This cluster includes the following access points.
Vetted resources educators can use to teach the concepts and skills in this topic.
| Name |
Description |
| Interpreting Basic Tax: | Students are asked to interpret the parts of an equation used to calculate the total purchase price including tax of a set of items. |
| Rewriting Numerical Expressions: | Students are asked to rewrite numerical expressions to find efficient ways to calculate. |
| Determine the Width: | Students are asked to find the width of a rectangle whose area and length are given as polynomials. |
| Quadratic Expressions: | Students are asked to identify equivalent quadratic expressions and to name the form in which each expression is written. |
| Finding Missing Values: | Students are asked to rewrite quadratic expressions and identify parts of the expressions. |
| Dot Expressions: | Students are asked to explain how parts of an algebraic expression relate to the number and type of symbols in a sequence of diagrams. |
| What Happens?: | Students are asked to determine how the volume of a cone will change when its dimensions are changed. |
| Name |
Description |
| Free Fall Clock and Reaction Time!: | This will be a lesson designed to introduce students to the concept of 9.81 m/s2 as a sort of clock that can be used for solving all kinematics equations where a = g. |
| Sorting Equations and Identities: | This lesson is intended to help you assess how well students are able to:
- Recognize the differences between equations and identities.
- Substitute numbers into algebraic statements in order to test their validity in special cases.
- Resist common errors when manipulating expressions such as 2(x – 3) = 2x – 3; (x + 3)2 = x2 + 32.
- Carry out correct algebraic manipulations.
It also aims to encourage discussion on some common misconceptions about algebra. |
| Manipulating Polynomials: | This lesson unit is intended to help you assess how well students are able to manipulate and calculate with polynomials. In particular, it aims to identify and help students who have difficulties in switching between visual and algebraic representations of polynomial expressions, performing arithmetic operations on algebraic representations of polynomials, factorizing and expanding appropriately when it helps to make the operations easier. |
| Math Is Exponentially Fun!: | The students will informally learn the rules for exponents: product of powers, powers of powers, zero and negative exponents. The activities provide the teacher with a progression of steps that help lead students to determine results without knowing the rules formally. The closing activity is hands-on to help reinforce all rules. |
| Name |
Description |
| Quadrupling Leads to Halving: | Students explore the structure of the operation s/(vn). This question provides students with an opportunity to see expressions as constructed out of a sequence of operations: first taking the square root of n, then dividing the result of that operation into s. |
| A Cubic Identity: | Solving this problem with algebra requires factoring a particular cubic equation (the difference of two cubes) as well as a quadratic equation. An alternative solution using prime numbers and arithmetic is presented. |
| The Physics Professor: | Students write explanations of the structure and function of a mathematical expression. |
| Equivalent Expressions: | This is a standard problem phrased in a non-standard way. Rather than asking students to perform an operation, expanding, it expects them to choose the operation for themselves in response to a question about structure. Students must understand the need to transform the factored form of the quadratic expression (a product of sums) into a sum of products in order to easily see a, the coefficient of the x2 term; k, the leading coefficient of the x term; and n, the constant term. |
| Throwing Horseshoes: | Students evaluate equivalent constructions of the same expression to determine which is the most useful for determining a maximum value. |
| The Bank Account: | Students explore an expression that calculates the balance of a bank account with compounding interest. |
| Radius of a Cylinder: | Students are asked to interpret the effect on the value of an expression given a change in value of one of the variables. |
| Mixing Fertilizer: | Students examine and answer questions related to a scenario similar to a "mixture" problem involving two different mixtures of fertilizer. In this example, students determine and then compare expressions that correspond to concentrations of various mixtures. Ultimately, students generalize the problem and verify conclusions using algebraic rather than numerical expressions. |
| Mixing Candies: | Students are asked to interpret expressions and equations within the context of the amounts of caramels and truffles in a box of candy. |
| Kitchen Floor Tiles: | This problem asks students to consider algebraic expressions calculating the number of floor tiles in given patterns. The purpose of this task is to give students practice in reading, analyzing, and constructing algebraic expressions, attending to the relationship between the form of an expression and the context from which it arises. The context here is intentionally thin; the point is not to provide a practical application to kitchen floors, but to give a framework that imbues the expressions with an external meaning. |
| Increasing or Decreasing? Variation 1: | Students examine variable expression that is a complex fraction with two distinct unit fractions in the denominator. Students are asked to consider how increasing one variable will affect the value of the entire expression. The variable expression is used in physics and describes the combined resistance of two resistors in parallel. |
| Delivery Trucks: | This resource describes a simple scenario which can be represented by the use of variables. Students are asked to examine several variable expressions, interpret their meaning, and describe what quantities they each represent in the given context. |
| Animal Populations: | In this task students interpret the relative size of variable expressions involving two variables in the context of a real world situation. All given expressions can be interpreted as quantities that one might study when looking at two animal populations. |
| Computations with Complex Numbers: | This resource involves simplifying algebraic expressions that involve complex numbers and various algebraic operations. |
| Seeing Dots: | The purpose of this task is to identify the structure in the two algebraic expressions by interpreting them in terms of a geometric context. Students will have likely seen this type of process before, so the principal source of challenge in this task is to encourage a multitude and variety of approaches, both in terms of the geometric argument and in terms of the algebraic manipulation. |
Vetted resources students can use to learn the concepts and skills in this topic.
