Core Mathematics Partnership Building Mathematical Knowledge and Story Problem Structures and Solution Methods: K-2 and Beyond Common Core State Standards for Mathematics Core Mathematics Partnership Building Mathematical Knowledge and High-Leverage Instruction for Student Success Wednesday, July 23, 2014

Learning Intentions & Success Criteria We are learning to… understand the CCSSM expectations for addition and subtraction situations and solution methods. Make a meaningful transition from concrete representations to abstract representation. We will be successful when we can… Identify the connection between story situations and solution methods. Use discrete and continuous models to illustrate word problems.

Addition and Subtraction Situations “Mathematizing real-world situations” (OA Progressions document, p. 8)

K-8 Progression of Word Problems & Real World Problems Grades K-2 Addition and Subtraction; Whole Numbers K.OA.2 (one step up to 10) (with support from K.OA.3) 1.OA.1 (one step up to 20) 2.OA.1 (two-step up to 100) Grades 3-5 All operations; Whole Numbers, Fractions, 3.OA.8 (two-step; whole numbers; all 4 operations) 4.OA.3 (multi-step; all 4 operations) 4.NF.3d (addition and subtraction of fractions) 5.NF.2 (addition and subtraction of fractions) Grades 6-8 Rational Numbers 6.NS.1 (Division of fractions by fractions) 7.NS.3 (all operations; rational numbers) 8.EE.8c (linear equations) Note story problem progression if participants do not surface it. Match up fluency expectations with story problem structures if participants do not surface it. Suggest that story problem structures that we will be examining today are launched in Grades K-2 and those same structures extend beyond Grade 2 as they are used in Grades 3-5 with whole numbers and fractions.

Four Kindergarten problem subtypes Skilled at solving in Grade 1 Experience at Grade 1 skilled at solving in Grade 2

Addition and Subtraction Word Problem Structures Add to Take From Put Together/Take Apart Compare Place each heading on a post it and sort problems according to the structure

Problem Sort Write each of the 4 problem categories on a sticky note and review its distinguishing characteristics. With a partner: Take a problem from the envelope. Read it out loud. Identify the problem type. Justify your reasoning. Place under the correct category. Then repeat the process.

Debrief Problem Situations What are some characteristics that distinguish one word problem type from another? Why is it important to understand what seem to be subtle distinctions in word problems?

Developmental Reasoning for Single-Digit Addition and Subtraction

Developmental Levels for Single-Digit Addition and Subtraction OA Progressions Appendix pages 36-39 Level 1: Direct Modeling by Counting All or Taking Away Level 2: Counting On Level 3: Convert to an Easier Equivalent Problem Jigsaw the levels at the tables. Each person reads a different level and summarize with an example for the group.

Relating Context and Solution Methods There were 5 cookies on the plate. Mom put 6 more cookies on the plate. How many cookies are on the plate now? How would a student approach this problem? Level 1 (Direct Modeling) Level 2 (Counting on) Level 3 (create an easier equivalent problem) (Use p. 36 as a reference.)

Intentional Use of Story Problem Structures In what ways might the intentional use of story problem structures during instruction support fluency with single digit addition and subtraction?

PRR: K-2 OA Overview Read from the OA Progression Document, pgs. 20-21. “Summary of K-2 Operations and Algebraic Thinking”. Summarize the shifts in expectation and understanding from Kindergarten to Grade 2. Consider the following: Story Problem Structures Expectations Solution Method Expectations

Instructional Sequence

“As children progress to Level 2 strategies they no longer need representations that show each quantity as a group of objects.” (OA Progressions, p. 16) As children leave behind the need to represent each quantity, what implications does that have for us as teachers?

Typical Instructional Sequence ? What’s missing from this sequence? Abstract Representing problem situations with equations Concrete Model with Objects

Ali has 4 toy cars. David has 3 toy cars Ali has 4 toy cars. David has 3 toy cars. How many toy cars do they have together? Concrete 4 + 3 = 7 Representational Abstract Use concrete objects to form two groups and put the two groups together. 1,2,3 4 1,2,3 1,2,3,4,5,6,7…7 cars

Representational Math Drawings Drawing pictures that represent concrete objects provides a bridge to help young children connect their concrete representations to the abstract world of mathematical symbols. “Math drawings facilitate reflection and discussion because they remain after the problem has been solved.” (OA Progressions, p. 8) Children need many opportunities to create such drawings.

