an overview of how gradpoint supports the publishers...

1

An Overview of How GradPoint Supports the Publishers’ Alignment Criteria of the Common Core State Standards for Math

www.pearsonschool.com/digitalApril 2013

IntroductionOver the past several months, GradPoint™ has undergone a number of enhancements in preparation for the Common Core State Standards. GradPoint currently offers high school math curriculum designed to fully meet the Common Core State Standards and leverages the powerful technology behind the program to support Common Core assessment priorities. These courses also address the standards from the International Association for K-12 Online Learning (iNACOL) as well as 21st Century Skills.

The Common Core State Standards for Mathematical Practice describe the mathematical skills and competencies that students must develop to be ready for college and work. These practices are informed by the National Council of Teachers of Mathematics (NCTM) process standards of problem solving, reasoning and proof, communication, representation, and connections. The practices also incorporate the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up:

adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy). GradPoint Math aligns to and supports these practices to ensure that students at every math level are working toward mastery of course standards while building a solid foundation to successfully tackle the challenges of future courses, college and work.

The following sections provide an overview of the key areas of the Common Core State Standards for Mathematical Practice with specific examples of how GradPoint supports them. While we have provided select examples of activities, the course provides learning objectives that address and support Common Core State Standards extensively.

2

Make Sense of Problems and Persevere in Solving ThemMathematically proficient students approach problems by first making conjectures about the form and meaning of the solution and planning 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, monitor and evaluate their progress and change course if necessary.

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.

They 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.

Support from GradPoint

Throughout the traditional online offerings, GradPoint presents different problem-solving strategies along with example problems. Some of the approaches include draw a diagram, look for a pattern, make a table, solve a simpler problem, use logical reasoning, and work backward. After many of the word problems are solved, students are guided to reflect on the answers to make sure the results are reasonable.

Both the online lessons and project-based lessons require students to support and explain their answers. They also provide opportunities for students to make connections among equations, tables, diagrams, and graphs.

Selected Examples from the Courseware

Within the Algebra lessons, many examples use the “Relate-Define-Write” format to guide students on writing symbolic models for real-world problems. For example, in the Algebra lesson “Using Variables,” students are led through the process of defining variables and translating a word problem into an equation:

Students are instructed to define variables and create equations in the Using Variable lesson.

3

Selected Examples from the Courseware, continued

In another example, an Algebra 2 project-based lesson (the Postage Cost Project) requires students to prepare data and then persist in diagnosing the results of that data, ultimately making predictions from the prepared data:

In the Postage Cost project, students evaluate and use data they have collected in creative ways to make predictions.

4

Reason Abstractly and QuantitativelyMathematically proficient students make sense of quantities and their relationships in problem situations bringing 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

• The ability to contextualize—to pause as needed during the manipulation process in order to probe into the referents for the symbols involved


Throughout the courses, the ability to decontextualize and contextualize problems is an integral part of solving problems. In the lessons, students have opportunities model situations and then solve equations using the properties of equality. As they move through the process, they reflect on the steps.


In the lessons, students are given word problems that must be translated into symbols and then solved for the given variable. Students must decontextualize the problem and use symbols to represent the situation. In the example below, the Algebra 2 lesson “Linear Models” uses “Relate-Define-Write” to decontextualize the problem. Later in the lesson, the equation is graphed:

In the Algebra 2 lesson Linear Models, students use the “Relate-Define-Write” strategy to decontexualize a problem.

5

Construct Viable Arguments and Critique the Reasoning of OthersMathematically 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 and are 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 that is.


Within the Algebra 1 and Algebra 2 lessons, students are asked to use deductive reasoning to justify steps when solving equation. Within the Geometry lessons, students develop an understanding of reasoning and proof, building naturally on the step-by-step processes of algebra. They prove theorems using paragraph proofs, flow proofs, and two-column proofs. Students further their understanding and ability to prove concepts not only by deduction, but also by using mathematical induction.


Within the online Algebra 1 lessons, like the one shown below, the example problems show the steps for isolating the variable and finding the solutions:

In this Algebra lesson, students are asked to solve multi-step equations by isolating the variables.

Many of the project-based lessons also require that the students present their results to the class, offering others opportunities to critique the reasoning presented. The student can then make modifications to his or her reasoning as a result of peer input.

6

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.


The majority of lessons start with an introductory screen that ties the concepts students will be learning

about to a real-world application. These examples allow students to connect the information they learn in the course to real-world experiences.

For the project-based lessons, students use hands-on activities to explore mathematical concepts and solve problems. They can use their results to revise their assumptions.


In the Plane Project, students will explore the different forces of flight and then use that information to construct a paper airplane. Students will then fly the paper airplane, collect the flight data, and use box-and-whisker plots to help them analyze the data:

In the Algebra 1 project called The Plane Project, students fly airplanes and collect related data.

7

Use Appropriate Tools StrategicallyMathematically proficient students consider the available tools when solving a mathematical problem and are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software.


In GradPoint, virtual scientific calculators and other tools needed to complete the lessons are provided.

For project-based lessons, there is a screen that includes information on materials needed. This information is specific to the project. Students may need a computer to access online activities. But they may also need paper, rulers, graph paper, compass, protractor, calculator, or a timer.


For example, in a lesson in which students are asked to create geometric constructions, a drawing pad may be available to draw figures virtually:

The drawing pad tool is available for students for geometric constructions.

8

Selected Examples from the Courseware, continued

In another example—the Postage Cost Project—the list of materials includes graph paper and a graphing calculator:

Students are given detailed instruction on the materials needed when starting a project-based lesson.

9

Attend to PrecisionMathematically proficient students try to communicate precisely using clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context.


For all example problems and practice problems in the traditional online lessons and the project-based lessons, students must define variables before solving equations. Students must also include units with their answers and provide the meaning of the answer. In some projects, students are asked to graph data on a coordinate plane following guidelines.


In the Data Journalist project, students analyze a data set and use appropriate, significant units of measure to accurately model the given real-world scenario. They will create a graph or chart that models the data, and write an interesting news story about the data:

Collecting and analyzing data is a common requirement in standard- and project-based lessons, like this one in which students act as a data journalist.

10

Look For and Make Use of Structure Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property.

They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects.


GradPoint reinforces the importance of mathematical patterns and structures in simplifying and solving problems. Some projects also provide opportunities to take a complicated problem and break it into simpler parts.


In the Expressions Project, students explore how parts of expressions relate to one another, how expressions can be displayed in multiple ways, and the effect of operations on expressions

In this Expressions project, students explore the many facets of expressions.

11

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. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x– 1)(x3 + x2 + 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.


To help students gain an understanding of mathematical processes, GradPoint provides full explanations of concepts before presenting shortcuts.


For example, online lessons address multiplying binomials, polynomials, and geometric series. There are lessons that teach the multiplication and division properties of exponents as well.

In the Top Tablet project, students review and practice multiplying polynomials:

Students evaluate repeated reasoning in the Top Tablet project.

an overview of how gradpoint supports the publishers...

Documents