# 0.1 Dividing Fractions

Size: px
Start display at page:

Transcription

1 0.. DIVIDING FRACTIONS Excerpt from: Mathematics for Elementary Teachers, First Edition, by Sybilla Beckmann. Copyright c 00, by Addison-Wesley 0. Dividing Fractions In this section, we will discuss the two interpretations of division for fractions, and we will see why the standard invert and multiply procedure for dividing fractions gives answers to fraction division problems that agree with what we expect from the meaning of division. The Two Interpretations of Division for Fractions Let s review the meaning of division for whole numbers, and see how to interpret division for fractions. The how many groups? interpretation With the how many groups? interpretation of division, means the number of groups we can make when we divide objects into groups with objects in each group. In other words, tells us how many groups of we can make from. Similarly, with the how many groups? interpretation of division, 5 means the number of groups we can make when we divide 5 of an object into groups with of an object in each group. In other words, 5 tells us how many groups of we can make from 5. For example, suppose you are making popcorn balls and each popcorn ball requires of a cup of popcorn. If you have = 5 of a cup of popcorn, then how many popcorn balls can you make? In this case you want to divide 5 of a cup of popcorn into groups (balls) with of a cup of popcorn in each group. According to the how many groups? interpretation of division, you can make popcorn balls. 5

2 The how many in one (each) group? interpretation With the how many in each group? interpretation of division, means the number of objects in each group when we distribute objects equally among groups. In other words, is the number of objects in one group if we use objects to evenly fill groups. When we work with fractions, it often helps to think of how many in each group? division story problems as asking how many are in one whole group?, and it helps to think of filling groups or part of a group. So in the context of fractions, we will usually refer to the how many in each group? interpretation as how many in one group?. With the how many in one group? interpretation of division, is the number of objects in one group when we distribute of an object equally among of a group. A clearer way to say this is: is the number of objects (or fraction of an object) in one whole group when of an object fills of a group. For example, suppose you pour of a pint of blueberries into a container and this fills of the container. How many pints of blueberries will it take to fill the whole container? In this case, of a pint of blueberries fills (i.e., is distributed equally among) of a group (a container). So according to the how many in one group? interpretation of division, the number of pints of blueberries in one whole group (one full container) is One way to better understand fraction division story problems is to think about replacing the fractions in the problem with whole numbers. For example, if you have pints of blueberries and they fill containers, then how many pints of blueberries are in each container? We solve this problem by dividing, according to the how many in each group? interpretation. Therefore if we replace the pints with of a pint, and the containers with of a container, we solve the problem in the same way as before: now becomes. Here is another way to think about the problem. Because of the container is filled, and because this amount is of a pint, therefore of the

3 0.. DIVIDING FRACTIONS number of pints in a full container is Therefore of a pint. In other words: number of pints in full container = number of pints in full container = Dividing by Versus Dividing in In mathematics, language is used much more precisely and carefully than in everyday conversation. This is one source of difficulty in learning mathematics. For example, consider the two phrases: dividing by, dividing in. You may feel that these two phrases mean the same thing, however, mathematically, they do not. To divide a number, say 5, by means to calculate 5. Remember that we read A B as A divided by B. We would divide 5 by if we wanted to know how many half cups of flour are in 5 cups of flour, for example. (Notice that there are 0 half-cups of flour in 5 cups of flour, not.) On the other hand, to divide a number in half means to find half of that number. So to divide 5 in half means to find of 5. One half of 5 means 5. So dividing in is the same as dividing by. The Invert and Multiply Procedure for Fraction Division Although division with fractions can be difficult to interpret, the procedure for dividing fractions is quite easy. To divide fractions, such as and 6 5 we can use the familiar invert and multiply method in which we invert the divisor and multiply by it: = = = 9 8

4 and 6 5 = 6 5 = 6 5 = 6 5 = 0 = 5 reciprocal Another way to describe this invert and multiply method for dividing fractions is in terms of the reciprocal of the divisor. The reciprocal of a fraction C is the fraction D. In order to divide fractions, we should multiply D C by the reciprocal of the divisor. So in general, A B C D = A B D C = A D B C Explaining Why Invert and Multiply is Valid by Relating Division to Multiplication The procedure for dividing fractions is easy enough to carry out, but why is it a valid method? Before we answer this question in general, consider a special case. Recall that every whole number is equal to a fraction (for example, 6 = 6 ). Therefore we can apply the invert and multiply procedure to whole numbers as well as to fractions. According to this procedure, = = = = Notice that this result, that =, agrees with our findings earlier in this chapter: that we can describe fractions in terms of division, namely that A = A B. B In general, why is the invert and multiply procedure a valid way to divide fractions? One way to explain this is to relate fraction division to fraction multiplication. Recall that every division problem is equivalent to a multiplication problem (actually two multiplication problems): A B =? is equivalent to (or B? = A). So? B = A =?

5 0.. DIVIDING FRACTIONS 5 is equivalent to? =. () Now remember that we want to explain why the invert and multiply rule for fraction division is valid. This rule says that ought to be equal to Let s check that this fraction works in the place of the? in Equation. In other words, let s check that if we multiply times, then we really do get : = ( ) ( ) = = ( ) ( ) = Therefore the answer we get from the invert and multiply procedure really is the answer to the original division problem. Notice that the line of reasoning above applies in the same way when other fractions replace the fractions and used above. It will still be valuable to explore fraction division further, interpreting fraction division directly rather than through multiplication. Class Activity 0A: Explaining Invert and Multiply by Relating Division to Multiplication Using the How Many Groups? Interpretation to Explain Why Invert And Multiply Is Valid Above, we explained why the invert and multiply procedure for dividing fractions is valid by considering fraction division in terms of fraction multiplication. Now we will explain why the invert and multiply procedure is valid by working with the how many groups? interpretation of division. Consider the division problem The following is a story problem for this division problem: How many cups of water are in of a cup of water?

