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Hover to see how Factors connect to Operations. Then click connected Factors to explore strategies related to multiple Factors.

Operations specify how quantities can be combined and transformed. There are four arithmetic Operations: **addition**, **subtraction**, **multiplication**, and **division**. Throughout the school years, students develop an understanding of how Operations work with different types of numbers, such as fractions and decimals.

Students typically begin by thinking about Operations as telling them which procedural calculations, or computations, to do (such as finding the sum or difference of two numbers). However, it is critical that students also develop a *conceptual* understanding of Operations. This conceptual foundation supports students in estimating approximate computations, alongside quickly and accurately calculating using procedures.

Students' skill with Operations is supported by these critical conceptual components:

- Understanding the
**algebraic properties**of the Operations involved in the problem: For instance, students should able to transform the harder sum 3+8+7 into the easier sum 3+7+8 (it's easy to see that 3+7 = 10, and then it's easy to add 8 to 10); - Understanding the
**Place Value**system and**decomposition**: For instance, a student could also transform 3+8+7 into 3+7+1+7 (making it easier to add up to 10, before adding 1 and 7); and - Understanding how
**relationships in a real world situation**can be expressed by Operations (i.e., modeling).

Providing math tasks with high cognitive demand conveys high expectations for all students by challenging them to engage in higher-order thinking.

As students solve problems in a group, they learn new strategies and practice communicating their mathematical thinking.

Students activate more cognitive processes by exploring and representing their understandings in visual form.

Thinking of and about patterns encourages learners to look for and understand the rules and relationships that are critical components of mathematical reasoning.

Teaching students to recognize common problem structures helps them transfer solution methods from familiar to unfamiliar problems.

Discussing strategies for solving mathematics problems after initially letting students attempt to problem solve on their own helps them understand how to organize their mathematical thinking and intentionally tackle problems.

Analyzing incorrect worked examples is especially beneficial for helping students develop a conceptual understanding of mathematical processes.

Math games allow students to practice many math skills in a fun, applied context.

Visual representations help students understand what a number represents as well as recognize relationships between numbers.

When teachers connect math to the students' world, students see how math is relevant and applicable to their daily lives.

Children's literature can be a welcoming way to help students learn math vocabulary and concepts.

When students create their own number and word problems, they connect math concepts to their background knowledge and lived experiences.

When students explain their thinking process aloud, they recognize the strategies they or others use and solidify their understanding.

Untimed tests provide students the opportunity to flexibly and productively work with numbers, further developing their problem-solving abilities.

Providing visuals to introduce, support, or review instruction activates more cognitive processes to support learning.

Analyzing and discussing solved problems helps students develop a deeper understanding of abstract mathematical processes.

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