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Semester A

Unit 1: Sequences & Series

Work with arithmetic and geometric sequences in both explicit and recursive forms. Explore series and sigma notation.

Section 1: Sequences

Write Explicit Formula for Sequence

Recursive Sequences

Write Recursive Formula for Sequence

Recursive to Explicit Sequence

Explicit to Recursive Sequence

Write Recursive Formula from Context

Section 2: Series

Sigma Notation Intro

Series to Sigma Notation

Arithmetic Series

Sum of Arithmetic Series (Sigma Notation)

Geometric Series (Level 1)

Sum of Geometric Series (Sigma Notation)

Sum of Geometric Series (Context)

Unit 2: Polynomials: Multiplying, Factoring, & Dividing

Master operations with polynomials and use those skills strategically to solve equations.

Section 1: Multiplying Polynomials

Multiply Binomials

Squaring Simple Binomials

Squaring Binomials

Multiply Polynomials (Level 1)

Multiply Polynomials (Level 2)

Section 2: Factoring Polynomials

Factor Cubics

Factor By Grouping

Factoring Quartic Trinomial

Mixed Multistep Factoring

Binomial as GCF

Factoring Diff / Sum of Cubes

Section 3: Factoring to Solve Equations

Solving Quadratic Binomials (Level 1)

Solving Quadratic Binomials (Level 2)

Solving Factored Equations

Factoring to Solve Cubic Equations

Factoring by Grouping to Solve Equations

Master Product Factoring to Solve Equations

Section 4: Polynomial Division

Polynomial Divided by Monomial

Polynomial Long Division (Level 1)

Polynomial Long Division (Level 2)

Synthetic Division (Level 1)

Polynomial Long Division (Level 3)

Synthetic Division (Level 2)

Synthetic Division (Level 3)

Unit 3: Complex Numbers & Quadratics

Explore how defining the square root of -1 as i creates an entire class of “complex numbers” and use them to solve problems.

Section 1: Complex Numbers

Simplifying Radicals with Negatives

Simplifying Radicals with Negatives (Algebraic)

Add Imaginary Numbers

Multiply Imaginary Numbers

Powers of i

Add/Subtract Complex Numbers

Multiply Complex Numbers

Section 2: Intro to Complex Roots

Solving Quadratic Binomials (Level 3)

Solving Factored Equations (Level 2)

Factoring by Grouping to Solve Equations (Level 2)

Find Complete the Square Constant

Complete the Square (Part 1 of Method)

Complete the Square (Part 2 of Method)

Complete the Square

Section 3: Roots of Quadratics

Quadratic Formula with Complex Roots (Level 1)

Quadratic Formula with Complex Roots (Level 2)

Finding the Discriminant

Nature of the Roots

Select Quadratic Equation Given Roots

Unit 4: Polynomial Functions

Connect zeros, roots, and x-intercepts to polynomial functions. Master graphing and analyzing polynomial functions in different forms.

Section 1: Factored Form, Roots, Zeros, & Intercepts

Polynomial Terminology

Determining the Number of x-Intercepts

Finding x-Intercepts in Unfactored Form

Finding y-Intercepts of Polynomials

Finding Roots from a Graph

Equation from Graph Roots (Level 1)

Equation from Graph Roots (Level 2)

Polynomial Function from Graph

Section 2: Graphing Polynomial Functions & Zeros

Remainder Theorem

Factor Theorem

Synthetic Division to Find Zeros (Level 1)

Sign Chart from Factored Polynomial (Level 1)

Sign Chart from Factored Polynomial (Level 2)

Sketching Graphs from a Sign Chart

Graph Factored Polynomial from Roots

Section 3: Analyzing Polynomial Functions

Identify Relative Extrema Graphically

Relative vs. Absolute Extrema

Increasing vs. Positive Graphically

Odd and Even Polynomial Functions

Even and Odd from a Graph

End Behavior Graphically

Determine Features of Polynomial Graph

Unit 5: Exponents & Radicals

Explore the relationship between exponents and radicals via fractional exponents.

Section 1: Exponent Rules

Multiplying Powers

Raising Powers to Powers

Dividing Powers

Negative Exponents (Level 1)

Negative Exponents (Level 2)

Exponent Rules (Level 1)

Exponent Rules (Level 2)

Section 2: Fractional Exponents

Exponential & Radical Form

Exponent Rules w/ Fractions

Negative & Fractional Exponents (MC)

Simplify Fractional Exponents (Type 1)

Solving Equations w/ Fractional Exponents

Section 3: Simplifying Radicals

Simplify Algebraic Square Roots

Simplify Basic Cube Roots

Simplify Algebraic Cube Roots

Simplify Nth Root Radicals (Level 1)

Simplify Nth Root Radicals (Level 2)

Simplifying Radicals (Nth Root)

Section 4: Radical Expressions

Radicals/Fractional Exponents on Integers

Combining Radicals of Different Roots

Combine Radicals & Fractional Exponents

Combining Like Terms w/ Radicals

Dividing Algebraic Radicals

Multiply Radical Binomials

Multiply Radical Binomials (Algebraic)

Rationalizing the Denominator

Semester B

Unit 6: Equations: Radical, Rational, and Systems

Solve more advanced equations with a variety of algebraic and graphical strategies.