6 6 Or, said another way: How many times will we need to pour cup of water into a container that holds cup of water in order to fill the container? From the diagram in Figure we can say right away that the answer to this problem is one and a little more because one half cup clearly fits in two thirds of a cup, but then a little more is still needed to fill the two thirds of a cup. But what is this little more? Remember the original question: we want to know how many cups of water are in of a cup of water. So the answer should be of the form so and so many cups of water. This means that we need to express this little more as a fraction of cup of water. How can we do that? By subdividing both the and the into common parts, namely by using common denominators. / cup / cup / cup = /6 cup Figure : How Many / Cups of Water Are in / Cup? / cup = /6 cup When we give and the common denominator of 6, then, as on the right of Figure, the cup of water is made out of parts ( sixths of a cup of water), and the cup of water is made out of parts ( sixths of a cup of water), so the little more we were discussing above is just one of those parts. Since cup is parts, and the little more is part, the little more is of the cup of water. This explains why = : there s a whole cup plus another of the cup in of a cup of water. To recap: we are considering the fraction division problem in terms of the story problem how many cups of water are in of a cup of water? If we give and the common denominator of 6, then we can rephrase the problem as how many of a cup are in of a cup? But in terms of 6 6 Figure??, this is equivalent to the problem how many s are in? which is the problem, whose answer is =. Notice that is exactly the same answer we get from the invert and multiply procedure for fraction division: = = =

7 0.. DIVIDING FRACTIONS 7 So the invert and multiply procedure gives the same answer to that we arrive at by using the how many groups? interpretation of division. The same line of reasoning will work for any fraction division problem A B C D Thinking logically, as above, and interpreting A C as how many C cups B D D of water are in A cups of water?, we can conclude that B A B C D = A D B D B C B D = (A D) (B C) = A D B C The final expression, A D, is the answer provided by the invert and multiply B C procedure for dividing fractions. Therefore we know that the invert and multiply procedure gives answers to division problems that agree with what we expect from the meaning of division. Class Activity 0B: How Many Groups? Fraction Division Problems Using the How Many in One Group? Interpretation to Explain Why Invert And Multiply Is Valid Above, we saw how to use the how many groups? interpretation of division to explain why the invert and multiply procedure for fraction division is valid. We can also use the how many in one group? interpretation for the same purpose. This interpretation, although perhaps more difficult to understand, has the advantage of showing us directly why we can multiply by the reciprocal of the divisor in order to divide fractions. Consider the following how many in one group? story problem for 5 : You used of can of paint to paint of a wall. How many cans 5 of paint will it take to paint the whole wall? This is a how many in one group? problem because we can think of the paint as filling of the wall. We can also see that this is a division problem 5 by writing the corresponding number sentence: 5 (amount to paint the whole wall) =

8 8 Therefore amount to paint the whole wall = 5 We will now see why it makes sense to solve this problem by multiplying by the reciprocal of, namely by 5. Let s focus on the wall to be painted, 5 as shown in Figure. Think of dividing the wall into 5 equal sections, of the / can of paint is divided equally among parts the amount of paint for the full wall is 5 times the amount in one part Figure : The Amount of Paint Needed for the Whole Wall is 5 of the Can Used to Cover of the Wall 5 which you painted with the can of paint. If you used a can of paint to paint sections, then each of the sections required or cans of paint. To determine how much paint you will need for the whole wall, multiply the amount you need for one section by 5. So you can determine the amount of paint you need for the whole wall by multiplying the can of paint by and then multiplying that result by 5, as summarized in Table. But to multiply a number by and then multiply it by 5 is the same as multiplying the number by 5. Therefore we can determine the number of cans of paint you need for the whole wall by multiplying by 5: 5 = 5 6 This is exactly the invert and multiply procedure for dividing. It 5 shows that you will need 5 of a can of paint for the whole wall. 6

9 0.. DIVIDING FRACTIONS 9 use can paint for 5 of the wall or or use 6 can paint for 5 of the wall use can paint for whole wall use in one step: can paint for of the wall 5 use can paint for whole wall Table : Determining How Much Paint to Use for a Whole Wall if Paint Covers of the Wall 5 Can of

10 0 The argument above works when other fractions replace and, thereby 5 explaining why A B C D = A B D C In other words, to divide fractions, multiply the dividend by the reciprocal of the divisor. Class Activity 0C: How Many in One Group? Fraction Division Problems Class Activity 0D: Are These Division Problems? Exercises for Section 0. on Dividing Fractions. Write a how many groups? story problem for 5. Use the story 7 problem and a diagram to help you solve the problem.. Write a how many in one group? story problem for. Use the situation of the story problem to help you explain why the answer is =.. Annie wants to solve the division problem story problem: by using the following I need cup of chocolate chips to make a batch of cookies. How many batches of cookies can I make with of a cup of chocolate chips? Annie draws a diagram like the one in Figure. Explain why it would be easy for Annie to misinterpret her diagram as showing that =. How should Annie interpret her diagram so as to conclude that =?. Which of the following are solved by the division problem? For those that are, which interpretation of division is used? For those that are not, determine how to solve the problem, if it can be solved. (a) of a bag of jelly worms make worms are in one bag? a cup. How many cups of jelly

11 0.. DIVIDING FRACTIONS / cup makes one batch / cup left Figure : How Batches of Cookies Can We Make With of a Cup of Chocolate Chips if Batch Requires Cup of Chocolate Chips? (b) of a bag of jelly worms make a cup. How many bags of jelly worms does it take to make one cup? (c) You have of a bag of jelly worms and a recipe that calls for of a cup of jelly worms. How many batches of your recipe can you make? (d) You have of a cup of jelly worms and a recipe that calls for of a cup of jelly worms. How many batches of your recipe can you make? (e) If of a pound of candy costs of a dollar, then how many pounds of candy should you be able to buy for dollar? (f) If you have of a pound of candy and you divide the candy in, then how much candy will you have in each portion? (g) If of a pound of candy costs \$, then how many dollars should you expect to pay for of a pound of candy? 5. Frank, John, and David earned \$ together. They want to divide it equally, except that David should only get a half share, since he did half as much work as either Frank or John did (and Frank and John worked equal amounts). Write a division problem to find out how much Frank should get. Which interpretation of division does this story problem use? 6. Bill leaves a tip of \$.50 for a meal. If the tip is 5% of the cost of the meal, then how much did the meal cost? Write a division problem to solve this. Which interpretation of division does this story problem use?

12 7. Compare the arithmetic needed to solve the following problems. (a) What fraction of a cup measure is filled when we pour in cup of water? (b) What is one quarter of cup? (c) How much more is cup than cup? (d) If cup of water fills of a plastic container, then how much water will the full container hold? 8. Use the meanings of multiplication and division to solve the following problems. (a) Suppose you drive 500 miles every half year in your car. At the end of years, how many miles will you have driven? (b) Mo used 8 ounces of liquid laundry detergent in 6 weeks. If Mo continues to use laundry detergent at this rate, how much will he use in a year? (c) Suppose you have a ounce bottle of weed killer concentrate. The directions say to mix two and a half ounces of weed killer concentrate with enough water to make a gallon. How many gallons of weed killer will you be able to make from this bottle? 9. The line segment below is of a unit long. Show a line segment that is 5 of a unit long. Explain how this problem is related to fraction division. unit Answers To Exercises For Section 0. on Dividing Fractions. A simple how many groups? story problem for 5 is how many of a cup of water are in cup of water? Figure shows cup of water and shows 5 of a cup of water shaded. The shaded portion is divided 7 into 5 equal parts and the full cup is 7 of those parts. So the full cup is 7 of the shaded part. Thus there are 7 of 5 of a cup of water in cup of water, so 5 =

13 0.. DIVIDING FRACTIONS cup each piece is 5 of the shaded of a cup 5 7 portion Figure : Showing Why 5 = 7 by Considering How Many Water are in Cup of Water of a Cup of. A how many in one group? story problem for is if ton of dirt fills a truck full, then how many tons of dirt will be needed to fill the truck completely full?. We can see that this is a how many in one group? type of problem because the ton of dirt fills of a group (the truck) and we want to know the amount of dirt in whole group. Figure 5 shows a truck bed divided into equal parts with of those parts filled with dirt. Since the parts are filled with ton of dirt, each of the parts must contain of a ton of dirt. To fill the truck completely, parts, each containing of a ton of dirt are needed. Therefore the truck takes = tons of dirt to fill it completely, and so =. the ton of dirt is divided equally among parts truck bed parts are needed to fill the truck; each part is / of a ton, so / tons of dirt are needed to fill the truck Figure 5: Showing Why = by Considering How Many Tons of Dirt it Takes to Fill a Truck if Ton Fills it Full. Annie s diagram shows that she can make full batch of cookies from

14 her of a cup of chocolate chips and that cup of chocolate chips will be left over. Because cups of chocolate chips are left over, it would be easy for Annie to misinterpret her picture as showing =. But the answer to the problem is supposed to be the number of batches Annie can make. In terms of batches, the remaining cup of chocolate chips makes of a batch of cookies. We can see this because quartercup sections make a full batch, so each quarter-cup section makes of a batch of cookies. So by interpreting the remaining cup of chocolate chips in terms of batches, we see that Annie can make batches of chocolate chips, thereby showing that =, not.. (a) This problem can be rephrased as if of a cup of jelly worms fill of a bag, then how many cups fill a whole bag?, therefore this is a how many in one group? division problem illustrating, not. Since = =, there are of a cup of jelly worms in a whole bag. (b) This problem is solved by, according to the how many in each group? interpretation. A group is a cup and each object is a bag of jelly worms. (c) This problem can t be solved because you don t know how many cups of jelly worms are in of a bag. (d) This problem is solved by, according to the how many groups? interpretation. Each group consists of of a cup of jelly worms. (e) This problem is solved by, according to the how many in one group? interpretation. This is because you can think of the problem as saying that of a pound of candy fills of a group and you want to know how many pounds fills whole group. (f) This problem is solved by, not. It is dividing in half, not dividing by. (g) This problem is solved by, according to the how many groups? interpretation because you want to know how many pounds are in of a pound. Each group consists of of a pound of candy. 5. If we consider Frank and John as each representing one group, and David as representing half of a group, then the \$ should be dis-

