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    • ▾Algebra 1
      • ▸Solving equations and inequalities
        • •Solving literal linear equations
        • •Solving and graphing multi-step inequalities in one-variable
        • •Solving and graphing compound linear inequalities
        • •Evaluating infinite series
      • ▸Working with units
        • •Rate conversion
        • •Word problems with multiple units
      • ▸Linear equations and graphs
        • •Intercepts from a graph
        • •Intercepts of a linear function from a table
        • •Solving for a variable given a linear equation in standard form
      • ▸Forms of linear equations
        • •Point-slope form
        • •Two point form
        • •Standard form for linear equations
        • •Intercepts from an equation
      • ▸Inequalities and systems of inequalities
        • •Graphing linear inequalities in two variables
        • •Graphing systems of linear inequalities in two variables
      • ▸Functions (algebra 1)
        • •Discrete vs continuous functions
        • •Evaluate a function from its graph
        • •Evaluating expressions with multiple variables
        • •Finding the domain and range of functions from graphs
        • •Interval notation
        • •Intro to multivariable functions
      • ▸Inverse functions (algebra 1)
        • •Determining if a discrete function is invertible
        • •Finding the inverse of discrete and linear functions
      • ▸Arithmetic series
        • •Intro to arithmetic sequences
        • •Find the explicit formula of an arithmetic series given terms
        • •Find the formula of an arithmetic sequence from two terms
        • •Univariate expressions representing consecutive terms
        • •Explicit to recursive formula for arithmetic series
      • ▸Geometric series
        • •Intro to geometric series
        • •Find the explicit formula of a geometric series given terms
        • •Explicit formula from two terms
        • •Expressions representing consecutive terms
      • ▸Absolute value, piecewise, and step functions
        • •Evaluating piecewise functions
        • •Graphs of piecewise functions
        • •Evaluating and graphing the floor function
        • •Solving floor and ceiling equations
        • •Absolute value function under transformations
        • •Solving absolute value equations and inequalities
      • ▸Exponents and radicals
        • •Product property for square roots
        • •Simplifying square root expressions
        • •Simplifying cube roots of integers
        • •Simplifying nested square roots
        • •Simplifying radicals of monomials
        • •Higher order roots
        • •Rationalization
      • ▸Exponential growth and decay
        • •Exponential vs. linear growth
        • •Graphing exponential functions under transformations
        • •Finding exponential functions from tables, graphs, and points
        • •Exponential growth and decay
      • ▾Polynomials
        • •Difference of squares
        • •Intro to polynomials
        • •Multiplying polynomials
        • •First and second differences of quadratics
        • •Proving polynomial identities
        • •Equating coefficients
      • ▸Factoring and solving quadratics
        • •Factoring monic quadratics using algebra tiles
        • •Factoring and solving monic quadratics
        • •Factoring non-monic quadratics by factoring out the GCD
        • •Factoring quadratics with difference of squares
        • •Identifying and factoring perfect square trinomials
        • •Choosing a factoring method (level 1)
        • •Factoring non-monic quadratics using algebra tiles
        • •Factoring non-monic quadratics
        • •Choosing a factoring method (level 2)
        • •Completing the square
        • •Quadratic formula
        • •Sign of the discriminant
        • •Golden ratio
        • •Po-Shen Loh method
      • ▸Quadratic equations
        • •Solving quadratics by u-substitution
        • •Solving equations in quadratic form using u-substitution
        • •Solving quadratic equations by taking square roots
        • •Intro to vertex form for quadratics
        • •Converting quadratics between standard form and vertex form
        • •Quadratic from the vertex and a point
        • •Finding the domain and range of quadratic functions from equations
        • •First and second differences
        • •Reducing to a linear equation
      • ▸Irrational numbers
        • •Sums and products of rational and irrational numbers
     › Algebra 1 › Polynomials

    Intro to polynomials

    First, students will learn the etymology and definition of polynomial. Then they'll apply the definition to determine whether some expressions are polynomials. They'll also learn how to determine the degree of univariate polynomials, and refer to them as monomial, binomial, trinomial, etc. Then they'll learn how to identify coefficients, terms, leading terms, leading coefficients, and constant terms. Further, they'll learn how to identify the degree of terms and polynomials. The terminology introduced here is important, as students will be seeing it extensively in the future.

    Conclude by giving your students these challenges:

    • Journeying in Numberland by NRICH
    • Different Sizes by NRICH
    • 1999 AMC 8, Problem 21
    • 2005 AMC 8, Problem 21

    Problem: Imagine the colored rectangular prisms, pictured below, as wooden blocks, which can be physically moved. What equation can be seen from this picture?

    One way you can lead students is to cover all but the bottom row. From this, students should see 1, 2, 3, 4. From here, you can lead them to \((1 + 2 + 3 + 4)^2\) as one way to represent the picture. To help them with the right side of the equation, you can tell them they must move the blocks in some way to find a second way of describing the picture. You should only provide hints when students are totally lost. Remember that struggle is essential to learning.

    Solution: Hopefully, at least some of your students will arrive at

    $$(1 + 2 + 3 + 4)^2 = 1^3 + 2^3 + 3^3 + 4^3$$

    Once your students have found the solution, or have struggled significantly, show them this animation.


    Notes: The idea came from this page. I modified an image I found on Wikipedia here to make the problem easier, as I think very few students would be capable of solving the original problem.

    Lessons and practice problems