CCSSM in the Second Grade Algebraic Thinking CCSSM in the Second Grade Oliver F. Jenkins MathEd Constructs, LLC www.mathedconstructs.com

Grade 2 CCSSM Domains Operations and Algebraic Thinking Represent and solve problems involving addition and subtraction. Add and subtract within 20. [Fluency standard] Work with equal groups of objects to gain foundations for multiplication. Number and Operations in Base Ten Understand place value. Use place value understanding and properties of operations to add and subtract.  Measurement and Data Measure and estimate lengths in standard units. Relate addition and subtraction to length. Work with time and money. Represent and interpret data.  Geometry Reason with shapes and their attributes. We’re going to focus on the first cluster of the first domain (Operations and Algebraic Thinking) – Represent and solve problems involving addition and subtraction. What distinguishes “Operations and Algebraic Thinking” from “Number and Operations”? Note the emphasis on representing problems and gaining foundations in the “Operations and Algebraic Thinking” domain. Are there other clusters in the grade to which this cluster might connect?

Algebraic Thinking Stream Number and Operations in Base Ten The Number System Algebra Number and Operations: Fractions Operations and Algebraic Thinking Expressions and Equations Note that two of the four second grade domains are part of the Algebraic Thinking Stream. Explain how domains in the stream are intended to develop understandings of algebra concepts and procedures. For example, the flow from “operations and algebraic thinking” to “expressions and equations.” K – 5 6 – 8 9 – 12 3 – 5

Domain: Operations and Algebraic Thinking Cluster: Represent and solve problems involving addition and subtraction Content Standard 2.OA.1: Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. Consider content standard 2.OA.1 in the cluster.

What must students know and understand to master this standard? Content Standard 2.OA.1: Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. Think-pair-share Whole-group discussion of responses List and categorize responses leading to the notions of concepts, procedures, practices, and products as representative of the knowledge related to the standard.

Unwrapping Content Standards Instructional Targets Knowledge and understanding (Conceptual understandings) Reasoning (Mathematical practices) Performance skills (Procedural skill and fluency) Products (Applications) Introduce and review worksheet for “unwrapping” content standards Assign task of “unwrapping” 2.OA.1 to small groups of 3 or 4 Whole-group discussion of small group analyses of the standard

What is the significance of . . . . . . problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions . . . Let’s take a look at what exactly is meant by this phrase. . . . in content standard 2.OA.1?

Problem Structures for Addition and Subtraction Adding to, Taking from, Putting together / Taking apart, and Comparing

Join Problems: Adding to Result Unknown Sandra had 8 pennies. George gave her 4 more. How many pennies does Sandra have altogether? Change Unknown Sandra had 8 pennies. George gave her some more. Now Sandra has 12 pennies. How many did George give her? Initial Unknown Sandra had some pennies. George gave her 4 more. Now Sandra has 12 pennies. How many pennies did Sandra have to begin with? Change Initial Result For the action of joining, there are three quantities involved: an initial or starting amount, a change amount (the part being added or joined), and the resulting amount (the total amount after the change takes place). [CCSSM language – adding to]

Separate Problems: Taking from Result Unknown Sandra had 12 pennies. She gave 4 pennies to George. How many pennies does Sandra have now? Change Unknown Sandra had 12 pennies. She gave some to George. Now she has 8 pennies. How many did she give to George? Initial Unknown Sandra had some pennies. She gave 4 to George. Now Sandra has 8 pennies left. How many pennies did Sandra have to begin with? Change Initial Result In the “separate” problems, the initial amount is the whole or the largest amount, whereas in the “join” problems, the result is the whole. In “separate” problems the change is that an amount is being removed from the initial value. [CCSSM language – taking from]

