Lesson objective: Interpret \(({a\over b})\times q\) as "a" parts of a partition of q into b equal parts.
Students bring prior knowledge of multiplying fractions by whole numbers, in which they typically replicated a given fraction some whole number of times. This prior knowledge is extended to multiplying whole numbers by fractions as students need to partition a distance into equal parts and then take some number of those parts to solve a problem. A conceptual challenge students may encounter is understanding that we can solve this problem by connecting division by a number n to multiplying the number by \(\frac1n\), and believing that these give the same result.
The concept is developed through work with a number line, which helps students see how making groups of equal shares can be easier to do after the original group has been partitioned.
This work helps students deepen their understanding of the connection between multiplication and division, as they may use division as part of their solution strategy for a multiplication problem.
Students engage in Mathematical Practice 4 (model with mathematics) as they use a number line to partition a whole into equal parts, so they can then go on to determine the amount of a certain number of those parts. Students will need support in connecting their model to multiplying a whole number by a fraction.
Key vocabulary:
- partition: divide into parts
- whole: the value to be partitioned