Mixed Numbers To multiply mixed numbers, first convert each to an improper fraction. Then multiply the fractions. For all problems on this page, show all work on a separate sheet of paper. Leave your answer as an improper fraction for all problems below. Find each product Solve each equation = s. 9 d =. a = 7. j = p = = t. 9 g = b = = x 7 = a 9 6 s = 7 c = 6 7. The weight of an object on Venus is about 9 0 Mars is about of its weight on Earth. The weight of an object on of its weight on Earth. What is the difference in the weight of a 70 pound person on Venus compared to Mars? A bulletin board measures feet wide and 8 feet high. Eight posters measuring feet wide 6. and feet high are placed on the bulletin board as well as a border covering a total area of square feet. What fraction of the bulletin board is still available for other items? 7. What fraction is located of the distance between and.

30 Activity -: Fraction Word Problems Semi-Sweet Chocolate Flour Cocoa Powder CHOCOLATE BANANA RAISIN COOKIES (Makes cookies) 8 of a pound Butter of a cup cups Sugar of a cup 8 cup Light-Brown Sugar of a cup Baking Powder teaspoons Chocolate Extract teaspoon Salt Mashed Bananas of a teaspoon Eggs large medium size Pecans cups White Chocolate 8 of a pound Raisins cup You gather the ingredients above. Assume each problem uses the ingredients as they start above (do not use the remaining amount for the next problem.) Determine whether the following problems require subtraction, multiplication, or both. Then solve. Show all work on a separate sheet of paper. Correct Answer Operation(s) Nicole ate of a pound of semi-sweet chocolate for breakfast. How much semi-sweet chocolate is left? Carter ate of the semi-sweet chocolate with his broccoli soup. How much semi-sweet chocolate did he eat? Brandon ate of the semi-sweet chocolate while standing on his head. How much semi-sweet chocolate was left? Grant accidentally threw of the flour in the trash. How much flour did he throw in the trash? Brooke gave of a cup of flour to her friend Eric. How much flour was left? Jared put of the flour on Connor s face. How much flour is not on Connor s face?

31 Bailey made 60 cookies, times the normal recipe, so that her sisters could have some also. How many mashed bananas did she use? Emily does not like pecans so she uses only of the 7 normal amount. How many cups of pecans does she use? One-third of a pound of white chocolate magically disappears, but some people say Crissy took it. How much white chocolate was left? Three-fourths of the butter somehow turns up sitting on Andrew s head. How much butter is left for the recipe? Chris has determined that eating small amount of cocoa powder can help you calculate pi more accurately. If Chris ate of a cup of the powder, how much was left to 6 use in the recipe? Chad loves pi also. He accidentally takes of the lightbrown sugar instead. How much light-brown sugar is left? 9 Justin took of a teaspoon of salt and threw it over his shoulder, accidentally hitting Valerie in the face. How much of the salt did not hit Valerie? Haley loves raisins. She took 8 of the raisins to create a 9 picture of Raisin Guy. How much of a cup of raisins did she use? Michael loves to mix things up so he put the pecans, butter, and both sugars together. He then threw of this mixture directly at Danielle s face. How much of the mixture was still left for Michael to throw at Christen? Brittany, like, loves to put extra butter on everything. So, like, she uses times the normal amount of butter in the recipe. How much, like, butter did she use? Heather lost of the mashed bananas while she was helping Crissy and Haley learn their dance. How many mashed bananas did she lose? Leslie found ants in 8 of a cup of flour. How many cups 9 of flour did not contain ants? Addition PLUS

32 Activity -6: Multiplying Fractions Visually & Mentally In order to demonstrate your understanding of the concept of multiplication of fractions, show how to find each of the following products visually. Make a diagram or sketch on graph paper... of of Write two interesting word problems that require finding the product of fractions. Solve each of your word problems using visual models. 9. I am a fraction in simplest form. One-sixth of me is the same as one-half of one-fourth. What fraction am I? Solve the following problems mentally Show how to use a sketch to solve the following problems After Kevin spent half of his week s pay on food, then spent one-third of what was left on rent, and then spent one-half of what was left on fun, he had $0 left of his paycheck. How much money was his week s check? Mr. Mangham has a class of students. Three-fourths of the students are weird and, of those, seven-eighths are clueless. What fraction of all the students are weird and clueless and how many students are neither weird nor clueless? (This assumes that if you are not weird, you are not clueless.) In an Algebra class, half of the students are boys. One-third of the students are wearing glasses. Half of the boys are wearing glasses. What fraction of the girls is wearing glasses? Evaluate each expression if a =, b =, c =, and d =. 6. 8b 6c. a + cd. 7 9d + 8. a( c + ). b( a + 8) 6. d( b + 6)

33 Activity -7: Muppets In Drew s class, one-fifth of the boys are absent and two-fifths of the girls are absent. Bert believes that you multiply, thus two twenty-fifths of the class is absent. Elmo believes that means that a total of three-fifths of the class is absent. Ernie believes three-tenths of the class is absent. Kermit can t figure out how much of the class is absent, but he believes it is somewhere between one-fifth and two-fifths. Analyze each character s answer to determine if they are correct or not. From this make a conclusion on which character knows the most math. A lot of students believe that you just add the two numbers together and get three-fifths. Look at the problem carefully. Do you add those two numbers together? One-fifth relates to the boys and two-fifths relates to the girls. The key missing part to this question is that it does not tell you how many boys are in the class or how many girls are in the class. Might those numbers make a difference in what the true answer is? To test this theory, one good method is to guess how many boys and girls might be in the class and then determine your answer. Based on these answers you can then see which character knows the most math. Total students Boys Girls Boys absent Girls absent Fraction of class absent Based on your results above, which character does know the most math? Why?

34 Activity: Dividing Fractions with Pictures Division Level 0 (draw pictures) Division Level (draw pictures) = = How many groups of are there in? How many groups of are there in? = NOTE: When divide by positive fraction, the answer gets bigger. Sample of how to read: = 8 How many groups of one-half are in two? Division Level (draw pictures) = = 6 = 6 Sample of how to read: How many groups of one-eighths are in foureighths? Division Level (draw pictures) Division Level (draw pictures) Division Level (draw pictures) = 6 6 Sample of how to read: 6 = How many groups of three-tenths are in nine-tenths? 8 8 = = = = 6 = = Level 6: Introduce reciprocal rule with mixed numbers. Level 7: Introduce reciprocal rule with negative numbers. 6 rules for dividing fractions:. Divide straight across.. Find a common denominator. Use the reciprocal to make the denominator one Sample of how to read: How many groups of one-forth are in nine-twelfths? Sample of how to read: How many groups of two-thirds are in three?

35 Activity -8: Dividing Fractions (Taken from TEXTEAMS Rethinking Middle School Mathematics: Numerical Reasoning). You can solve a division problem by subtracting.. To divide two numbers, a b, you can think, How many b s are in a?. You can check a division problem by multiplying.. The division sign means into groups of.. The quotient tells how many groups of there are. 6. You can break the dividend apart to make dividing easier. 7. Remainders can be represented as whole numbers or fractions. 8. If you divide a number by itself, the answer is one. 9. If you divide a number by one, the answer is the number itself. 0. If you reverse the order of the dividend and the divisor, the quotient will be different unless the dividend and divisor are the same number. Mr. Mangham plans to make small cheese pizzas to sell during the weekends to make some extra money. He has 9 bars of cheddar cheese. How many pizzas can he make if each takes the amount listed in the table? On a piece of notebook paper or computer paper create the five columns shown below. Answer the following questions by completing the row for each question.... Situation bar of cheese bar of cheese bar of cheese Think about it (what is the question asking) How many s are in 9? Picture It Write it in symbols Process it (What do you actually do?) 9 9 Solve it Mr. Mangham still needs more money so he is going to sell small bags of coffee. He buys a large twelve-pound bag. How many small bags can he make based on the following situations? Situation Think about it (what is the question asking) Picture It Write it in symbols Process it (What do you actually do?) Solve it. pound. 6 pound 6. pound

