Addition and Subtraction
Grades 0-4
Algebra
Grades 6-12
Calculus
Grades 10-12
Circles
Grades 6-12
Complex Numbers
Grades 10-12
Data/Graphing
Grades 1-9
Exponents
Grades 5-12
Factors/Primes
Grades 4-10
Fractions/Decimals
Grades 1-11
Functions
Grades 10-12
Geometry 2D
Grades 2-12
Geometry 3D
Grades 6-11
Matrices
Grades 9-12
Metric Units
Grades 6-12
Multiply/Divide
Grades 1-9
Numbers, Divisibility, Negatives
Grades 5-9
Numeracy
Grades 0-4
Patterning
Grades 5-12
Percentages
Grades 6-10
Place Value
Grades 0-6
Probability
Grades 5-12
Pythagoras
Grades 7-11
Radicals
Grades 8-12
Rates/Ratios
Grades 5-10
Scientific Notation
Grades 6-12
Shapes and Angles
Grades 0-6
Slope/Linear Equations
Grades 9-12
Speed/Distance/Time
Grades 6-12
Statistics
Grades 5-12
Time
Grades 2-8
Trigonometry
Grades 10-12
Visual Patterning
Grades 1-4
Select a unit to see its details
Pinch or scroll to zoom in and out
Drag to move the map around
This math unit develops students' abilities in factoring polynomials, varying in progression from third-order (cubic) to fourth-order (quartic) equations. Starting with factorizing quartic polynomials by interpreting them as products of quadratic factors, students initially manage simpler coefficients before tackling more complex coefficients. These introductory problems focus heavily on recognizing patterns and implementing the correct factoring techniques without additional hints, utilizing multiple choice and true/false formats to solidify understanding and verify accuracy. The unit then transitions into factoring third-order polynomials by employing specific methods such as the sum of cubes formula. Here, students encounter both basic and intricate coefficients, further enhancing their comprehension of polynomial identities and structural forms across different settings. Towards the later stages, the emphasis shifts towards factorization by grouping in cubic polynomials. These exercises refine students' skills in identifying and grouping common factors, initially through multiple-choice questions and later through true/false formats. This helps to ensure a deeper understanding of polynomial manipulation techniques, culminating in a comprehensive grasp of higher order polynomial factoring.Skills you will learn include:
This math unit covers essential aspects of function transformations, starting from understanding how simple transformations affect the vertex of a function, to more complex manipulations that alter the domain and range. It begins by teaching students how to identify and apply transformations like shifts and reflections that lead to changes in the vertex of functions. The unit progresses to more intricate transformations affecting the domain and range, requiring students to reverse-engineer transformations from changes in these properties. The unit further expands into function composition, focusing on determining and representing the domain of composite functions, which is complemented by ensuring understanding of conditions around real roots in quadratic functions nested within other functions. Additionally, the course transitions into determining inverses specifically between exponential and logarithmic functions, and thoroughly explores sinusoidal functions where students progressively learn to identify and relate parameters like amplitude, phase shifts, and vertical shifts from both algebraic forms and their corresponding graphs. This comprehensive approach not only strengthens algebraic manipulation skills but also enhances visual and theoretical understanding of various function transformations and their applications in greater math contexts.Skills you will learn include:
This math unit begins by teaching students how to factor quadratic equations with a leading coefficient by splitting terms and identifying common factors. As the unit progresses, learners tackle higher-order polynomials, specifically quartic and cubic polynomials, by initially treating them as quadratic expressions. Students are guided with hints and multiple-choice questions to simplify the factorization of these polynomials. Later, the complexity increases as they factor third-degree polynomials using formulas like the sum of cubes and techniques such as grouping. This process extends to determining whether provided factorizations are correct in true/false formats, enhancing their understanding of polynomial identities and their ability to apply polynomial factoring techniques systematically. The unit prepares students to handle polynomial equations by breaking them down into more manageable quadratic or simpler forms, advancing from basic quadratic factoring to more complex high-order polynomial manipulation.Skills you will learn include:
This math unit progresses through a focused exploration of sinusoidal functions, starting by understanding how individual parameters like amplitude, period, and phase shift influence the function's graph. It begins with basic exercises that require students to match sinusoidal functions and their parameters to corresponding graphs. As the unit progresses, the complexity increases as students deal with multiple parameters simultaneously, analyzing how these combine to transform functions graphically and algebraically. Students then move to reverse tasks, where they identify sinusoidal functions from graphs and deduce parameters from functions. Ultimately, the unit covers up to four parameters, enhancing the learners' ability to handle more complex transformations and deepening their understanding of how variations in amplitude, period, phase shift, and vertical shift affect the mathematical expressions and visual representations of sinusoidal functions. The unit is grounded in developing analytical skills necessary for function transformations, composition, and inversion, which is crucial for higher mathematics and applied sciences.Skills you will learn include:
This math unit focuses extensively on analyzing and determining the domain of various functions involving fractions with numerators and denominators that include quadratic expressions and their roots. Initially, students engage in identifying the domains of functions with quadratic denominators, both with real and complex roots, utilizing number lines and set notation to express these domains. As the unit progresses, complexities increase with the introduction of radical expressions in the denominators and numerators, challenging students to consider additional constraints like the non-negativity of radicands (expressions under square roots) and ensuring denominators are non-zero. The skills evolve from simple domain identification on number lines to more nuanced definitions involving complex roots and square roots, requiring a deeper understanding of function behavior and restrictions, particularly in the context of real and complex numbers. Students learn to map accurate domain representations and handle functions with various root conditions, enhancing their problem-solving abilities in algebraic contexts.Skills you will learn include:
This math unit develops proficiency in factoring quadratic equations with leading coefficients through a structured progression of skills. It begins with the basic method of splitting the middle term to identify pairs of numbers that adhere to specific conditions related to the coefficients, a foundational skill in factoring quadratics. Students then practice this skill across various equations, initially by selecting the correct factored form from multiple choices that directly apply this concept. As the unit progresses, it introduces more complex tasks where students must identify and factor out common factors within the terms of the quadratic expression. This includes handling expressions where common groups appear, leading to factoring the expression into a product of binomials. The practice evolves to include tasks that require recognizing common factors in grouped terms, thereby enhancing their capability to simplify quadratic expressions thoroughly. By the end of the unit, students are expected to confidently solve quadratic equations by correctly identifying and applying the appropriate factors, a crucial competency in polynomial manipulation and algebraic problem solving.Skills you will learn include:
This math unit focuses on developing a comprehensive understanding of function transformations. Initially, students learn to identify and describe single transformations from graphs or algebraic expressions, including shifts, reflections, and stretches or compressions. They progress to recognizing how specific transformations affect functions, both graphically and in terms of function notation. The unit also introduces complex scenarios involving double transformations, requiring identification of the effects from a combination of transformations on functions. As the unit advances, the focus shifts to specific attributes of functions such as the vertex, and domain and range. Students learn how transformations affect the vertex of a function, with exercises that include calculating new vertex positions and reverse-engineering transformations from a given vertex position. Similarly, they explore how transformations influence the domain and range of functions, identifying transformed domain or range values from given transformations and vice versa. This comprehensive approach solidifies their understanding of how algebraic manipulations impact function properties.Skills you will learn include:
This math unit starts with the basics of function composition by identifying roles within composite functions, such as input, output, inner, outer, first, and second functions. It progresses to practical applications, requiring students to substitute functions into each other to evaluate composite function outputs. The unit then advances to a more complex skill set involving the determination of domains for composite functions, combining different types of functions such as linear, root, integer, and quadratic operations. This part teaches students how different functions influence the constraints on domain values, often visualized on number lines to help with conceptual understanding. Towards the end, the unit explores function inversion, challenging students to recognize and verify the inverses of given functions, including more complex scenarios involving exponential and logarithmic functions, fostering a deep understanding of functions and their interrelationships in algebra.Skills you will learn include:
This math unit focuses on progressively developing students' understanding and skills in identifying the domain of various functions, beginning with simpler quadratic functions and moving toward more complex rational functions involving polynomials, square roots, and complex roots. Initially, students learn to determine and visually represent the domain of quadratic functions on a number line. They then explore polynomial functions of higher powers, followed by functions involving square roots of linear and quadratic expressions, requiring careful analysis of conditions that ensure real number outputs. The unit advances into exploring the domain of functions where integers and linear expressions are in the numerator, and quadratic expressions, including those with complex roots, are in the denominator. Further complexities involve understanding and representing domains of functions combining square roots and quadratic components in both the numerator and the denominator, culminating in an analysis of conditions that maintain function validity, particularly focusing on real numbers and avoiding undefined behaviors like division by zero.Skills you will learn include:
In this math unit, students progress through a focused exploration of transforming quadratic equations into vertex form by mastering the method of completing the square. Initially, they learn to manipulate quadratic expressions by factoring out coefficients, both positive and negative, to set the stage for effectively completing the square. The practice evolves from handling simpler cases with a leading coefficient of 1 or -1, gradually advancing to more complex scenarios involving arbitrary coefficients denoted by N or -N, reflecting both positive and negative values. As the unit progresses, students refine their skills from partially to fully completing the square, acknowledging how different coefficients impact the transformation. The unit culminates in an in-depth practice of converting standard form quadratic equations into vertex form for a range of coefficients. Each step enhances students' algebraic manipulation abilities, fosters a deeper understanding of quadratic transformations, and prepares them to analyze quadratic functions in terms of their graphs and vertices. This sequence henceforward enhances their problem-solving skills and algebraic thinking in the context of quadratic equations.Skills you will learn include:
This math unit starts with the fundamental skills of expanding algebraic expressions and applying the distributive property to simplify polynomial terms. Students initially practice multiplying a constant or variable with binomial expressions, moving towards identifying equivalent expressions with a focus on polynomials and quadratic functions. The complexity gradually increases as learners manipulate expressions involving same or different variables. Further along in the unit, students delve into more sophisticated tasks such as multiplying and removing variables from bracketed terms, including applications of the FOIL method and reinforcing the correct handling of signs when dealing with squared variables and constants. The unit transitions into quadratic equations, where students factor and simplify quadratic expressions, including those with coefficients, thus enhancing their algebraic manipulation skills. Towards the end of the unit, advanced concepts such as completing the square are introduced, focusing on transforming quadratic expressions into perfect square trinomials. This cements a deeper understanding of polynomials and quadratic equations, preparing students for more complex algebraic problems.Skills you will learn include:
This math unit focuses on the end behavior of polynomial functions, progressing from basic understanding to more complex application and analysis. Initially, students learn to interpret the end behavior of polynomials through graphical analysis, identifying trends as \(x\) approaches positive or negative infinity. They then progress to matching these behaviors with appropriate graphical representations and deducing rules based on graph trends. As the unit advances, students deal with the correlation between the highest power and the leading coefficient of polynomials and how these influence the end behavior. They learn to determine these elements from graphs, rules, and directly from polynomial functions. Finally, the unit covers predicting polynomial graphs based on a given set of highest power and leading coefficient values, wrapping up with an ability to match polynomial functions to their respective end behaviors and rules, essentially preparing them for practical applications and deeper studies in polynomial functions.Skills you will learn include:
This math unit progresses through a series of skills, focusing on understanding and defining the domain and range of functions through various representations and conditions. Initially, the unit addresses basic visual and symbolic representations involving the translation of inequalities and verbal descriptions into number lines and set builder notation, utilizing unions to depict composite intervals. As students progress, they learn to convert these representations into interval notation, further solidifying their understanding of function behavior over specified domains and ranges. The unit advances into more complex scenarios where students must determine function domains involving higher polynomials, linear and quadratic expressions under square roots, and functions defined by fractional expressions over linear, quadratic, and radical denominators. The tasks require careful analysis to avoid undefined expressions and adhere to the conditions provided by each mathematical model, enhancing proficiency in translating between different mathematical representations of domains, such as number lines and algebraic conditions. The increasing complexity aids in mastering the algebraic and graphical understanding necessary to analyze functions critically within defined parameters.Skills you will learn include:
This math unit begins by teaching students how to visually interpret and define functions through relation maps and progresses to detailed explanations of domain and range using various mathematical notations and representations. Starting with basic function definitions, students learn how to find domain and range using number lines and inequalities, gradually moving to translating these into set builder and interval notations without unions. The complexity increases as students learn to navigate between different representations, including verbal descriptions and symbolic expressions. Towards the end of the unit, the scope expands to include unions in domains and ranges, enhancing their ability to handle more complex scenarios by interpreting inequalities, set builder notations, and intervals that involve combining separate mathematical intervals. This progression builds a foundational understanding of functions, crucial for further study in calculus and algebra.Skills you will learn include:
In this math unit, students begin by learning to simplify algebraic functions involving the multiplication of a single variable with bracketed terms, setting a foundation for understanding polynomials and quadratics. Initially, they focus on mastering basic expansion of expressions where the variable is identical, such as y(y+3). The unit progresses to include more complexity by introducing expressions with different variables, enhancing understanding through exercises like \((z + 3)(m + 7)\). As learners advance through the unit, they tackle increasingly sophisticated problems that demand deeper conceptual understanding and manipulation skills. They move from multiplying simple binomials to handling expressions involving squared terms and the distribution of different variables across sums and differences within parentheses. Towards the end of the unit, students work on identifying and simplifying expressions to bracketed terms with different variables and coefficients and factoring quadratic equations. This progress from simple expansions to more complex operations prepares them for future studies in higher-level algebra, including the distinct skills of recognizing, manipulating, and simplifying polynomial and quadratic forms in various mathematical contexts.Skills you will learn include:
This math unit begins by familiarizing students with the basics of the quadratic formula, specifically identifying coefficients \(a\), \(b\), and \(c\) in quadratic equations. Progressing, it explores solving for complex roots using the quadratic formula, deepening the understanding of discriminants' impact on root nature. The unit then transitions into practical applications, starting with translating word problems about geometrical contexts into quadratic expressions and equations. Focus shifts to solving real-world quadratic equation problems concerning the height of objects over time, optimizing variables like time and height, and concluding with maximizing or analyzing real-life economic scenarios like profit and revenue based on changes in production volume or price. Throughout, students incrementally build proficiency in framing, manipulating, and solving quadratic equations in various practical and theoretical contexts, enhancing their analytical and algebraic skills.Skills you will learn include:
This math unit initiates with the basic fundamentals of factoring quadratic equations, specifically focusing on transforming them into their factored forms. Initially, students practice factoring quadratics where a common factor is removed, enhancing their ability to simplify these expressions into binomials. As they progress, the unit shifts towards more complex applications, requiring students to identify and factor out the greatest common factor from quadratics presented in standard form. Further advancing, the unit tackles the challenge of splitting quadratic expressions with a leading coefficient by identifying numbers that appropriately add up to and multiply into particular values derived from the coefficients. Students practice these skills through a combination of split-term techniques and identifying common factors across different terms of the quadratic expressions. Towards the end of the unit, the emphasis lies on consolidating and applying various factoring skills to not only solve but also to correctly identify the most simplified forms of quadratic equations. This progression ensures a deepened understanding of polynomial factoring in quadrics, particularly in manipulating and solving expressions with leading coefficients.Skills you will learn include:
This math unit progresses from basic to advanced skills in algebraic functions, focusing particularly on polynomials and quadratics. Initially, students learn to find the greatest common factor (GCF) within algebraic expressions, a foundational skill for simplifying and manipulating expressions. The unit progresses to more complex maneuvers such as removing and manipulating different or same variables from bracketed terms. As students progress, the focus shifts to factoring quadratic equations, first without coefficients and then with coefficients, where they apply their skills to split quadratic terms and identify correct factorizations. Later worksheets introduce the relationships between quadratic roots, coefficients, and their arithmetic properties (sum and product), culminating in an ability to solve and factorize quadratic expressions based on specific numerical criteria. This learning path enhances understanding of polynomial behavior and quadratic relationships, preparing students for deeper studies in algebra and related mathematics fields. The structured approach develops from identifying simple common factors to complex factorizations and understanding polynomial roots, reflecting a comprehensive learning curve within the algebraic scope.Skills you will learn include:
This math unit focuses on developing a comprehensive understanding of quadratic functions through a progression of skills. Initially, students learn to determine and analyze the range of quadratic functions from their graphs and equations based on vertex form. These skills involve understanding the effects of vertex position and the direction a parabole opens (upward or downward). As students advance, they shift focus towards manipulating and translating quadratic equations and their graphs. They learn to convert graphs to vertex form equations, match equations to their respective graphs, and understand how the algebraic structure of these equations affects their graphical characteristics and range. Further into the unit, students apply techniques to complete the square, essential for converting quadratic equations into vertex form. This begins with partial completion when the leading coefficient is -1 or a negative number and progresses to fully completing the square to understand vertex position and to discern the properties like maximum and minimum values of the quadratic functions. This sequence deepens students' ability to manipulate and understand quadratic transformations, heading towards mastery in handling quadratic equations in various forms and contexts.Skills you will learn include:
This math unit delves into the fundamentals of graphing circles on a coordinate plane, beginning with basic identification and placement of circle centers. Students first learn to graph circles given the center coordinates and then advance to calculating circle graphs based on specified radii. Progressing further, the unit teaches conversion of a circle's equation into its graphical representation by interpreting the meaning of value coefficients in the standard circle equation \( (x-h)^2 + (y-k)^2 = r^2 \). Students continue to refine their skills by extracting center coordinates and radii directly from the circle equations, enhancing their ability to visualize and plot these geometric forms accurately. The subsequent lessons expand these concepts by guiding students to derive a circle's standard equation using both center coordinates and radius. Towards the end of the unit, learners are equipped to transform a complete circle graph back into its algebraic equation, solidifying their understanding of the relationship between a circle’s graphical and algebraic representations. This step-by-step progression effectively builds foundational knowledge crucial for solving geometry and coordinate graphing problems involving circles.Skills you will learn include:
This math unit begins with students learning how to expand and simplify algebraic expressions involving squared terms. As the unit advances, learners systematically delve into quadratic functions, starting with identifying whether quadratic polynomials are perfect squares. They practice the technique of completing the square, initially with simpler forms where the leading coefficient is 1 or -1, and subsequently handling quadratic equations with variable coefficients (both positive and negative). These exercises prepare students to transform quadratic expressions into vertex form, gradually moving from partial completion to fully completing the square. By the end of the unit, students become proficient in manipulating and transforming quadratic functions, developing key skills in rewriting equations to reveal maximum and minimum values and other properties critical for understanding the geometry of parabolas. This progression deepens their comprehension of algebraic structures and polynomial behavior in quadratic equations.Skills you will learn include:
This math unit guides students through advanced aspects of polynomial and quadratic algebra, focusing on skills ranging from basic operations to complex problem-solving. Initially, students learn to expand and simplify products of binomials containing either distinct or identical variables, setting a foundation in polynomial manipulation. Progression is evident as they delve into the difference of squares, first using variables and later integrating integers, enhancing their ability to factorize and simplify expressions. As the unit progresses, the focus shifts to handling squared expressions under square roots and solving for variables in contexts involving quadratic-like structures. Students also solve expressions using the difference of squares formula and exponents, which further sharpens their skills in simplifying and determining algebraic identities and equivalencies. Ultimately, students apply these concepts to more intricate problems, such as solving polynomial algebra where squared variables with coefficients are involved, culminating in solving rational equations, thereby achieving a comprehensive understanding of polynomial and quadratic equations.Skills you will learn include:
This math unit progresses through various skills related to quadratic functions, focusing extensively on vertex form. It begins by identifying different forms of quadratic equations. It then delves into the use of vertex form to solve practical problems related to the vertex, understanding min/max y-values, and determining the range of quadratic functions. This leads to exercises around graph interpretation, where students match quadratic equations to graphs, and finally, complete the square to transform equations into vertex form. Early exercises focus on recognizing and manipulating the vertex form to understand graphical properties like the vertex and the range. Later exercises are more algebraically intensive, where students practice the technique of completing the square with varying coefficients, ultimately facilitating a deeper understanding of the algebraic structures and transformations of quadratic equations. The unit caps off with mastering completing the square for different coefficients, solidifying an essential algebraic skill with quadratic functions.Skills you will learn include:
This math unit begins by developing students' ability to identify and verify the midpoint of a line segment using graphical representations. As the unit progresses, it adds complexity by introducing the concept of perpendicular slopes and the calculation of slopes that are perpendicular to a given line segment. Students start with true/false questions to confirm their understanding of midpoints and perpendicular slopes and then move on to using formulas. The unit evolves to focus on finding the equation of the perpendicular bisector of line segments, both from graphical representations and by analyzing points directly. Towards the end, the unit consolidates these skills by requiring students to verify if equations presented to them correctly represent the perpendicular bisector of specified line segments, using both graphical and point-based information. Overall, this unit builds from basic midpoint validation to advanced application of formulas for constructing perpendicular bisectors, enhancing students' analytical geometry skills.Skills you will learn include:
This math unit begins with introducing the basics of quadratic formulas, transitioning from naming and identifying their purposes to delving deeper into characteristics such as the discriminant and its implications on quadratic graphs and roots. Students start by linking quadratic equations to their respective formulas and understanding the functions they describe. The unit progresses to interpreting the discriminant values to analyze how they influence the graph's appearance, the nature of the roots, and the count of roots in quadratic equations, further evaluating whether these roots are real or complex. The exercises escalate to involve calculating specific aspects like the radical roots and the x-coordinate of the vertex using quadratic formulas. Towards the end, the unit focuses on the comprehensive application of quadratic discriminants, including identifying discriminant values from graphs and examining the consistency of root types according to discriminant values, reinforcing a deeper understanding of the interplay between quadratic equations, their graphs, discriminants, and roots.Skills you will learn include:
This math unit begins by teaching students how to multiply constants and single variables by bracketed terms, foundational for understanding polynomials and quadratics. It progresses to more complex skills such as multiplying different or same variables by bracketed terms, reinforcing the distributive property and FOIL method. As students advance, they encounter problems involving expanding and simplifying expressions of increasing complexity, including those with negative numbers. The unit culminates in advanced manipulations including identifying integer pairs that meet specific summative and multiplicative conditions and solving squared bracketed terms. Fundamentally, this unit furnishes students with a deep understanding of algebraic expressions crucial for tackling polynomials, quadratics, and advanced algebraic functions effectively.Skills you will learn include:
This math unit begins with the foundational concept of identifying y-intercepts from linear equations in slope-intercept and standard forms using integer coefficients. As the unit progresses, it introduces the concept of x-intercepts, requiring students to manipulate equations set to zero in either variable while still using integer values. The complexity increases as the unit shifts to equations involving decimal coefficients. This additional challenge tests the students' ability to work with more precise values and enhances their algebraic manipulation skills. Towards the end of the unit, the focus shifts to finding intersection points between different types of lines including horizontal, vertical, and other linear equations demonstrating both integer and decimal solutions. This progression from basic intercept identification to solving for intersections between various lines helps students understand the graphical behavior of linear equations and their points of intersection.Skills you will learn include:
This math unit begins by building foundational skills in understanding slopes of line segments, both graphically and through coordinates, initially focusing on finding and verifying the slope values of line segments. Students practice calculating the slope using the formula and assess given slopes for accuracy in true/false formats. The unit progresses to finding the midpoints of segments, utilizing both graphical presentations and coordinate data, again using true/false questions to confirm understanding. Later, the unit introduces more complex concepts like perpendicular slopes and bisectors. Students learn to determine perpendicular slopes by applying the negative reciprocal rule and verify their correctness in true or false questions. The unit advances to constructing equations for perpendicular bisectors using both graphical and coordinate data, enriching students' skills in applying geometric concepts to solve problems involving midpoints, slopes, and perpendicular relations in line segments. The sequence moves from basic slope and midpoint identification to more intricate applications, such as creating and verifying perpendicular bisectors.Skills you will learn include:
This math unit begins with understanding algebraic functions in the context of finding integer pairs that satisfy conditions of sum and product in quadratic expressions, incorporating negatives for complexity. Students progress to manipulating quadratic equations directly by identifying relationships between their roots, sums, and products. Initially, they focus on the fundamentals such as solving and understanding polynomials and quadratic equations, moving towards intermediate skills like identifying the greatest common factor (GCF) and factoring quadratic expressions efficiently. As the unit advances, students delve deeper into polynomial algebra by systematically transforming quadratic equations through removal of common factors and simplification processes. They practice converting from standard form to simplified expressions, mastering the skill of distinguishing correct factorization and manipulation strategies. Furthermore, the unit integrates the application of these foundational and intermediate algebraic skills in various problem-solving scenarios involving polynomial expressions, thereby enhancing understanding of advanced polynomial properties and manipulations within the broader context of algebraic functions.Skills you will learn include:
This math unit focuses on understanding and applying the concept of perpendicular slopes within different contexts of linear equations. It begins with basic calculations to find the negative reciprocal of given integer slopes and progresses to handling fractional and decimal slopes to identify perpendicular lines. The unit further develops by having learners convert and determine perpendicular slopes between different forms of linear equations, such as slope-intercept form, zero-intercept form, and standard form. Additionally, learners practice converting these equations for graphical representation, aiding in visual understanding and verification of perpendicular relationships. The depth of the unit increases as students move from initially identifying perpendicular slopes in simpler formats to manipulating complex algebraic forms and graphing them, thus building a comprehensive skill set in analyzing and constructing perpendicular lines within coordinate geometry. Throughout the unit, the primary emphasis remains on mastering the concept that the product of the slopes of perpendicular lines is -1, and applying this understanding in various mathematical scenarios.Skills you will learn include:
This math unit begins by developing students' abilities to interpret and identify different types of function graphs based on their corresponding algebraic equations. Initially, students match specific graphs to their equations and recognize the type of function each equation represents. The unit progresses to enhance understanding of function domains and ranges through visual representations, starting from identifying domains and ranges, to determining whether given relations qualify as functions based on their visual mappings. Further along, the unit delves deeper into defining and describing domains and ranges more formally. Students learn to translate number line visuals into verbal descriptions and vice versa, convert verbal descriptions into mathematical inequalities without unions, and then express these inequalities using set builder notation. Finally, the unit teaches students to transform set builder notations into interval notations and accurately represent these intervals on a number line, further solidifying their comprehension of domain and range concepts in the context of functions.Skills you will learn include:
This math unit develops the understanding and skills related to slopes and equations of lines, with a specific focus on parallelism. Initially, students learn to recognize and convert line equations between different forms, starting from understanding simple forms such as slope-zero intercept and slope-intercept forms, to more complex transformations involving standard forms and decimal representations of slope. As the unit progresses, the emphasis shifts to applying these foundational skills to understand parallel lines. Students practice identifying parallel slopes by converting equations between various formats including zero-intercept, slope-y-intercept, fraction form, and graph representation to standard forms. Through these exercises, students enhance their ability to interpret and manipulate different algebraic expressions of linear equations, deepening their grasp of how slopes indicate parallelism and how lines can be graphically and algebraically analyzed and compared for this property.Skills you will learn include:
This math unit covers a comprehensive range of skills in understanding and utilizing line equations and graphing. Initially, students begin by learning how to determine the slope of a line directly from a graph, setting the foundation for deeper exploration of linear relationships. They progress to calculating the rise (change in y-values) and run (change in x-values) between two points on a Cartesian plane, essential skills for understanding the slope of a line. The unit advances into more complex tasks that involve selecting the correct linear equation based on the slope, y-intercept, and visual information from graphs. Students practice how to analyze linear graphs and match them to their equations, ultimately enhancing their ability to interpret graphical data into algebraic expressions. This includes identifying lines that pass through the origin and understanding the impact of different slopes and y-intercepts. Towards the end of the unit, the focus shifts to applying these skills to solve for intercepts from equations presented in standard form and slope-intercept form. This progression solidifies students' understanding of linear equations, graph interpretation, and algebraic manipulation, ensuring comprehensive knowledge in constructing and analyzing line equations in various forms.Skills you will learn include:
This math unit opens with foundational algebraic concepts, beginning with solving basic linear equations with one variable. As the unit progresses, the focus shifts towards more complex operations involving algebraic fractions, where students first learn to solve and simplify equations with fractions and eventually handle advanced fraction manipulations, including those with multiple variables. The unit proceeds to expand into polynomial manipulation, targeting skills from expanding expressions with a single variable multiplied by bracketed terms to handling polynomials involving multiple variables. Students practice distributing variables across terms and simplifying the resulting expressions—a vital skill for more advanced studies in algebra. Towards the end of the unit, the emphasis is on multiplying bracketed terms—both with the same and different variables—to reinforce understanding of the distributive property and improve the ability to expand and manipulate polynomial expressions. The unit concludes with exercises that involve solving for integer pairs that meet specific conditions, synthesizing earlier concepts with integer properties and polynomial reasoning.Skills you will learn include:
This math unit begins by introducing students to basic algebraic functions, starting with the multiplication of a single value with bracketed terms and progressing to more complex polynomial expressions. Initially, students practice distributing a single number across terms within parentheses, helping them recognize and simplify equivalent expressions. As the unit advances, the focus shifts to more intricate operations involving the same or different variables within bracketed terms, enhancing their ability to expand and simplify such expressions further. Midway through the unit, the emphasis is placed on removing values from these terms, reinforcing students' understanding and manipulation of algebraic expressions across various contexts. The complexity increases as students learn to factor quadratic equations and recognize equivalent expressions involving operations with variables, addition, subtraction, multiplication, and powers. Towards the end, the unit dives into the relationships between the roots of quadratic equations, exploring how sums and products of integers relate to polynomial coefficients, including exercises with negative numbers. This structured progression sharpens students' problem-solving skills in polynomials and quadratics, preparing them for advanced topics in algebra.Skills you will learn include:
This math unit begins with foundational practices in understanding and calculating the slope of a line through various methods and progressively moves towards applying these concepts to broader topics in linear equations and graphing. Initially, students explore the concept of slope using fact families and simple rise/run calculations from graphs. Progression occurs when students calculate the slope from specific points on a graph and ultimately advance to deriving slopes directly from rise and run values presented in equations. As the unit progresses, students take on tasks such as extrapolation of points from graphed lines based on linear equations and mathematical analysis to find specific points on a graph from given linear equations. The unit culminates with students identifying and manipulating linear equations based on slopes and intercepts from graphical representations and equations in standard form, enhancing their overall understanding of the relationship between algebraic expressions and their graphical manifestations in coordinate geometry.Skills you will learn include:
This math unit begins by teaching students how to recognize and calculate the slope of linear equations presented in various forms. Starting with converting standard form equations to slope-intercept form and directly finding slopes, the unit progresses to applying these skills by interpreting graphed lines and determining their standard form equations. Increasing complexity is introduced as students learn to calculate negative reciprocal slopes to identify perpendicular lines, initially focusing on understanding the negative inverse property through converting integer and decimal slopes. The unit further delves into graphical interpretations, allowing students to visually analyze lines on graphs to identify perpendicular slopes and convert these observations into different numerical and algebraic representations. The advanced topics cover converting between different forms of slope equations and understanding the relationship between slopes of perpendicular lines within different forms of linear equations, emphasizing practical applications and manipulation of slope and perpendicularity in various contexts.Skills you will learn include:
This math unit primarily develops an understanding of slopes and their applications in determining parallelism between lines. It starts by teaching students how to graph linear equations from the slope-intercept form and progresses to converting these equations between different formats, reinforcing their understanding of slope as a critical element in linear equations. The central theme evolves around identifying parallel lines through various representations of slope, including fractional, decimal, and zero forms, along with graph interpretations. Students are guided through recognizing parallel slopes directly from graphs, as well as determining them through algebraic equation conversions involving both the slope-intercept form and graphical representations. Additionally, the unit enhances skills in manipulating and understanding equations, fostering an in-depth comprehension of how slopes establish relationships between parallel lines, crucial for graphing and algebraic problem solving in coordinate geometry. The progress from basic graph plotting and slope identification to detailed analysis of slopes in different forms and their corresponding graphical interpretations encapsulates the unit’s comprehensive approach to understanding linear relationships.Skills you will learn include:
This math unit begins by introducing students to the fundamental skills of substituting numbers and variables into linear equations. Initially, students practice simple substitutions where numbers are replaced in equations with one defined variable, advancing to solve for unknown variables using these substitutions. As the unit progresses, the complexity increases as students learn to apply the substitution method to systems of linear equations, where they must substitute entire equations to simplify and solve for variables. The unit deepens understanding by requiring students to manipulate and simplify algebraic expressions to isolate variables and solve equations. Multiple choice questions are included to help verify their solutions. Towards the end of the unit, the focus shifts to practical applications, employing algebraic manipulations in balance scales scenarios where substitution and subtraction are used to solve more visually presented equations, enhancing problem-solving skills in real-world contexts. Finally, the unit circles back to simpler algebraic operations such as addition within systems of equations, ensuring students consolidate their understanding of basic operations within the context of linear systems. This approach builds a robust foundation in algebra, preparing students for more complex mathematical concepts.Skills you will learn include:
This math unit begins by developing foundational algebraic skills through the multiplication of bracketed terms with different variables, establishing an understanding of polynomial manipulations. It progresses to solving linear equations, starting with simpler forms involving one variable with three terms, and gradually increasing in complexity to equations with four terms. The unit then transitions to the manipulation of algebraic fractions, increasingly focusing on solving equations that contain variables within fractions and reinforcing the reduction of fractions to their simplest forms. The latter part of the unit introduces solving problems presented in a visual format with balance shapes, which require the use of substitution and subtraction to formulate and solve equations. This specialized focus aims to enhance understanding of how algebraic principles apply to practical and abstract mathematical problems, culminating in the ability to simplify complex algebraic expressions and solve advanced algebraic equations.Skills you will learn include:
This math unit progresses through various aspects of understanding and calculating the slope of a line, as well as deriving other line characteristics such as rise and run using the slope formula. Initially, the unit introduces the basic terms "rise" and "run" and the concept of slope by identifying these components on a graph. Students then learn to compute the slope using the rise over run formula expressed as an equation. Progressively, they apply this understanding to determine the rise and the run from known values of slope and one of the other variables. Through practice, the unit strengthens students' ability to manipulate and solve equations involving slope, rise, and run. Later, this evolves into more complex tasks where slope is calculated between specific points or derived from graphical representations, enhancing skills in interpreting and analyzing linear relationships by applying algebraic methods and critical thinking.Skills you will learn include:
In this math unit, students begin by expanding and simplifying algebraic expressions focusing on multiplying bracketed terms with the same variable, setting a foundational understanding of polynomial operations. They then progress to evaluating algebraic expressions through variable substitution, dealing initially with simple and negative terms, and advancing to more complex situations involving squared terms and negative coefficients. The practice intensifies as students substitute values into multiple fractional squared and bracketed squared terms, reinforcing their capacity to manage and compute expressions under specified conditions. Further advancing their algebraic skills, students practice solving basic linear equations to isolate variables, first with two terms and progressing to three terms, enhancing their handling of various algebraic functions. The unit culminates with advanced algebra concepts where students translate balanced shapes into equations, focusing on establishing and solving ratios, and involves visual and analytical skills by substituting and subtracting to find solutions in complex algebraic contexts. This flow from foundational polynomial operations to complex variable substitution and application in real-world contexts challenges students to deepen their understanding and proficiency in algebra.Skills you will learn include:
In this math unit, students progress through a sequence of topics that build foundational to advanced skills in algebra. The unit starts with basic skills such as expanding and simplifying algebraic expressions when multiplying a variable by a bracketed term, followed by solving linear equations with increasing complexity—from three to four terms. It then advances to manipulating algebraic fractions, where students solve and simplify equations that involve fractions with variables. The complexity in fraction manipulation progresses across orientations until students deal with comprehensive problems that require reducing fractions that involve variables to their simplest forms. Towards the end of the unit, the focus shifts to applying algebra in practical contexts using balance shapes. Students learn to analyze image-based problems and to use substitutions and subtraction to solve for the equations and answer visually represented through balance beams. This culminates in understanding complex ratios, substitutions, and algebraic manipulations through symbolic and visual interpretations, rounding out their algebraic skills with both numerical and real-world problem-solving abilities.Skills you will learn include:
This math unit starts with foundational algebra concepts, encouraging students to develop algebraic thinking through balance problems and simple substitutions without explicitly introducing variables and equations. As the unit progresses, students move on to solving linear equations by isolating one variable and manipulating three-term equations using basic arithmetic operations including addition and subtraction. The unit further deepens comprehension of algebra by introducing variable substitution in simple algebraic expressions and balancing equations from visual cues. Later, students engage with the manipulation and evaluation of algebraic functions involving negative integers, fractions, and bracketed squared terms. The curriculum culminates in calculating exponents, solidifying an understanding of advanced algebraic operations. Throughout this progression, the focus shifts from intuitive problem-solving and basic operations to complex algebraic manipulations and computational skills in various algebraic contexts.Skills you will learn include:
This math unit introduces and develops foundational algebra skills, advancing from simple to more complex concepts. Initially, students balance shapes using simple ratios, exploring basic equality principles and ratio application. They delve further into understanding algebra by interpreting the meaning of a dot as multiplication in algebraic functions and applying simple substitution in balancing more complex scenarios with three beams. Progressively, the unit tackles linear equations, first by solving two-term equations and advancing to three-term equations, enhancing student skills in basic arithmetic operations and variable isolation. Concurrently, students learn to interpret algebraic expressions, distinguishing between operations like multiplication, addition, and exponentiation, with focus on how numbers and variables interact in an equation. Towards the latter stages, the unit emphasizes on translating verbal descriptions into algebraic equations and practicing variable substitution in algebraic expressions, both unbracketed and bracketed. These exercises reinforce understanding of algebra's fundamental concepts, setting a robust basis for advanced topics.Skills you will learn include: