Integer Rod Operations

1. Integer Rod Operations Adding, Subtracting, Multiplying, and Dividing

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

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

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

5. Do the Operation: Addition • Use one common denominator bar • Place both numerators (in order, from left to right) directly above the common denominator • Total of 2 rows

6. Simplify the Representation: Addition • Use one common denominator bar • Represent all with the least number of rods possible • Total of 2 rows

7. Addition – Semi-Concrete A. B. C. D. E. F. A. B. C. D. E. F.

8. Addition – Semi-Concrete A. B. C. D. E. F. A. B. C. D. E. F.

9. Adding – Semi-Abstract

10. Do the Operation: Subtraction • Use one common denominator bar • Place the minuend (the sum) directly above the common denominator • Place the subtrahend (addend) directly above the minuend (the sum) • Use dashed lines to indicate the difference (missing addend) next to the subtrahend • Total of 3 rows

11. Simplify the Representation: Subtraction • Use one common denominator bar • Place the difference (missing addend) directly above the common denominator bar • Represent all with the least number of rods possible • Total of 2 rows

12. Subtraction – Semi-Concrete A. B. C. D. E. F. A. B. C. D. E. F.

13. Subtraction – Semi-Concrete A. B. C. D. E. F. A. B. C. D. E. F.

14. Subtraction – Semi-Abstract

15. Race Representation: Multiplication • Use one common denominator bar • The numerator will represent the SECONDfactoronly • Do NOT represent the first factor

16. 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

17. 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

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

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

20. Multiplication – Semi-Abstract

21. 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

22. 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

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

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

25. Division – Semi-Abstract

26. 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