Daily Fraction Five. Fraction Five ~ Fifth Grade. Laying the Foundation for a Conceptual Understanding of Fractions. Fifth Grade 5.

Transcription

1 Daily Fraction Five Fraction Five ~ Fifth Grade Laying the Foundation for a Conceptual Understanding of Fractions Fifth Grade

2 FRACTION FIVE RATIONAL Fractions present a considerable challenge to most kids. The lack of developmental understanding of fractions sometimes forces students to give up and resort to rules. However, if a firm fractional foundation is built, students stand a greater chance of understanding. This is a three month precursor to the fraction standards. The questions laid out in this activity are meant for pre-teaching purposes. It is called Fraction Five because it is meant to be about 5 minutes of conceptual fraction discussion/understanding, daily. The questions continually ask students to make sense of the numerator and denominator. This part to whole relationship is vital for students to understand. Through these discussions, teachers are able to formatively assess their students understanding of the part to whole relationship. By starting to have these discussions with students about fractions early and frequently, throughout the beginning year, students are ready to conceptually understand more easily, when direct instruction begins. The questions can be copied for each student or displayed on the board each morning. These questions are not meant for grading purposes. They are intended to foster rich discussions and to be used as a formative assessment, to enable teachers to plan accordingly, to their students level of understanding and depth of knowledge of the foundations of fractions.

3 Daily Fraction Five 5 th Grade October 1. Explain the relationship between the numerator and denominator in the fraction ¾. 2. Is ¾ closer to zero or closer to one? Explain your answer. 3. Explain in your own words what 5/8 means. Can you make up a story to show your understanding? 4. How many students in your class have on tennis shoes today? Can you make that into a fraction? Is it closer to one, half or zero? 5. Draw a model of 1 and 5/6. Explain your model. 6. Give an equivalent fraction for 3/5. Explain how you got your answer. 7. Using a rectangle as a whole, draw a model of: one-fourth, two thirds and five sixths. Explain your drawings. 8. What fraction of this set is black? Can you name a fraction that does not have a denominator of nine?

4 9. Is each fraction closer to zero, half or one: three-fourths two-sixths one-eighth five-eights three-sevenths two-thirds 10. What fraction of the big square does the small square represent? 11. What fraction of students in your class brought their lunch today? Is that fraction closer to zero, half or one? Explain your answer. 12. If the blue rectangle represents the whole, what fraction does the red represent? Explain how you decided. BLUE RED 13. Robin has 60 jellybeans. How can she separate them in to thirds? 14. Four children are going to share 10 brownies. How much will each child get? Draw a picture to explain your thinking.

5 15. Draw a number line below on your paper. Label the number line in thirds Sam sold ¼ of his raffle tickets. He has 60 left to sell. How many tickets has he sold? 17. If 12 counters are three-fourths of a set, how many counters are in the set? How do you know? 18. Jesse painted 5/7 of his garage. Is he almost done or just beginning? Explain. 19. Draw a model of 2 and 4/ How do you know that 1/3 = 2/6. Can you draw a picture to show this equation is true?

6 Answers: 1. ¾ - The 4 represents to total number of parts that the whole is divided into. 3 represents how many parts we are talking about or dealing with. 2. ¾ is closer to one. This fraction is one away from being a whole. If it were 4/4 it would be one whole. 3. 5/8 is just over half. 4/8 is exactly half. 5/8 means that there are 8 parts total. The five tells us that we are only dealing with 5 of those parts. I have read about 5/8 of my class reading assignment. I have read a little over half of the assignment. 4. The total amount of students in your room will be the denominator and the number of kids that have on tennis shoes is the numerator. Discuss with you students whether this fraction is closer to zero, half or one. Even with larger denominators, we can have the discussion with students about whether the fraction is closer to zero, half or one. Once they understand the part to whole relationship, they can analyze any fraction. 5. Make a model of 1 5/6. This model represents one whole and then 5/6. 6. An equivalent fraction for 3/5 is 6/10. I am able to multiply both the numerator and denominator by a form of one to create an equivalent fraction without changing the value. I multiplied by 2/2. When we multiply any number by one, it does not change its value. This is a discussion that needs to take place with students. In the past we have given them the rule of, Whatever you do to the top, you must do to the bottom. However, explaining that multiplying by one doesn t change the value, it gives them a tool to use in other problems. 7. One fourth one part out of four shaded Two thirds two parts out of three shaded Five Sixths five parts out of six shaded 8. 3/9 of the triangles are shaded. 1/3 is also an equivalent fraction that represents how many triangles are shaded.

