### Background information

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This background information belongs to both Numberboard and Number Factory

These two applets focus on the operations of addition, subtraction, multiplication, division (by means of ratio), and parentheses. In both versions of this game, players are given 4 numbers and are asked to get as close as possible to the target number using all four numbers and any of the provided operations. Students score more points the closer they can get to the target number.

A natural outcome of these games is that students are likely to perform many different calculations and create many different equations before coming to a final conclusion. For example, if students are provided with the numbers 6, 8, 2, and 1 and a target of 24, students might first try to add them all together and find that the sum is only 17. Next, students might try multiplying the 6 and the 2 and then adding the 8 and the 1 to get 21 - closer, but not 24. Because they recognize that their solution is still too low, students might then try multiplying the 2 and the 8 and then adding the 2 and the 1 for a total of 23 - again closer, but still not 24. Should students continue their search, they might recognize that they can multiply the 6 and the 8 and then divide this product by the product of 2 and 1 to reach their target of 24. Indeed, a long process to reach the answer. However, students gain much from their experience. Students learn how to alter their equation to either increase or decrease their solution and how to create formal equations that match their thinking.

In addition to providing students with a fun way to review the basic operations, this applet also helps to sharpen students' skills related to the order of operations. Because the game provides four numbers and the use of four operations and parentheses, students must consider the order of operations when constructing their equations. Considering the earlier example of how students might reach the target of 24, some students might be challenged by getting close to the target, but have even more trouble constructing their equation using the formal symbols. The use of parentheses and careful consideration of the order of operations will be of paramount importance. One of the real benefits of the interactive calculator is that students have the ability to test whether their anticipated answer matches the answer provided by the calculator given the equation they constructed. When answers do not match, such discrepancies should create cognitive conflict and a search for resolution.

### Teaching Considerations

To bring forward some of the issues related to the order of operations, teachers might want to pose carefully selected problems to their students that are similar to those found on the applets and that will provide opportunities to discuss some of the challenges. For example, teachers might provide students with the numbers 2, 4, 6, and 8 and ask them to come as close to the target of 16 as possible. The solution to this problem involves several operations being performed in a very specific order. Students might recognize that they must add together 6+2 to get 8 and then multiply this sum by the 8 to get 64 and then divide by 4 to get 16. However, creating an equation that will perform the various operations as intended in one equation is no easy task (e.g., in this example students would need to write ((2+6)x8):4).

Teachers might want to encourage students to try to do some of the calculations in their heads before trying to create an equation. This will help the students to not rely on the interactive calculator, except when checking their solutions and equations against the computer.

Frog seems like a good applet to serve as a warm-up for this game - helping students to review their multiplication facts. In the same way, Speedy Pictures would work well for reviewing addition.

### Suggestion

I believe that it would be very helpful to allow students the opportunity to go back and try the same problem over again (same 4 numbers and same target number) if the student recognized that his score was not high enough (suggesting that there must have been a strategy for getting closer to the solution than what he did). The benefit here is that the student would be encouraged to go back and redo problems (a positive and worthwhile habit). Is there some way that the student can also know if her strategy was the closest possible based on the points provided? For example, if the solution is the closest, then the student receives a 10, if within 0.5 of the answer then 9, etc. Something that would help the student know if his strategy was the closest… Just a thought.