| Title |
Description |
| Quadrupling Leads to Halving: | Students explore the structure of the operation s/(vn). This question provides students with an opportunity to see expressions as constructed out of a sequence of operations: first taking the square root of n, then dividing the result of that operation into s. |
| A Cubic Identity: | Solving this problem with algebra requires factoring a particular cubic equation (the difference of two cubes) as well as a quadratic equation. An alternative solution using prime numbers and arithmetic is presented. |
| Equivalent Expressions: | This is a standard problem phrased in a non-standard way. Rather than asking students to perform an operation, expanding, it expects them to choose the operation for themselves in response to a question about structure. Students must understand the need to transform the factored form of the quadratic expression (a product of sums) into a sum of products in order to easily see a, the coefficient of the x2 term; k, the leading coefficient of the x term; and n, the constant term. |
| Radius of a Cylinder: | Students are asked to interpret the effect on the value of an expression given a change in value of one of the variables. |
| Mixing Fertilizer: | Students examine and answer questions related to a scenario similar to a "mixture" problem involving two different mixtures of fertilizer. In this example, students determine and then compare expressions that correspond to concentrations of various mixtures. Ultimately, students generalize the problem and verify conclusions using algebraic rather than numerical expressions. |
| Mixing Candies: | Students are asked to interpret expressions and equations within the context of the amounts of caramels and truffles in a box of candy. |
| Kitchen Floor Tiles: | This problem asks students to consider algebraic expressions calculating the number of floor tiles in given patterns. The purpose of this task is to give students practice in reading, analyzing, and constructing algebraic expressions, attending to the relationship between the form of an expression and the context from which it arises. The context here is intentionally thin; the point is not to provide a practical application to kitchen floors, but to give a framework that imbues the expressions with an external meaning. |
| Delivery Trucks: | This resource describes a simple scenario which can be represented by the use of variables. Students are asked to examine several variable expressions, interpret their meaning, and describe what quantities they each represent in the given context. |
| Animal Populations: | In this task students interpret the relative size of variable expressions involving two variables in the context of a real world situation. All given expressions can be interpreted as quantities that one might study when looking at two animal populations. |
| Computations with Complex Numbers: | This resource involves simplifying algebraic expressions that involve complex numbers and various algebraic operations. |
| Seeing Dots: | The purpose of this task is to identify the structure in the two algebraic expressions by interpreting them in terms of a geometric context. Students will have likely seen this type of process before, so the principal source of challenge in this task is to encourage a multitude and variety of approaches, both in terms of the geometric argument and in terms of the algebraic manipulation. |
Vetted resources caregivers can use to help students learn the concepts and skills in this topic.
| Title |
Description |
| Quadrupling Leads to Halving: | Students explore the structure of the operation s/(vn). This question provides students with an opportunity to see expressions as constructed out of a sequence of operations: first taking the square root of n, then dividing the result of that operation into s. |
| A Cubic Identity: | Solving this problem with algebra requires factoring a particular cubic equation (the difference of two cubes) as well as a quadratic equation. An alternative solution using prime numbers and arithmetic is presented. |
| Equivalent Expressions: | This is a standard problem phrased in a non-standard way. Rather than asking students to perform an operation, expanding, it expects them to choose the operation for themselves in response to a question about structure. Students must understand the need to transform the factored form of the quadratic expression (a product of sums) into a sum of products in order to easily see a, the coefficient of the x2 term; k, the leading coefficient of the x term; and n, the constant term. |
| Radius of a Cylinder: | Students are asked to interpret the effect on the value of an expression given a change in value of one of the variables. |
| Mixing Fertilizer: | Students examine and answer questions related to a scenario similar to a "mixture" problem involving two different mixtures of fertilizer. In this example, students determine and then compare expressions that correspond to concentrations of various mixtures. Ultimately, students generalize the problem and verify conclusions using algebraic rather than numerical expressions. |
| Mixing Candies: | Students are asked to interpret expressions and equations within the context of the amounts of caramels and truffles in a box of candy. |
| Kitchen Floor Tiles: | This problem asks students to consider algebraic expressions calculating the number of floor tiles in given patterns. The purpose of this task is to give students practice in reading, analyzing, and constructing algebraic expressions, attending to the relationship between the form of an expression and the context from which it arises. The context here is intentionally thin; the point is not to provide a practical application to kitchen floors, but to give a framework that imbues the expressions with an external meaning. |
| Delivery Trucks: | This resource describes a simple scenario which can be represented by the use of variables. Students are asked to examine several variable expressions, interpret their meaning, and describe what quantities they each represent in the given context. |
| Animal Populations: | In this task students interpret the relative size of variable expressions involving two variables in the context of a real world situation. All given expressions can be interpreted as quantities that one might study when looking at two animal populations. |
| Computations with Complex Numbers: | This resource involves simplifying algebraic expressions that involve complex numbers and various algebraic operations. |
| Seeing Dots: | The purpose of this task is to identify the structure in the two algebraic expressions by interpreting them in terms of a geometric context. Students will have likely seen this type of process before, so the principal source of challenge in this task is to encourage a multitude and variety of approaches, both in terms of the geometric argument and in terms of the algebraic manipulation. |
as the product of P and a factor not depending on P.