Part-Part-Whole Problems: Putting together/Taking apart Whole Unknown George has 4 pennies and 8 nickels. How many coins does he have? George has 4 pennies and Sandra has 8 pennies. They put their pennies into a piggy bank. How many pennies did they put into the bank? Part Unknown George has 12 coins. Eight of his coins are pennies, and the rest are nickels. How many nickels does George have? George and Sandra put 12 pennies into the piggy bank. George put in 4 pennies. How many pennies did Sandra put in? Part Whole Part-part-whole problems involve two parts that are combined into one whole. The combining may be a physical action, or it may be a mental combination where the parts are not physically combined. [CCSSM language – putting together, taking apart]

Compare Problems: Comparing Difference Unknown George has 12 pennies and Sandra has 8 pennies. How many more pennies does George have than Sandra? (more) George has 12 pennies. Sandra has 8 pennies. How many fewer pennies does Sandra have than George? (fewer) Larger Unknown George has 4 more pennies than Sandra. Sandra has 8 pennies. How many pennies does George have? (more) Sandra has 4 fewer pennies than George. Sandra has 8 pennies. How many pennies does George have? (fewer) Smaller Unknown George has 4 more pennies than Sandra. George has 12 pennies. How many pennies does Sandra have? (more) Sandra has 4 fewer pennies than George. George has 12 pennies. How many pennies does Sandra have? (fewer) Difference Compare problems involve the comparison of two quantities. The third amount does not actually exist but is the difference between the two amounts. There are three way to present compare problems, corresponding to which quantity is unknown (smaller, larger, or difference). For each of these, two examples are given; one problem where the difference is stated in terms of more and another in terms of less. [CCSSM language – comparing] Large set Small set

Progressions of Note Kindergarten Add to and take from, result unknown Put together/take apart, total unknown Grade 1 Addition and subtraction problems of all three types within 20 Competence levels not expected for: Add to or take from, start unknown Compare, “fewer version” of bigger unknown and “more” version of smaller unknown Grade 2 One- and two-step situational problems of all three types involving addition and subtraction within 100 Students do not work with two- step problems in which both steps involve the most difficult problem sub-types and variants Grade 3 Transition to multiplication and division Compare most difficult because one the quantities (the difference) is not present in the situation physically, and must be conceptualized and constructed in a representation Distribute and discuss Table 2: Addition and subtraction situations by grade level.

Problem Solving Tasks Desirable Features, Types, and Examples with Reference to Applicable Problem Situations

Desirable Features of Problem-Solving Tasks Genuine problems that reflect the goals of school mathematics Motivating situations that consider students’ interests and experiences, local contexts, puzzles, and applications Interesting tasks that have multiple solution strategies, multiple representations, and multiple solutions Rich opportunities for mathematical communication Appropriate content considering students’ ability levels and prior knowledge Reasonable difficulty levels that challenge yet not discourage

Problem Types Contextual Problems. Context problems are connected as closely as possible to children’s lives, rather than to “school mathematics.” They are designed to anticipate and to develop children’s mathematical modeling of the real world. Model Problems. The model is a thinking tool to help children both understand what is happening in the problem and a means of keeping track of the numbers and solving the problem. Contextual Problems. Context problems are connected as closely as possible to children’s lives, rather than to “school mathematics.” They are designed to anticipate and to develop children’s mathematical modeling of the real world. Contextual problems might derive from recent experiences in the classroom, a field trip, a discussion you have been having in art, science, or social studies, or from children’s literature. Model-Based Problems. The model is a thinking tool to help children both understand what is happening in the problem and a means of keeping track of the numbers and solving the problem.

Sample Problems Apple Farm Field Trip A Pencil and a Sticker For each task, participants solve the problems and identify the applicable problem situation sub-types and variants. Follow up with whole-group discussions of solutions and classifications; address the relative difficulty of the tasks and not how problem context determines sub-type and variant. Demonstrate how to navigate the IMP website to find tasks illustrative of CCSSM content standards.

Implications for Instruction

Bruner’s Stages of Representation Enactive: Concrete stage. Learning begins with an action – touching, feeling, and manipulating. Iconic: Pictorial stage. Students are drawing on paper what they already know how to do with the concrete manipulatives.  Symbolic: Abstract stage. The words and symbols representing information do not have any inherent connection to the information.

