FRAMEWORKS:
NO.3.6.2
Develop and analyze algorithms for computing with fractions (including mixed numbers) and decimals and demonstrate, with and without technology, computational fluency in their use and justify the solution
NO.3.6.3
Solve, with and without appropriate technology, multistep problems using a variety of methods and tools (i.e., objects, mental computation, paper and pencil)
Objectives:
Grade Level(s): 4, 5, 6
Subject(s):
Description: This lesson is the result of work completed in the class Mathematics and Science for Elementary Teachers at Elon College.
Lessons were prepared for and implemented in 4th grade classrooms at Haw River Elementary School, Haw River, NC.
Goals: Use concrete pictorial models to represent fractions and mixed numbers; relate symbols to the models. Use models and pictures to compare fractions including equivalent fractions and mixed numbers; explain the comparison.
Objectives:
1. Describe the importance of fractions in everyday life.
2. Define equivalent fractions and give examples.
3. Recognize fractions that are equivalent.
4. Manipulate materials to show several examples of equivalent fractions.
Materials: 5 Large poster board pizzas (one cut into 3rds, 4ths, 6ths, 8ths, and 12ths.) Small paper pizzas sliced into 12ths (for each student) set of fraction cards(1/4, 1/3, 2/3, 2/6, 1/12)/group pizza boxes for storing materials/group (I received free small
Procedure: Write the words equivalent fractions on the board. Have the students discuss what fractions are, and then discuss what they think equivalent means. Have students discuss real world situations where equivalent fractions may be important. (Our class has many Spanish speaking students, so we included Spanish terms, Fraccion Equivalente)
Three of the large pizzas (4th, 8th, 12ths) should be placed on the board. Have three students come to the board and stand in front of each pizza. The student with 4 slices should be asked to take 1/4 of their pizza, the student with 8 slices should be asked to take 2/8 of their pizza, and the student with 12 slices should be asked to take 3/12 of the pizza. They will then write the fraction of the pizza that they "ate" or took away.
This a great discovery activity where the class can compare the three pizzas and talk about what they observe. The class will discuss how they can tell that the fractions are equivalent, and strategies, such as grouping, that will help them recognize equivalent fractions.
Next, replace the pizzas with 4 and 8 slices with the pizzas with 3 and 6 slices. Have three more students come to the board and have these students each take 1/3 of their pizza. Students should be using the grouping strategies introduced to them.
Next, the class will be introduced to the activity of munching fractions through demonstration on the board using the large pizza cut into 12 slices. Teacher can draw fraction cards and then have students take that fraction from the pizza on the board. For example, if a student has the fraction card 2/3, then the student would group the pizza slices into 3 groups and see that they would take 8/12 of the pizza, which is equivalent to 2/3. Explain that the object of the activity is to "eat" all of the pizza quicker than your partners.
Next, divide students into groups of two or three and give each group a pizza box with a set of fraction cards and a bag of pizza pieces (each student should have 12 slices) Have students place the fraction cards in their pizza boxes.
Students take turns shaking the pizza box and drawing out fraction cards. The student must decide how much of their pizza is equivalent to the fraction on the card and "eat" that part of the pizza. Have students check each other. After each student decides how much of their pizza to eat, have them write down the equivalent fractions and draw a pictorial representation of each fraction. Then it is the next students turn. The activity continues in this manner until one student has "eaten" their pizza. Then students start over again.
Remind students that they must keep drawing at the end of the activity until they find a fraction that is equivalent. For example, if they have 1/12 of their pizza left, they must keep drawing until they select a 1/12 piece, or a fraction that is smaller.
Once students have finished the activity, have students place all materials back into pizza boxes.
(This activity can be extended to include other equivalent fractions such as 5ths, etc, by using an equivalent cake eating contest. In this activity a rectangle would be divided into equal parts. The possibilities are endless!)
Assessment: Have students review the equivalent fractions that were presented in the lesson: 1/4=3/12, 1/3=4/12, 2/3=8/12, 2/6=4/12, 1/12=1/12. Students will turn in the sheets that they were writing down their equivalent fractions on.
Teacher may also ask students concrete situational examples such as:
Tonight you are having pizza for dinner. The pizza has _____ slices. You mother gave you enough money for 1/4 of the pizza. How many pieces would you get? What fraction of the pizza would this represent?
Notes on the objectives:
If the fractions have the same denominator, their sum is the sum of the numerators over the denominator. If the fractions have the same denominator, their difference is the difference of the numerators over the denominator. We do not add or subtract the denominators! Reduce if necessary.
