The Teachers' Lounge |

The Math behind MathDice |

By Tom Rowan

The Math Dice games provide direct support for students’ development of a variety of very valuable mathematics concepts and skills. Beyond those concepts and skills that are directly supported, there are additional mathematics concepts that can be brought into the games through questions and follow-up discussions. This paper discusses the mathematics benefits of playing the games. While it is intended to address most of the benefits, you may find others that are not included. Here are some of the mathematics concepts and skills that are directly or indirectly supported by the games:

- Basic computational facts
- Understanding of the operations of addition, subtraction, multiplication, division, and exponents
- Estimation
- Mental computation, including combinations of operations and strategies for efficiency of mental computation
- Writing equations to represent mental computation
- Reasoning and problem solving
- Commutative, associative, and distributive properties
- Order of operations

Proficiency with basic computational facts

As students play the games they use basic computational facts for addition, subtraction, and multiplication. For example, if the target number is 38 and the scoring numbers are 3, 5, and 6, students will constantly use and re-use combinations of 3, 5, and 6 as they attempt to get as close to 38 as they can. They may say 3 + 6 = 9 and 5 x 9 = 45, thus using an addition fact and a multiplication fact. They will likely try other combinations to get still closer, such as 5 x 6 = 30, and 30 + 3 = 33. As can be seen, the facts that are practiced will involve both the original scoring numbers and numbers that can be obtained by adding, subtracting, or multiplying the original scoring numbers, as was the case in the first example where 9 was not one of the original scoring numbers but was obtained by first adding two of the scoring numbers.

Understanding the operations of addition, subtraction, multiplication, division, and the use of exponentsAll too often the only experience that many students have with these operations is through the use of paper and pencil procedures that are given to them. Students who learn the operations this way often have no real understanding of how the operations apply in real life. They are not able to use the operations flexibly and they may make computational errors and have no way to know that their answers are wrong. They do not have what many mathematics educators call “operations sense”. Students who have operations sense understand in a general way the effect that an operation has on the numbers that they are using. They know that multiplication of whole numbers generally has a much greater effect on the numbers than addition, for example. They often have a general idea of what the size of the answer will be before they do any actual computation. This is highly desirable. Playing the Math Dice games helps students to build an understanding of the effect an operation will have, as well as to develop flexibility about how to use the operations. Knowing that 4 x (5 + 3) gives the same result as 4 x 8 or (4 x 5) + (4 x 3) is very useful in the games. It is also very useful when doing computation in other settings, whether with paper and pencil, mentally, or with a calculator. Playing the games reinforces the relationships among the various operations and helps students think in creative ways about numbers and operations. Another example would be having 2, 3, 6 with a target of 18. A student might first think that 3x6 equals 18, but the 2 must then be used and cause the result to move away from the target. But by exploring a bit further into the use of the different operations, it can be seen that by adding the 3+6 you can reach 9, which allows the 2 to come into play through multiplication to reach the target exactly rather than being a "tail" that has to plus or minus away from the target. Again, the effect of the operations is reinforced.

When students are at a level where they can use exponents, they not only do the computation that is involved with exponents, but they also reinforce the concepts underlying the relationship of exponents to multiplication. Many students have difficulty understanding exponents (and the related concept of roots). The use of exponents in the games helps to reinforce the understanding that is so important for consistent use of these important mathematical operations. Using exponents in the Math Dice games constantly reinforces the understanding that the exponent tells you how many times the base number appears as a factor. All too many students confuse the exponent as being a factor.

EstimationAs students play the game most will recognize that estimation can be used to check various possible answers rather than actually doing the computation. If the various possible answers are too close each other after doing the estimation, then the different possibilities will need to be computed, but estimation becomes a valuable tool for an initial check that may save time. The estimation in this case is for finding difference between the target and the estimate, thus also involving either subtraction or addition. Estimation is used with the initial computation to check number combinations that yield results close to the target and may involve any operation. If the student sees by estimating that multiplying all three scoring numbers gives an answer that is obviously much too great or too small, then that procedure is discarded and another is tried. For example, in the example used earlier when discussing basic computational facts, students might initially think of multiplying the three target numbers, but by estimation without actually carrying out the multiplication see that the result would be far too great. A quick estimation has occurred, perhaps without the student even having made a conscious decision to do it. Such quick estimation facility is very valuable. As in the case of mental mathematics, estimation also uses and reinforces place value understanding and number sense.

Mental computationStudents constantly use mental computation with basic facts and numbers beyond basic facts while playing Math Dice. For example, the numbers rolled to establish a target number may be 7 and 12. Multiplication of a 1-digit by a 2-digit number is immediately required to ascertain the target number of 84. Some students may think 7 x 10 is 70 and 7 x 2 is 14 to get the target. Others may think that 6 x 7 is 42 and then double 42 to get the target. One strategy reinforces place value understanding while the other uses and extends number sense. The scoring dice in this case may show 5, 6, and 4. Basic facts are used initially with these scoring numbers, but since the target is a relatively large number, the computation must go well beyond basic facts. A student may try 4 x 6 = 24 and 5 x 24 = 120, thinking that multiplication is the only way to get close to such a large target. The student may also try to get closer by adding 6 and 4 to get 10 and then multiplying by 5 to get 50. Now the student must decide which of those is closer to 84, again using mental computation to do 120 – 84 and 84 – 50 to see which is closer. Both of these computations may involve place value understanding and/or number relationships. A wide variety of mental computation will be used during the course of a single game. Such mental computation activities are powerful tools for reinforcing place value concepts and number sense. When doing mental computation, students often use the place value of the numbers to carry out the computation, since algorithms that can be done without attention to place value are very difficult to carry out mentally. For example, in the preceding situation 5 x 24 is highly unlikely to be done by the traditional multiplication algorithm of multiplying 5 x 4 to get 20, putting down the 0 and carrying the 2. It is much easier to see 24 as 20 + 4 and then to multiply 5 x 20 to get 100 and 5 x 4 to get 20, and then to add 100 + 20. Some students may even use 10 more directly by thinking 10 x 24 = 240 and half of 240 is 120. Without ever realizing it, these students have used place value concepts to carry out the computation. Even with the subtraction of 84 from 120 that was mentioned, mental computation is likely to be done by place value understanding rather than the algorithm. The traditional algorithm would have the student regrouping the 2 tens so that the 4 can be subtracted, a very cumbersome approach. Most students will quickly say 84, 94, 114, then 6 more to get to 120 so the difference is 26. Here again, place value understanding and the relationship between subtraction and addition both come into play reinforcing a level of number sense that is very empowering for the student.

Writing equations to represent mental computationOnce students have done the mental computation to arrive at a solution they feel is close to the target, it is necessary for them to write the equation that represents what they have done. This is essential for at least two reasons. The first is that it may be the only way to check that the solution can in fact be obtained using the number combinations described orally by the student. The second reason to write the equation is to have a record of the different responses that have been given so that the computational accuracy can be checked. In order to write the equation that represents the computation the student did, order of operations must be understood and used. This more often than not requires the use of parentheses to indicate the part of the computation that is to be done first. Writing and manipulating equations is, of course, an essential part of any higher mathematics course that uses algebraic notation and reasoning. Examples here would include having 3, 5, and 6 with a target of 33. Here the student may see that 5 x 6+3 and (5+6) x 3) will work, but may write 5 + 6 x 3 without realizing that this gives 23 by order of operations, not 33. Another example is 2, 3, and 5 with a target of 13 where the student may write either 2 x 5 + 3 or 3 + 5 x 2 without having to use parentheses to indicate the order, but may realize that the second expression may confuse someone who is not familiar with the order of operations.

Reasoning and problem solvingAs students play these games they constantly use their reasoning and problem solving skills. If the games are played in a competitive way, then students don’t want to give the first solution that comes to their mind, they want to try to reason their way to a solution that is as close as possible so their opponent can’t get closer. This uses a variety of problem solving skills that have to do with the nature of the numbers in play. For example, if the target number is odd and the scoring numbers are all even, is it possible to get a combination that will hit the target exactly? Say, for example, 2, 4, and 6 with a target of 7; we can't do much with addition, subtraction, multiplication or even powers in this case. Are there any other ideas? Well, what if we consider division? This situation may help students realize that division has some pretty amazing properties; if we divide 6 by 2 an odd number pops right out! What if there are two odd and one even scoring number? Is that different from having one odd and two even? Can this same type of reasoning be used if the target is a multiple of 3? Does having a target that ends in 5 or 0 make one think about factors of 5 or 10?

Commutative, associative, and distributive propertiesHere again, the mathematics comes about in a natural way. Students realize quickly that reversing the order of the addends in the case of addition, or the factors in the case of multiplication, will not change the solution. For example 9, 3, and 2 with a target of 21, is easier to arrive at by multiplying 2 x 9, than 9 x 2. It is important to realize this in order to save time and be first with a solution that is as close to the target as possible. The distributive property may be used, but may not be clear if the equations are not written. It is important to write the equations for the solutions and to use parentheses in many cased, as was indicated previously in the discussion of writing equations.

Order of operationsMany students struggle with the order of operations because it is a rule that is imposed because of the need to have the operations act as functions. In other words, for any given expression, there must be only one answer. Without the order of operations rule, this would not be the case. For example, 4 + 3 x 5; is the solution 19 or 35? If we do the addition first and the multiplication last, the solution is 35; but if we multiply first and then add, the solution is 19. Both solutions cannot be true or we have an ambiguous system. Should we make the rule be to always do the operations in order from left to right as we read? This could be very confusing, since 4 + 3 x 5 would give a solution of 35, but 3 x 5 + 4 would give a solution of 19. The same numbers and operations are used and the operations of multiplication and addition are both commutative, but this rearrangement gives two different solutions with the left to right rule. Mathematicians have decided that this ambiguity is not acceptable. The order of operations rule thus tells us that multiplication and division should be done before addition and subtraction unless parentheses are used. If parentheses are used, then the operations in the parentheses are done before the operations outside of parentheses. Further, operations by exponents are done before multiplication, division, addition, and subtraction. So the rule that students must use for the order in which the operations must be carried out is parentheses, exponents, multiplication and division, addition and subtraction. Because it is an arbitrary rule, students are often told to recall it using the mnemonic: __p__lease __e__xcuse __m__y __d__ear __a__unt __s__ally, where the first letter of each word indicates the operation.

Many of the benefits that are discussed above may be missed if the teacher doesn’t ask students good questions about their decisions and solution strategies during the game. Questions such as, “Why do you need to have the parentheses there?” will cause the students to think about the order of operations at a conscious level, thus reinforcing the understanding of that important rule. “What if …?” questions are also very useful, especially if students have failed to think of a scoring number in some way such as, for example, by using exponents. Questions such as, “If you could trade one of the scoring dice for some other number, what would it be?” extend the reasoning and problem solving characteristic of the game and may also introduce a mathematical idea that was not likely to occur without such a lead. This might be true for the even – odd situation that was discussed above. Questions that simply ask for a sharing of reasoning are always useful, such as, “Why did you choose that particular way of combining the scoring numbers?” Even the simplest questions, such as, “What was your thinking?” are powerful in causing the student to reflect on what he or she did, and perhaps even why it was done.