Lesson #2
Objective: Students will learn to add and subtract fractions using a number line.
Timing: 2-3 – 50 minute periods
Materials: Rulers, graph paper, pre – made number lines (optional)
Opening: Write 1/3+3/4 on the board or the overhead. Have students discuss in their groups different ways to solve this problem. Groups should attempt to come up with at least 2 different ways. (algorithm, array(s), number line, manipulatives, drawings, etc…) Make note of the methods used. Keep chart if possible. Have groups share out in front of the class.
Student Experimentation/Documentation:
Give each table group rods and cubes and ask them to represent the problem with the manipulatives. Teacher should do the same on the overhead. Discuss what the problems with this representation might be. (Unlike denominators). Discuss the idea of like terms and why adding these two fractions is difficult without common denominators. (Be open to the possibility that a student will show how to do this without finding a common denominator. Using an array will allow a student to shade 1/3 of a rectangle horizontally and 3/4 of a rectangle vertically. If they count all the shaded area (count 2 for each overlapping grid piece) they will have solved the problem. The interesting part of this problem is when the students discover that they did actually find a common denominator, (12 grid pieces when dividing a rectangle into thirds horizontally, then into fourths vertically)).
1/3 is denoted by diagonal lines going from the upper left to lower right of each cell in the top row. 3/4 is denoted with diagonal lines going from the lower left to the upper right in the first 3 columns of the array. If I add the number of cells that have "fill" I get the answer 13/12.
Now have the students work with you when transitioning to number lines. If they divide a line into thirds (black lines below), then subdivide each third into fourths (blue lines between each third), they will have created twelfths and can add the two fractions together. It will be important initially to draw a second number line representing fourths (black) and then subdivide each fourth into thirds (blue), they will see that 1/3 = 4/12 and 3/4 = 9/12 then summing these 2 fractions becomes a matter of simple addition. This model works will with subtraction also. Take 2/3-1/4, if you approached the number line in the same way and subdivided it into twelfths, then you would merely start at 8/12 and move left on the line 3/12 spaces.
Practice: Have students practicing subdividing number lines to create common denominators. Begin with a series of related fractions on the first day. Then move into fractions that are not as clearly connected. Focus on length when adding or subtracting to continue giving a context to what we are doing.
Lesson #3
Objective: Students will multiply fractions using arrays and explore the difference between products of whole numbers and products of their unit fraction reciprocals.
Timing: 2 - 50 minute periods
Materials: Rulers, graph paper
Opening: Write 8 x 4 on the board or the overhead. Have students discuss in their groups different ways to solve this problem. Groups should attempt to come up with at least 2 different ways. (algorithm, array(s), number line, manipulatives, drawings, etc…) Make note of the methods used. Keep chart if possible. Have groups share out in front of the class. Discuss the relationship between a product and its factors.
Choose a representation of each type of solution/process to display in front of the class. End the displays with an array. If no one has drawn one, do it yourself. Ask the students if they recognize this method. Go into detail explaining how the array represents a multiplication problem. The array below is an example of this problem.
To solve a problem like this, students will need to create a grid by subdividing a rectangle into 8 parts vertically, (this can be done with precision by measuring out an 8" long line, then subdividing each inch,) and then dividing the rectangle horizontally into 4 parts( each 1 in. long ). This gives you a grid of 32 squares, or 8 x 4 = 32. Have students practice this type of solution a few more times and share drawing in front of the class.
Ask them to compare and contrast the similarities and differences between the 8 x 4 array and the 1/8 x 1/4 array. What can you say about the individual factors we are multiplying? (reciprocals). What do you notice about the products. Continue questioning until someone gets to the idea that the product of the fraction problem is smaller than each of it's factors and the product of the whole number problem is larger than the factors. Spend time discussing why this is. It will help students understand this concept, (a very important one, by the way).
Continue to work with whole numbers and their reciprocals to reinforce the idea that the product of a fraction multiplied by a fraction will give a product that is less than each of the factors being multiplied.
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