Quantitative Relationships

This engaging math tutorial breaks down the specific skill of calculating a percentage when given a part and a whole (e.g., "What percent is 20 of 50?"). The video begins by reviewing the relationship between parts, totals, and percentages, establishing that a percentage is simply a ratio out of 100. It categorizes percentage problems into three distinct types—finding a part, finding a percent, and finding a total—and explicitly focuses on the second type for this lesson. The video explores two primary methods for solving these problems. The first method involves using equivalent fractions to scale the denominator to 100, which works well for "friendly" numbers. The second, more universal method involves treating the fraction as a division problem to find a decimal, then converting that decimal to a percentage. The narrator uses humorous skits involving family members giving gifts (travel souvenirs and cookies) to provide concrete word problems that illustrate the math concepts. For educators, this video serves as an excellent bridge between fractions, decimals, and percents. It is particularly useful for helping students transition from simple scaling strategies to the more robust division method required for real-world numbers. The clear visualization of the "Part / Total = Percent" structure helps demystify word problems, while the distinction between the three types of percent questions provides a solid framework for algebraic thinking in middle school math.

This advanced algebra video explores the relationship between mathematical sequences and their domains, challenging the standard convention that sequences must always begin with the first term at n=1. The narrator demonstrates that sequences can be defined using different starting points for the domain (specifically n=0 vs n=1) as long as the formula is adjusted accordingly. Through clear examples of both arithmetic and geometric sequences, viewers learn how to manipulate explicit and recursive formulas to match specific domain constraints. The video breaks down three specific examples: a doubling geometric sequence starting at 4, a decreasing arithmetic sequence starting at 52, and an alternating geometric sequence starting at 1. For each, the narrator compares valid and invalid formulas, showing how changing the starting value of 'n' changes the structure of the equation. A key segment involves an "imposter" activity where students must analyze four different formulas to identify the one that does not produce the correct sequence based on its defined domain. This resource is highly valuable for high school algebra classrooms as it moves students beyond rote memorization of formulas into a deeper conceptual understanding of functions and domains. It addresses the common student struggle of reconciling different notations for the same pattern. Teachers can use this video to introduce zero-indexing (common in computer science) or to reinforce the importance of checking work by substituting values. It promotes critical thinking by asking students to verify formulas rather than just generate them.

This comprehensive mathematics tutorial explores the fundamental differences between arithmetic and geometric means, followed by a series of practice problems involving ratios, proportions, and geometric applications. The video begins by conceptually defining arithmetic mean as the middle term of an arithmetic sequence and geometric mean as the middle term of a geometric sequence. It progresses through eight distinct problems ranging from basic mean calculations to complex word problems involving scale factors, geometric shapes, and algebraic manipulation.

This educational math video features 'Mr. J' guiding students through a mastery check on analyzing numerical patterns and relationships, specifically designed for a 5th-grade level. The video begins by presenting four practice problems where students are tasked with generating two number patterns based on specific rules (e.g., 'Start at 0 and add 4') and then determining the mathematical relationship between the corresponding terms of those two patterns. The format is interactive, encouraging viewers to pause the video, solve the problems independently, and then resume to check their work against the instructor's detailed explanations. The key themes explored include generating numerical sequences using addition and subtraction rules, aligning sequences to identify corresponding terms, and discerning the functional relationship (add, subtract, multiply, divide) between two sets of numbers. The video emphasizes precision in language, requiring students to write full sentences describing the relationships, such as 'Multiply the terms in Pattern X by 7 to get the corresponding terms in Pattern Y.' It covers various relationship types, including additive, subtractive, multiplicative, and divisive relationships. For educators, this video serves as an excellent formative assessment tool or independent practice station. It models not just the calculation, but the organizational skills required to solve these problems, such as lining up numbers vertically to see the pairs clearly. The instructor also demonstrates how to troubleshoot, correcting a formatting error in real-time, which models resilience and attention to detail. This resource effectively bridges the gap between simple arithmetic and early algebraic thinking by focusing on input-output relationships.