15 0.. DIVIDING FRACTIONS 5 tributed equally among groups. Therefore, this is a how many in one group division problem. Each group should get = 5 = 5 = 8 5 = 5 5 = = 5.60 dollars. Therefore Frank and John should each get \$5.60 and David should get half of that, which is \$ According to the how many in one group? interpretation, the problem is solved by \$ because \$.50 fills 0.5 of a group and we want to know how much is in whole group. So the meal cost \$ = \$ = \$ = \$50 5 = \$0 7. Each problem, except for the first and last, requires different arithmetic to solve it. (a) This is asking: equals what times? We solve this by calculating, which is. We can also think of this as a division problem with the how many groups? interpretation because we want to know how many of a cup are in of a cup. According to the meaning of division, this is. (b) This is asking: what is of? We solve this by calculating =. (c) This is asking: what is? The answer is which happens to be the same answer as in part (b), but the arithmetic to solve it is different. (d) Since cup of water fills of a plastic container, the full container will hold times as much water, or = of a cup. We can also think of this as a division problem with the how many in one group? interpretation. cup of water is put into of a group. We want to know how much is in one group. According to the meaning of division it s, which again is equal to. 8. (a) The number of years in years is. There will be that many groups of 500 miles driven. So after years you will have

16 6 driven miles. ( ) 500 = (5 ) 500 = =, 750 (b) Since one year is 5 weeks there are 5 6 groups of 6 weeks in a year. Mo will use 8 ounces for each of those groups, so Mo will use (5 6 ) 8 = (5 ) 8 = 0 8 =, 0 ounces of detergent in a year. (c) There are groups of ounces in ounces. Each of those groups makes gallon. So the bottle makes = gallons 5 of weed killer. 9. One way to solve the problem is to determine how many units are in 5 units. This will tell us how many of the unit long segments to lay end to end in order to get the 5 unit long segment. Since 5 = 5 =, there are segments of length units in a segment of length 5 units. So you need to form a line segment that is times as long as the one pictured, plus another as long: Problems for Section 0. on Dividing Fractions. A bread problem: If one loaf of bread requires cups of flour, then how many loaves of bread can you make with 0 cups of flour? (Assume that you have enough of all other ingredients on hand.) (a) Solve the bread problem by drawing a diagram. reasoning. Explain your

21 0.. DIVIDING FRACTIONS (a) How many cups of flour are in one muffin? (b) How many muffins does cup of flour make? (c) If you have cups of flour, then how many batches of muffins can you make? (Assume that you can make fractional batches of muffins and that you have enough of all the ingredients.) 8. Write a how many in one group? story problem for and use your story problem to explain why it makes sense to solve by inverting and multiplying, in other words by multiplying by. 9. Write a how many in one group? story problem for and use your story problem to explain why it makes sense to solve by inverting and multiplying, in other words by multiplying by. 0. Write a how many in one group? story problem for 9 and use your story problem to explain why it makes sense to solve 9 by inverting and multiplying, in other words by multiplying 9 by.. Write a how many in one group? story problem for and use your story problem to explain why it makes sense to solve by inverting and multiplying, in other words by multiplying by.. Write a how many in one group? story problem for and use your story problem to explain why it makes sense to solve by inverting and multiplying.. Give an example of either a hands-on activity or a story problem for elementary school children that is related to a fraction division problem (even if the children wouldn t think of the activity or problem as fraction division). Write the fraction division problem that is related to your activity or story problem. Describe how the children could solve the problem by using logical thinking aided by physical actions or by drawing pictures.. Buttercup the gerbil drank of a bottle of water in days. Assuming Buttercup continues to drink water at the same rate, how many bottles of water will Buttercup drink in 5 days? Use multiplication and division to solve this problem, explaining in detail why you can use multiplication when you do and why you can use division when you do.

22 5. If you used truck loads of mulch for a garden that covers of an acre, then how many truck loads of mulch should you order for a garden that covers acres? (Assume that you will spread the mulch at the same rate as before.) Use multiplication and division to solve this problem, explaining in detail why you can use multiplication when you do and why you can use division when you do. 6. If pints of jelly filled jars, then how many jars will you need for pints of jelly? Will the last jar of jelly be completely full? If not, how full will it be? (Assume that all jars are the same size.) Use multiplication and division to solve this problem, explaining in detail why you can use multiplication when you do and why you can use division when you do.

### Fraction Problems. Figure 1: Five Rectangular Plots of Land

Fraction Problems 1. Anna says that the dark blocks pictured below can t represent 1 because there are 6 dark blocks and 6 is more than 1 but 1 is supposed to be less than 1. What must Anna learn about

### Unit 2 Number and Operations Fractions: Multiplying and Dividing Fractions

Unit Number and Operations Fractions: Multiplying and Dividing Fractions Introduction In this unit, students will divide whole numbers and interpret the answer as a fraction instead of with a remainder.

### Fractions Packet. Contents

Fractions Packet Contents Intro to Fractions.. page Reducing Fractions.. page Ordering Fractions page Multiplication and Division of Fractions page Addition and Subtraction of Fractions.. page Answer Keys..

### Mathematics Pacing Resource Document 5.AT.2 Standard: 5.AT.2: Teacher Background Information:

Standard: : Solve real-world problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators (e.g., by using visual fraction models and equations

### Grade 7 Mathematics. Unit 5. Operations with Fractions. Estimated Time: 24 Hours

Grade 7 Mathematics Operations with Fractions Estimated Time: 24 Hours [C] Communication [CN] Connections [ME] Mental Mathematics and Estimation [PS] Problem Solving [R] Reasoning [T] Technology [V] Visualization

### PAYCHEX, INC. BASIC BUSINESS MATH TRAINING MODULE

PAYCHEX, INC. BASIC BUSINESS MATH TRAINING MODULE 1 Property of Paychex, Inc. Basic Business Math Table of Contents Overview...3 Objectives...3 Calculator...4 Basic Calculations...6 Order of Operation...9

### Math Refresher. Book #2. Workers Opportunities Resources Knowledge

Math Refresher Book #2 Workers Opportunities Resources Knowledge Contents Introduction...1 Basic Math Concepts...2 1. Fractions...2 2. Decimals...11 3. Percentages...15 4. Ratios...17 Sample Questions...18

