1. The line segment represents 1 2 / 3 km. Extend the line to represent 3 3 / 4 km. Be as accurate as you can without measuring the actual length of original line segment. A Problem for You 1 2 / 3 km

2. What do you think is the math concept for which this task is trying to provoke? What key ideas are necessary for solving this problem? Follow-up Questions 1 2 / 3 km 1 km How? Cut into 5 pieces. 1 km 3 / 4 km Why 5? 1 2 / 3 = 5 / 3 Referent unit (1km) Mixed num. - improper fraction conversion

3. A Second Solution Follow-up Questions 2 1 / 4 units 1 unit How is 3 3 / 4 related to 1 2 / 3 ? How many times 1 2 / 3 fit into 3 3 / 4 related to 1 2 / 3 ? 3 3 / 4  1 2 / 3 = 9 / 4 = 2 1 / 4 1 unit 1 / 4 unit Fraction as quotient of a division Division as repeated-subtraction What can we learn from this solution?

4. A Third Solution Follow-up Questions What can we learn from this solution? What is the ratio of 3 3 / 4 to 1 2 / 3 ? 3 3 / 4 : 1 2 / 3 15 / 4 : 5 / 3 45 / 12 : 20 / 12 45 : 20 9 : 4 9 / 4 : 1 9 : 4 9 segments Ratio as multiplicative comparison Common denominator 4 segments

5. A Third Solution Follow-up Questions What can we learn from this solution? What is the ratio of 3 3 / 4 to 1 2 / 3 ? 3 3 / 4 : 1 2 / 3 15 / 4 : 5 / 3 45 / 12 : 20 / 12 45 : 20 9 : 4 9 / 4 : 1 9 : 4 Ratio as multiplicative comparison Common denominator Any ratio can be written as a unit ratio, e.g. 9 / 4 : 1 1 unit 2 1 / 4 units