These two applets focus on the operations of addition, multiplication, and subtraction, each in isolation. Players can select a large grid (10x10) or a small grid (5x5) and with either one or two players. Make Three is virtually the same although players have to find only 3 in a row rather than five (although also found it with five when using the large grid). Directions for Make Three state that players can select Addition or Multiplication, although only the Addition and Subtraction options are available.
One of the benefits of this game is that it encourages students to identify all of the possible solutions before making their selection, moving them beyond simply finding one solution and helping them to recognize the unique solution pattern. Because the goal of the game is for students to achieve five in a row (or three in a row depending on the version being played), students must constantly consider which problem to select given its location on the grid. For example, if a student is asked to find a problem that corresponds to the target number 8, then the student has many options to choose from (e.g., 5+3, 4+4, 3+5, 2+6, etc.). However, if the problem 3+5 is located next to four other selected problems, then this would clearly be the best problem to select since it will give the player the desired five in a row. To maximize students' options and success, it is in their best interest to find as many of the problem solutions as possible before making a selection.
These games provide students with opportunities to recognize important number relationships and patterns, such as how they can find the other solutions by adding 1 to one of the values while also subtracting 1 from the other value. For example, given the target number of 16, a student might recognize that if he were to take one away from the 10 and give it to the 6 the target answer of 16 would not change, but the problem would now become 9+7. Depending on students' mathematical sophistication, teachers might want to further formalize students' thinking by pointing out that the pattern of adding 1 to one value and subtracting 1 from the other value is, in essence, adding 0 (since 1 + (-1) = 0). Same could be true when considering a fixed starting point, such as 4+4=8. Then students could recognize that adding 1 and subtracting 1 yields 5+3=8; adding 2 and subtracting 2 yields 6+2=8, etc. In each case the sum of what is being added is 0 since as one value increases by one (or two, or three, etc.), the other value decreases by the same amount. More formally, this concept connects nicely to the Additive Identity and Additive Inverse properties.
An interesting observation is to note that the solution set for the addition problem solutions follows along a positive 1 slope on the grid but for the subtraction problem solutions the solution set follows along a negative 1 slope. This would make an interesting question for teachers to ask their students as to why this is the case… Indeed, the reason for this difference is due to the fact that the sum of the two values always remains constant as you move along the +1 slope since you decrease the first value by one while increasing the second value by one. For example, when trying to find the solution options for 7, the students can select 6+1 or 5+2 or 4+3, etc., whereby holding the sum constant at 7. However, for the subtraction problems we are interested in holding the difference constant. In our previous example with the target of 7, the sum remains the same while the difference between the two values changes. By contrast, when trying to hold the difference constant, say our target number for subtraction is 4, then we want the difference to always be 4. If we started with 8-4 as one solution, we could find another solution by adding one to the first term, giving us 9. To keep the difference constant we must now subtract 5 from 9 to get our solution of 4.
Unlike for addition and subtraction, the patterns that emerge from this game are not as straight forward and obvious for students to identify. Because of this, students will likely require some degree of teacher input before they will be able to develop important generalizations and connections. One important consideration is that for any given target number the students are essentially finding the factors. A strong connection between the problems and the factors could easily be made through this applet, helping students to recognize that it is only the combination of these factors that will yield solutions. A strong connection to division could also be made here when discussing factors (and is similar to the "Hit the Safe" game where student try to unlock the safe by finding the factors or the "Devil" game where students try to identify which value is the multiple of the given number). For example, when provided a target number of 16, students should recognize that the factors are 1, 2, 4, 8, and 16. Working backward, students could select one of the factors and then divide it into the target number to determine that problem solution. Cognitively, this is a much different mental process and approach than simply having students use their memory to recall the problem solutions.
An interesting pattern that exists with multiplication is when considering the number of factors or problems for any given target number. It is interesting for students to note that when a target number is a perfect square, such as 1, 4, 9, 16, etc., then the total number of solutions will always be an odd number. If the target number other than a perfect square, then the number of solutions will always be even. This makes for an interesting discovery lesson with students as to why this is the case. Unfortunately, due to space limitations, these games are not able to represent all of the possible combinations (factors). For example, if the target number is 24 the students are able to represent all of the combinations using the grid except 2x12, 12x2, 1x24 and 24x1 since each axis only goes through 10. Teachers must then move beyond the limitations of the game in the classroom in order to fully develop these ideas.
To encourage students to reflect on their work and to further assist them in making important connections, teachers might want to consider posing carefully constructed questions to their students following their work with these applets. Teachers should first consider what connections they want their students to form and what observations they want their students to formalize. For example, the following questions are provided as examples of the types of questions teachers might want to consider posing to their students following experience with these applets.