Activities
Below is a tentative overview of 21 days of math instruction. Staying on the schedule will depend on the students' abilities to grasp the skills. Certain skills may require more or less time.
I plan to develop fraction concepts by providing the students with unitizing and number line work throughout the year. I use homeroom time as an opportunity to review skills in all subjects. Several times per week I have my students complete a math review. It includes a variety of problems and really helps maintain and sharpen their math skills. I plan to include a unitizing picture and number lines regularly on these review sheets.
Day 1: The Unit
Day 2: Unit Fractions – paper folding
Day 3: Unit Fractions – number lines
Day 4: Unit Fractions – Cuisenaire rods
Day 5: General Fractions – paper folding
Day 6: General Fractions – comparing (same size parts and same number of parts)
Day 7: General Fractions – comparing (comparing to one half and one whole)
Day 8: Test
Day 9: Mixed Numbers and Improper Fractions – concrete square tiles
Day 10: Mixed Numbers and Improper Fractions – paper folding
Day 11: Equivalent Fractions – paper folding
Day 12: Equivalent Fractions – Cuisenaire rods
Day 13: Quiz
Day 14: Adding/Subtracting with like denominators – paper folding
Day 15: Adding/Subtracting with like denominators – pictorial
Day 16: Adding/Subtracting with related denominators – paper folding
Day 17: Adding/Subtracting with related denominators – pictorial
Day 18: Adding/Subtracting with unlike denominators – paper folding
Day 19: Adding/Subtracting with unlike denominators – paper folding
Day 20: Review
Day 21: Test
Activity 1:
The first activity is designed to use fraction strips to compare general fractions. This would be day 5 of the unit outlined above. The activity is designed to help students understand how to compare fractions. The key concepts include when the denominators are the same, when the numerators are the same and how to compare fractions to one half and one.
The students will use fraction strips to complete this activity. I would prepare a set of fractions strips for each student (see appendix 1). (I will include an extra strip of thirds for each student.) For this activity I recommend keeping the unit fractions attached. Students could also fold rectangular strips of paper to make their own representations.
Throughout this activity, I will present students with problems that increase in difficulty according to the scaffolding laid out previously in this curriculum unit. I would begin with some examples involving fractions with the same denominators. For example, I might ask students to compare 3/8 and 5/8. The students would find their strip of eighths and compare three to five. I would have the students share how they found 3/8 and 5/8 and model it on the board with fraction strips. I would ask, "Which is bigger? Do you agree? How do you know?" Then I would give another slightly more difficult problem with like denominators, such as the problem of comparing 4/6 and 2/6. I might ask, "Which strip do I need? Who can help me model 4/6? Who can help me model 2/6? Are these fractions modeled correctly? Which one is smaller?" The correct model would use the sixths strip with four sixths shaded. The second strip would also be sixths, but only two would be shaded. The students should recognize that four sixths (4/6) is greater than two sixths (2/6). You could also include the greater than (>), less than () and equal to (=) signs if you would like to further emphasize the symbolic aspects at play. I would give one more difficult example such as comparing 7/12 and 9/12. Again I would have students model this independently and then have the class review this at the board. I would ask questions such as, "How do I show 7/12? 9/12? Which is bigger? How do you know?"
The next part of this activity involves comparing fractions with like numerators. In a similar way to the exercises described above, I would ask students to work with easier to compare fractions such as 2/3 and 2/5, and ask questions such as "What strips do I need? Can you find 2/3? Can you find 2/5? Which one is smaller?" I would have students model on the board how they depicted each fraction and how they reached their answer. Next, I would move to a more difficult type of problem and ask them to compare 7/10 and 7/8. I might ask, "Which strips did I need? How many tenths do I have? Do I need another strip? Yes. Which one? Eighths. How many eighths do I need? Which one is bigger? 7/8. Is it correct? Yes." Finally, I would move on to an even more advanced type of comparison, asking them to compare 4/3 and 4/6. (You may need to have extra strips of thirds available.) Again, I would ask questions like, "How many thirds do I need? 4. Do I have enough on my strip? No. What should I do? Get another strip of thirds. How many sixths do I need? 4. Is it modeled correctly? Yes. Which is bigger? 4/3." After this, I will have the students complete the worksheet in Appendix 2 at their desks.