7 9. Three fourths One Five eights Half Two sixths Zero Three sevenths Half One eighth Zero Two thirds One 10. The small square is about ¼ of the larger square. Have students draw in lightly the lines of the small box in the bigger box. 11. The total amount of students in your room will be the denominator and the number of kids that brought their lunch is the numerator. Discuss with you students whether this fraction is closer to zero, half or one. 12. The red bar is about ¾ of the blue bar. Kids can tear paper if they need to in order to prove this to themselves. 13. Robin can put her group of 60 jelly beans into 3 equal piles. There would be 20 in each pile. 14. A drawing helps students understand this problem so much easier. Child One Child Two Child Three Child Four Child One Child Two Child Three Child Four These two last brownies can be divided into fourths or halves. Each child would get 2/4 or ½ of the remaining brownies. Each child would receive 2 ½ brownies total. 15. This number line goes beyond one. Children need to see this. So many times we only show them a number line that is zero to one. Help students understand that they will number in mixed numbers or improper fractions once that go past one. 1/3, 2/3, 3/3 or one, 4/3, 5/3, 6/3 or two. 16. Sam has sold ¼ of his tickets. 60 tickets are the remaining threefourths. We divided 60 by 3 because it is the remaining 3 parts. Sixty divided into three parts is 20. We then, add the last part of 20 to show that there are 80 tickets total. Help students see this through a model. ¼ ½ or 2/4 ¾ One or 4/ sold not sold not sold not sold

8 17. If 12 counters are three-fourths of a set, how many counters are in the set? Help students to decompose this into fourths. They divide 12 by 3 because 12 equals 3 parts of the 4 parts. Modeling this will be critical for students. 4 = ¼ of the counters 8 = ½ of the counters 12 = ¾ of the counters 16 = 4/4 or 1 whole 18. If Jesse painted 5/7 of his garage, he has 2/7 left to paint, so he is almost finished We can show that 1/3 = 2/6 by showing the form of one that we multiplied by to create the equivalent fraction. 1 x 2 = Daily Fraction Five 5 th Grade November 1. If four marbles are half of the set, how big is the set? Explain your answer. 2. Susan has cleaned five-sixths of her room. Is she just beginning to clean or almost through? Explain. 3. How many students brought their lunch today? Is it closer to 0, ½ or 1? Explain your answer. 4. Susan has eaten 1/3 of her lunch. Is she almost finished or just starting to eat? Explain your answer.

9 5. Draw a model of 2 and 1/5. Explain. 6. Decompose the fraction 5/8. Decompose 2 into thirds. Explain your answers. 7. Draw a picture of how to solve 4 x 1/3 = Explain your answer. 8. What fraction of this set is not black? Can you name a fraction that does not have a denominator of twelve? 9. 6 students will share 10 cookies. How many cookies will each student receive? 10. About what fraction of the big rectangle does the small square represent? 11. Which is larger: ½ of a large pizza or ½ of a small cookie? Explain your answer.

10 12. If the blue rectangle represents the whole, what fractions does the red represent? Explain how you decided. BLUE RED 13. Draw a picture that represents 5/4 and 6/3. Which is the larger fraction? Explain. 14. Mr. Davis is buying clay for a project for a group project. Each student needs ¾ of a package to complete their project. If the group has 4 students, how many packages of clay does Mr. Davis need to buy? 15. Each teacher gets 1/3 pack of paper for our spring art project. We gave out 6 packs of paper. How many teachers got paper? 16. Decompose 6/6 into sixths. How many sixths are there in 2 wholes? 17. Which fraction is greater 1/3 or 1/4? Explain your answer? 18. Which fraction is greater? 4/6 or 5/8 Explain your answer?