This concise and clear instructional video teaches students how to translate verbal phrases into numerical expressions, a foundational skill for algebra and problem-solving. "Math with Mr. J" guides viewers through eight distinct examples, demonstrating how to identify key mathematical vocabulary—such as "sum," "decreased by," "product," and "quotient"—and convert them into appropriate mathematical operations using numbers and symbols. The video systematically explores all four basic operations: addition, subtraction, multiplication, and division. It pays special attention to common stumbling blocks, such as the specific ordering required for phrases like "less than," and introduces students to various notational forms for multiplication (parentheses, dots, asterisks) and division (fraction bars). This variety helps build mathematical fluency and prepares students for higher-level math where notation changes. For educators, this resource serves as an excellent direct instruction tool or review for 5th and 6th graders transitioning from arithmetic to algebraic thinking. It directly addresses the Common Core standards regarding writing and interpreting numerical expressions. The clear visual format and step-by-step narration make it ideal for introducing the concept of "math as a language" and helping students decode word problems.

This educational video teaches students how to identify and complete number patterns using clear visual aids and narration. Through four distinct examples involving colorful balloons, the video demonstrates arithmetic sequences involving both addition (skip counting forward) and subtraction (counting backward). It systematically breaks down the process of finding the rule between consecutive numbers to determine the next number in the sequence. Key themes include pattern recognition, basic arithmetic (addition and subtraction), and critical thinking skills related to algebraic reasoning. The video covers skip counting by 2s and 5s, as well as counting backwards by 1s and 2s, reinforcing mental math strategies in a step-by-step format. Ideally suited for early elementary classrooms, this video provides a solid foundation for algebraic thinking. Teachers can use it to introduce the concept of "rules" in sequences, support lessons on skip counting, or use the pause points as checks for understanding. The clear visual representation of the "jump" between numbers helps students visualize the mathematical operation occurring at each step.

A comprehensive math tutorial solving 10 diverse practice problems related to conic sections. The video systematically covers circles, ellipses, hyperbolas, and parabolas, teaching viewers how to identify shapes from equations, graph them, and calculate key features. The narrator demonstrates essential techniques such as completing the square to convert general equations into standard form and using formulas to find vertices, foci, directrices, and asymptotes. Key themes include the algebraic definitions of conic sections, the geometric relationship between centers, foci, and vertices, and the process of sketching graphs from equations. The video contrasts the four types of conics, highlighting the specific differences in their standard equations, such as sign changes and coefficient differences. It also delves into analytic geometry concepts like the distance formula and midpoint formula within the context of conics. This video is highly valuable for Algebra II and Pre-Calculus classrooms as a review tool or a guided practice session. Teachers can use specific segments to model problem-solving strategies for individual conic sections or assign the entire video as a study guide for unit tests. The step-by-step explanations provide clear models for students to check their work and understand the logic behind geometric definitions.

This educational math video provides a comprehensive introduction to the concepts of ratios and proportions, specifically tailored for upper elementary and middle school students. It begins by defining a ratio as a comparison of two quantities and demonstrates three different ways to write them: word form (is to), colon form (:), and fraction form. Using clear visual aids like fruits and bakery items, the video explains how to simplify ratios to their lowest terms and identify equivalent ratios. The second half of the video transitions into proportions, defining them as statements of equality between two ratios. It introduces specific vocabulary such as "means" and "extremes" and teaches students how to verify if two ratios form a proportion using cross-multiplication. The video concludes by demonstrating how to solve for a missing term (variable) within a proportion using algebra-lite steps, providing both real-world word problems and abstract numerical examples. This resource is highly valuable for teachers introducing pre-algebraic thinking. The step-by-step visual demonstrations make abstract concepts concrete, while the clear procedures for cross-multiplication provide students with a reliable tool for checking their work. The video effectively scaffolds learning from basic identification to solving for unknown variables, making it excellent for direct instruction or review stations.

This comprehensive algebra tutorial provides a deep dive into direct, inverse, and joint variation word problems. The video begins by systematically teaching students how to translate written sentences into algebraic equations using the constant of variation (k). It covers simple variations as well as more complex scenarios involving squares, square roots, and cubes. The instructional approach is notable for teaching two distinct methods for every problem: a conceptual method using proportional reasoning (e.g., "if x doubles, y must quadruple") and a standard algebraic method (solving for k, then substitution). This dual approach helps students verify their answers and develop a stronger intuitive understanding of how variables relate to one another. Ideal for Algebra 1 and Algebra 2 curriculums, this video serves as an excellent resource for teaching students how to model real-world relationships mathematically. It progresses from basic direct variation to complex mixed variation problems, offering clear, step-by-step mental math strategies for simplifying calculations along the way.

This video bridges the gap between abstract mathematical concepts and real-life application by exploring domain restrictions through everyday scenarios. Hosted by Justin, the lesson moves beyond theoretical definitions to demonstrate how physical reality imposes natural limits on mathematical functions. The video specifically examines three distinct situations—rolling a die, pricing old smartphones, and buying groceries—to illustrate how context determines valid input values. The content covers key algebraic concepts including input/output relationships, discrete versus continuous functions, and inequality notation. It guides viewers through the logical process of determining a domain, starting with a mathematical assumption and then refining it based on reality (e.g., recognizing that time and money cannot be negative). The distinction between discrete data (countable integers like die rolls) and continuous intervals (time and currency) is visually represented using set notation and number lines. For educators, this video is an excellent tool for scaffolding the difficult concept of domain and range in word problems. It addresses the common student struggle of translating text into mathematical notation and provides a clear model for error analysis—specifically showing why a mathematically correct inequality might be practically incorrect without real-world context. This resource helps students develop the critical thinking habits necessary for modeling with mathematics.

This instructional math video bridges the gap between theoretical graphing concepts and real-world problem solving. It demonstrates how to interpret key features of linear graphs—specifically slope, x-intercepts, and y-intercepts—to answer practical questions about distance, profit, and physical measurements. The video uses three distinct scenarios to illustrate these concepts: a hiker walking home, an ice cream vendor seeking a profit, and a river's water level decreasing over time. The video focuses on three primary mathematical themes: (1) Understanding slope as a unit rate and using it for unit conversion (dimensional analysis), (2) Identifying the x-intercept as a "break-even point" where a value reaches zero, and (3) Recognizing the y-intercept as an initial value or starting condition (when x=0). It emphasizes distinguishing between mathematical values (like a negative slope) and their physical interpretations (speed is positive). For educators, this video is an excellent resource for Applied Algebra or Pre-Algebra classes. It moves beyond rote calculation to conceptual understanding, helping students see *why* we calculate slope or intercepts. The visual overlays showing dimensional analysis provide a strong cross-curricular link to science, while the business application offers a practical financial literacy moment. It effectively models how to extract hidden information from a visual representation.

This video provides a comprehensive introduction to infinite geometric series, distinguishing between those that grow indefinitely and those that approach a specific limit. It guides students through the conceptual shift from calculating finite sums to analyzing infinite behavior, using both numerical examples and geometric visualizations to illustrate how adding infinite terms can result in a finite sum. The narration explains the critical role of the common ratio in determining the behavior of the series. The content covers key concepts including summation notation, the definitions of convergent and divergent series, and the specific conditions for the common ratio ($r$) that dictate these behaviors. It walks viewers through the derivation of the infinite geometric series formula ($S_\infty = \frac{a}{1-r}$) by applying limits to the finite series formula. The video specifically addresses the misconception that all infinite series can be summed, emphasizing the necessity of checking if the absolute value of $r$ is less than 1. For educators, this resource serves as an excellent bridge between Algebra II and Pre-Calculus concepts. It offers visual proofs of convergence using geometric shapes (fractal-like triangles and subdivided squares) which helps concrete thinkers grasp abstract limits. The included practice problems and "trap" questions—where students must identify a divergent series rather than blindly applying a formula—encourage critical thinking and procedural fluency.