Strategies
The "Unit"
Understanding the unit is also an important part of the process of unitizing or chunking.
Using the image above, I would want my students to be able to identify or "see" in the image one half, one fourth, and one eighth. Developing the students' ability to unitize will help them to be more flexible in their thinking and recognition of the unit. Once the students were skilled in this concept I could provide them with some more challenging unitizing pictures. There are many great examples in Susan Lamon's book that I have referenced in the bibliography. I could also vary this activity by providing the fraction sometimes and other times asking an open-ended question about the different units or chunks that the image could be seen in the image. For example, given a unitizing image, I might ask students "who can see a chunk? Can you show the class the chunk you found and name it? Who sees another chunk? Are there any more chunks?" By providing a more open-ended question, the students have to apply their knowledge of fractions and unitizing more deeply than when a fraction is given.
Unit Fractions
I would model unit fractions with fraction strips, circular fraction pieces, a collection of objects, and a number line. Once students have practiced counting on the number line, I would then move into comparing different size fractions on separate number lines. It is important to keep the size of one whole unit consistent, so students can build their conceptual knowledge of the relative size of fractions. By comparing the number lines below, students can see that one third is larger than one fourth.
The number line is a particularly valuable model since it encourages students to think about the linear representation. Providing my students with different visual representations of fractions will help to develop the flexibility in which they are able to think about fractions.
Improper fractions will be included in all of the different models from the start, so the students understand that fractions can be bigger than one whole. Counting out loud and numbering number lines using different unit fractions is important to help students understand how multiple copies of the unit fraction work.
General Fractions
It is important for students to understand unit fractions before moving on to general fractions. Students need to have a good grasp of the role the denominator plays in understanding the relative size of unit fractions. This understanding is the starting point for conceptualizing the relative size of general fractions.
There are several key concepts and strategies to help students compare the relative size of fractions. The numerator tells the number of copies and the denominator tells the size of the part. The simplest types of fractions to compare are those with the same denominators. In that case, students need only to look at the numerators to compare. For example 4/8 and 6/8, the fact that these two fractions have the same denominators means that the size of the pieces, or the unit, is the same. Thus, they just need to compare four pieces to six pieces, and so can conclude that 6/8 is bigger than 4/8. It is important for the students to be comfortable with this concept.
The next simplest types of fractions to compare are those whose numerators are the same. For example with 5/6 and 5/12, both numerators are five. Now the students need to look to the denominator. Remember, the denominator tells the size of the pieces. Therefore, a larger number in the denominator means the size of each piece is smaller. So 5/6 is bigger than 5/12 because I have the same number of pieces and sixths are bigger.
Another key concept is for students to compare a fraction to one half or one. If one fraction is larger than one half and the other fraction is less than one half, then it is very easy to determine which is bigger. Take for example 2/6 and 8/12. Since 2/6 is less than 1/2 and 8/12 is greater than one half, it follows that 2/6 is less than 8/12. However, if both fractions are either smaller than one half or larger than one half, then the students must consider the part size. For example if I am comparing 5/6 and 7/8, both fractions are greater than one half. I notice that both fractions are missing one piece. Now I must look at the part size. I know that sixths are larger than eighths. I can determine that 5/6 is less than 7/8.
As I am working with my students, I know that they must grasp these concepts to really develop a strong understanding of what fractions represent. As I teach I must keep these notions in mind and give my students many hands-on opportunities so they can develop this type of deep understanding.
Equivalence
Again I plan to allow the students to explore this concept primarily through paper folding activities based on the Singapore Math model. An example of this strategy would be to give each student a square or rectangular piece of paper. I would ask the students to fold the paper into four equal parts. I would ask them to name the fraction that two of the parts represent. Then I would ask the students to further fold the paper to produce eight equal parts. Then I would ask them how many eighths is equal to two fourths.
I would provide my students with many opportunities for paper folding activities. I would be very careful to scaffold the paper folding gradually moving from easy to more challenging problems. An easier problem would be 2/4 equals 4/8. A more difficult problem would be 6/8 equals 9/12 because the answer is not as obvious. However, the answer can be figured out in the same way by folding paper or by drawing bars.
Partitioning strategies will also be used to develop the concept of equivalent fractions. For example, I would present the students with the bars below and ask them to partition them into equal parts.
Example A:
In example A, each segment of the first bar would need to be subdivided into two equal parts. The second bar would remain the same. I would again provide my students with multiple opportunities to partition. Initially the problems would be with concrete objects such as a strip of paper. Then I would move the students to a pictorial representation. I would make sure that the concrete paper strip matched the pictorial representation exactly.
Example B:
In example B, the problem depicts bars divided into halves and thirds. This would be a good introduction to partitioning both bars at the same time. I also would scaffold the example with questions to help the students. I might ask the students, "Can you see where one half would be on the bottom bar? Can you see where one third and two thirds would be on the top bar? Can you finish breaking each bar into equal parts? Into how many parts did you break each bar? Each bar would be partitioned into sixths.
Example C:
In example C, the partitioning strategy is more challenging. Each segment of the first bar would have to be further subdivided into five equal parts. Each part of the second bar would have to be subdivided into four equal parts. Thus both bars will end up divided into twenty equal parts. This type of partitioning will lead to the students' conceptual development of common denominators. The same partitioning strategies are also used when solving word problems using Singapore bar models.
