Last week in one of our math mini lessons, students were challenged to determine if 30 x 4 = 15 x 8. After a few minutes of finding the product of each expression, students determined that each had a product of 120. Students were also challenged to see if they noticed anything else that these two problems had in common. After a few moments, many students' faces lit up! Many students noticed that half of 30 is 15 and 4 doubled is 8! After some conversation about this, students were challenged to explore this concept during the work period to *prove if this theory is true all the time. (Seeking to make a generalization or proof across all problems is one of the cornerstones of algebra. Without even knowing it, our third grade students are working on their algebraic thinking every day. How cool! Now, back to doubling and halving...) The following chart displays the two "must do" problems that they students were expected to complete during the work period.
In closing, students determined that doubling and halving could work for all problems; however, it might not be the most efficient for all problems. For example, we discussed 11 x 23. When halving either one of these factors (11 or 23), the mathematician will be challenged to work with fractional parts. Is this impossible? Certainly not. Is it the most efficient way to solve? Probably not.
Student Challenge:
Try solving the following using the "Doubling and Halving" strategy: 16 x 25, 15 x 32, 12 x 30.
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