Pathways to Teacher Leadership in Mathematics Monday, July 7, 2014 Story Problem Structures and Solution Methods: K-2 and Beyond Common Core State Standards for Mathematics Pathways to Teacher Leadership in Mathematics Monday, July 7, 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.2 (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. 38 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?

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.

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

Grade 1 Additive Comparison Task Clare has 15 red bears and 9 blue bears. How many more red bears than blue bears does Clare have? Solve this problem using a discrete or a continuous tape diagram. Record the strategy you used to find the answer.

Sample 1st grade work Solved using a matching strategy and counting red bears by 1’s Solved with a counting up strategy.

Grade 4 Multiplicative Comparison Task There are tulips, daisies, and daffodils growing in the garden. There are 2 times as many daisies as tulips and 3 times as many daffodils as tulips. If there are 12 daisies, how many flowers are in the garden? Solve this problem using a tape diagram. Record the strategy you used to find the answer.

Grade 3 Student Work

Grade 4 Fraction Problem Zaya is moving to another town and decided to give 2/4 of her book collection to her friend Kelly and 1/8 of her book collection to her friend Mike. How much of her book collection does Zaya have left? Solve this problem using a tape diagram. Record the strategy you used to find the answer.

Grade 4 Student Work

Grade 6 Ratio Problem The ratio of marbles in Molly’s bag to those in Hope’s bag was 6:1. After Molly gave half of her marbles to Hope, Hope had 11 more marbles than Molly. How many marbles did Molly have at first? Solve this problem using a tape diagram. Record the strategy you used to find the answer.

Grade 6 Student Work

Grade 7 EE Task I am thinking of two numbers. The sum of my numbers is 147. The larger number is 39 more than the smaller number. What is the larger number? Solve this problem using a tape diagram. Record the strategy you used to find the answer.

Grade 7 Student Work

Reflection 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.

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.

Disclaimer Pathways to Teacher Leadership in Mathematics Project University of Wisconsin-Milwaukee, 2014-2017   This material was developed for the Pathways to Teacher Leadership in Mathematics 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 Improving Teacher Quality Program.