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Proportional Reasoning involves an understanding of ratio and the relationships between ratios. Ratios express a part-to-whole relationship that can be represented in the form of fractions, percentages, or rates. Proportional Reasoning is an essential component of arithmetic proficiency in elementary school and provides entry to higher math skills including algebra, geometry, and probability and statistics.

Students begin with an informal understanding of proportional relationships (e.g., partitioning or fair sharing, or relating proportions in shaded areas of shapes). This informal knowledge develops over several years into a more formal conceptual understanding as they connect different aspects of Proportional Reasoning including:

- Learning to make comparisons based on multiplication rather than addition;
- Learning which aspects of a proportion can be changed (i.e., the specific values) and which must be constant (i.e., the multiplicative relationship between the values); and
- Learning to make composite units -- that is, learning to reconceptualize a proportion as a single entity.

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

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

Continual use of foundational skills with different problems reinforces a conceptual understanding of math skills.

Daily review strengthens previous learning and can lead to fluent recall.

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 the structures of algebraic representations 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 Algebraic Thinking and intentionally tackle problems.

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

Adding motions to complement learning activates more cognitive processes for recall and understanding.

In guided inquiry, teachers help students use their own language for constructing knowledge by active listening and questioning.

Spending time with new content helps move concepts and ideas into Long-term Memory.

Learning about students' cultures and connecting them to instructional practices helps all students feel like valued members of the community.

Math centers with math games, manipulatives, and activities support learner interests and promote the development of more complex math skills and social interactions.

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

Rhyming, alliteration, and other sound devices reinforce math skills development by activating the mental processes that promote memory.

When students have meaningful discussions about math and use math vocabulary, they develop the thinking, questioning, and explanation skills needed to master mathematical concepts.

By talking through their thinking at each step of a process, teachers can model what learning looks like.

Visualizing how ideas fit together helps students construct meaning and strengthens recall.

Providing physical and virtual representations of numbers and math concepts helps activate mental processes.

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

Having students teach their knowledge, skills, and understanding to their classmates strengthens learning.

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

Math games and manipulatives for vision differences support math development for learners with visual needs.

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.

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

Writing that encourages students to articulate their understanding of math concepts or explain math ideas helps deepen students' mathematical understanding.

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