Examples:
3/8 + 2/8 = 5/8
9/2  5/2 = 4/2 = 2
If the fractions have different denominators:
1) First, find the least common denominator.
2) Then write equivalent fractions using this denominator.
3) Add or subtract the fractions. Reduce if necessary.
Example:
3/4 + 1/6 = ?
The least common denominator is 12.
3/4 + 1/6 = 9/12 + 2/12 = 11/12.
Example:
9/10  1/2 = ?
The least common denominator is 10.
9/10  1/2 = 9/10  5/10 = 4/10 = 2/5.
Example:
2/3 + 2/7 = ?
The least common denominator is 21
2/3 + 2/7 = 14/21 + 6/21 = 20/21.
To add or subtract mixed numbers, simply convert the mixed numbers into improper fractions, then add or subtract them as fractions.
Example:
9 1/2 + 5 3/4 = ?
Converting each number to an improper fraction, we have 9 1/2 = 19/2 and 5 3/4 = 23/4.
We want to calculate 19/2 + 23/4. The LCM of 2 and 4 is 4, so
19/2 + 23/4 = 38/4 + 23/4 = (38 + 23)/4 = 61/4.
Converting back to a mixed number, we have 61/4 = 15 1/4.
The strategy of converting numbers into fractions when adding or subtracting is often useful, even in situations where one of the numbers is whole or a fraction.
Example:
13  1 1/3 = ?
In this situation, we may regard 13 as a mixed number without a fractional part. To convert it into a fraction, we look at the denominator of the fraction 4/3, which is 1 1/3 expressed as an improper fraction. The denominator is 3, and 13 = 39/3. So 13  1 1/3 = 39/3  4/3 = (394)/3 = 35/3, and 35/3 = 11 2/3.
Example:
5 1/8  2/3 = ?
This time, we may regard 2/3 as a mixed number with 0 as its whole part. Converting the first mixed number to an improper fraction, we have 5 1/8 = 41/8. The problem becomes
5 1/8  2/3 = 41/8  2/3 = 123/24  16/24 = (123  16)/24 = 107/24.
Converting back to a mixed number, we have 107/24 = 4 11/24.
Example:
92 + 4/5 = ?
This is easy. To express this as a mixed number, just put the whole number and the fraction side by side. The answer is 92 4/5.
To multiply a fraction by a whole number, write the whole number as an improper fraction with a denominator of 1, then multiply as fractions.
Example:
8 × 5/21 = ?
We can write the number 8 as 8/1. Now we multiply the fractions.
8 × 5/21 = 8/1 × 5/21 = (8 × 5)/(1 × 21) = 40/21
Example:
2/15 × 10 = ?
We can write the number 10 as 10/1. Now we multiply the fractions.
2/15 × 10 = 2/15 × 10/1 = (2 × 10)/(15 × 1) = 20/15 = 4/3
When two fractions are multiplied, the result is a fraction with a numerator that is the product of the fractions' numerators and a denominator that is the product of the fractions' denominators.
Example:
4/7 × 5/11 = ?
The numerator will be the product of the numerators: 4 × 5, and the denominator will be the product of the denominators: 7 × 11.
The answer is (4 × 5)/(7 × 11) = 20/77.
Remember that like numbers in the numerator and denominator cancel out.
Example:
14/15 × 15/17 = ?
Since the 15's in the numerator and denominator cancel, the answer is
14/15 × 15/17 = 14/1 × 1/17 = (14 × 1)/(1 × 17) = 14/17
Example:
4/11 × 22/36 = ?
In the solution below, first we cancel the common factor of 11 in the top and bottom of the product, then we cancel the common factor of 4 in the top and bottom of the product.
4/11 × 22/36 = 4/1 × 2/36 = 1/1 × 2/9 = 2/9
To add fractions with different denominators, you must first change both denominators to be the same. To do this you will need the skills you learned in Unit 5. Once you have both denominators the same, then you just add the numerators. Sometimes one denominator does not divide evenly into the other, as they did in Unit 6. In this case you must change BOTH denominators into a number that both original denominators will divide into evenly. An easy way to find this new denominator is to multiply the 2 original denominators together to get you new one. EXAMPLES:

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To add fractions with different denominators, you must first change both denominators to be the same.  EXAMPLES: 
To do this you will need the skills you learned in Unit 5. Once you have both denominators the same, then you just add the numerators. If the denominator of one fraction divides evenly (no remainder) into the other, then change the fraction with the smaller denominator to a fraction with the larger denominator, then add numerators. The denominator will be the LARGER of the 2 denominators. 
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29)  30)  This problem is hard! 
31)  This problem is hard!  32)  This problem is hard! 