### [This page is designed for duplication as Transparency #1.]

[This page is designed for duplication as Transparency #1.] The following problems require you to add or subtract fractions. Remember, the denominators (bottom numbers) must be the same, and they don t

### **Unedited Draft** Arithmetic Revisited Lesson 4: Part 3: Multiplying Mixed Numbers

. Introduction: **Unedited Draft** Arithmetic Revisited Lesson : Part 3: Multiplying Mixed Numbers As we mentioned in a note on the section on adding mixed numbers, because the plus sign is missing, it

### Multiplying and Dividing Signed Numbers. Finding the Product of Two Signed Numbers. (a) (3)( 4) ( 4) ( 4) ( 4) 12 (b) (4)( 5) ( 5) ( 5) ( 5) ( 5) 20

SECTION.4 Multiplying and Dividing Signed Numbers.4 OBJECTIVES 1. Multiply signed numbers 2. Use the commutative property of multiplication 3. Use the associative property of multiplication 4. Divide signed

### + = has become. has become. Maths in School. Fraction Calculations in School. by Kate Robinson

+ has become 0 Maths in School has become 0 Fraction Calculations in School by Kate Robinson Fractions Calculations in School Contents Introduction p. Simplifying fractions (cancelling down) p. Adding

### Math and FUNDRAISING. Ex. 73, p. 111 1.3 0. 7

Standards Preparation Connect 2.7 KEY VOCABULARY leading digit compatible numbers For an interactive example of multiplying decimals go to classzone.com. Multiplying and Dividing Decimals Gr. 5 NS 2.1

### Math. Fraction Word Problems. Answers. Name: Solve each problem. Write your answer as a mixed number (if possible).

1) Robin needed 3 2 3 feet of thread to finish a pillow she was making. If she has 2 times as much thread as she needs, what is the length of the thread she has? 2) A single box of thumb tacks weighed

### Division of whole numbers is defined in terms of multiplication using the idea of a missing factor.

32 CHAPTER 1. PLACE VALUE AND MODELS FOR ARITHMETIC 1.6 Division Division of whole numbers is defined in terms of multiplication using the idea of a missing factor. Definition 6.1. Division is defined

### Using Proportions to Solve Percent Problems I

RP7-1 Using Proportions to Solve Percent Problems I Pages 46 48 Standards: 7.RP.A. Goals: Students will write equivalent statements for proportions by keeping track of the part and the whole, and by solving

### 1 st Grade Math Do-Anytime Activities

1 st Grade Have your child help create a number line (0-15) outside with sidewalk chalk. Call out a number and have your child jump on that number. Make up directions such as Hop to the number that is

### Five Ways to Solve Proportion Problems

Five Ways to Solve Proportion Problems Understanding ratios and using proportional thinking is the most important set of math concepts we teach in middle school. Ratios grow out of fractions and lead into

### Solving Algebra and Other Story Problems with Simple Diagrams: a Method Demonstrated in Grade 4 6 Texts Used in Singapore

The Mathematics Educator 2004, Vol. 14, No. 1, 42 46 Solving Algebra and Other Story Problems with Simple Diagrams: a Method Demonstrated in Grade 4 6 Texts Used in Singapore Sybilla Beckmann Out of the

### Capacity quiz. 4. How much milk is in this jug? A) 20 ml B) 100 ml C) 200 ml D) 2 000 ml. 5. What do you think is the capacity of a small wine glass?

Level A 1. Is this statement true or false? There are 100 millilitres in a litre. A) True B) False 2. Is this statement true or false? ml is the short form of millilitre. A) True B) False 3. How much fruit

### Understanding Ratios Grade Five

Ohio Standards Connection: Number, Number Sense and Operations Standard Benchmark B Use models and pictures to relate concepts of ratio, proportion and percent. Indicator 1 Use models and visual representation

### Building Concepts: Dividing a Fraction by a Whole Number

Lesson Overview This TI-Nspire lesson uses a unit square to explore division of a unit fraction and a fraction in general by a whole number. The concept of dividing a quantity by a whole number, n, can

### Solving Equations by the Multiplication Property

2.2 Solving Equations by the Multiplication Property 2.2 OBJECTIVES 1. Determine whether a given number is a solution for an equation 2. Use the multiplication property to solve equations. Find the mean

### Multiplying Fractions by Whole Numbers

Multiplying Fractions by Whole Numbers Objective To apply and extend previous understandings of multiplication to multiply a fraction by a whole number. www.everydaymathonline.com epresentations etoolkit

### Fraction Models Grade Three

Ohio Standards Connection Number, Number Sense and Operations Benchmark C Represent commonly used fractions and mixed numbers using words and physical models. Indicator 5 Represent fractions and mixed

### Measurements 1. BIRKBECK MATHS SUPPORT www.mathsupport.wordpress.com. In this section we will look at. Helping you practice. Online Quizzes and Videos

BIRKBECK MATHS SUPPORT www.mathsupport.wordpress.com Measurements 1 In this section we will look at - Examples of everyday measurement - Some units we use to take measurements - Symbols for units and converting

### NS6-50 Dividing Whole Numbers by Unit Fractions Pages 16 17

NS6-0 Dividing Whole Numbers by Unit Fractions Pages 6 STANDARDS 6.NS.A. Goals Students will divide whole numbers by unit fractions. Vocabulary division fraction unit fraction whole number PRIOR KNOWLEDGE

### Models for Dividing Fractions Grade Six

Ohio Standards Connection Number, Number Sense and Operations Benchmark H Use and analyze the steps in standard and nonstandard algorithms for computing with fractions, decimals and integers. Indicator

### NF5-12 Flexibility with Equivalent Fractions and Pages 110 112

NF5- Flexibility with Equivalent Fractions and Pages 0 Lowest Terms STANDARDS preparation for 5.NF.A., 5.NF.A. Goals Students will equivalent fractions using division and reduce fractions to lowest terms.

