While we often think of fluency as the goal, the study of basic math facts begins with a focus on understanding. Exploring basic math facts is our opportunity to build number sense, strengthen students’ understanding of operations, and support computational fluency.

Students make sense of math facts by looking at patterns in the facts, connecting fact sets to each other (e.g., in addition they connect doubles and doubles plus 1 facts or in multiplication they connect x2 facts with x4 or x8 facts), creating models to show the facts, and discussing the facts through problem contexts that help them understand what each number represents.

The understanding of inverse operations is critical for students’ ultimate fluency with subtraction and division facts.

- Creating models of the equations allows our students to see the connection between addition and subtraction as they move 4 and 5 cubes together to make 9 and then separate the 9 cubes into a group of 4 and a group of 5.
- Students might construct an array by building 4 rows of 3 tiles to see 12 total tiles (4 x 3 = 12) and then divide that array by separating the 12 tiles into 4 rows and find that 3 are in each row (12 ÷ 4 = 3).

And then observing the related equations allows them to make observations about the connections between the numbers in the equations.

Once understanding occurs and students can utilize strategies to find a set of sums or products (e.g., 4 + 5 = 4 + 4 + 1, so the sum is 8+1 or 9 since it is like thinking about doubles plus 1), students begin to work on fluency (automaticity). Through interactive games that repeatedly pose doubles plus 1 facts, students begin to gain fluency with these addition facts, but students often have a difficult time moving from addition (or multiplication) fluency to subtraction (or division) fluency.

For most people, subtraction facts are recalled by “thinking addition” which means subtraction fluency is grounded on both

- addition fluency
- the understanding of inverses.

Students who know 5 + 4 = 9 and understand inverses through their earlier study using models, problems, and discussions, simply think about 9 – 5 = ? as “what plus 5 equals 9”. To strengthen students’ ability to make that connection, consider the power of exposure to missing addend equations rather than simply moving from sum unknown to difference unknown.

4 + 5 = __ (sum unknown) | I find the sum (an initial level of fluency). |

__ + 5 = 9 (sum known, missing addend) | I see the sum, but what addend is missing? I need to find the missing part (continuing to strengthen addition fluency). |

9 – 5 = __ (difference unknown) | After work with missing addend equations, I can better see the connection to those equations. I know the sum (total), but need to identify the missing part. I realize I was doing this same thing in missing addend equations. |

Having students practice fluency with missing sums, then missing addends, then moving to the related subtraction facts provides a progression that allows our students to see connections between the 3 numbers in the equations, helps them make sense of the idea of inverse operations, and builds fluency. This progression works when supporting students with division fluency as well, beginning with equations with missing products, then posing equations with unknown factors, and then exploring equations with unknown quotients.

For more ideas, lessons, and classroom resources related to mastering math facts, see *Mastering the Basic Math Facts in Addition and Subtraction* (http://www.heinemann.com/products/E07476.aspx) or *Mastering the Basic Math Facts in Multiplication and Division (**http://www.heinemann.com/products/E05965.aspx**). *