## Description

- Overview:
- This lesson unit is intended to help teachers assess how well students are able to solve quadratics in one variable. In particular, the lesson will help teachers identify and help students who have the following difficulties: making sense of a real life situation and deciding on the math to apply to the problem; solving quadratic equations by taking square roots, completing the square, using the quadratic formula, and factoring; and interpreting results in the context of a real life situation.

- Level:
- Lower Primary, Upper Primary, Middle School, High School
- Grades:
- Kindergarten, Grade 1, Grade 2, Grade 3, Grade 4, Grade 5, Grade 6, Grade 7, Grade 8, Grade 9, Grade 10, Grade 11, Grade 12
- Material Type:
- Assessment, Lesson Plan
- Provider:
- Shell Center for Mathematical Education, U.C. Berkeley
- Provider Set:
- Mathematics Assessment Project (MAP)
- Date Added:
- 04/26/2013

- License:
- Creative Commons Attribution-NonCommercial-NoDerivs 3.0
- Media Format:
- Downloadable docs, Text/HTML

# Comments

## Standards

# Common Core State Standards Math

Grades 9-12,Algebra: Reasoning with Equations and InequalitiesCluster: Understand solving equations as a process of reasoning and explain the reasoning

Standard: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Reason abstractly and quantitatively. Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Reasoning with Equations and InequalitiesCluster: Solve equations and inequalities in one variable

Standard: Solve quadratic equations in one variable.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Reasoning with Equations and InequalitiesCluster: Solve equations and inequalities in one variable

Standard: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Modeling with GeometryCluster: Apply geometric concepts in modeling situations

Standard: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Modeling with GeometryCluster: Apply geometric concepts in modeling situations

Standard: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Geometry: Modeling with GeometryCluster: Apply geometric concepts in modeling situations

Standard: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Reasoning with Equations and Inequalities

Standard: Solve equations and inequalities in one variable

Indicator: Solve quadratic equations in one variable.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Reasoning with Equations and Inequalities

Standard: Solve equations and inequalities in one variable

Indicator: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Modeling with Geometry

Standard: Apply geometric concepts in modeling situations

Indicator: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Modeling with Geometry

Standard: Apply geometric concepts in modeling situations

Indicator: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?"ť They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Reasoning with Equations and Inequalities

Standard: Understand solving equations as a process of reasoning and explain the reasoning

Indicator: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Reason abstractly and quantitatively. Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize"Óto abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents"Óand the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and"Óif there is a flaw in an argument"Óexplain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Geometry: Modeling with Geometry

Standard: Apply geometric concepts in modeling situations

Indicator: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).*

Degree of Alignment: Not Rated (0 users)

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# Tags (11)

- Mathematics
- Algebra and Calculus
- CCSS
- Common Core Math
- Common Core PD
- Factoring
- ODE Learning
- Quadratic Equations
- Quadratic Formula
- Real World Math
- Square Roots

Lesson unit serves as a formative assessment tool to determine how well students are able to solve quadratics in one variable. Cutting Corners provides a real-life situation using mathematical principles to analyze and interpret results. Students work individually and in small cooperative learning groups to build conceptual understanding through the use of a range of mathematical practices.

Materials provide instructional strategies to address common misconceptions.

A-REI.2 – Rating: 3

A-REI.3 – Rating: 2

A-REI.4 – Rating: 2

G-MG.1 – Rating: 3

G-MG.3 – Rating: 2