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Modeling Math for Mastery. Libby Serkies, MA Ed, PhD Candidate Central Illinois Adult Education Service Center L-Serkies@wiu.edu. Number Sense is….
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Modeling Math for Mastery Libby Serkies, MA Ed, PhD Candidate Central Illinois Adult Education Service Center L-Serkies@wiu.edu
Number Sense is… …an awareness and understanding about what numbers are, their relationships, their magnitude, the relative effect of operating on numbers, including the use of mental mathematics and estimation. Fennel & Landis, 1994
According to the experts… It is imperative to teach students the meaning of what they are doing when they manipulate numbers during arithmetic computations. Meaning not only increases the chances that information will be stored in long-term memory, but also gives the learner the opportunity to change procedures as the nature of the problem changes. Without meaning, students memorize procedures without understanding how and why they work. As a result, they end up confused about when to use which procedure. - Dr. David Sousa
Adding fractions Fractions with like denominators:
Adding fractions Fractions with like denominators:
Adding fractions Fractions with like denominators: 4 5
Adding fractions Fractions with unlike denominators: How do we combine these two fractions when they don’t have the same terms?
Step 1: Create 2 fraction bars of the same size, and divide one with horizontal lines to denote the fraction we are using. Divide the other with vertical lines to denote the fraction we are using.
Step 2: Color in the fractions to represent the numerators of both fractions:
Step 3: Transfer the marks (lines) from the right fraction to the left:
Step 3 - part II: Then transfer the marks from the left fraction to the right: Now we have common terms!Each box in both fractions is the same size.
Step 4: Next count & move all the colored boxes from one fraction to the other.
Step 5: Next, count the number of colored boxes. This is your new numerator. 3 1 2 4 5 8 6 7 9 10 11 12 13 13
Step 6: Next, count the total number of boxes. This is your new denominator. 3 1 2 4 5 8 6 7 9 10 11 12 13 14 15 16 17 18 19 20 13 20
Extension Ask your students: • What can you do to 5 to get 20? • (Multiply by 4) • What can you do to 4 to get 20? • (Multiply by 5)
Subtracting Fractions Begins the same way adding fractions does - look for or create common terms:
Step 1: Create models to show fractions:
Step 2: Color in the representations of each fraction:
Step 3: Construct common terms - transfer marks:
Step 4: Remove the smaller number of colored boxes from the larger:
Step 5: Count the # of originally colored boxes left from the first fraction. This is your new numerator: 4 3 2 4 1
Step 6: Count the # of total boxes in the first fraction. This is your new denominator: 4 15 3 2 4 5 1 8 7 6 10 9 11 12 13 14 15
Step 6: Solve: Simplify if necessary Remember to ask your students what they NOW know about subtracting fractions with unlike denominators! 4 15
Step 1: Draw representations of the fractions we are using:
Step 2: Transfer marks and portions of one fraction to the other:
Step 3: The denominator of the product is the total number of pieces our diagram has: 6 2 3 1 5 4 6
Step 4: The numerator of the product is the number of pieces that are shaded twice: 2 6 2 3 1 5 4 6
Step 5: Finish the problem: 1 2 or 6 3
By extension… ASK: What do you now know about multiplying fractions?
Dividing fractions (modeling) DIVIDEND DIVISOR
Step 1: Draw a representation of the dividend-- 1/4 and the divisor-- 2/3.
Step 2: Draw a representation of the dividend -- 1/4 and the divisor -- 2/3.
Step 3: Transfer the markings from one fraction to the other:
Step 4: Number the originally shaded tiles - this is your new numerator: 3 = 1 2 3
Step 5: Number the tiles shaded by the divisor - this is your new denominator: 3 4 2 = 3 1 8 5 7 6 8
Step 6: Combine - 123 4 5 678 1 2 3
Finally: Check your answer with “keep-change-flip” Keep Change FLIP