### Decimals and other fractions

Chapter 2 Decimals and other fractions How to deal with the bits and pieces When drugs come from the manufacturer they are in doses to suit most adult patients. However, many of your patients will be very

### Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving

Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words

### AR State PIRC/ Center for Effective Parenting

Helping Young Children Learn to Read What Parents Can Do Reading is one of the basic building blocks for your child s learning and school success. It is a skill on which most other learning is built. Children

### MATH Student Book. 5th Grade Unit 7

MATH Student Book th Grade Unit Unit FRACTION OPERATIONS MATH 0 FRACTION OPERATIONS Introduction. Like Denominators... Adding and Subtracting Fractions Adding and Subtracting Mixed Numbers 0 Estimating

### More Multi Step Problems. Miscellaneous Themes

More Multi Step Problems Miscellaneous Themes Jelly beans cost \$2.00 for 4 packages. There were 25 students. How much money will be needed so that each student can get an equal amount of jellybeans? How

### 4. Write a mixed number and an improper fraction for the picture below.

5.5.1 Name Grade 5: Fractions 1. Write the fraction for the shaded part. 2. Write the equivalent fraction. 3. Circle the number equal to 1. A) 9 B) 7 C) 4 D) 7 8 7 0 1 4. Write a mixed number and an improper

### QM0113 BASIC MATHEMATICS I (ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION)

SUBCOURSE QM0113 EDITION A BASIC MATHEMATICS I (ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION) BASIC MATHEMATICS I (ADDITION, SUBTRACTION, MULTIPLICATION AND DIVISION) Subcourse Number QM 0113 EDITION

### Introduction to Fractions, Equivalent and Simplifying (1-2 days)

Introduction to Fractions, Equivalent and Simplifying (1-2 days) 1. Fraction 2. Numerator 3. Denominator 4. Equivalent 5. Simplest form Real World Examples: 1. Fractions in general, why and where we use

### Lesson 15 Teacher Page A

Lesson 5 Teacher Page A Overview Students explore fraction equivalence using chips. Materials Chips for students Display chips for teacher Student Pages A-E Teaching Actions. Tell the students this story:

### My Math Chapter 8 Lesson 3. Use the two pictures below to answer the questions. Engage NY Lessons 5.4 (Appendix C)

7 Flex Day (Instruction Based on Data) Recommended Resources: Decomposing Fractions Pizza Share Give Em Chocolate 8 Use cuisinaire rods and tape diagrams to model equivalent fraction values. Determine

### 1.6 Division of Whole Numbers

1.6 Division of Whole Numbers 1.6 OBJECTIVES 1. Use repeated subtraction to divide whole numbers 2. Check the results of a division problem 3. Divide whole numbers using long division 4. Estimate a quotient

### Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.

Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material

### 3.4 Multiplication and Division of Rational Numbers

3.4 Multiplication and Division of Rational Numbers We now turn our attention to multiplication and division with both fractions and decimals. Consider the multiplication problem: 8 12 2 One approach is

### Commutative Property Grade One

Ohio Standards Connection Patterns, Functions and Algebra Benchmark E Solve open sentences and explain strategies. Indicator 4 Solve open sentences by representing an expression in more than one way using

### Today, my view has changed completely. I can no longer imagine teaching math without making writing an integral aspect of students' learning.

October 2004 Volume 62 Number 2 Writing! Pages 30-33 Writing in Math Marilyn Burns Innovative teachers can make writing an invaluable part of math instruction. One reason I chose mathematics for my undergraduate

### Grade 4 Mathematics Measurement: Lesson 3

Grade 4 Mathematics Measurement: Lesson 3 Read aloud to the students the material that is printed in boldface type inside the boxes. Information in regular type inside the boxes and all information outside

### Activity 1: Using base ten blocks to model operations on decimals

Rational Numbers 9: Decimal Form of Rational Numbers Objectives To use base ten blocks to model operations on decimal numbers To review the algorithms for addition, subtraction, multiplication and division

### DIMENSIONAL ANALYSIS #2

DIMENSIONAL ANALYSIS #2 Area is measured in square units, such as square feet or square centimeters. These units can be abbreviated as ft 2 (square feet) and cm 2 (square centimeters). For example, we

### Lesson 17 Teacher Page A

Overview Students name fractions greater than with fraction circles. Students name fractions using both mixed numbers and improper fractions. Materials Fraction Circles for students and teacher Transparency

### Decimal Fractions. Grades 6 and 7. Teacher Document. We acknowledge the valuable comments of Hanlie Murray and Sarie Smit

Decimal Fractions Grades 6 and 7 Teacher Document Malati staff involved in developing these materials: Therine van Niekerk Amanda le Roux Karen Newstead Bingo Lukhele We acknowledge the valuable comments

### FRACTIONS. The student will be able to: Essential Fraction Vocabulary

FRACTIONS The student will be able to:. Perform basic operations with common fractions: addition, subtraction, multiplication, and division. Common fractions, such as /, /, and /, are used on the GED Test

### This explains why the mixed number equivalent to 7/3 is 2 + 1/3, also written 2

Chapter 28: Proper and Improper Fractions A fraction is called improper if the numerator is greater than the denominator For example, 7/ is improper because the numerator 7 is greater than the denominator

### Algebra EOC Practice Test #4

Class: Date: Algebra EOC Practice Test #4 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. For f(x) = 3x + 4, find f(2) and find x such that f(x) = 17.

