Additive Relationships

Additive relationships are mathematical connections in which one quantity can be found by adding to or subtracting from another, making them fundamental to reasoning about change, comparison, and parts and wholes. They work by representing an unknown or changing quantity with an expression such as a + b = c; addition combines quantities, while subtraction reverses that operation to reveal a missing value, difference, or starting amount. Students and researchers use these relationships to interpret word problems, complete tables, analyze patterns, and build equations that model measurable situations. Understanding them supports arithmetic fluency and provides a foundation for algebraic reasoning, quantitative analysis, and more complex mathematical models.

Additive Relationships - Related Videos

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JoVE Core - Organic Chemistry

Conjugate Addition (1,4-Addition) vs Direct Addition (1,2-Addition)

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2025

α,β-Unsaturated carbonyl compounds with two electrophilic sites, the carbonyl carbon, and the β carbon, are susceptible to nucleophilic attack via two modes: conjugate or 1,4-addition and direct or 1,2-addition. Conjugate addition results in a thermodynamically stable product. The reaction retains the stronger C=O bond at the expense of the weaker C=C π bond. The process is slow as the β carbon is less electrophilic than the carbonyl carbon. Direct addition products are formed faster owing to...

Relationship Formation

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2020

What do you think is the single most influential factor in determining with whom you become friends and whom you form romantic relationships? You might be surprised to learn that the answer is simple: the people with whom you have the most contact. This most important factor is proximity. You are more likely to be friends with people you have regular contact with. For example, there are decades of research that shows that you are more likely to become friends with people who live in your dorm,...

Evolutionary Relationships - Concepts

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2019

Humans have been attempting to properly classify living things since Aristotle made the first attempt during the 4th century BC. Aristotle’s system was improved upon during the Renaissance and then, subsequently, by Carolus Linnaeus in the mid 1700’s. These more formal classification and organization systems grouped species by their physical similarity to one another. For example, all vertebrates have a backbone, but invertebrates do not. Traits like the backbone are called synapomorphies,...

Evolutionary Relationships - Student Protocol

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2019

Forming an Evolutionary Hypothesis ExpandNOTE: A cladogram is an important tool for forming an evolutionary hypothesis. A cladogram is a tree-shaped chart used to depict the hypothetical genealogical relationships between species. The tips or leaves of the chart represent specific species and the branches of the tree are different lengths. The different lengths represent the degree of change between each of the species. The common ancestor of all of the species that a specific line branches...

Conjugate Addition of Enolates: Michael Addition

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2023

The attack of a nucleophile at the β carbon of an α,β-unsaturated carbonyl compound is called conjugate addition. Conjugate addition reactions of active methylene compounds, such as β-diketones, β-keto esters, β-keto nitriles, and α-nitro ketones, are called Michael addition reactions. The reaction is catalyzed by a base that abstracts the acidic methylene hydrogen, generating a doubly-stabilized enolate ion that serves as the nucleophile or the Michael donor. The base employed depends on the...

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