Section 1: Radical Equations

Radical Equations Intro

Radical Equations w/ Multiple Solutions

Radical Equations w/ Extraneous Solutions

Radical Equations w/ Two Radicals

Section 2: Rational Expressions & Equations

Simplifying Rationals (Level 1)

Simplifying Rationals (Level 2)

Solving Quadratic Proportions

Rational Equations (Level 1)

Rational Equations (Level 2)

Rational Equations (Level 3)

Rational Equations (Level 4)

Section 3: Systems of Equations

Function Equality Graphically

Quadratic / Linear Systems (Level 1)

Quadratic / Linear Systems (Level 2)

Systems of Circles & Lines

Unit 7: Functions: Transformations, Inverses, & Average Rate of Change

Apply familiar strategies to shift, reflect, stretch, and compress different types of functions. Includes inverse functions and average rate of change.

Section 1: Transforming Functions

Translate Functions (Basic)

Translate Functions

Write Translated Function (Basic)

Write Translated Function

Reflect Functions (Level 1)

Reflect Functions (Level 2)

Mixed Function Transformations (Level 1)

Stretch / Compress Functions (Level 1)

Stretch / Compress Functions (Level 2)

Mixed Function Transformations (Level 2)

Write Transformed Function (Level 1)

Write Transformed Function (Level 2)

Transformations of Functions

Section 2: Inverse Functions & Average Rate of Change

Multi-Step Function Inverses

Inverse of Rational Functions

Inverse of Various Functions (MC)

Identifying Inverses Graphically

Average Rate of Change from a Graph

Average Rate of Change from an Equation

Average Rate of Change Function Comparison

Unit 8: Exponential Functions, Equations, & Logarithms

Explore the inverse relationship between exponents and logarithms and apply properties of logarithms to solve problems where a variable is in the exponent, including real-world contexts.

Section 1: Exponential Functions

Plugging into Exponential Formulas

Exponential Functions: Input & Output

Half Life & Doubling Time

Convert Form of Exponential Function

Compounding: Solve for Future Value

Section 2: Exponential Rate of Change

Find Growth/Decay Percentage (Level 1)

Find Growth/Decay Percentage (Level 2)

Growth/Decay: Write Function / Find % Change

Interpret Exponential Function Constant

Rewrite Exponential Function w/ Exponent Rules

Section 3: Logarithms

Evaluating Logarithms (Intro)

Evaluating Logarithms (Negatives & Fractions)

Evaluating Natural Logarithms

Logs & Exponentials as Inverses (Numerical)

Logs & Exponentials as Inverses (Algebraic)

Section 4: Solving Equations: Exponents & Logarithms

Solving Equations Using Logs

Solving Using a Common Base (Level 1)

Solving Using a Common Base (Level 2)

Evaluating Logarithms

Section 5: Applying Logarithms

Solving for Time in Exponential Models

Half Life & Doubling (Solve for t)

Continuous Interest: Solve for Future Value

Compounding: Solve for Time

Unit 9: Trigonometry

Revisit trig ratios. Introduction to radian measure. Explore graphs of trig functions and use them to model problems in context.

Section 1: Right Triangles & Trig Ratios

Pythagorean Theorem (Radical Answers)

Pythagorean Theorem (Radical Answers)

Special Right Triangles (Radical Answers)

Estimating Line Ratios

Identifying Trig Ratios in a Diagram

Identifying Reciprocal Trig Ratios

Rotate Angle Through Point & Find Ratio

Section 2: Rotations & The Unit Circle

Estimating Angle Measurements

Rotating Small Angles

Rotating Large Angles

Finding Coterminal Angles

Finding Reference Angles

Finding Common Reference Angles

Find Coterminal, Reference, & Quadrant

Find Trig Ratio from a Point

Find Coordinate on Unit Circle

Exact Trig Values Graphically (Sine & Cosine)

Exact Trig Values Graphically (Tangent)

Signs of Trig Functions

Signs of All Six Trig Functions

Section 3: Advanced Trig Ratios

Find Sine or Cosine Using Identity (Q1)

Find Sine or Cosine Using Identity (Q1)

Find Trig Ratio Given Another (All Six)

Find Trig Ratio Given Another (All Quadrants)

Section 4: Radians

Fraction of a Semicircle / Degrees to Radians

Rotate Radians & Find Value

Radians to Degrees

Degrees to Radians

Finding Reference Angles (Radians)

Find Coterminal, Reference, & Quadrant (Radians)

Find Angle (Radians) Given Reference/ Quadrant

Exact Value of Trig Functions (Radians)

Section 5: Intro to Trig Graphs

Midline from a Trig Graph

Amplitude from a Trig Graph

Period from a Trig Graph

Determine Min & Max from Trig Equation

Determine Period from Trig Equation

Write Trig Function Given Features

Trig Equation from a Graph

Section 6: Graphing & Modeling with Trig Functions

Trig Graphs: Amplitude & Period

Trig Graphs: Amplitude, Period, Midline

Graphing Trig Functions

Interpret Trig Graphs in Context

Interpret Trig Equations in Context

Unit 10: Statistics & Probability

Find area under a normal curve. Use a variety of function types to create nonlinear regressions and make predictions. Develop an intuition for confidence intervals and margin of error.

Section 1: Measuring Variation & Normal Distributions

Interpreting Calculator Output (Stats)

Calculate Measures of Dispersion

Normal Distribution: Empirical Rule

Normal Distribution: Find Area

Normal Distribution w/ Population Size

Section 2: Nonlinear Regression

Calculate Regressions

Exponential Regression

Power Regression

Quadratic Regression

Section 3: Simulations & Confidence Intervals

Margin of Error & Confidence Intervals

Analyzing Confidence Intervals

Create Confidence Interval from a Simulation

Proportion Inferences from Simulations

Significance of Difference in Means

Section 4: Probability

Probability from a Two Way Table

Conditional Probability from a Table

Probability with Or / And

Symbolic Probability Rules

Determine Independence from a Table

✨ 🍋 ✨ Why Lemons-Aid? ✨ 🍋 ✨

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A BIBLICAL WORLDVIEW: The Bible, infallible and inerrant, is the very written word of God, who has revealed Himself to man. The Bible is like the light we cast on all content areas in order to understand it, whether that be literature, physical science, history, or geometry. Students learn all content through a Biblical lens. Theology is important for understanding all subject areas. We carefully curate courses that capture learners’ imagination while pointing them to God through sound doctrine. THIS is most important!

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RICH CONTENT / CORE KNOWLEDGE: While other schools and systems try to align their content to broad standards that are vague and open to wild interpretations, we focus our content on what students should know and be able to do so they see the world biblically and head into their adult lives filled with knowledge, wisdom, and mastery of skill such as computing and writing. For over a century, progressive education reform has been “anti-content,” which means they de-emphasize rich content and focus instructional time on things such as self-esteem and “skills” they hope will benefit a learner in the future. This is why American kids do so poorly in testing compared to nations with content-rich curricula. We want our learners to increase in knowledge and grow in wisdom, which our content-area experts foster while teaching.

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EXPLICIT TEACHING: We understand the skills and concepts students need to learn and know how to teach them. Lemons-Aid’s materials are top-notch, organized, and clear for students and parents to understand. We are especially skilled at breaking down a complicated process into understandable parts. Further, explicit instruction is “a structured, systematic, and effective methodology for teaching academic skills. It is called explicit because it is an unambiguous and direct approach to teaching that includes both instructional design and delivery procedures. Explicit instruction is characterized by a series of supports or scaffolds, whereby students are guided through the learning process with clear statements about the purpose and rationale for learning the new skill, clear explanations and demonstrations of the instructional target, and supported practice with feedback until independent mastery has been achieved.”

• Explicit Instruction: Effective and Efficient Teaching by Anita L. Archer and Charles A. Hughes.

Anita Archer trained Mrs. Lemons in workshops, and it changed her teaching. Read a little more about the research behind explicit teaching here and here.

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STUDENT ACCOUNTABILITY = ACHIEVEMENT: Students master skills with us and make gains. We have a high degree of accountability. Since we make promises here and parents are paying good money, we understand you trust us to work! Students have to work too, and let’s be honest: they’re kids and don’t always want to. We push it. We teach them how to stay engaged, we cold-call on kids, we tell them to use the chatbox, and we want them to use emojis! If they are resistant, we contact the student through the teacher tab first. If that doesn’t work, we call in the big guns–Mom and Dad. We want kids to learn. We don’t want them to pass through our classes without gaining skills and doing great learning.

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DO HARD THINGS. Boost your confidence, master new skills, learn new concepts. This takes a commitment to do hard things. Like the standards we have for our teachers, we also expect our learners to do hard things, whether that means they stand firm in their convictions, learn geometry, write an essay, or give an oral presentation. You can do hard things!

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HEALTHY RELATIONSHIPS: To balance our high expectations for their learning and behavior, we build relationships with them. We want them to know we care about and know them. We’ll ask about their play last weekend or the new trick they’re trying to master on the skateboard. We also want students to get to know each other and encourage community engagement.

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DEPENDABLE: Multiple teachers are teaching this class, and we have an entire year of lessons planned and scheduled. Since we are a mission-driven organization, we protect our brand and the relationships with our families. We are accountable to our learners. When things come up for teachers, we work to get substitutes and do everything we can before canceling a class. We do not like canceling or changing, and we often teach classes at a loss to give others a chance to join. We have limits, of course, but we are not flippant or irresponsible about canceling! When things come up for students, since we have multiple sections, they can transfer from section to section. All our teachers teach the same content the same week, giving families even more flexibility!

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TEACHER FEEDBACK: The back-and-forth work between a student and teacher significantly benefits a student if done well. We follow best practices in designing class time, assignments, and routines. According to Pennington Publishing, effective writing feedback (or grading) is:

• Specific, not general
• Immediate, not postponed
• Routine with a revision / feedback cycle
• Explanatory
• The right amount
• Targeted to the most critical issues
• Varied (written, audio, and video comments)
• Holding students accountable

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WORKSHOP TIME: We use “workshop time” so students will work while the teacher answers questions, gets them started, and holds them accountable. In a writing class, the teacher “visits” learners on their Google Documents and watches and helps them write. The immediacy of the feedback/revision cycle with the instructor allows writers to improve rapidly. Additionally, once we started using this method in writing classes, we saw nearly a 100% completion rate in student essays!

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GRAPHIC ORGANIZERS: Students need graphic organizers to help them see the structure and breakdown of a concept or process. For example, we use them to help learners understand how to write a paragraph or essay and to use the writing process. This is how they learn to develop coherent ideas. They don’t figure out how to do this magically; the graphic organizers and the intentional, explicit teaching help them learn the skills!

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STUDENT MASTERY: Each class includes explicit, direct instruction with teacher modeling. Students are guided toward mastery of skills and understandings to grasp the concepts and become independent. Students are held to a high standard of academic work, including often ignored skills like the use of grammar and neatness in math.

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STUDY THE BEAUTIFUL

We are surrounded by the mediocre, which is not good! We see this in expectations at some schools, the poor customer service at a store, and even architecture like in a gray, uninspiring complex of high-occupancy housing.

In contrast, we are surrounded by the beautiful, which is good! We see the beautiful in classic literature, music, and beautiful architecture like pictured here.

The mediocre demoralizes learners while the beautiful inspires.

At Lemons-Aid Learning, we study the beautiful: classic literature, artful sentence construction, art, poetry, maths, God’s hand in all of history, and God’s very creation. His creation glorifies Him, and in our study of all content areas, we learn about who God is.

We do not compromise. This means we don’t choose a graphic novel of Shakespeare’s A Midsummer Night’s Dream. We read the original play. We know how to make the complexity and beauty of classic study approachable and understandable to a modern audience. It’s more difficult, but worth the effort!

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CUSTOMER SERVICE

We serve the Lord and we work hard for families. We work to give quick responses to questions, authentic and careful feedback, and to solve any conflict. As home educators ourselves, familiar with the joys and struggles of teaching our own children, we can relate! We are supporting families, equipping learners, and serving Christ. We are 100% devoted to Him and to you!

To read more about our teaching and learning methods, read our blogs, written by our teachers and staff.

The Lemons-Aid Team

Lemons-Aid teachers have a few things in common.
❤️ They love their students and value each of their unique strengths and personalities that make our classes special. Our classes can be described as fun, personal, academic, challenging, and supportive.
🤩 We work to keep learners engaged, so there is always a degree of student accountability for their attention and focus, whether that be through asking them direct questions or by using the chatbox.
💭 We know all kids can learn, but sometimes things are hard! To support students, we teach them how to develop effective thinking and learning habits that will bring them success in class and in life.
🌟 Building relationships with students so they know we care about them helps us balance the high expectations we have for them regarding their effort, work quality, and behavior. Our students are encouraged, cared for, and they achieve!

Christian Teachers on Outschool

Lemons-Aid Coupon Code:
We want to serve you on Lemons-Aid! For first-time learners on Lemons-Aid, you can use the coupon code Newbie20 to get $20 off your first class.

Outschool Coupon Code:
If the schedule on this platform doesn’t work for you, we will happily teach you on Outschool, but we can’t talk about Jesus. Use this referral code and get $20 off your first class on Outschool: LEMONSA2020

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