### FRACTIONS, DECIMALS AND PERCENTAGES

Fractions Fractions Part FRACTIONS, DECIMALS AND PERCENTAGES Fractions, decimals and percentages are all ways of expressing parts of a whole. Each one of these forms can be renamed using the other two

### Working with Equivalent Fractions, Decimals & Percentages

Virtual Math Girl: Module 3 (Student) and 4 (Teacher) Working with Equivalent s, Decimals & Percentages Activities and Supplemental Materials The purpose of Virtual Math Girl Module 3 (Student) and 4 (Teacher)

### Understanding Division of Fractions

Understanding Division of Fractions Reteaching - Reteaching - Divide a fraction by a whole number. Find _. Use a model to show _. Divide each eighth into equal parts. Each section shows _ ( ). _. Divide

### Unit 5 Length. Year 4. Five daily lessons. Autumn term Unit Objectives. Link Objectives

Unit 5 Length Five daily lessons Year 4 Autumn term Unit Objectives Year 4 Suggest suitable units and measuring equipment to Page 92 estimate or measure length. Use read and write standard metric units

### Arithmetic Reasoning. Booklet #2

Booklet #2 Arithmetic Reasoning The CSEA Examination Preparation Booklet Series is designed to help members prepare for New York State and local government civil service examinations. This booklet is designed

### Chapter 4 Online Appendix: The Mathematics of Utility Functions

Chapter 4 Online Appendix: The Mathematics of Utility Functions We saw in the text that utility functions and indifference curves are different ways to represent a consumer s preferences. Calculus can

### Ratios and Proportional Relationships: Lessons 1-6

Unit 7-1 Lesson 1-6 Ratios and Proportional Relationships: Lessons 1-6 Name Date Classwork Book Math 7: Mr. Sanford Lessons 1-6: Proportional Relationship Lesson 1-1 Lesson 1: An Experience in Relationships

### Session 7 Fractions and Decimals

Key Terms in This Session Session 7 Fractions and Decimals Previously Introduced prime number rational numbers New in This Session period repeating decimal terminating decimal Introduction In this session,

### EXTRA ACTIVITy pages

EXTRA FUN ACTIVITIES This booklet contains extra activity pages for the student as well as the tests. See the next page for information about the activity pages. Go to page 7 to find the Alpha tests. EXTRA

### Test 4 Sample Problem Solutions, 27.58 = 27 47 100, 7 5, 1 6. 5 = 14 10 = 1.4. Moving the decimal two spots to the left gives

Test 4 Sample Problem Solutions Convert from a decimal to a fraction: 0.023, 27.58, 0.777... For the first two we have 0.023 = 23 58, 27.58 = 27 1000 100. For the last, if we set x = 0.777..., then 10x

### Math 110 Worksheet #4. Ratios

Ratios The math name for a fraction is ratio. A ratio is just a fancy way to say a fraction. In the Culinary Arts, you use ratios all the time. The Working Factor is a ratio. working factor = new yield

### Fractions. Chapter 3. 3.1 Understanding fractions

Chapter Fractions This chapter will show you how to find equivalent fractions and write a fraction in its simplest form put fractions in order of size find a fraction of a quantity use improper fractions

### Gluten-Free Baking: Tips & Recipes

Gluten-Free Baking: Tips & Recipes From the National Foundation for Celiac Awareness Webinar: Holiday Special: Gluten-Free Baking Featuring Chef Richard Coppedge, Jr., CMB, Professor, Baking and Pastry

### Working with whole numbers

1 CHAPTER 1 Working with whole numbers In this chapter you will revise earlier work on: addition and subtraction without a calculator multiplication and division without a calculator using positive and

### REVIEW SHEETS BASIC MATHEMATICS MATH 010

REVIEW SHEETS BASIC MATHEMATICS MATH 010 A Summary of Concepts Needed to be Successful in Mathematics The following sheets list the key concepts that are taught in the specified math course. The sheets

### Copyright 2013 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or

A W weet Copyright 203 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without

### Understanding Income and Expenses EPISODE # 123

Understanding Income and Expenses EPISODE # 123 LESSON LEVEL Grades 4-6 KEY TOPICS Entrepreneurship Income and expenses Cash flow LEARNING OBJECTIVES 1. Understand what your income and expenses are. 2.

### Comparing and Ordering Fractions

Comparing and Ordering Fractions Compare and order fractions on number lines.. Compare. Write >, d) < 6 b) = e) 0 < c) 7 f) > > 0. Order each set of numbers from least to greatest. Use a

### Sample Fraction Addition and Subtraction Concepts Activities 1 3

Sample Fraction Addition and Subtraction Concepts Activities 1 3 College- and Career-Ready Standard Addressed: Build fractions from unit fractions by applying and extending previous understandings of operations

### Pre-Algebra Lecture 6

Pre-Algebra Lecture 6 Today we will discuss Decimals and Percentages. Outline: 1. Decimals 2. Ordering Decimals 3. Rounding Decimals 4. Adding and subtracting Decimals 5. Multiplying and Dividing Decimals

### Solving Proportions by Cross Multiplication Objective To introduce and use cross multiplication to solve proportions.