Activity 2:
This second activity corresponds to day 6 of the unit described above. The activity continues to explore the different strategies used to compare general fractions. For this activity the students will need a set of fraction strips, as in the previous activity. For the first group of problems the students will compare each fraction to one half. In these problems, two fractions are given to students, chosen so that one fraction is greater than one half, and the other is less than one half. I would have the students find their halves strip and keep it out for the first few problems. For example, I might say, "Let's compare 4/9 and 6/10. How many ninths do I need? 4." I would then model 4/9 on the board using fraction strips and show the students the 1/2 strip. I would ask, "Is 4/9 greater than or less than 1/2?" and then model the comparison of 4/9 to 1/2 on the board. I would continue to ask questions such as, "Do you agree? Yes. How many tenths do I need? Six. Is 6/10 modeled correctly? Yes." Now using the 1/2 strip, I would ask, "Is 6/10 greater than or less than 1/2? Greater than. Do you agree? Yes." I would follow with a slightly more advanced problem by having the students compare 4/10 and 5/6. Students will work with the fraction strips at their desks. I would probe, "Can you see 4/10? Can you see 1/2? Is 4/10 greater than or less than 1/2? Can you see 5/6? How does 5/6 compare to 1/2? So which fraction is greater?"
Now I need some help with this next problem because I do not understand it. Compare 4/10 and 2/6. (Continue modeling the problem on the board with fractions strips as students work with the strips at their desks.) I would say, "Is 4/10 greater than or less than 1/2? Less than. Is 2/6 greater than or less than 1/2? Less than. So how can I tell which one is greater? Line up the strips. Does anyone have any other ideas? See which fraction is closer to 1/2. How many tenths equals 1/2? Five. How many more tenths do I need to make 1/2? One. How many sixths do I need to make 1/2? Three. How many more sixths would I need to make 1/2? One. Do you know how I can decide which fraction is bigger now? Compare the part size. Are sixths or tenths smaller? Tenths. Do you agree? Yes. So which fraction is closer to 1/2? 4/10. Do you agree? Yes."
Now compare 2/8 and 4/10. (Continue modeling the problem on the board with fraction strips while students work with their strips at their desks.) Ask the students, "Is 2/8 greater than or less than 1/2? Is 4/10 greater than or less than 1/2? Can I tell which is larger yet? How many eighths equals 1/2? How many more eighths would I need? How many tenths equals 1/2? How many more tenths would I need to make 1/2? Are eighths or tenths smaller? So if I need two more eighths or one more tenth, can you see which one is closer to 1/2?"
Now have the students compare 5/6 and 3/4. (Continue modeling the problem on the board with fraction strips while students work with their strips at their desks.) I would ask, "Is 5/6 greater than or less than 1/2? Is 3/4 greater than or less than 1/2? Do you agree that they are both greater than 1/2? Should I compare them to 1/2 or is there another number to which I can compare them? Do you agree that I can compare them to one whole? How close is 5/6 to one whole? How close is 3/4 to one whole? Do you agree that I need one more unit of each to make one whole? Are sixths or fourths smaller units? So is 5/6 or 3/4 closer to one whole?" Now I would have the students complete Appendix 3 at their desks.
Activity 3:
The third activity is designed to develop the concept of equivalent fractions. It will utilize paper folding as a method of partitioning. Give the students a rectangular piece of paper. Establish that an identical rectangle represents one whole. Ask the students to fold it to make halves. The students may fold the paper different ways as long as the two halves have equal area. Share the different ways the students created their halves by drawing a similar representation on the board. Then ask the students to identify 1/2. Tell the students to now fold the paper again to make fourths. Again share the different ways the students made fourths. Then ask the students how many fourths are equal to one half. They should tell you two. How many fourths would it take to make one whole? They should tell you four. Ask your students to work with their neighbor and show you how many fourths there are in 3/2. They should tell you six, as illustrated with the following paper folds:
The problem mentioned and modeled above is important to familiarize your students with the notion that fractions can be larger than one whole. A good way to accomplish this goal is to include a problem or two that is greater than one whole as you work through all of the different concepts with fractions.
Next I would ask the students to work independently again with their own rectangle. I would ask them to show me how many eighths are in one fourth. The students would have to fold their rectangle again. They should show you two. I would check for understanding by asking, "How many eighths are in two fourths? How many eighths are in three fourths?" Then I would give the students a copy of the worksheet in Appendix 4. The students should be able to move from the concrete materials to a pictorial representation. Notice how the pictures in the exercise match the rectangular squares used in the lesson. Different materials can be used on different days, but within a lesson be consistent with the models and representations. Notice at the bottom of the worksheet, that the students are challenged to determine the algorithm on their own.
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