- Domains Overview
- Level A https://www.vaccrsframeworks.org/math/algebraic-thinking-level-a/
- Level B https://www.vaccrsframeworks.org/math/algebraic-thinking-level-b/
- Level C https://www.vaccrsframeworks.org/math/algebraic-thinking-level-c/
- Level D https://www.vaccrsframeworks.org/math/algebraic-thinking-level-d/
- Level E https://www.vaccrsframeworks.org/math/algebraic-thinking-level-e/
- Level A https://www.vaccrsframeworks.org/math/geometry-level-a/
- Level B https://www.vaccrsframeworks.org/math/geometry-level-b/
- Level C https://www.vaccrsframeworks.org/math/geometry-level-c/
- Level D https://www.vaccrsframeworks.org/math/geometry-level-d/
- Level E https://www.vaccrsframeworks.org/math/geometry-level-e/
Measurement and Data
- Level A https://www.vaccrsframeworks.org/math/measurement-and-data-level-a/
- Level B https://www.vaccrsframeworks.org/math/measurement-and-data-level-b/
- Level C https://www.vaccrsframeworks.org/math/measurement-and-data-level-c/
- Level D https://www.vaccrsframeworks.org/math/measurement-and-data-level-d/
- Level E https://www.vaccrsframeworks.org/math/measurement-and-data-level-e/
- Level A https://www.vaccrsframeworks.org/math/number-sense-level-a/
- Level B https://www.vaccrsframeworks.org/math/number-sense-level-b/
- Level C https://www.vaccrsframeworks.org/math/number-sense-level-c/
- Level D https://www.vaccrsframeworks.org/math/number-sense-level-d/
- Level E https://www.vaccrsframeworks.org/math/number-sense-level-e/
(CCSS Grades K-1 / Beginning ABE Literacy)
At Level A, students are engaging with mathematics at the most basic, fundamental level. Here, the focus is on conceptual understanding of whole numbers as representations of quantities and of concepts such as such as length as well as on the operations addition and subtraction.
Instructional activities should include:
- Challenging students to represent and recognize whole number quantities. These activities should include drawing, hands-on manipulation of objects, discussion, and writing numbers.
- Recognizing addition and subtraction with whole numbers. This should include developing addition or subtraction equations given an illustration or manipulative and vice versa. It should also include introducing students to the use of variable representation in addition and subtraction.
- Engaging in discussions around addition and subtraction based on mathematical equations, illustrations, and hands-on manipulatives. These discussions include possible starting points for problem solving, basic analysis to make sense of the problem, and an understanding of the quantities involved. Activities should include representing length using addition and subtraction.
- Recognizing common errors in mathematical processes in addition or subtraction mathematical equations, illustrations, and hands-on manipulatives.
Recognizing, evaluating, and comparing
(CCSS Grades 2-3 / Beginning Basic Education)
At level B, students’ engagement with mathematical practice has two core components.
The initial component is simply an extension of all of the specific practices represented in Level A, building towards complete fluency. Instructional activities should include:
- Representing whole number quantities using spoken, hands-on, and written techniques. This includes an understanding of place value of whole numbers.
- Recognizing addition and subtraction with whole numbers including similar activities to those described at Level A.
- Engaging in discussions around addition and subtraction based on mathematical equations, illustrations, and hands-on manipulatives. These discussions should include possible starting points for problem solving, basic analysis to make sense of the problem, and an understanding of the quantities involved. Activities should include representing length and/or measurement using addition and subtraction. At this point, standard units of length should be introduced and used in situations.
- Recognizing common errors in mathematical processes involving addition or subtraction mathematical equations, illustrations, and hands-on manipulatives.
The second component introduces the operations multiplication and division and expands content within geometry, algebra, and number sense. Instructional activities should include:
- Recognizing multiplication of whole numbers and division of whole numbers using single digit divisors. This should include development of multiplication or division equations given an illustration or manipulative. It should also include algebraic representation of multiplication or division. Students should recognize and understand division as the inverse of multiplication.
- Engaging in discussions around multiplication and division based on mathematical equations, illustrations, and hands-on manipulatives. These discussions should include possible starting points for problem solving, basic analysis to make sense of the problem, an understanding of the quantities involved, and questions to clarify the mathematical process.
- Recognizing common errors in mathematical processes involving multiplication and division given mathematical equations, illustrations, and hands-on manipulatives.
- Developing conceptual understanding of fractions, particularly unit fractions and their connection to division as well as the use of single-digit-denominator fractions to define parts of a whole.
- Using the four basic operations to solve problems involving area and perimeter.
- Using geometry and basic geometric terms to more precisely define and compare shapes introduced at Level A. Students should identify and describe relationships among these basic shapes. They should also be able to construct complex shapes from basic shapes, identify them, and explain how such construction/deconstruction increases or decreases dimensions.
(CCSS Grades 4-6 / Low Intermediate Adult Basic Education)
At Level C, students use developed foundational skills to become more advanced mathematical thinkers. Instructional work at Level C should build students’ familiarity with and comfort taking an inquiry approach, which they will need to apply at Levels D and E. While foundational skills are important, at level C instructional emphasis should shift towards development of critical thinking skills around math practice as opposed to fluency within specific computational content. Math concepts and processes should be presented in applied contexts that address life and general workforce skills.
Instructional activities should include:
- Developing complete understanding of the relationship among fractions, decimals, percents, and proportions. Given different scenarios, students should be able to decide which format is best. Students may develop a preference for one approach over another.
- Composing short multi-step problems across all domains. Students should be able to illustrate the steps to be taken and decode the elements within multi-step problems. Students should be able to represent problems numerically and verbally explain, using key algebraic terms, why steps occur at different points in a process. Students should be able to model mathematical situations using drawings, digital media, and hands-on manipulation of real objects.
- Understanding that there is more than one way to solve a problem. Estimation should be used to check the reasonableness of answers.
- Expanding the use of precise mathematical vocabulary. Students should be using specific mathematical terms whenever applicable.
- Selecting and using specific mathematical tools when problem solving. These include a scientific calculator, GED® math formula sheet, charts, tables, and measuring devices.
(CCSS Grades 6-8 / High Intermediate Adult Basic Education)
Level D completes the foundational numeracy skills that adults need to fully engage in college and career readiness. Instruction is more rigorous than at any of the previous levels. This level focuses on life skills and may extend beyond generalized workforce skills into a college and career specific application. Instructional activities should include:
- Analyzing and solving complex problems where the solution(s) are not directly derived from the information provided.
- Discussing different methods of approaching problems. Student-led and small group discussions should be common; students should be able to use estimation to reduce the amount of arithmetic in a problem.
- Problems that require algebraic thinking may be used as a tool applied to a problem from various math content areas.
- Comprehending information presented in a variety of formats, such as charts, graphs, tables, illustrations, printed articles, instruction sets and manuals, and media presentations. Students should also be able to present information in a variety of formats.
- Challenging students to explain and justify steps taken in a complex problem-solving process. Students should be expected to use specific mathematical terms at the appropriate times.
- Leading students to assess their work at specific points during the problem-solving process and redirect if necessary.
- Solving problems that require creating a mathematical model to simplify the situation. Students should be challenged to recognize mathematics in indirect applications or contexts.
(CCSS Grades 9-12 / Adult Secondary Education, Low and High)
At Level E, students prepare to engage in college-level or specialized workplace mathematics. This involves industry- or career-specific math skills delivered in an on-the-job or project-based environment. Instructional practice should include:
- Student selection of mathematical tools that best fit a context or application. These include digital tools and applications such as using technology to build graphic representations from data. Students should use mathematical reasoning to explain the tools they have chosen.
- Used of algebraic thinking and functions to create mathematical models for an application or context.
- Justification of solutions or mathematical pathways using advanced mathematical reasoning.
- Instructional delivery by career-specific mentors or professionals with significant field experience.