Foundational Skills in Mathematics K-2 (#5012005) 

Clarification 2: Instruction includes emphasis on using the properties to make a ten when adding three or more numbers. 

Clarification 3: Addition is limited to sums within 20.

Clarification 2: Problems include the unknown on either side of the equal sign.

Clarification 3: Addition and subtraction are limited to sums within 20 and related subtraction facts. Refer to Situations Involving Operations with Numbers (Appendix A).

Example: 

Example: ?-4=8

Clarification 2: Data sets include geometric figures that are categorized using their defining attributes and data from the classroom or school.

Clarification 3: Pictographs are limited to single-unit scales.

Clarification 2: Instruction includes figures given in a variety of sizes, orientations and non-examples that lack one or more defining attributes.

Clarification 3: Within this benchmark, the expectation is not to sort a combination of two- and three-dimensional figures at the same time or to define the attributes of trapezoids.

Clarification 4: Instruction includes using formal and informal language to describe the defining attributes of figures when comparing and sorting.

Clarification 2: Composite figures are composed without gaps or overlaps.

Clarification 3: Within this benchmark, it is not the expectation to compose two- and three- dimensional figures at the same time.

Example: A semi-circle and a triangle can be composed to create a two-dimensional representation of an ice cream cone.

Clarification 2: When estimating length, the expectation is to give a reasonable number of inches for the length of a given object.

Clarification 2: Two objects can be compared indirectly by directly comparing them to a third object.

Clarification 2: Instruction includes the connection to partitioning circles into halves and to semi-circles.

Clarification 2: Within this benchmark, the expectation is not to use decimal values.

Clarification 2: Instruction focuses on the connection to place value and skip counting.

Clarification 3: Within this benchmark, the expectation is not to use decimal values or to find the value of a combination of coins and dollars.

Clarification 3: Instruction includes recognizing counting sequences using visual charts, such as a 120 chart, to emphasize base 10 place value.

Clarification 2: Within this benchmark, the expectation is to use terms (e.g., less than, greater than, between or equal to) and symbols (<, > or =).

Clarification 2: Instruction includes situations involving adding to, putting together, comparing and taking from.

Example: Ten more than 23 is 33.

Clarification 2: Instruction includes the use of manipulatives, number lines, drawings or models.

Clarification 2: Instruction includes the use of manipulatives, drawings or equations to decompose tens and regroup ones, when needed.

Clarification 2: Problems include creating real-world situations based on an equation.

Clarification 3: Addition and subtraction are limited to sums up to 100 and related differences. Refer to Situations Involving Operations with Numbers (Appendix A).

Clarification 2: Problem types are limited to an equation with three or four terms. The sum or difference can be on either side of the equal sign.

Clarification 3: Addition and subtraction are limited to sums up to 100 and related differences.

Clarification 2: Problems include having the unknown on either side of the equal sign.

Clarification 3: Addition and subtraction are limited to sums up to 100 and related differences. Refer to Situations Involving Operations with Numbers (Appendix A).

Clarification 2: Addends are limited to whole numbers less than or equal to 12.

Example: The number 9 is odd because it can be represented as two equal groups with one left over or as the expression 4+4+1.

Clarification 2: The total number of objects is limited to 25.

Clarification 2: Data displays can be represented both horizontally and vertically. Scales on graphs are limited to ones, fives or tens.

Clarification 2: Problems include mathematical and real-world context.

Clarification 2: Problem types include being given an image and determining whether a given line is a line of symmetry or not.

Clarification 2: Instruction includes real-world objects, such as picture frames or desktops.

Clarification 2: Within this benchmark, the expectation is not to use a formula to find perimeter.

Clarification 3: Instruction includes cases where the side lengths are given or measured to the nearest unit.
Clarification 4: Perimeter cannot exceed 100 units and responses include the appropriate units.

Clarification 2: Instruction focuses on recognizing that when an object is measured in two different units, fewer of the larger units are required. When comparing measurements of the same object in different units, measurement conversions are not expected.

Clarification 3: When estimating the size of an object, a comparison with an object of known size can be used.

Clarification 2: Within this benchmark, the expectation is not to understand military time.

Clarification 2: Addition and subtraction problems are limited to sums within 100 and related differences. Refer to Situations Involving Operations with Numbers (Appendix A).

Example: The number seven hundred nine written in standard form is 709 and in expanded form is 700+9.

Clarification 2: Within this benchmark, the expectation is to use terms (e.g., less than, greater than, between or equal to) and symbols (<, > or =).

Example: The difference 87-25 can be found by subtracting 20 from 80 to get 60 and then 5 from 7 to get 2. Then add 60 and 2 to obtain 62.

Clarification 2: Instruction focuses on composing and decomposing ones, tens and hundreds when needed.

Example: The difference 1,000-17 can be found by using a number line and making a “jump” of 10 from 1,000 to 990 and then 7 “jumps” of 1 to 983.

Clarification 3: Addition and subtraction are limited to sums within 10 and related subtraction facts. Refer to Situations Involving Operations with Numbers (Appendix A).

Clarification 2: Problem types are limited to an equation with two or three terms. The sum or difference can be on either side of the equal sign.
Clarification 3: Addition and subtraction are limited to sums within 20 and related subtraction facts.

Clarification 2: Instruction includes geometric figures that can be categorized using their defining attributes. 

Clarification 3: Within this benchmark, it is not the expectation for students to construct formal representations or graphs on their own.

Clarification 2: Instruction focuses on using informal language to describe relative positions and the similarities or differences between figures when comparing and sorting.

Clarification 2: Instruction focuses on using informal language to describe relative positions and the similarities or differences between figures when comparing and sorting.

Clarification 2: Language to compare length includes short, shorter, long, longer, tall, taller, high or higher. Language to compare volume includes has more, has less, holds more, holds less, more full, less full, full, empty, takes up more space or takes up less space. Language to compare weight includes heavy, heavier, light, lighter, weighs more or weighs less.

Clarification 2: Instruction includes counting objects and pictures presented in a line, rectangular array, circle or scattered arrangement. Objects presented in a scattered arrangement are limited to 10.

Clarification 3: Within this benchmark, the expectation is not to write the number in word form.

Clarification 2: Within this benchmark, the expectation is not to use the relational symbols =,> or <.

Clarification 2: Within this benchmark, the expectation is to recognize and count to 100 by the end of Kindergarten.

Clarification 2: Instruction focuses on the connection that addition is “putting together” or “counting on” and that subtraction is “taking apart” or “taking from.” Refer to Situations Involving Operations with Numbers (Appendix A). 

Clarification 3: Within this benchmark, it is the expectation that one problem can be represented in multiple ways and understanding how the different representations are related to each other.

Example: The numbers 3, 5 and 8 make a fact family (number bonds). It can be represented as 5 and 3 make 8; 3 and 5 make 8; 8 take away 5 is 3; and 8 take away 3 is 5.

• Cultivate a community of growth mindset learners. 
• Foster perseverance in students by choosing tasks that are challenging. 
• Develop students’ ability to analyze and problem solve. 
• Recognize students’ effort when solving challenging problems.

• Help students make connections between concepts and representations.
• Provide opportunities for students to use manipulatives when investigating concepts.
• Guide students from concrete to pictorial to abstract representations as understanding progresses.
• Show students that various representations can have different purposes and can be useful in different situations. 

• Provide students with the flexibility to solve problems by selecting a procedure that allows them to solve efficiently and accurately.
• Offer multiple opportunities for students to practice efficient and generalizable methods.
• Provide opportunities for students to reflect on the method they used and determine if a more efficient method could have been used. 

• Establish a culture in which students ask questions of the teacher and their peers, and error is an opportunity for learning.
• Create opportunities for students to discuss their thinking with peers.
• Select, sequence and present student work to advance and deepen understanding of correct and increasingly efficient methods.
• Develop students’ ability to justify methods and compare their responses to the responses of their peers. 

• Help students recognize the patterns in the world around them and connect these patterns to mathematical concepts.
• Support students to develop generalizations based on the similarities found among problems.
• Provide opportunities for students to create plans and procedures to solve problems.
• Develop students’ ability to construct relationships between their current understanding and more sophisticated ways of thinking.

• Have students estimate or predict solutions prior to solving.
• Prompt students to continually ask, “Does this solution make sense? How do you know?”
• Reinforce that students check their work as they progress within and after a task.
• Strengthen students’ ability to verify solutions through justifications. 

• Provide opportunities for students to create models, both concrete and abstract, and perform investigations.
• Challenge students to question the accuracy of their models and methods.
• Support students as they validate conclusions by comparing them to the given situation.
• Indicate how various concepts can be applied to other disciplines.

2-3 Students include relevant textual evidence in their written and oral communication. Students should name the text when they refer to it. In 3rd grade, students should use a combination of direct and indirect citations.

4-5 Students continue with previous skills and reference comments made by speakers and peers. Students cite texts that they’ve directly quoted, paraphrased, or used for information. When writing, students will use the form of citation dictated by the instructor or the style guide referenced by the instructor. 

6-8 Students continue with previous skills and use a style guide to create a proper citation.

9-12 Students continue with previous skills and should be aware of existing style guides and the ways in which they differ.

In grades 1-2, students build upon these skills by justifying what they are thinking. For example: “I think ________ because _______.” The collaborative conversations are becoming academic conversations.

In grades 3-12, students engage in academic conversations discussing claims and justifying their reasoning, refining and applying skills. Students build on ideas, propel the conversation, and support claims and counterclaims with evidence.