In mathematics, diamond problems are practice problems that aid in skill development. Unlike many mathematical tools that focus on building a single skill, diamond problems actually build two skills at the same time. The unique nature of the problem helps students figure out how to find two numbers that add together to form a specific sum while also using the numbers to find a specific multiplication product. While some students may feel that this is little more than busywork, being able to create products and sums from the same set of numbers is an essential skill that's used heavily in high school math and beyond.

## What Are Diamond Problems?

Diamond problems are also referred to as "diamond math" due to the unique way that they are constructed. Most diamond problems are drawn in an actual four-sided diamond, with a large X in the middle of it that separates it into four smaller diamonds. One number is written in the diamond at the bottom, while another number is written in the diamond at the top. The diamonds on the left and right are left empty, as these are the two fields that the student has to fill in. Keep in mind that not all diamond problems are drawn in this exact way; you'll sometimes see them with just a large X to create the four sections without the diamond shape surrounding it. Either method is fine, but the drawn diamond is the more standard version.

#### Tips

There is also a method of factoring quadratic equations known as the box method. It focuses more on the full process of factoring trinomials, whereas the diamond method tries to emphasize the skills to find factors quickly.

The rules of a diamond math problem are simple: The student has to place numbers in the two empty cells. When added together, the two numbers have to equal the number in the bottom cell. When multiplied together, they have to equal the number in the top cell. Depending on the skill level of the students, both positive and negative numbers may be required (which would result in negative numbers in the top or bottom cells, a big hint to the students.) If students are still at an early point of developing this skill, however, it's recommended that you stick with all positive numbers to start.

## How Is This Used?

Diamond math trains people to recognize possible factors that also equal a specified sum. This is very important when factoring quadratic equations using the FOIL method in algebra, since a problem such as

requires both multiplication and addition to come up with the factor pairs of

for simplification. This skill carries on beyond just polynomial factoring as well, since algebra plays an important part in more advanced mathematics. Developing the skill now using tools such as diamond problems will make it much easier for students to identify proper factors in the future. Having a good mastery of diamond problems, and also the concepts behind them, can allow students to then understand more advanced concepts like the quadratic formula and solve quadratic equations much more quickly.

### Solving Diamond Problems

The easiest way to solve diamond problems is to factor the top number and determine how many possibilities there are for the empty cells. Starting with the bottom number is much harder since there are a huge number of combinations of whole numbers that can be added to create a sum; if negative numbers are allowed, that number is actually infinite. Make a list of all of the combinations of numbers that create the desired product when multiplied together (such as 3 and 4 if the product is 12.) Once you have your list, try adding the two numbers together to see if they equal your desired sum (such as 3 + 4 if the sum is 7.) Once you find a match, write those two numbers in the two empty cells. It doesn't matter which order the numbers are written in, since the numbers in the diamond problem are only in a collection and not actually in a mathematical problem. Even if they were, they are only used in addition and multiplication, which allow you to place numbers in any order and still get the same result.

#### Tips

The diamond method can be used to factor trinomial equations very efficiently, however they work best on trinomials with a degree of 2, especially for beginning students.

## Solving Diamond Problems

The easiest way to solve diamond problems is to factor the top number and determine how many possibilities there are for the empty cells. Starting with the bottom number is much harder since there are a huge number of combinations of whole numbers that can be added to create a sum; if negative numbers are allowed, that number is actually infinite. Make a list of all of the combinations of numbers that create the desired product when multiplied together (such as 3 and 4 if the product is 12.) Once you have your list, try adding the two numbers together to see if they equal your desired sum (such as 3 + 4 if the sum is 7.) Once you find a match, write those two numbers in the two empty cells. It doesn't matter which order the numbers are written in, since the numbers in the diamond problem are only in a collection and not actually in a mathematical problem. Even if they were, they are only used in addition and multiplication, which allow you to place numbers in any order and still get the same result. If you find subtracting a number could work as a solution, it is best to write that number as a negative number in the diamond.

## Resources for Instructors

There are many great resources for generating diamond problem solver sheets, and you can also create your own very easily. Worksheet works provides an amazing resource to generate random problems with answer keys. Try to encourage students to solve these exercises without the use of diamond problem calculators online. Sometimes a step-by-step solution for a couple example problems can help students succeed.

It can also be very useful to design word problems where the diamond method can be used to find the answer. Look to examples of gravity and exponential growth for inspiration. However, try to keep solutions and problems limited to integers, even for more experienced students, because diamond problems are more focused on understanding the concepts of factoring instead of complex arithmetic, fractions, or decimals.

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About the Author

Holding a BS in computer science and several years of experience building, repairing and maintaining computers and electronics, Jack Gerard has had a love of science and mathematics for years. When not working on writing projects as part of his 15+ year career, he also works as a programmer writing gaming and accessibility software.

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