If you were to write instructions for getting ready for school, it would matter very much which instruction came first: putting on your socks, putting on your shoes, etc.

Sometimes we need multiple expressions in mathematics, and the order matters there, too! Mathematicians didnâ€™t always agree on the Order of Operations, but at some point it became important to develop rules to help them work together.

To help us organize our math into something we can trust, we can diagram a math expression using the Circles of Evaluation. For example, the expression ( 1 - 4 ) ÷ ( 10 × 7 )$\displaystyle ( 1 - 4 ) \div ( 10 \;\times\; 7 )$ can be diagrammed as shown below.

/
​-
 1 4
​*
 10 7
(/ (​- 1 4) (​* 10 7))

Order of Operations is important when programming, too!

To convert a Circle of Evaluation into code, we walk through the circle from outside-in, moving left-to-right. We type an open parenthesis when we start a circle, and a close parenthesis when we end one. Once we’re in a circle, we write whatever is on the left of the circle, then the operation at the top, and then whatever is on the right. The circle above, for example, would be programmed as (​(​1 - 4​) / (​10 * 7​)​).

These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, and 1738598). Bootstrap by the Bootstrap Community is licensed under a Creative Commons 4.0 Unported License. This license does not grant permission to run training or professional development. Offering training or professional development with materials substantially derived from Bootstrap must be approved in writing by a Bootstrap Director. Permissions beyond the scope of this license, such as to run training, may be available by contacting contact@BootstrapWorld.org.