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Q&A #6232 |
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Hi Beverly, I think its great that you are looking for models to help you show "why dividing a negative fraction by a negative fraction will give a positive number." And I think that manipulatives and drawings are excellent tools. They are effective for use when teaching some of the concepts leading up to the example you cited in the subject line. However, tools have limitations. And so, I also believe this specific problem and others of this degree of difficulty should come at a time when these tools are no longer necessary. Prior to reaching this level of complexity, students should already have had experience using manipulatives, drawings, and pattern analysis in multiplying and dividing integers, and with multiplying and dividing positive fractions. Here are some lesson ideas I use with my students... Multiplication/Division of integers. Idea #1: My favorite manipulative for teaching the arithmetic of integers is tile spacers. (Tile spacers are used to keep the distance between tiles equal when someone is laying tiles for a floor, or a countertop, or a wall. You can buy them in bulk very inexpensively from hardware stores.) I like them because they are shaped like plus signs. One can trim a plus sign with a pair of scissors to make a negative. The spacers are soft plastic. I start my kids with the concept of making zero. There a lots of ways to make zero: one "+" and one "-" , two "+" and two "-", three "+" and three "-", ... I have them start with several zeroes on their working area to begin with. Example: 3 times -2 "3 times -2" can be interpreted to mean "put in 3 groups of -2." So their work area could look like + + + + + + + + + + - - - - - - - - - - (--) (--) (--) <----Put in So, 3 times -2 equals -6. Example: -3 times -2. If "3 time -2" means "put in 3 groups of -2," then "-3 times -2" can be interpreted to mean "take out 3 groups of -2." + + + + + + + + + + - - - - (- -) (- -)(- -) ---->Take away So, -3 times -2 equals +6. Idea #2: Pattern analysis. The "negative times a negative equals a positive" is a bit strange for many students, so I like to have more than one way of having kids look at this concept. Start with: Then multiply each integer by 2: . . . . . . . . . I ask them to look at the sequence 4 4 * 2 = 8 of numbers generated by the products. 3 3 * 2 = 6 Without too much trouble they say 2 2 * 2 = 4 numbers are even and the sequence is 1 1 * 2 = 2 decreasing. 0 0 * 2 = ? I ask what comes next? 0 -1 -1 * 2 = ? ...what comes next? -2 -2 -2 * 2 = ? ...what comes next? -4 -3 -3 * 2 = ? -4 -4 * 2 = ? With these patterns we support the idea . . . that a negative times a positive is . . . negative. . . . Now start with: And multiply each integer by -2: . . . . . . . . . 3 3 * -2 = -6 I ask them to look at the sequence 2 2 * -2 = -4 of numbers generated by the products. 1 1 * -2 = -2 With a little help they say the 0 0 * -2 = ? numbers are even and the sequence is -1 -1 * -2 = ? increasing. -2 -2 * -2 = ? I ask what comes next? 0 -3 -3 * -2 = ? ...what comes next? 2 -4 -4 * -2 = ? ...what comes next? 4 -5 -5 * -2 = ? ...what comes next? 6 . . . . . . With these patterns we support the idea . . . that a negative times a negative is positive. Division by fractions: Idea #3: Pictures or fraction squares. Example: 3 divided by 1/2 "3 divided by 1/2" can be interpreted to mean "how many 1/2 are in 3 wholes?" Start with 3 wholes. ------- ------- ------- | | | | | | | | | | | | | | | | | | ------- ------- ------- Divide each whole in half. ------- ------- ------- | | | | | | |-------| |-------| |-------| | | | | | | ------- ------- ------- There are 6 halves in 3. So 3 divided by 1/2 equals 6. Example: 3/4 divided by 1/2. "3/4 divided by 1/2" can be interpreted to mean "how many 1/2 are in 3/4?" If this is a whole,.........then this is a picture of 3/4. ------- ------- --- | | | | | | | | | |---|---| --> |---|--- | | | | | | | | ------- ------- ------- How many half's are in the 3/4? ------- | | ------- Answer: 1 complete half and 1/2 of a half. or 1 1/2. Idea #4: Unit analysis. Before we go on, we need to take a side trip into the analysis of the words, numerator and denominator. The word, numerator, comes from a Latin word which means to count. The word, denominator, comes from a Latin word which means to name. So, 3/4 tells us that we have 3 of a part called "fourth" and can be written as 3 fourths. Example: 3/4 divided by 1/2 3 fourths ---------- Since these fractions don't have the same name, we need 1 half to change the name of at least one of them. 3 fourths 3 fourths ---------- -----> ----------- The units (or names) divide out... 1 half 2 fourths So, 3/4 divided by 1/2 equals 3/2 (or 1 1/2). Example: 2/3 divided by 3/4. 2 thirds 2 (4 twelfths) 8 twelfths ---------- -----> ---------------- -----> ------------ 3 fourths 3 (3 twelfths) 9 twelfths So, 2/3 divided by 3/4 equals 8/9. This method, of course, requires that the students have a great deal of experience in equivalent fractions Now back to your original problem... -2/3 divided by -3/4. When my students and I do a problem of this complexity, I ask them to break this problem down into subproblems: signs and division. Then we combine the results of solving the subproblems to get a final answer. Hope this helps. -Jeanne, for the T2T service |
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