Solving Proportions by Cross Multiplication Objective To introduce and use cross multiplication to solve proportions. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop

### Simplifying Improper Fractions Poster

Simplifying Improper Fractions Poster Congratulations on your purchase of this Really Good Stuff Simplifying Improper Fractions Poster a reference tool showing students how to change improper fractions

### Student Outcomes. Lesson Notes. Classwork. Discussion (10 minutes)

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 Student Outcomes Students know the definition of a number raised to a negative exponent. Students simplify and write equivalent expressions that contain

### JobTestPrep's Numeracy Review Decimals & Percentages

JobTestPrep's Numeracy Review Decimals & Percentages 1 Table of contents What is decimal? 3 Converting fractions to decimals 4 Converting decimals to fractions 6 Percentages 6 Adding and subtracting decimals

### Securing number facts, calculating, identifying relationships

1 of 19 The National Strategies Primary Year 4 Block E: Three 3-week units Securing number facts, calculating, identifying relationships Tables 10 10; multiples Written methods: TU U; TU U; rounding remainders

### Chapter 11 Number Theory

Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications

### GRADE 6 MATH: SHARE MY CANDY

GRADE 6 MATH: SHARE MY CANDY UNIT OVERVIEW The length of this unit is approximately 2-3 weeks. Students will develop an understanding of dividing fractions by fractions by building upon the conceptual

### Fractional Part of a Set

Addition and Subtraction Basic Facts... Subtraction Basic Facts... Order in Addition...7 Adding Three Numbers...8 Inverses: Addition and Subtraction... Problem Solving: Two-Step Problems... 0 Multiplication

### UTILITY AND DEMAND. Chapter. Household Consumption Choices

Chapter 7 UTILITY AND DEMAND Household Consumption Choices Topic: Consumption Possibilities 1) The level of utility a consumer can achieve is limited by A) prices only. B) income only. C) the consumer

### CBA Fractions Student Sheet 1

Student Sheet 1 1. If 3 people share 12 cookies equally, how many cookies does each person get? 2. Four people want to share 5 cakes equally. Show how much each person gets. Student Sheet 2 1. The candy

### Revision Notes Adult Numeracy Level 2

Revision Notes Adult Numeracy Level 2 Place Value The use of place value from earlier levels applies but is extended to all sizes of numbers. The values of columns are: Millions Hundred thousands Ten thousands

### Mathematics Task Arcs

Overview of Mathematics Task Arcs: Mathematics Task Arcs A task arc is a set of related lessons which consists of eight tasks and their associated lesson guides. The lessons are focused on a small number

### Section 4.1 Rules of Exponents

Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells

### 2016 Practice Test Grade 53

English Mathematics Language Arts 2016 Practice Test Grade 53 Session 1 Directions: Today, you will take Session 1 of the Grade 35 Mathematics Test. You will not be able to use a calculator in this session.

### Grade 4 - Module 5: Fraction Equivalence, Ordering, and Operations

Grade 4 - Module 5: Fraction Equivalence, Ordering, and Operations Benchmark (standard or reference point by which something is measured) Common denominator (when two or more fractions have the same denominator)

### GRADE 6 MATH: RATIOS AND PROPORTIONAL RELATIONSHIPS

GRADE 6 MATH: RATIOS AND PROPORTIONAL RELATIONSHIPS UNIT OVERVIEW This 4-5 week unit focuses on developing an understanding of ratio concepts and using ratio reasoning to solve problems. TASK DETAILS Task

### Ratios (pages 288 291)

A Ratios (pages 2 29) A ratio is a comparison of two numbers by division. Ratio Arithmetic: to : Algebra: a to b a:b a b When you write a ratio as a fraction, write it in simplest form. Two ratios that

### Phonics. High Frequency Words P.008. Objective The student will read high frequency words.

P.008 Jumping Words Objective The student will read high frequency words. Materials High frequency words (P.HFW.005 - P.HFW.064) Choose target words. Checkerboard and checkers (Activity Master P.008.AM1a

### A booklet for Parents

By the end of Year 2, most children should be able to Count up to 100 objects by grouping them and counting in tens, fives or twos; explain what each digit in a two-digit number represents, including numbers

### Preliminary Mathematics

Preliminary Mathematics The purpose of this document is to provide you with a refresher over some topics that will be essential for what we do in this class. We will begin with fractions, decimals, and

### WRITING PROOFS. Christopher Heil Georgia Institute of Technology

WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this

### Christmas Theme: The Greatest Gift

Christmas Theme: The Greatest Gift OVERVIEW Key Point: Jesus is the greatest gift of all. Bible Story: The wise men brought gifts Bible Reference: Matthew 2:1-2 Challenge Verse: And we have seen and testify

### MATH COMPUTATION. Part 1. TIME : 15 Minutes

MATH COMPUTATION Part 1 TIME : 15 Minutes This is a practice test - the results are not valid for certificate requirements. A calculator may not be used for this test. MATH COMPUTATION 1. 182 7 = A. 20

### Decimal Notations for Fractions Number and Operations Fractions /4.NF

Decimal Notations for Fractions Number and Operations Fractions /4.NF Domain: Cluster: Standard: 4.NF Number and Operations Fractions Understand decimal notation for fractions, and compare decimal fractions.

### Lesson Plan -- Percent of a Number/Increase and Decrease

Lesson Plan -- Percent of a Number/Increase and Decrease Chapter Resources - Lesson 4-11 Find a Percent of a Number - Lesson 4-11 Find a Percent of a Number Answers - Lesson 4-12 Percent of Increase and

### Measurement. Customary Units of Measure

Chapter 7 Measurement There are two main systems for measuring distance, weight, and liquid capacity. The United States and parts of the former British Empire use customary, or standard, units of measure.

### THEME: DEPRECIATION. By John W. Day, MBA

THEME: DEPRECIATION By John W. Day, MBA ACCOUNTING TERM: Depreciation Depreciation is defined as a portion of the cost that reflects the use of a fixed asset during an accounting period. A fixed asset