Numbers and Operations 3

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(K-4) Elementary School Learning Targets	(5-8) Middle School Learning Targets	(9-12) High School Learning Targets
E.NO-3 Use reasoning to support solutions and informal arguments and to develop metacognitive skills: Use estimation and rounding to support informal arguments; Develop both additive and multiplicative thinking; Test, model, and explain solutions.	M.NO-3 Develop metacognitive skills through making conjectures and justifying mathematical solutions and arguments: Use estimation and rounding to support reasonableness of arguments/justifications; Apply multiplicative and proportional reasoning; Make, test, and justify conjectures using mathematical concepts and models.	H.NO-3 Develop metacognitive skills through use of mathematical arguments to justify reasonableness of outcomes, to support formal proofs (including technology applications), and to develop metacognitive skills.

Grade Differentiation

Elementary School Progress Indicators

Progress Indicator: E.NO.3a exploring and explaining answers to questions, such as "Does this answer make sense?"
Core Content Connectors: K- 1 -2	Potential cluster for demonstration	Common Core: Standards for Mathematical Practice1
1-2.NO.3a1 Identify the problem within a word problem (e.g., "Bill had two apples. Sam gave Bill two more apples. How many apples does Bill have now?" – student underlines "How many apples does Bill have now?"	1.OA Represent and solve problems involving addition and subtraction	CCSS.Math.Practice.MP1 Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	CCSS.Math.Practice.MP5 Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
1-2.NO.3a2 Make sense of and solve story word problems	2.OA Represent and solve problems involving addition and subtraction	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
1-2.NO.3a3 Use a given strategy to check solutions (e.g., use addition to check an answer to a subtraction problem)	2.NBT Use place value understanding and properties of operations to add and subtract	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
Progress Indicators: E.NO.3b constructing arguments using concrete referents such as objects, diagrams, tables, actions (e.g., clapping, movement) and estimating
Core Content Connectors: K- 1 -2	Potential cluster for demonstration	Common Core: Standards for Mathematical Practice (p. 6-8 in Mathematics document)
1-2.NO.3b1 Use an appropriate tool to help solve a given problem (e.g., use a ruler to measure in inches)	2.MD Measure and estimate lengths in standard units	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
1-2.NO.3b2 Use a strategy, rule, or identified characteristic to solve a given routine or non-routine problem (e.g., sort given shapes based upon the rule that triangles have three corners; inches are smaller units than feet so use inches to measure smaller items such as a pencil)	1G Reason with shapes and their attributes	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.

Progress Indicator: E.NO.3c evaluating the reasonableness of answers using mental computation, arithmetic patterns, and estimation strategies, including rounding to the nearest 10 or 100
Core Content Connectors: 3-4	Potential cluster for demonstration	Common Core: Standards for Mathematical Practice2
3-4.NO.3c1 Solve real world problems (e.g., determine how many nickels needed to be the equivalent of a quarter) by using mental math (such as skip counting by 2s within 20, 5s within 50, 10s within 100)	4.OA Use the four operations with whole numbers to solve problems	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
3-4.NO.3c2 Evaluate the reasonableness of answers after making computations	3.OA Solve problems involving the 4 operations and identify and explain patterns in arithmetic	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
3-4.NO.3c3 Determine the reasonableness of answers using estimation	3.OA Solve problems involving the 4 operations and identify and explain patterns in arithmetic	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
3-4.NO.3c4 Determine the reasonableness of answers using arithmetic patterns	3.OA Solve problems involving the four operations and identify and explain patterns in arithmetic	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
Progress Indicator: E.NO.3d constructing arguments and explaining reasonableness of outcomes using a variety of concrete supports (e.g., models, diagrams, tables)
Core Content Connectors: 3-4	Potential cluster for demonstration	Common Core: Standards for Mathematical Practice3
3-4.NO.3d1 Use modeling or diagrams to construct or select a viable argument for a given problem (e.g., "what is the difference between equilateral, scalene, and isosceles triangles?" and use models of triangles to illustrate correct answer)	3G Reason with shapes and their attributes	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
3-4.NO.3d2 Use models to find equivalencies or solutions	4.NF Extend understanding of fraction equivalence and ordering	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
3-4.NO.3d3 Justify and label solutions as a result of making a comparison (e.g., compare 15 and 25 and label 25 as greater: 15<25)	4NBT Generalize place value understanding for multi-digit whole numbers.	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
3-4.NO.3d4 Explain why a tool or strategy would be used to solve a given problem	3.OA Solve problems involving the four operations, and identify and explain patterns in arithmetic	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
Explanations and clarifications:

Middle School Progress Indicators

Progress Indicator: M.NO.3a using informal and rule-based arguments, evidence, and examples (e.g., estimation, rounding, arrays, visual models, diagrams) to justify mathematical solutions
Core Content Connectors: 5-6	Potential cluster for demonstration	Common Core: Standards for Mathematical Practice4
5-6.NO.3a1 Justify the use of a strategy, rule, or identified characteristic to solve a given problem (e.g., why use a formula to find the area of rectangles)	6.EE Reason about and solve one-variable equations and inequalities	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
Progress Indicator: M.NO.3b critiquing the mathematical arguments provided by others
Core Content Connectors: 5-6	Potential cluster for demonstration	Common Core: Standards for Mathematical Practice5
5-6.NO.3b1 Use up to two rules to extend a pattern and verify provided responses or select correct answers (e.g., Rules: +3, +2 and table lists pairs, 4:5, 7:7, and 10:9)	5.OA Analyze patterns and relationships	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
5-6.NO.3b2 Explain the selection of rule(s) to use to verify a response Explain the rules used to verify a solution	5.OA Analyze patterns and relationships	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
5-6.NO.3b3 Verify provided solutions within word problems (e.g., Sally wanted to give her brother 1/2 of her books. Her brother only took 1/4 of the 1/2 she offered. Sally gave him 1/8 of all of her books. Is this true?)	6.NS Apply and extend previous understandings of numbers to the system of rational numbers	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
5-6.NO.3b4 Critique a set of solutions for a given problem and determine any viable answers	5.NBT Perform operations with multi-digit whole numbers and with decimals to hundredths	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
5-6.NO.3b5 Evaluate the accuracy of statements made based upon provided data	6.SP Develop understanding of statistical variability	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
Explanations and clarifications:

Progress Indicator: M.NO.3c using stated assumptions, definitions, patterns, and previously established results in constructing mathematical arguments
Core Content Connectors: 7-8	Potential cluster for demonstration	Common Core: Standards for Mathematical Practice6
7-8.NO.3c1 Use the rules for mathematical operations to verify the results when more than one operation is required to solve a problem	7.EE Solve real-life and mathematical problems using numerical and algebraic expressions and equations	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
7-8.NO.3c2 Explain the selection of rule(s) to use to verify a response	7.NS Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
7-8.NO.3c3 Analyze provided information (e.g., a graph) to describe the relationship between two quantities	7.RP Analyze proportional relationships and use them to solve real-world mathematical problems	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
7-8.NO.3c4 Construct an argument using established data and any given pattern within that data	8.SP Investigate patterns of association in bivariate data	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
7-8.NO.3c5 Explain each step to solve a problem (e.g., explain how to solve a multi-step equation)	7.EE Solve real-life and mathematical problems using numerical and algebraic expressions and equations	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
Explanations and clarifications: It was decided to not address M.NO.3d making conjectures and building a logical progression of statements to explore the truth of conjectures due to the complex nature of the PI for this population of students.

High School Progress Indicators

Progress Indicator: H.NO.3a comparing the effectiveness of two plausible arguments, distinguishing correct logic or reasoning from that which is flawed, and if there is a flaw in an argument, explaining it
Core Content Connectors: 9-12	Potential cluster for demonstration	Common Core: Standards for Mathematical Practice7
H.NO.3a1 Verify data displays are interpreted accurately within a response	S.ID Summarize, represent and interpret data on a single count or measurement variable	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
H.NO.3a2 Rewrite mathematical statements (e.g., an expression) in multiple forms	A.SSE Write expressions in equivalent forms to solve problems	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
H.NO.3a3 Identify an appropriate argument based upon provided data	S.IC Make inferences and justify conclusions from sample surveys, experiments, and observational studies	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
H.NO.3a4 Compare the steps using different strategies to solve a problem (compare two strategies to decide best way to solve problem)	A.REI Understand solving equations as process of reasoning and explain the reasoning	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
H.NO.3a5 Evaluate provided arguments or logic based upon provided data	S.IC Understand and evaluate random processes underlying statistical experiments	Make sense of problems & persevere in solving them. Reason abstractly & quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.	Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
Explanations and clarifications: