# Push and Support Cards: A Bridge to Advanced Mathematics for the Multi-Level Adult Education Classroom Curriculum Guide

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## Description

- Overview:
- Adult education classrooms are commonly comprised of learners who have widely disparate levels of mathematical problem-solving skills. This is true regardless of what level a student may be assessed at when entering an adult education program or what level class they are placed in. Providing students with differentiated instruction in the form of Push and Support cards is one way to level this imbalance, keeping all students engaged in one high-cognitive task that supports and encourages learners who are stuck, while at the same time, providing extensions for students who move through the initial phase of the task quickly. Thus, all

students are continually moving forward during the activity, and when the task ends, all students have made progress in their journey towards developing conceptual understanding of mathematical ideas along with a productive disposition, belief in one’s own ability to successfully engage with mathematics.

- Subject:
- Mathematics
- Level:
- Lower Primary, Upper Primary, Middle School, High School, Community College / Lower Division, Adult Education
- 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:
- Lesson Plan, Module, Teaching/Learning Strategy
- Author:
- Patricia Helmuth
- Date Added:
- 05/24/2018

- License:
- Creative Commons Attribution-NonCommercial-ShareAlike 4.0
- Language:
- English
- Media Format:
- Graphics/Photos, Text/HTML

# Comments

## Standards

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: 2.5 Superior (2 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: 2.5 Superior (2 users)

Cluster: Analyze and solve linear equations and pairs of simultaneous linear equations

Standard: Analyze and solve pairs of simultaneous linear equations.

Degree of Alignment: 2.3 Strong (3 users)

Cluster: Mathematical practices

Standard: Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x –1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x^2 + x + 1), and (x – 1)(x^3 + x^2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Degree of Alignment: 2 Strong (2 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: 2 Strong (2 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: 2 Strong (2 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Understand the concept of a function and use function notation.

Standard: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Degree of Alignment: 2 Strong (2 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Analyze functions using different representations

Standard: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*

Degree of Alignment: 2 Strong (2 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Reasoning with Equations and InequalitiesCluster: Solve systems of equations

Standard: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Degree of Alignment: 2 Strong (2 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Analyze functions using different representations

Standard: Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01)^(12t), y = (1.2)^(t/10), and classify them as representing exponential growth and decay.

Degree of Alignment: 2 Strong (2 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Reasoning with Equations and InequalitiesCluster: Represent and solve equations and inequalities graphically

Standard: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Degree of Alignment: 2 Strong (2 users)

# Common Core State Standards Math

Grades 9-12,Number and Quantity: QuantitiesCluster: Reason quantitatively and use units to solve problems

Standard: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.*

Degree of Alignment: 2 Strong (2 users)

# Common Core State Standards Math

Grades 9-12,Number and Quantity: QuantitiesCluster: Reason quantitatively and use units to solve problems

Standard: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.*

Degree of Alignment: 2 Strong (2 users)

# Common Core State Standards Math

Grades 9-12,Functions: Linear, Quadratic, and Exponential ModelsCluster: Construct and compare linear, quadratic, and exponential models and solve problems

Standard: Distinguish between situations that can be modeled with linear functions and with exponential functions.*

Degree of Alignment: 2 Strong (2 users)

Learning Domain: Expressions and Equations

Standard: Analyze and solve linear equations and pairs of simultaneous linear equations

Indicator: Analyze and solve pairs of simultaneous linear equations.

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: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01)^(12t), y = (1.2)^(t/10), and classify them as representing exponential growth and decay.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Linear, Quadratic, and Exponential Models

Standard: Construct and compare linear, quadratic, and exponential models and solve problems

Indicator: Distinguish between situations that can be modeled with linear functions and with exponential functions.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Understand the concept of a function and use function notation.

Indicator: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Reasoning with Equations and Inequalities

Standard: Solve systems of equations

Indicator: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Number and Quantity: Quantities

Standard: Reason quantitatively and use units to solve problems

Indicator: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Number and Quantity: Quantities

Standard: Reason quantitatively and use units to solve problems

Indicator: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y - 2)/(x -1) = 3. Noticing the regularity in the way terms cancel when expanding (x - 1)(x + 1), (x - 1)(x^2 + x + 1), and (x - 1)(x^3 + x^2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Reasoning with Equations and Inequalities

Standard: Represent and solve equations and inequalities graphically

Indicator: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*

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: 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)

## Evaluations

# Achieve OER

Average Score (3 Points Possible)Degree of Alignment | 2.1 (3 users) |

Quality of Explanation of the Subject Matter | 3 (3 users) |

Utility of Materials Designed to Support Teaching | 3 (3 users) |

Quality of Assessments | 2.5 (2 users) |

Quality of Technological Interactivity | 2.5 (2 users) |

Quality of Instructional and Practice Exercises | 3 (2 users) |

Opportunities for Deeper Learning | 3 (3 users) |

# Tags (18)

- Adult Basic Education
- Adult Education
- Differentiation
- Exponential Functions and Graphs
- Functions and Graphs
- High School Equivalency
- Inequalities
- Linear Equations
- Notice/Wonder
- Number and Quantity
- Problem-Solving
- Push and Support
- Standards of Mathematical Practices
- Student Centered Instruction
- Systems of Linear Equations
- GED
- Mathematics
- Multilevel

This is a very though resource, it provides the teacher an abundance of information and resources. There are lots of opportunities to apply and understand the concepts presented. Great!!

Great lesson! Guide provides much opportunity for deeper learning!

on Jun 03, 05:58am Evaluation

## Opportunities for Deeper Learning: Superior (3)

Appropriate scaffolding and direction? Yes, this curriculum guide has got that! Visit any of these tasks for a rich collaborative experience for your students.

on Jun 03, 05:58am Evaluation

## Quality of Technological Interactivity: Not Applicable (N/A)

These tasks can be used in any adult education classroom without technology access. Teachers will need to plan ahead if they print the guide and include any links they want to access if they do not have access to technology in their class.

on Jun 03, 05:58am Evaluation

## Quality of Assessments: Not Applicable (N/A)

Notice and Wonder is used to connect to what students already know and can be a formative assessment. Formal sharing is also included with each task. None of the tasks are designed to be an assessment in and of themselves.

on Jun 03, 05:58am Evaluation

## Utility of Materials Designed to Support Teaching: Superior (3)

While there is no materials list provided, it doesn't seem appropriate for this guide. Materials can be collected for each task the teacher chooses to use with students.

on Jun 03, 05:58am Evaluation

## Quality of Explanation of the Subject Matter: Superior (3)

With the teacher as the audience for this guide, the explanation is superior. Multiple links are provided for additional reading. The guide provides concrete examples, but also guidance on ways to create Push and Support Cards for our own problems.