Please be back in 10 minutes.

Tape Diagrams

CCSSM Suggested Math Drawing: Tape Diagram What is a tape drawing? A drawing that looks like a segment of tape, used to illustrate number relationships. Also known as strip diagrams, bar model, fraction strip, or length model. (CCSSM Glossary, p. 87)

CCSSM Suggested Math Drawing: Visual Models Through the Grades Grade 1: Math Drawings (1OA1, 1OA2) Grade 2: Math Drawings (2OA1, 2OA2, 2MD5) Grade 3: Visual Fraction Model (3NF3a-d) Grade 4: Visual Fraction Model (4NF3, 4NF4, 4OA2) Grade 5: Visual Fraction Model (5 standards) Grade 6: Tape Diagrams (6RP3) Visual Fraction Model (6NS1) Grade 7: Visual Model for Problem Solving (7RP1-3) Number Line Diagram (7NS1) Why are they important? Next level of abstraction Serves as a model of thinking to support transitioning from Level 1 to Level 2

Visual Models for Ratios & Multiplicative Comparison Visual Fraction Model for Thirds Part Whole Model Part Whole 3 pieces of size one-third Visual Models for Ratios & Multiplicative Comparison Additive Comparison Model You may choose to display this slide as participants work to create their diagrams. Blocks for the parts may be resized to proportionally match the quantities of each part. larger quantity smaller quantity difference 4 times as many as… 1:4 ratio

Diagrams: Tape, Part-Whole, and Number-Bond OA Progressions, page 16 Review the three different diagrams found in the margin. Discuss similarities and differences with your shoulder partner.

Ali has 4 toy cars. David has 3 toy cars Ali has 4 toy cars. David has 3 toy cars. How many toy cars do they have together? Representational Concrete 7 cars Abstract 4 + 3 = 7 4 5,6,7 Use concrete objects to form two groups and put the two groups together. Abstracting to another level. 7 cars 4 5, 6, 7 1,2,3 4 1,2,3 1,2,3,4,5,6,7…7 cars

Practicing Tape Diagrams Return to the problems used for the earlier card sort. Remove the Compare problems. For each problem: Solve using a Level 1, Level 2 and Level 3 strategy. Create a tape diagram (both discrete and continuous). As you work, refer to the diagrams in the margin on page 16 of OA Progressions.

These diagrams are a major step forward because the same diagrams can represent the adding and subtracting situations for all kinds of numbers students encounter in later grades (multi-digit whole numbers, fractions, decimals, variables). (OA Progressions, p. 17)

Grounding Our Thinking In Representations

Learning Intentions & Success Criteria We are learning to… understand the CCSSM expectations for addition and subtraction situations and solution methods. Make a meaningful transition from concrete representations to abstract representation. We will be successful when we can… Identify coherence within story situations and solution methods. Use discrete and continuous models to illustrate word problems.

Summary and Reflection Grades K-2 Addition and Subtraction; Whole Numbers K.OA.2 (one step up to 10) (with support from K.OA.3) 1.OA.1 (one step up to 20) 2.OA.1 (two-step up to 100) How do you see real world contexts and tape diagrams as helping students gain a deep understanding of numerical equations and algebraic equations? Write two ideas on an index card. Pass your card two places to the left. Read your new card and in turn expand on one of the ideas. Repeat.

Focus Topics or Standards Reflection Summary Summarize some key points and classroom ideas related to the topics or focus standards in this session. Session Focus Topics or Standards Summary of Key Points Classroom Ideas to Try K.OA.2, 1.OA.1, 2.OA 1., MP 2, MP 5

Disclaimer Core Mathematics Partnership Project University of Wisconsin-Milwaukee, 2013-2016   This material was developed for the Core Mathematics Partnership project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Other use of this work without prior written permission is prohibited—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors. This project was supported through a grant from the Wisconsin ESEA Title II, Part B, Mathematics and Science Partnerships.