11 19. Which fraction is less? 2/3 or 2/4 Explain your answer? 20. Which fraction is greater? ¾ or 5/6 Explain your answer? Answers 1. If 4 marbles are half of the set, then the set has 8 marbles. Half of eight is four. 2. 5/6 is one sixth away from a whole or one. So if Susan has cleaned 5/6 of her room, she is almost finished. 3. The total amount of students in your room will be the denominator and the number of kids that brought their lunch is the numerator. Discuss with you students whether this fraction is closer to zero, half or one. 4. Susan has eaten one third of her lunch. She still has 2/3 to eat so she has just begun eating. 5. Students model should be three wholes. Two should be colored in solid and the last whole should be divided into fifths. One of the fifths should be colored in. 6. Decompose 5/8. 1/8 + 1/8 + 1/8 + 1/8 + 1/8 = 5/8 Decompose 2 into thirds. 1/3 + 1/3 + 1/3 + 1/3 + 1/3 + 1/3 = 2 7. Students can draw a model of 1/3 four times. Multiplication has been taught in 4 th grade as repeated addition. 1/3 x 4 is 4/3 or 1 and 1/3 8. 8/12 of the set is not black. An equivalent fraction would be 2/ students will share 10 cookies evenly. Each student will receive one cookie. The remaining 4 cookies will be cut into sixths and each student will receive 4 sixths. The remaining cookies could also be cut into thirds and each student gets one third. 10. The smaller square is about 1/3 the size of the larger square. 11. ½ of a large pizza has a greater quantity that ½ of a small cookie. They are both halves, but their whole is a different size. 12. I compared the two bars and drew a box for the piece that is missing between the two bars. That piece is about 1/6 because I

12 measure it against the whole blue piece. So the red piece is about 1 and 1/ /3 should be the largest picture. 6/3 = 2 wholes. 5/4 is only 1 whole and ¼. 14. Students will probably add to solve this problem. They can add ¾, 4 times to get the number of packages to buy. They can also multiply ¾ times 12. Resist the urge to show the algorithm. Have them to model with a math drawing. Using this model, students can see the fourths and can count them out individually. There are 12/4 or 3 wholes. When students use models, they can move the fourths around to fill the 3 wholes. 15. If 6 packs of paper are used and each teacher got 1/3 of a pack then 18 teachers got paper. There are three, thirds in each pack and there were six packs given out. 6 x 3 = /6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = There are 12 sixths in 2 wholes. The next three questions are about the size of fractions. Students should not be taught to cross multiply. Students should reason about the size of the denominators and the number of pieces that they are dealing with (the numerators). This type of reasoning gives students the flexibility to work with fractions easily. 17. There is one piece of each fraction. Thirds are larger pieces than fourths so 1/3 is the greater fraction 18. 4/6 is the greatest fraction. Both fractions are one over half. Since sixths pieces are larger than eighth pieces 4/6 would be the greatest.

13 19. There are two pieces of each of these fractions. Thirds are larger pieces than fourths and since there is two pieces of each, 2/4 is the smaller fraction /6 is greater because both fractions are one away from being a whole. Since the sixths pieces are smaller than the fourth pieces; 5/6 is closer to one or the greatest. The smaller the piece the closer it can get to one.

14 Daily Fraction Five - 5 th Grade December 1. Write an equivalent fraction for three fourths, one and two sixths, and 3/5. Explain your answer. 2. Divide the number line in to twelfths. 0 1/3 2/ Draw a picture showing that 3/8 is less than ¾ cookies are 1/3 of the box. How many are in the box. Explain your answer students are sharing ten pizzas. How much of a pizza will each student receive? Draw a picture to prove your answer. 6. Draw a picture of 2 and 3/8. Be ready to explain your drawing. 7. Closer to Zero, Half, or One? Explain your reasoning. 6/8 2/9 4/9 7/10 4/6 5/6 8. Our class has collected $50 for our fundraiser. That is 1/3 of the amount we are supposed to collect. How much money is our class responsible for collecting?

15 9. 6 students are going to share 21 cookies. How much will each student receive? 10. Expain how I can determine that 1/8 is less than 1/3? Can you think of more than one way to know this? 11. Write in your own words why 5/5 is equal to Draw a picture of 3 x ¼. How many different ways can you solve this? 13. Write any equivalent fraction for 8/10. Explain how your chose your answer out of 14 students turned in their homework last night. Describe that fraction. 15. Explain why 9/5 is equal to 18/10? 16. Draw a picture to explain how you know that 2/4 = 4/ Put each fraction in order from least to greatest. 3/8 6/11 5/10 1/7 1/4 8/9 Explain your reasoning. (No calculators ) 18. Susan has completed 1/8 of her math problems. She has done 5 problems. How many problems will Susan complete in all?