36 Activity -9: Dividing Fractions (Taken from TEXTEAMS Rethinking Middle School Mathematics: Numerical Reasoning) Mr. Mangham is making ribbons for all of his students to wear in honor of his former student, Marci Holden. It takes 6 of a yard to make a ribbon for each student. How many badge ribbons can he make from the lengths listed below? For each answer that has a remainder, some ribbon left over, tell what fractional part of another badge ribbon you could make with the amount left over. On a piece of notebook paper or computer paper create the six columns shown below. You will answer the following questions by completing the row for each question. The first one has been done for you..... Situation yard yard 8 yard yard Think about it (what is the question asking) yd. divided into pieces of 6 yd. 6 Picture It Write it in symbols 6 Process it (What do you actually do?) Divide yard in half, whole into sixths. How many sixths in a half? Solve it Next, Mr. Mangham is making bows for the math students that are not in his classes so that he can easily recognize them. It takes yard of ribbon to make one bow. How many bows can Mr. Mangham make from each of the following amounts of ribbon? Situation yard 8 9 yard yard Think about it (what is the question asking) Picture It Write it in symbols Process it (What do you actually do?) Solve it 8. yard

37 Activity -0: Painting Avery s father donated 6 gallons of paint to her school. The teachers at Avery s school have decided to paint tabletops with the paint. It will take of a gallon of paint to cover a small-sized table. It will take of a gallon of paint to cover a medium-sized table, and of a gallon of paint for a large-sized table. Solve. Show all work on a separate sheet of paper.. How many small-sized tables can the teachers paint if they use all of the paint? Show a model and write an expression that relates to your model.... How many medium-sized tables can the teachers paint if they use all of the paint? Show a model and write an expression that relates to your model. How many large-sized tables can the teachers paint if they use all of the paint? Show a model and write an expression that relates to your model. Based on your observations, write a rule that makes sense for dividing any whole number by a unit fraction. Avery found some green tables and blue tables that needed painting in the cafeteria. Her father agreed to donate 6 more gallons of paint. It will take of a gallon to repaint the green tables and of a gallon to repaint the blue tables. Solve. Show all work on a separate sheet of paper.. If the teachers use all the paint, how many green tables can they paint? Use a model and write an expression that relates to the model If the teachers use all the paint, how many blue tables can they paint? Use a model and write an expression that relates to the model. In question # you described a method for dividing fractions. Does your method work for questions # and #6? If needed, revise your rule so that it does work. 8. Use your rule to solve the following problems. What do you notice about your answers? 9. Based on your answers above can you predict the answer to your prediction correct? 8? Solve the problem. Was

38 Activity -: Dividing Fractions Any number times its reciprocal equals. The reciprocal of is. So we need to multiply both the numerator and denominator by. To divide fractions, multiply the first fraction by the reciprocal of the second number. Reciprocal comes from re meaning backward and pro meaning forward. In writing the reciprocal you have gone back and forth and returned to the identity for multiplication. Find the reciprocal of each number Find each quotient. Show all work on a separate sheet of paper. Simplify, but improper fractions are okay Solve each equation. Show all work on a separate sheet of paper j = 9. 7 = b = s 6. = p. s =. 6 a = 6

39 Activity -: Dividing Mixed Numbers Name the reciprocal (also called the multiplicative inverse) for each rational number You may leave your answer as a simplified improper fraction for all problems below. Solve each equation. Show all work on a separate sheet of paper = k 6. 7 c = 0 0 = k = x. 9 7 = y 0. 6 = b 9 7 = d g = 8 = r = p Reg Morris holds the world s crawling record (true story). He crawled 8 miles on a measured. course miles long. To set the record, he crawled without stopping for 9 hours. How many laps did Morris complete? What was his crawling speed in mi/h? 6. A turtle walked mile at the rate of mile per hour. How long did it take? A certain math book is feet wide? of an inch thick. How many of these books will fit on a shelf that is How many complete games of chess can Yuri play in hour if playing 8 games takes him hours? 0 Evaluate each expression. Show all work on a separate sheet of paper y z, if y = and z = 0. c d, if c = and d = 8. f g, if f = and g = 9 w z, if w = 9 and z = 7

