Standard #: MAFS.912.A-SSE.2.3 (Archived Standard)

Students are asked to rewrite a quadratic expression in vertex form to find maximum and minimum values.

Students are asked to find the zeros of a quadratic function in the context of a modeling problem.

Students are asked to transform an exponential expression so that the rate of change corresponds to a different time interval.

Students are asked to use the properties of exponents to show that two expressions are equivalent and compare the two functions in terms of what each reveals.

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.

Learn the process of completing the square of a quadratic function to find the maximum or minimum to discover how high a dolphin jumped in this interactive tutorial.

This is part 2 of a 2 part series. Click HERE to open part 1.

Learn the process of completing the square of a quadratic function to find the maximum or minimum to discover how high a dolphin jumped in this interactive tutorial.

This is part 1 of a 2 part series. Click HERE to open Part 2.

Learn to find the zeros of a quadratic function and interpret their meaning in real-world contexts with this interactive tutorial.

Learn to complete the square of a quadratic expression and identify the maximum or minimum value of the quadratic function it defines. In this interactive tutorial, you'll also interpret the meaning of the maximum and minimum of a quadratic function in a real world context.

There are many different ways to write exponential expressions that describe the same quantity, in this task the amount of a radioactive substance after t years. Depending on what aspect of the context we need to investigate, one expression of the quantity may be more useful than another. This task contrasts the usefulness of four equivalent expressions. Students first have to confirm that the given expressions for the radioactive substance are equivalent. Then they have to explain the significance of each expression in the context of the situation.

In this resource, a method of deriving the quadratic formula from a theoretical standpoint is demonstrated. This task is for instructional purposes only and builds on "Building an explicit quadratic function."

This task compares the usefulness of different forms of a quadratic expression. Students have to choose which form most easily provides information about the maximum value, the zeros and the vertical intercept of a quadratic expression in the context of a real world situation. Rather than just manipulating one form into the other, students can make sense out of the structure of the expressions.

(From Algebra: Form and Function, McCallum et al., Wiley 2010)

The purpose of this task is to help students see manipulation of expressions as an activity undertaken for a purpose.

Variation 1 of this task presents a related more complex expression already in the correct form to answer the question.

The expression arises in physics as the reciprocal of the combined resistance of two resistors in parallel. However, the context is not explicitly considered here.

This task illustrates the process of rearranging the terms of an expression to reveal different aspects about the quantity it represents, precisely the language being used in standard MAFS.912.A-SSE.2.3. Students are provided with an expression giving the temperature of a container at a time t, and have to use simple inequalities (e.g., that 2^t>0 for all t) to reduce the complexity of an expression to a form where bounds on the temperature of a container of ice cream are made apparent.

Students compare graphs of different quadratic functions, then produce equations of their own to satisfy given conditions.

This exploration can be done in class near the beginning of a unit on graphing parabolas. Students need to be familiar with intercepts, and need to know what the vertex is. It is effective after students have graphed parabolas in vertex form (y=a(x–h)²+k), but have not yet explored graphing other forms.

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.

This tutorial demonstrates how to use the power of a power property with both numerals and variables.

Learn the process of completing the square of a quadratic function to find the maximum or minimum to discover how high a dolphin jumped in this interactive tutorial.

This is part 2 of a 2 part series. Click HERE to open part 1.

Learn the process of completing the square of a quadratic function to find the maximum or minimum to discover how high a dolphin jumped in this interactive tutorial.

This is part 1 of a 2 part series. Click HERE to open Part 2.

Learn to find the zeros of a quadratic function and interpret their meaning in real-world contexts with this interactive tutorial.

Learn to complete the square of a quadratic expression and identify the maximum or minimum value of the quadratic function it defines. In this interactive tutorial, you'll also interpret the meaning of the maximum and minimum of a quadratic function in a real world context.

There are many different ways to write exponential expressions that describe the same quantity, in this task the amount of a radioactive substance after t years. Depending on what aspect of the context we need to investigate, one expression of the quantity may be more useful than another. This task contrasts the usefulness of four equivalent expressions. Students first have to confirm that the given expressions for the radioactive substance are equivalent. Then they have to explain the significance of each expression in the context of the situation.

In this resource, a method of deriving the quadratic formula from a theoretical standpoint is demonstrated. This task is for instructional purposes only and builds on "Building an explicit quadratic function."

This task compares the usefulness of different forms of a quadratic expression. Students have to choose which form most easily provides information about the maximum value, the zeros and the vertical intercept of a quadratic expression in the context of a real world situation. Rather than just manipulating one form into the other, students can make sense out of the structure of the expressions.

(From Algebra: Form and Function, McCallum et al., Wiley 2010)

The purpose of this task is to help students see manipulation of expressions as an activity undertaken for a purpose.

Variation 1 of this task presents a related more complex expression already in the correct form to answer the question.

The expression arises in physics as the reciprocal of the combined resistance of two resistors in parallel. However, the context is not explicitly considered here.

This task illustrates the process of rearranging the terms of an expression to reveal different aspects about the quantity it represents, precisely the language being used in standard MAFS.912.A-SSE.2.3. Students are provided with an expression giving the temperature of a container at a time t, and have to use simple inequalities (e.g., that 2^t>0 for all t) to reduce the complexity of an expression to a form where bounds on the temperature of a container of ice cream are made apparent.

Students compare graphs of different quadratic functions, then produce equations of their own to satisfy given conditions.

This exploration can be done in class near the beginning of a unit on graphing parabolas. Students need to be familiar with intercepts, and need to know what the vertex is. It is effective after students have graphed parabolas in vertex form (y=a(x–h)²+k), but have not yet explored graphing other forms.

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.

This tutorial demonstrates how to use the power of a power property with both numerals and variables.

There are many different ways to write exponential expressions that describe the same quantity, in this task the amount of a radioactive substance after t years. Depending on what aspect of the context we need to investigate, one expression of the quantity may be more useful than another. This task contrasts the usefulness of four equivalent expressions. Students first have to confirm that the given expressions for the radioactive substance are equivalent. Then they have to explain the significance of each expression in the context of the situation.

In this resource, a method of deriving the quadratic formula from a theoretical standpoint is demonstrated. This task is for instructional purposes only and builds on "Building an explicit quadratic function."

This task compares the usefulness of different forms of a quadratic expression. Students have to choose which form most easily provides information about the maximum value, the zeros and the vertical intercept of a quadratic expression in the context of a real world situation. Rather than just manipulating one form into the other, students can make sense out of the structure of the expressions.

(From Algebra: Form and Function, McCallum et al., Wiley 2010)

The purpose of this task is to help students see manipulation of expressions as an activity undertaken for a purpose.

Variation 1 of this task presents a related more complex expression already in the correct form to answer the question.

The expression arises in physics as the reciprocal of the combined resistance of two resistors in parallel. However, the context is not explicitly considered here.

This task illustrates the process of rearranging the terms of an expression to reveal different aspects about the quantity it represents, precisely the language being used in standard MAFS.912.A-SSE.2.3. Students are provided with an expression giving the temperature of a container at a time t, and have to use simple inequalities (e.g., that 2^t>0 for all t) to reduce the complexity of an expression to a form where bounds on the temperature of a container of ice cream are made apparent.

Students compare graphs of different quadratic functions, then produce equations of their own to satisfy given conditions.

This exploration can be done in class near the beginning of a unit on graphing parabolas. Students need to be familiar with intercepts, and need to know what the vertex is. It is effective after students have graphed parabolas in vertex form (y=a(x–h)²+k), but have not yet explored graphing other forms.

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.

This tutorial demonstrates how to use the power of a power property with both numerals and variables.