Daily Fraction Five. Fraction Five ~ Laying the Foundation for a Conceptual Understanding of Fractions. Fourth Grade 4.NF

Transcription

1 Daily Fraction Five Fraction Five ~ Fourth Grade Laying the Foundation for a Conceptual Understanding of Fractions Fourth 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 3 rd grade 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 4 th Grade October 1. How many students are here today? How many have on red? 2. How many students are here today? How many have brown hair? 3. How many students are here today? How many brought their lunch? 4. How many students are here today? How many rode the bus to school? 5. How many students are here today? How many ate breakfast? 6. How many students are here today? How many like broccoli? 7. Which is greater ¼ or 1/6? 8. How many students are here today? How many like chocolate ice cream the best? 9. How many students are here today? How many have a brother?

4 10. How many students are here today? How many have on the color blue? 11. Make a fraction out of how many students in your class have on tennis shoes today. Is that closer to 0, ½ or 1? Explain 12. I rode ¾ of the rides at the fair. Did I ride most of the rides? How many did I not ride? 13. Draw a model for 2 ¼. Be ready to explain your drawing. 14. Had you rather have 2/3 of a pizza or ½ of a pizza? Explain your answer. 15. Label the number line by fourths Which is larger ~ ¼ or 1/6? Explain your answer. 17. Susan was so proud that she had complete ¼ of her homework. Was she almost finished or just beginning? Explain your answer.

5 18. 4 children want to share 2 sandwiches. How much sandwich would each child get? 19. Draw a model for 2 ¾. Can you show how to determine the improper fraction? 20. Which is larger: ¼ or 5/6? How do you know?

6 Answers: 1. Questions 1-10 ~ What does the denominator mean? What does the numerator mean? Is the fraction closer to 0, ½ or 1? How do you know? The focus should be the understanding of the numerator and denominator relationship. On number 5, if they all ate breakfast, the use this time to discuss a whole. 24/24 = 1 2. This question asks students to create a fraction from their class. They should have the understanding from the previous 10 days that they will need to know how many students are there to determine the denominator (the whole), in order to create the fraction. 3. Almost all of the rides have been ridden because 3 is only one away from 4. The closer the numerator and denominator are the closer to one the fraction is. 4. Models may vary. Encourage students to think of different models. 5. How can we tell that 2/3 has a greater value than ½. 2/3 means two pieces out of three. How does this relate to half? Is it greater than half? You may need to help students draw a model of 2/3 and ½ ¼ ½ or 2/4 ¾ 1 Emphasize that ½ and 2/4 are equivalent fractions because they have the same value. 7. There is a one in each numerator. That tells me that I have one piece of each fraction. The fourths are larger pieces that sixths. That tells me that ¼ is the larger fraction. 8. One is just one piece of the fraction ¼. It can be read as one out of four. That lets me know that she has just started her homework. 9. Encourage students to draw this out. Two sandwiches shared between four students can be represented as: Each student would receive 2 fourths. Help students to see that each student would receive a half. 2/4 = ½.

7 10. 1/4 2/4 5/4 6/4 9/4 3/4 4/4 7/4 8/4 9/4 = 2 ¼ 11. 5/6 is the larger fraction. The numerator and denominator are one away from each other so that means that I have almost a whole. ¼ is just one piece of ¼. It is just a little more than zero. The numerator and denominator are as far apart as they can be which lets me know that that the fraction value is closer to zero. 12. Fourths are decomposed as ¼ + ¼ + ¼ + ¼ = 4/4 ¾ is almost a whole. So, if I rode ¾ of the rides, then I have almost ridden them all ¼ is two wholes and ¼ more. It can also be represented as 9/4. Some students may represent draw it with all of the 9/4s. That would be a great discussion point of improper fractions and mixed numbers. 14. Two thirds of a pizza is more than ½ of a pizza. If I draw a model, I can see that two thirds is more than half. 1/2 1/3 2/3 15. Make sure students have spaced the fourths evenly apart and have each dot labeled. They should also have 4/4 on one. 16. One fourth is larger. There is one piece in each fraction. I know this because there is a one in each of the numerators. The sizes of the pieces are known because of the denominators. Fourths pieces are larger than sixths pieces. The larger the denominator goes, the smaller the pieces become. 17. ¼ or one part out of 4 is closer to zero. 18. If four children want to share 2 sandwiches, each child would get a half or ½ Sandwich One Sandwich Two 19. 1/4 2/4 5/4 6/4 3/4 4/4 7/4 8/4

8 9/ /6 is larger than ¼. 5/6 is almost a whole. It is five parts out of six parts and more than half. ¼ is closer to zero. It is one part out of four parts and less than half.

