1. Integer Rod Operations Multiplying and Dividing

2. Representing Fractions Using Bars  How do we represent fractions using integer bars? Part to whole Whole changes as necessary to make equivalents  A train is two rods put together – ALL trains must have at least one E in them  We will ALWAYS use the least number of bars possible to make a representation  Do NOT draw more lines on representations than necessary

3. Six Steps Required 1.Represent the fraction with the smallest and least number of rods possible 2.Race the denominators to a tie. This will ALWAYS take 3 rows – the new common denominator is at the bottom

4. Six Steps Required - Continued 3.Represent the fraction using the “race” as a guide using the common denominator rod and the least number of rods possible for the numerator 4.Do the operation

5. Six Steps Required - Continued 5.Simplify the representation – least number of rods possible 6.Interpret the representation in #5 as a fraction number answer

6. Race Representation: Multiplication  Use one common denominator bar  The numerator will represent the SECOND factor only  Do NOT represent the first factor

7. Do the Operation: Multiplication  Use one common denominator bar  Place the numerator of the second factor directly above the common denominator  Look at the first factor in the problem Treat the numerator of the second factor as the denominator of the first factor Place a bar above it that represents the numerator for the first factor  Total of 3 rows

8. Simplify the Representation: Multiplication  Use one original common denominator bar  Place the top bar from the step above directly above the common denominator bar  Represent all with the least number of rods possible  Total of 2 rows

9. Multiplication – Concrete A. B. C. D. E. F. A. B. C. D. E. F.

10. Multiplication – Semi-Concrete A. B. C. D. E. F. A. B. C. D. E. F.

11. Multiplication – Semi-Abstract

12. Do the Operation: Division  Use one common denominator bar  Place the divisor (the factor) directly above the common denominator  Place the dividend (the product) directly above the divisor (the factor)  Total of 3 rows

13. Simplify the Representation: Division  Use the divisor (the factor) as the new common denominator  Place the dividend (the product) directly above the divisor (the factor)  Represent all with the least number of rods possible  Total of 2 rows

14. Division – Concrete A. B. C. D. E. F. A. B. C. D. E. F.

15. Division – Semi-Concrete A. B. C. D. E. F. A. B. C. D. E. F.

16. Division – Semi-Abstract