Mixed Numbers and Improper Fractions
The next task would be to teach the students mixed and improper numbers. To develop understanding I will use paper folding, square tiles, pictorial images, number lines, and Cuisenaire® rods. Cuisenaire® rods are wooden or plastic rod-shaped pieces of different lengths and color. The length of the rod ranges from one unit to ten units. Providing a variety of different contexts will enable the students to better generalize the information and apply it. While developing proficiency changing mixed numbers to improper fractions back and forth, students will also hone their skills adding unit fractions with like denominators. Each gray segment below represents one third. So if I add 1/3 +1/3 +1/3 +1/3 +1/3, the result is five thirds (5/3) or one and two thirds (1 2/3), both of which can be pointed out in the rods illustrated below. I would provide several examples that did not require simplifying.
A slightly more difficult problem would be to understand say, 6/4. Following with more difficult problems like this one will help scaffold students' learning and understanding. In this problem, the students would have to transform the improper fraction into a mixed number (1 2/4) and then simplify it (1 1/2). Again, several examples of this type would be required to build understanding.
Adding and Subtracting Fractions
The subsections below provide an outline of how to scaffold instruction for adding and subtracting fractions. The problems are carefully sequenced and should be taught in order. I recommend using paper folding, Cuisenaire® rods or fractions pieces as concrete materials. Rectangles and squares will work well for pictorial representations.
Adding and Subtracting with Like Denominators
The first concept to develop in fraction addition and subtraction is adding and subtracting fractions with like denominators. Through the paper folding activities previously used with unit fractions, the students should already be adept in adding unit fractions. I would carefully sequence the problems that I used in class and begin with examples where the sum is less than one whole such as three eighths plus two eighths:
This problem requires straight addition without any simplification. The next type of problems would have a sum of less than one, but would require regrouping as in one sixth plus three sixths:
Finally, I would then move to a problem with a sum greater than one without regrouping such as three fourths plus two fourths. After providing several examples at this level, I would reveal the most difficult type of problem in this sequence. It would require the students to calculate a mixed number sum with regrouping. An example would be nine twelfths plus six twelfths, which can be illustrated using the bars below:
By looking at the shaded bars above, if you add the shaded segments the problem represented would be:
I could either work on subtraction problems of similar difficulty interchangeably with the addition, or I could work on addition first, and then scaffold the subtraction problems in the same manner. A sample sequence of subtractions problems would begin with a one step problem such as 5/6 - 4/6 = 1/6. The next problem would be slightly more difficult, requiring simplification, such as 7/8 - 3/8 = 4/8 = 1/2. An example of a challenging problem is 1 6/8 – 4/8 = 1 2/8 = 1 1/4. I would I believe either order would be effective. If addition and subtraction were handled in isolation I would provide some mixed practice for my students once both operations were covered.
Adding and Subtracting with Related Denominators
The next concept to tackle is adding and subtracting fractions with related denominators. In this type of problem while the denominators are different they are related in such a way that the smaller one can be transformed into the larger one with knowledge of equivalent fractions. In particular, the denominator of one fraction is a multiple of the other, such as is the case with 3/5 and 7/10. Once again, paper folding, pictorial representations and Cuisenaire rods will be used to model and depict the problems. Additionally, I would use the strategy of listing out equivalent fractions until a common denominator was discovered. Then the two fractions would have like denominators and could easily be added or subtracted together. I would begin with two unit fractions. In this type of problem, only one unit fraction would need to be transformed into a general fraction. An example is:
I would proceed to a similar problem requiring one of the fractions to be transformed, but that would also include simplifying the answer. Two sixths plus one third would fit nicely sequentially.
Once the students were successful with this level of problem, I would provide a similar problem, but where the answer was an improper fraction that had to be changed into a mixed number.
Remember to begin with concrete modeling such as fraction pieces, followed by pictorial representations such as rectangles that match the fraction pieces. Finally, students should solve the algorithm using paper and pencil.
The last type of problem in the sequence would include an improper fraction that when changed into a mixed number required simplification.
Again, the problems would be carefully sequenced to gradually increase the complexity.
Adding and Subtracting Fractions with Unlike Denominators
The students should by now feel pretty comfortable with making like denominators. In the next series of problems, the denominators are not related. Therefore both fractions will have to be renamed with a common denominator. An example is:
I would provide the students with several similar examples in which both fractions had to be renamed, but further simplification was not required.
Then I would have the students solve similar problems that also required simplifying the answer. For example:
Again, it is important to provide several problems of this type. The next step would be to solve a problem where the sum was greater than one, but did not require simplification.
The most difficult kind of problem will include changing both fractions to a common denominator, changing the resulting improper fraction to a mixed number and simplifying the answer. An example is:
Once again, providing your students with several problems of each type is very important for them to conceptual the process. With the many variations of problem types, it is important to develop a deep understanding, so the students can fully understand how to change denominators, when to change an improper fraction to a mixed number and when to simplify.
By carefully scaffolding the problems in a natural progression from easy to more challenging, the students will use their prior knowledge to help them find a solution. The structure of the unit is designed around the philosophy of Singapore Math. It is designed to ensure the success of struggling and average students, while offering opportunities to differentiate the problems for the high achieving students.
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