40 Activity -: Dividing Mixed Numbers Write an expression for each word problem and solve on a separate sheet of paper.. Draw a diagram to show how many -ft. pieces of string can be cut from a piece of string four and a half feet long.. How many -c servings are there in a 6 c package of rice?. George cut oranges into quarters. How many pieces of orange did he have?. Anna bought a package of ribbon 0 yd long. She needs many pieces can Anna cut from the ribbon?. Using #, what if Anna decided to use -yd 6. A bulletin board is 6 in. wide and 6 in. high. How many yd pieces for a bulletin board. How pieces? How many pieces can she now cut? -in There are boys and girls in the Krunch family. Mr. Krunch bought columns can be created? pounds of candy to divide equally among them. How much candy did each child get? It takes cup of liquid fertilizer to make 7 gallons of spray. How much liquid fertilizer is needed to make 80 gallons of spray? Darlene has hours to complete three household chores. If she divides her time evenly, how many hours can she give to each? 0. Elizabeth bought pounds of tomatoes for dollars. How much did she pay per pound?. Dad paid $.00 for a pound box of candy. How much is that per pound?.... The runner ran miles in hour. At that rate, how many miles could he run in hour? That is, what is his speed in miles per hour? Farmer Brown measured his remaining insecticide and found that he had two and a quarter gallons. It takes three-fourths gallon to make a tank of mix. How many tankfuls can he make? Linda has requires yards of material. She is making baby clothes for her niece. If each pattern 6 yards of material, how many patterns will she be able to make? On Sunday, Mr. Underwood swam Underwood s swimming pace in miles per hour? miles in. hours. As a mixed number, what was Mr.

41 Activity -: Rectangles Determine the missing dimension of each of the following rectangles. All area units are in square inches... A = A = 8. A = A =. 7. A = A = 0 8 A = A =. Length =. Width =. Length = 7. Width = 6 inches Width =. Length = 9 inches Length =. Width = 0 inches Width = 6. Length = 8 inches Length = 8. Length = inches Width = 8 inches Length = inches Width = 6 inches Width =

42 Activity -: A Camping Trip You and your friends are planning a camping trip. You are in charge of making and bringing enough trail mix. The trail mix needs to provide each person with one serving for each day of the trip. Here is the recipe for trail mix: cups raisins cup peanuts cup dried pineapple pieces cup walnuts cup cashew nuts 8 cups coconut pieces 8 cup chocolate chips Combine all ingredients and mix them.. How many cups in all does the recipe make?. If one serving is cup, how many servings does the recipe make?. Suppose people go camping for days. How many cups in all should you make? Raisins Cashews. For #, how much of each ingredient should you use to make enough trail mix for each person to have a -cup serving on each day? Peanuts Pineapple Coconuts Chocolate Walnuts. Suppose people are going on the camping trip for days and each serving of trail mix is cup. How many cups in all should you make? Raisins Cashews 6. For #, how much of each ingredient should you use? Peanuts Pineapple Coconuts Chocolate Walnuts

43 Activity -6: Fraction Word Problems Write an expression for each word problem and solve on a separate sheet of paper. Liz Ann and Bryan are running a race. Bryan only ran miles before he dropped out. Liz Ann ran times farther than Bryan. How far did Liz Ann run and how many more miles did Liz Ann run than Bryan? George Bush was eating his daily jelly beans. He decided he would share his jar of pounds of jelly beans amongst himself and seven other world leaders. How many pounds did each leader get? Trent bought 6 pounds of tomatoes. An average tomato weighs of a pound. Approximately 8 how many tomatoes did Trent buy? The Honey Baked Ham Store is busy during the holiday season. The average ham they sell weighs 9 pounds. If they say each ham will make servings, how much does one serving weigh? Bob is very interested in knowing exactly how fast he drives. On his last trip, he figured he drove 6 miles in hours. How fast was he driving? 7 Bill owns a bakery that makes really good sugar cookies. It takes complete batches and Garrett can make make? In a recipe you need cups of sugar to bake of another batch. How many cups of sugar is used for each batch? pies in an hour. If he works for 7 hours, how many pies can Garrett cups of milk and you want to use all cups of milk in the refrigerator. How many batches of the recipe can you make? Larry, who lives in London, won the equivalent of is equal to Becky has eats million dollars in the lottery. If one dollar pounds (money in England), how many pounds did he win? pizzas left over from a party. The next day each person who visits Becky s house of a pizza. How many people visit before the pizza is all gone? Lizzy Lou had of a pumpkin pie. She decided to split it between herself, Emily Sue, Billy Bob, Joe Jill, and Danny Dude. How much of a whole pie will each person get?