Lessons Built on Context or Story Problems Students should not just solve the problems but also use words, pictures, and numbers to explain: How they went about solving the problem Why they think they are correct Explanations provided by students should communicate what they did well enough to allow someone else to understand their thinking Children should be allowed to use whatever physical materials they feel are needed Lessons Built on Context or Story Problems. What might be a good lesson for second graders that is built around word problems look like? The answer comes more naturally if you think about students not just solving the problems but also using words, pictures, and numbers to explain how they went about solving the problem and why they think they are correct. Children should be allowed to use whatever physical materials they feel they need to help them, or they can simply draw pictures. Whatever they put on their paper should explain what they did well enough to allow someone else to understand their thinking.

Introducing Symbolism Guidelines Examples / Issues The minus sign should be read as “minus” or “subtract” but not as “take away” The plus sign is easier since it is typically a substitute for “and” Use the phrase “is the same as” in place of or in conjunction with “equals” as you read equations with students or think of the equal sign as a balance 24 – 5 Read as 24 minus 5 or 24 subtract 5 Do not read as 24 take away 5 Students often come to interpret the equals sign as the symbol that tells you that “the answer is coming up” 11 + 8 = 19 + 5, for example, is judged to be a true statement with a final result of 24 The minus sign should be read as “minus” or “subtract” but not as “take away.” The plus sign is easier since it is typically a substitute for “and.” Some care should be taken with the equal sign. The equal sign means “is the same as.” However, most children come to think of it as a symbol that tells you that the “answer is coming up.” A good idea is to often use the phrase “is the same as” in place of or in conjunction with “equals” as you read equations with students. Another approach is to think of the equal sign as a balance; whatever is on one side of the equation “balances” or equals what is on the other side.

Lessons Built on Model-Based Problems Addition Use counters with the two parts kept in separate piles, in separate sections of a mat, or in two distinct colors Carefully structure representations of addition based on a number line; it can present some real conceptual difficulties for first and second graders Subtraction a Think-Addition Thinking about subtraction as “think-addition” rather than “take-away” is extremely significant for mastering subtraction facts Comparison Models Have children make two amounts, perhaps with two bars of connecting cubes; discuss the difference between the two bars to generate the third number Have children make up story problems that involve two amounts, say 10 and 6; discuss which equations go with the problems that are created Addition. Use counters with the two parts kept in separate piles, in separate sections of a mat, or in two distinct colors. A number line presents some real conceptual difficulties for first and second graders.   Subtraction as Think-Addition. Thinking about subtraction as “think-addition” rather than “take-away” is extremely significant for mastering subtraction facts. Comparison Models. Have children make two amounts, perhaps with two bars of connecting cubes. Discuss the difference between the two bars to generate the third number. For example, if the children make a bar of 10 and a bar of 6, ask, “How many more do we need to match the 10 bar?” The difference is 4. “What equations can we make with these three numbers?” Have children make up story problems that involve the two amounts of 10 and 6. Discuss which equations go with the problems that are created.

Problem-Solving Lesson Format Pose a problem Students’ problem solving Whole-class discussion Summing up Exercises or extensions (optional) Review purpose of each phase of the lesson format. Discuss the importance of teacher questions at each phase Teacher monitoring and prompting during the “students’ problem solving” phase; decisions regarding the order in which student solutions will be presented during the “whole-class discussion” Carefully structured presentation of student solutions during the “whole-class discussion” with emphases placed on valid mathematical arguments, balanced participation, and critiquing the reasoning of others Link lesson objectives to conclusions drawn during whole-class discussion as part of the “summing up” phase

Solving problems is fun! Your Turn Build a problem-solving lesson based on either the “Apple Farm Field Trip” or the “A Pencil and a Sticker” task Identify lesson objectives aligned with standard 2.OA.1 Describe how the class and lesson materials will be organized Write three questions that you will ask students during each of the first four lesson phases Solving problems is fun! Assign this lesson planning exercise to small groups. Time permitting, ask groups to share their lesson plans and rationale for the instructional decisions made.