16 19. Sam will run 10 laps at track practice. He has run 2 laps so far. What fraction of the laps has Sam ran? What fraction does he have left to complete? Explain your answers. 20. Kerri has 7 brownie bars that she wants to give to 4 friends. How much will each friend get. Draw a picture to show your thinking.

17 Answers 1. Equivalent fraction for three fourths, one and two sixths, and 3/5. ¾ x 2/2 =6/8 One and two sixths = 8/6 3/5 x 2/2 = 6/10/14 I can create an equivalent fraction by multiplying by a form of one. 2. The number line should be divided equally in to twelve spots. 1/3 = 3/12 2/3 = 9/ /4 3/8 1/ cookies are 1/3 of the box. How many are in the box? There are 18 cookies in the box. 6 is 1/3. That means that 12 is 2/3 and 18 is 3/ students are sharing ten pizzas. How much of a pizza will each student receive? One way for students to think about this problem is that they each could share each pizza. That would mean that each pizza would be cut into 15 pieces. Each student would end with 10 slices that are cut into fifteenths. Therefore 10/15 or 2/3 of a pizza is what each student would receive. 6. Draw a model for 2 and 3/8. Students should draw two wholes and then a model of 3/ /8 closer to one 2/9 - closer to zero 4/9 closer to half 7/10 closer to half 4/6 closer to half 5/6 closer to one is 1/3 of our total. What is our total? The total = 150 1/3 = 50 1/3 = 50 1/3 = students are going to share 21 cookies. The fraction is 21/6. 21 divided by six is 3 with 3 cookies left over. Those three cookies will be divided into sixths also. So each student will receive 3 cookies and 3/6 or ½ of a cookie. 10. Expain how I can determine that 1/8 is less than 1/3? There is one piece of each of them. I know that because the numerator is one in each of the fractions. I know that thirds are larger than eighths. So, if I have one of each, 1/3 will be larger /5 means 5 parts out of a total of 5 parts. This equals one whole. When drawn in a model. All 5 places will be colored in, creating a whole.

18 12. Draw a picture of 3 x ¼. ¼ + ¼ + ¼ = ¾ Students should draw a picture of ¼, 3 times. 13. Write any equivalent fraction for 8/10. 8/10 x 3/3 = 24/30 I multiplied 8/10 by 3/3, or any form of one /14 is close to half. Half would be 7/14. 8/14 is more than half. 15. Explain why 9/5 is equal to 18/10? The fraction was multiplied by a form of one of 2/2. Even though it is improper, we can still multiply it by a form of one. 16. Draw a picture to explain how you know that 2/4 = 4/8. One fourth Two fourths This model shows fourths and eights. 2/4 = 4/8 Least to greatest /7 ¼ 3/8 5/10 6/11 8/9 One seventh is less than one fourth because I have one piece of each and sevenths are smaller pieces than fourths. Three eights is less than five tenths because I know that five tenths is equivalent to one half and 3/8 is one away from being half. Six elevenths is just a little over half. 8/9 is the greatest because it is one ninth from being one. 18. The problem tells us that one eight of her math problems is equal to 5 problems. I need to figure how what 8/8 would be to know how many problems. 1/8 =5 1/8 =5 1/8 =5 1/8 =5 1/8 =5 1/8 =5 1/8 =5 1/8 =5 5x8=40 There are 40 problems in Susan s assignment. 19. Sam ran two of the ten laps that he needs to run. That means that he has ran 2/10 of the laps he is going to run. Some students may say 2/10 = 1/5. Sam has 8/10 or 4/5 of his laps to run. 20. Kerri is sharing 7 brownie bars among 4 people. Have students draw a picture to solve this problem. Friend 1 Friend 2 Friend 3 Friend 4 The three remaining brownies will be split into fourths because we are sharing with four friends. There will be 12 fourths to share after they are divided. Each friend will get 3 of those fourths. Each friend will receive 1 ¾ of a brownie bar each. Students may need to draw the fourths and actually break down how they are given out to be able to understand this.