### Background information

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This background information belongs to both Falling Sums and Estimate

These applets help students develop their estimation skills and strategies for adding and subtracting two and three digit numbers quickly in their head. With Falling Sums, students are able to select the level of difficulty, (e.g., 0-20, 0-200, or 0-2000), as well as the operation(s) to focus (e.g., addition, subtraction, or both) and are asked to simply determine the broad range of their solution (e.g., with the 0-200 students must recognize if the solution is between 0-100 or 100-200). With Estimate, students must come within 5-10% of the actual solution in a very short period of time

Perhaps the most important aspect of these two applets is the development of estimation strategies. Given the subtle differences, these applets would work nicely together as a package. Falling Sums would seem to be a perfect introduction, beginning with sums less than 20 and asking students to simply estimate if the sum is greater than or less than a given value. The strategies that students must develop for the Falling Sums applet are different than those needed for the Estimate applet. For example, if working with sums less than 20 with Falling Sums, students must determine if the sum is greater than 10 or less than 10. One effective strategy students might construct is to simply recognize that when the two values are each less than 5 then the sum will always be less than 10. Likewise, if the two values are each greater than 5 then the sum will always be greater than 10. The only challenge to this strategy occurs when one is greater and one is less than 5. However, the resultant strategy for this situation is worthwhile since it develops their understanding of differences away from 5 (similar to developing an understanding of the mean). For example, if given 7+4 the student might recognize that 7 is 2 greater than 5 whereas 4 is only 1 less than 5, therefore the solution must be greater than 10. For the Estimate applet, students are asked to be more precise by typing in an actual value. Students' estimation must be within 5-10% or the students will be asked to refine their estimate. Because students are asked to type in a value within 5-10% of the actual sum, students are not able to employ the strategy used for Falling Sums. Instead, students must develop a strategy for adding 3 digit numbers. Indeed, a much more challenging task.

In addition to developing estimation skills, Falling Sums would seem to be a wonderful activity to help students commit to memory the sum of various addition problems, especially those under 20, through timed repetition. It seems that with a little practice students could become quite good at finding the sums for a large number of problems in a short period of time. These games are both fun and challenging, it is easy to see how students could spend lots of time playing with these applets and learning lots of math in the process.

Teaching Considerations
Because of the importance of students developing rich estimation strategies, these applets would be perfect for having the class create a best strategies booklet. After working with these applets for a considerable amount of time, students could share the strategies that they use with the other students. Some students might find it difficult to identify the strategies that they use and would benefit from teacher led assistance. For example, teachers might want to present various problems from the applets to the whole class and ask them to solve on their own, while also considering the strategy they are using. The class could then openly discuss the various strategies students used to find their solution.
Given the similarities and differences in these two games (and student ability level), it seems like a natural progression for students to begin with the "Falling Sums" applet before moving on to the more challenging and more precise "Estimate" applet.