**Math Topics**- Common Core
- Initiatives
- Methodology
- Resources
- Projects
- Manipulatives
- Software

**Learning Support**- Standardized Test Prep
- Technology Integration
- Assisting Readers

Select the cluster for resources on this page:

**Standards**:

- SSE-A.1. Interpret expressions that represent a quantity in terms of its context.
- SSE-A.1.a. Interpret parts of an expression, such as terms, factors, and coefficients.
- SSE-A.1.b. Interpret complicated expressions by viewing one or more of their parts as a single
entity.
*For example, interpret P (1 + r)*^{n}as the product of P and a factor not depending on P.

- SSE-A.2. Use the structure of an expression to identify ways to rewrite it.
*For example, see x*^{4 }– y^{4}as (x^{2})^{2}– (y^{2})^{2}, thus recognizing it as a difference of squares that can be factored as (x^{2}- y^{2})(x^{2}+ y^{2}).

**Technology-enhanced investigations:**

Mathwords.com: Math Dictionary: Key vocabulary for this domain. Use with SSE-A, SSE-B:

coefficient | factoring rules | maximum of a function | term |

exponent rules | factor of a polynomial | minimum of a function | zero of a function |

expression | geometric series |

National Library of Virtual Manipulatives: Algebra Tiles: Visualize multiplying and factoring algebraic expressions using tiles. Several activities are provided.

Purple Math: Special Factoring: Factoring differences of squares, sums and differences of cubes, recognizing patterns. Use with SSE-A.2.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives:

- US Common Core State Standard Math HSA-SSE.A.1: 3 manipulatives: Binomial cube, sums of consecutive integers, integer polynomial
- US Common Core State Standard Math HSA-SSE.A.2: 9 manipulatives: Binomial cube, sums of consecutive integers, integer polynomial, multiplying a monomial and a linear polynomial, squaring a binomial, square of the sum of two numbers, multiplying binomials, difference of cubes, difference of squares

YouTube video lessons:

- Translating Words into Algebraic Expressions: Aligns with SSE-A.1
- Translating Words into Expressions, Equations, and Inequalities: Aligns with SSE-A.1
- Expressions and Equations: Interpreting Expressions: Aligns with SSE-A.1
- Factoring the Difference of Two Perfect Squares (like x
^{2}-4): Aligns with SSE-A.2. - Factoring difference of squares (like x
^{2}-49y^{2}): Aligns with SSE-A.2. - Factoring a Difference of Two Perfect Squares using an Area Model (like
16p
^{4}- 9 or n^{4}- 9m^{4}): Aligns with SSE-A.2. - Factoring the Difference of Two Squares - Ex
2 (like 16x
^{2}-25y^{2}or 64x^{2}-81y^{2}): Aligns with SSE-A.2. - Factoring the Difference of Two Squares - Ex 3 (like (x-3)
^{2}-49 or (x+7)^{2}-25): Aligns with SSE-A.2. - Factoring Trinomials by Using the Punnett Square-Box
Method
(like x
^{2}+ 7x + 12): Aligns with SSE-A.2. - Factoring
a Perfect Square Trinomial (like 16x
^{2}+ 56x + 49): Aligns with SSE-A.2. - Factoring Perfect Square Trinomials - Ex 1
(like 9x
^{2}+ 24xy + 16y^{2}) : Aligns with SSE-A.2. - Factoring Perfect Square Trinomials - Ex 2
(like 81x
^{2}- 36xy + 4y^{2}): Aligns with SSE-A.2. - Factor by Grouping and Factoring Completely: Aligns with SSE-A.2.
- Factoring Trinomials: Factor by Grouping - ex 1
(like 12x
^{2}+ 34x + 10) : Aligns with SSE-A.2. - Factoring Trinomials: Factor by Grouping - ex 2 (like 6x
^{2}+ 15x - 21) : Aligns with SSE-A.2. - Factoring Trinomials: Factor by Grouping - ex 3
(like -81x
^{2}+ 60x - 28) : Aligns with SSE-A.2.

**Multiple Choice:**

Khan Academy: Practice questions with videos. Use the following with SSE-A.1:

Khan Academy: Algebra 1: Polynomial Factorization and Algebra 2: Polynomial Functions are comprehensive units with practice questions and videos. Use with SSE-A.2.

**Constructed-response:**

Khan Academy: HSA-A-SSE includes HSA-SSE.A.1 and HSA-SSE.A.2 Common Core aligned problems.

Also see:

- Algebra 1: Polynomial Factorization and Algebra 2: Polynomial Functions are comprehensive units with practice questions and videos. Use with SSE-A.2.

