*This information below was taken from documents prepared by the CCPS Elementary Math Department.*

**Invented Strategy** for Adding and Subtracting Two-Digit Numbers

**What is an invented strategy?**

An

*invented strategy*is any method used in calculation that does not use physical materials, counting by ones, or a clearly defined set of steps or procedure. Invented strategies are flexible methods for computation that involve taking apart and combining numbers in a variety of ways. This flexibility is based on place value or numbers that work easily together. A student, a peer, or a class can invent strategies. Teachers may also suggest ideas.

**Invented Strategies vs. Standard Algorithms**

·

*Invented strategies are number-oriented rather than digit-oriented.*

When using the standard algorithm for 36 + 43, children may not think of 30 and 40 but rather 3 + 4. The idea of the value of the entire number is lost.

·

*Invented strategies are left handed, not right handed.*

Invented strategies begin with the largest parts of numbers (represented by the leftmost digits). For 75 + 26, an invented strategy might begin with 70 + 20. Starting on the left provides a sense of the size of the eventual answer in just one step. The standard algorithm, however, begins on the right with 5 + 6 is 11. Starting with the rightmost digits hides the result until the end of the computational procedure.

·

*Invented strategies are flexible.*

Unlike standard algorithms, invented strategies use different entry points to begin solving a problem.

**What are the benefits of using invented strategies in the elementary classroom?**

· Students make fewer mistakes because they develop and understand their own computational strategies.

· Students develop deep understanding of place-value because invented strategies are number-oriented.

· As students develop proficiency with invented strategies, they are able to use them mentally without having to record their thinking.

· Invented strategies are often faster than the standard algorithm because they take less time than the steps to the standard algorithm.

· According to international measures of proficiency, students who use invented strategies perform as well or better than their peers who are taught only standard algorithms.

**When are students taught standard algorithms?**

Students first develop conceptual understanding of operations. Next, they begin to develop, discuss, and look for efficient, accurate, and generalizable computational methods (invented strategies). This stage of development is extensive, requiring months, not weeks, of work. Students do not invent flexible methods of computation spontaneously. Teachers must carefully create learning situations and environments that allow children to develop their own methods of invented computation. Finally, students are introduced to the standard algorithm. Rather than seeing it as a series of steps to follow, students better understand that the standard algorithm, like every computational method they have used, must makes sense mathematically.

Source: Van de Walle, J.A., Louvin, L.H., Karp, K.S., Bay-Williams, J.M. (2014).

*Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades Pre-K – 2, 2nd edition.*Boston, MA: Pearson.

## Open Number Line

**What is an open number line?**

An

*open*or

*empty number line*is a number line with no numbers or markers. Students may use open number lines to visually record their thinking during the process of mental computation. When students record their thinking using an open number line, they do not need to follow rules of scale or spatial distance. They must, however, respect the order of numbers.

**Why should students use open number lines?**

Open number lines allow students to see the variety of ways that the same question can be solved. As students share their solutions and different ways of thinking, more efficient methods of using the number line and computation develop.

**How can open number lines be used?**

Number lines may be used to solve many different types of problems including addition and subtraction of whole numbers, decimals, fractions, money, and measurement; or in calculating elapsed time.