9 Daily Fraction Five 4 th Grade November 1. Which is larger 3/8 or 7/8? Place these fractions on a number line to show your thinking. 2. Which is greater?: 1/6 or 1/8? 3. Which is larger: 2/5 or 4/5? 4. Which is larger: 5/8 or 4/6? 5. Which is less: 6/10 or 3/8? 6. Which is less: 1/3 or 1/5? Explain your answer. 7. Which is less: ¼ or ¾. 8. Which is greater: 2/3 or 4/6? 9. Which is less: 4/10 or 4/6? 10. After the pizza party, 1/8 of the pepperoni pizza was left and 3/8 or the cheese pizza was left. How much pizza was eaten during the party? Draw a picture to show your thinking.

10 11. Name two fractions between ½ and one. Place them on a number line. 12. Name two fractions between 0 and half. Place them on a number line. 13. Draw a number line that represents 1 ¼. Label all of the fourths 14. Estimate the sum of ¼ and 1/5. Explain your thinking with a drawing. 15. Estimate the sum of 7/8 and 5/6. Explain your thinking with words or a drawing. 16. Estimate the sum of 4/5 and ½. Explain your thinking with words or a drawing. 17. Estimate the differene of 1 and ¾. Explain your thinking with words or a drawing. 18. Susan baked 24 cookies. She wants to share them with 8 students. Write this as a fraction and solve. 19. Robin bought 7bags of candy that she wants to split between 3 friends. How much candy will each friend receive? Draw a picture to support your answer.

11 20. Kim has 9 candy bars that she wants to share between 4 students equally. How much will each student receive? Write this as a fraction and solve. Draw a picture to show your thinking.

12 Answers: /8 2/8 3/8 4/8 5/8 6/8 7/8 8/8 or 1 3/8 is greater than 7/8. 2. Help students to reason about this question without teaching them to cross multiply. Use fraction sense to solve this problem. There is one in each numerator. Sixths are larger than eights so 1/6 is the greater piece. 3. When the denominators are the same, the pieces are the same size and can be compared by looking at how many pieces there are. (numerator) 4. Use fraction sense to solve this problem verses teaching students to cross multiply. Both fractions are one over half. The sixths are the larger pieces. 4/6 is the greater fraction because it is more over half than 5/8. 5. Use fraction sense to solve this problem. 6/10 is one over half. 3/8 is one under half. 3/8 is the smaller fraction. 6. There is a one in each numerator so students can use the denominator size. 1/5 is the smaller piece because fifths are less than thirds. There is one piece of each so 1/5 is the smaller fraction. 7. The denominators are the same so that tells me that the size of the pieces is the same. ¾ > ¼. 8. Each fraction is one over half. Thirds are larger pieces than sixths, so 2/3 is the larger fraction. 9. 4/6 is one piece over half and 4/10 is one under half, therefore 4/10 is the smaller fraction /8 was left so 7/8 was eaten. 3/8 was left so 5/8 was eaten. 12 pieces were eaten or 12/8 pizza was eaten. 11. Name two fractions between half and one. Students should be encouraged to name any fraction that is over half and less than one. Ask them how they know it is over half. 12. Name two fractions between zero and half. Students should be encouraged to name any fraction that is greater than zero and less than half. Ask them how they know it is less than half.

13 13. Name two fractions between half and one. Students should be encouraged to name any fraction that is over half and less than one. Ask them how they know it is over half. 14. ¼ 2/4 ¾ 4/4 or 1 1 ¼ 1 2/4 15. I estimate the sum to be close to half because I know that ¼ + ¼ = ½.!/5 is just a little bit smaller than ¼ so I know the sum is close to ½. 16. I know that 4/5 is almost one and ½ is half so I estimate the sum is 1 ½. 17. I know that one is 4/4 so 4/4-3/4 = ¼ /8. Students must understand that a fraction is a division problem. Each student would receive 3 cookies. 19. Build from yesterday s problem. Remind students that fractions are division problems. 7 bags of candy split between 3 friends is equivalent to 7/3or 2 1/ Nine candy bars split 4 ways. 9/4 Use the picture from the previous problem as a basis for your math drawings. 2 ¼

14 Daily Fraction Five 4 th Grade December 1. About what part of the big rectangle is the smaller rectangle? How do you know? 2. Draw a model for 3 2/4. Explain your drawing out of 12 (4/12) students sold raffle tickets. How many did not sell raffle tickets? 4. Even though this is divided into 4 parts, why can t we label this picture as the fraction 1/4? 5. What fraction of the figure is shaded? Explain your answer. 6. Decompose 1 into eights. Draw a picture to support your thinking.

15 7. Decompose 2 into thirds. Draw a picture to support your thinking. 8. Decompose 1 ¼ into fourths. Draw this on a number line. 9. Decompose 2 ½ into halves. Draw this on a number line students share 9 large cookies. How many cookies does each student receive? Draw a picture to show this. 11. What number is 1/3 of 18? How can you show with arrays? students shared 9 brownies. How many brownies did each student receive? Draw a picture to show this. 13. Which is greater, 1/6 or 1/3? Explain you answer with a picture. 14. What is ½ of 1 dollar? What is ½ or 100 dollars? Why are those different amounts? 15. What is ¼ of 100 dollars? What is ¾ of one dollar?

16 16. ¼ of the seats in our gym are filled. ¼ of the seats in our classroom are filled. Which room has more seats filled. Explain your answer. (Emphasize the size of the units) 17. Write in your own words what one-fourth means. Draw a picture to support your definition. 18. Write in your own words what three-fourths means. Draw a picture to support your definition. 19. I had two birthday cakes that were cut into 8ths. 1 piece of the first one was left and two pieces of the second one was left. What fraction of cake was left? Draw a picture to show your answer students are going to share 5 small pizzas. How much pizza does each student get? Draw a picture to show this.

17 Answers 1. The smaller rectangle is about ¼ the size of the larger rectangle. 2. Models for 3 and 2/4 Students should realize that 2/4 is equivalent to 1/2, if not; help them to make that connection. There model should have 3 wholes and 2/4 shaded in on the fourth whole. 3. 4/12 of the students sold raffle tickets. That means that 8/12 did not sell raffle tickets. 4. Fractions must be divided equally. The rectangle is divided into 4 pieces, but they are not equal pieces. 5. About 1/6 of the figure is shaded. I estimated by drawing in the box in the rest for the rest of the shape. 6. Decompose 1 into eights. There are eight, eights in one whole. 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8, 8/8. 7. Decompose 2 into thirds. There are three, thirds in one whole. 1/3, 2/3, 3/3 or one, 4/3, 5/3, 6/3 or two wholes. 8. Decompose 1 ¼ into fourths. ¼, 2/4, ¾, 4/4, 5/4 or 1 ¼. 9. Decompose 2 into halves: ½, 1 or 2/2, 1 ½, 2 or 4/2 10. Four students share 9 large cookies Each student will get 2 and ¼ cookies /3 of 18 means that I am putting 18 into 3 groups. The denominator tells me how many groups to put the whole number in. I will arrange them in an array for the students to see them better.

18 I put 18 into 3 groups of six. I shaded the first group. That first group is 1/3 of 18 which equals Emphasize that each student gets one brownie and then the other 3 are cut in half. So each student gets 1 and 1/2 brownie. They could also cut the other 3 into sixths. Each student would get 3/6 which is equivalent to ½. 13. One third is greater. I have one piece of each and I know that thirds are greater than sixths. 14. One half of one dollar is.5 cents and one half of one hundred dollars is $5 0. One dollar is less than one hundred dollars. However, I see a pattern with the 50. One less of one thousand dollars would be $ Discuss and help them understand that even though ¾ is a larger fraction, the unit is smaller. You must have like units to compare accurately. One-fourth of a hundred dollars is 25 dollars. ¾ of One dollar is.75 cents. 16. Emphasize the size of the units. We are looking at ¼ of each unit, but one is a classroom and one is the gymnasium. 17. One fourth is one part out of four. The whole is shaded into 4 parts and one of the parts is shaded. It is closer to zero than 2/4 or ¾. ¼ 2/4 ¾ 18. Three Fourths is three parts out of four. The whole is shaded into 4 parts and three of the parts are shaded. It is closer to one whole. 19. If I had two birthday cakes cut into eights. That means that I have 16 eighths (16/8) because I have two cakes cut that way. One piece of the first cake was left and two pieces of the second was left. That is three pieces all together. That means that I have 3/8 of a cake left over. 20. If 4 students share 5 pizzas then each student will get one whole pizza and ¼ of the fifth pizza. I divided the fifth pizza in to fourths because that is how many students I am sharing it between.