L. Spector: The Math Page: Skill in Algebra. Students see worked examples of concepts and following the presentation within each section, students are presented problems in constructed response form. They would solve using paper-pencil and then by using their mouse to roll over a box, they see the solution to problems presented. Use the following with SSE-A.2:

- Lesson 15: Common Factor: There are two levels, including factoring polynomials and factoring by grouping.
- Lesson 17: Factoring Trinomials: includes factoring where step 1 is to remove the common factor.
- Lesson 18: The Square of a Binomial: Perfect Square Trinomials
- Lesson 19: The Difference of Two Squares: There are two levels, including a summary of multiplying/factoring and factoring by grouping, the sum and difference of odd powers, and the difference of even powers.

**Performance tasks:**

Illustrative Mathematics: Algebra:

- The Bank Account: Involves an exponential. Aligns with SSE-A.1.
- Profit of a Company, assessment variation: Involves three
equivalent quadratic equations for
*P(x)*. Aligns with SSE-A.1 and SSE-B.3. - Mixing Candies: Aligns with SSE-A.1.a.
- Throwing Horseshoes: Involves a quadratic. Aligns with SSE-A.1.a., SSE-B.3, and mathematical practice standards 4 and 7. See NCTM's Reasoning and Sense Making Task Library: Horseshoes in Flight for the task overview and teacher notes for its use in the classroom.
- Seeing Dots Aligns with SSE-A, and SSE-B.3.a and SSE-B.3.b.

Mathematics Assessment Project: Standards: High School: Algebra:

- Task 221: Interpreting Algebraic Expressions
- Task 426: Comparing Investments
- Task 436: Representing Polynomials
- Task 437: Manipulating Polynomials
- Task 441: Sectors of Circles
- Task 218: Sorting Equations and Identities

Mathematics Vision Project, Secondary 1 Student Edition:

- Getting Ready Module: This module contains 10 classroom tasks. Task 1: Checkerboard Borders; Task 2: Building More Checkerboard Borders, and Task 3: Serving Up Symbols align to algebra standard SSE-A.1.
- Module 3: Linear and Exponential Functions: This module contains 10 classroom tasks. Task 7: Making My Point, Task 8: Efficiency Experts, and Task 9: Up a Little, Down a Little align to algebra standard SSE-A.1.

**Standards**:

- SSE-B.3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
- SSE-B.3a. Factor a quadratic expression to reveal the zeros of the function it defines.
- SSE-B.3b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
- SSE-B.3c. Use the properties of
exponents to transform expressions for exponential functions.
*For example the expression 1.15*.^{t}can be rewritten as (1.15^{1/12})^{12t}≈ 1.012^{12t}to reveal the approximate equivalent monthly interest rate if the annual rate is 15% - SSE-B.4 Derive the formula for the sum of
a finite geometric series (when the common ratio is not 1), and use
the formula to solve problems.
*For example, calculate mortgage payments.*

**Technology-enhanced investigations:**

MIT BLOSSOMS: Video lessons with additional teacher and learner resources. Descriptions are from the video summaries.

- The Quadratic Equation: It’s Hip to Be Squared: "The lesson begins with three
quadratics and their graphs (three parabolas): y = x
^{2}- 2x + (0 or 1 or 2). The prerequisite or co-requisite is some working experience with algebra, like factoring x^{2}-2x into x(x-2). The objective is to connect four things: the formula for y, the graph of y (a parabola), the roots of y and the minimum or maximum of y." Aligns with Algebra standards SSE-B.3.a, CED-A.2, and REI-B.4.b and Function standards IF-B.4, IF-C.7.a, IF-C.8.a, BF-B.3. - Flu Math Games: "This video lesson shows students that math can play a role in understanding how an infectious disease spreads and how it can be controlled." Additional simulations are included. Aligns with Algebra standards SSE-B.3.c and REI-A.1; Function standards IF-C.8.b, BF-B.4.a, and LE-A-1.(a, c); and Statistics and Probability standards ID-B.6.a, IC-A.1, IC-B.4, CP-A.2, and MD-A.1.
- The Power of Exponentials, Big and Small: "Exponential growth is keenly applicable to a variety of different fields ranging from cell growth in biology, nuclear chain reactions in physics to computational complexity in computer science. In this lesson, through various examples and activities, [the presenters] have tried to compare exponential growth to polynomial growth and to develop an insight about how quickly the number can grow or decay in exponentials." Aligns with Algebra standards SSE-B.3.c and CED-A.2 and Function standards IF-A.3, IF-C.7.(a, c, e), IF-C.8.b, and LE-A.1.(a, c).
- The Towers of Hanoi: Experiential Recursive Thinking: "This lesson is about the Towers of Hanoi problem, a classic famous problem involving recursive thinking to reduce what appears to be a very large and difficult problem into a series of simpler ones. The learning objective is for students to begin to understand recursive logic and thinking, relevant to computer scientists, mathematicians and engineers." Aligns with Algebra standard SSE-B.3.c, and Function standards IF-C.8.b and LE-A.1.a.
- Tragedy of the Commons: "This lesson introduces students to the “Tragedy of the Commons,” an extended metaphor for problems of shared environmental or man-made resources that are overused and eventually depleted. In this metaphor, shared resources are compared to a common grazing pasture, or “commons,” on which any dairy farmer can graze as many cows as he/she wishes. If too many cows are added to the commons, they will overeat the grass in the pasture and the shared resource will become depleted – a disadvantage to everyone. In this lesson, students will be inspired to think about possible solutions to this problem." Aligns with Algebra standard SSE-B.3.c and Function standards IF-B.4, IF-C.8.b, LE-A.1.a, LE-A.3, and LE-B.5.

NCTM Illuminations: Algebra Tiles: Use virtual manipulative algebra tiles "to represent variables and constants, learn how to represent and solve algebra problems. Solve equations, substitute in variable expressions, and expand and factor."

National Library of Virtual Manipulatives: Grapher: A tool for graphing and exploring functions

Imagine Learning Classroom (formerly LearnZillion) subscription needed:

- Lesson set: Factor a quadratic expression to reveal the zeros of the function it defines: This is a series of five video lessons that also include guided practice on this topic. Aligns with SSE-B.3.a.
- Lesson set: Complete the square to find the maximum/minimum value of a quadratic function: This is a series of three video lessons that also include guided practice on this topic. Videos demonstrate how to determine if the quadratic function has a maximum or minimum value, complete the square, and identify the coordinate of the maximum or minimum. Aligns with SSE-B.3.b.

Khan Academy: Finite geometric series formula: video lesson. Aligns with SSE-B.4.

Purple Math: Aligns with SSE-B.4:

- Derivation of Finite Geometric Summation Formula
- Geometric Series: finite and infinite; examples of using the formulas

Saltire Software: Common Core Nuggets: There are three Quadratic Zeroes Applets, which align with SSE-B.3.a.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives:

- US Common Core State Standard Math HSA-SSE.B.3: 17 manipulatives. Among those: Completing the square, continuous exponential growth, exponential decay, investment returns, simple versus compound interest, half-life of a radio active element, Newton's law of cooling, carbon dating, geometric series, and more.
- US Common Core State Standard Math HSA-SSE.B.4: 7 manipulatives. Among those: Plot of a geometric sequence and its partial sums, sum of a geometric series, geometric series, graphical representation of geometric series, visual computation of three geometric sums, and more.

**Multiple Choice:**

Khan Academy: Sequences, series intro includes a section on finite geometric series. Practice questions with videos.

**Constructed-response:**

Khan Academy: HSA-A-SSE includes HSA-SSE.B.3 and HSA-SSE.B.4 Common Core aligned problems.

Also see:

- Algebra 1: Quadratics with methods for solving quadratic equations and Algebra 1: Sequences are comprehensive units with practice questions and videos. Use with SSE-B.3 and SSE-B.4.

MathWarehouse:

**Performance tasks:**

Illustrative Mathematics: Algebra:

- Seeing Dots Aligns with SSE-A, and SSE-B.3.a and SSE-B.3.b.
- Taxes and Sales Aligns with SSE-B.
- Profit of a Company, assessment variation: Involves three
equivalent quadratic equations for
*P(x)*. Aligns with SSE-A.1 and SSE-B.3. - Throwing Horseshoes: Involves a quadratic. Aligns with SSE-A.1.a., SSE-B.3, and mathematical practice standards 4 and 7. See NCTM's Reasoning and Sense Making Task Library: Horseshoes in Flight for the task overview and teacher notes for its use in the classroom.
- Building a General Quadratic Function Aligns with SSE-B and function standard BF-B.3.
- Graphs of Quadratic Functions: Aligns with SSE-B.3 and function standard IF-C.7.
- Course of Antibiotics: Aligns with SSE-B.4.
- Triangle Series: Aligns with SSE-B.4.
- Cantor Set: Aligns with SSE-B.4.
- YouTube Explosion: Aligns with SSE-B.4.
- A Lifetime of Savings: Aligns with SSE-B.4.

Inside Mathematics: MARS Task: Two Solutions: This task addresses standard SSE-B.3. Students find solutions to equations and inequalities. Some problems involve factoring a quadratic expression to reveal the zeros of the function it defines.

Mathematics Assessment Project: Standards: High School: Algebra:

- Task 405: Modeling Conditional Probabilities 2
- Task 218: Sorting Equations and Identities
- Task 224: Forming Quadratics
- Task 426: Comparing Investments

Common Core Math:
Intro | HS Algebra Domain: **SSE** | APR | CED | REI |