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OPERATIONS AND ALGEBRAIC THINKING - OA K-2 PUTTING IT TOGETHER: NUMBER STRATEGIES AND STORY PROBLEM TYPES Math Teacher Leader Meeting January 10, 2012 Beth Schefelker Connie Laughlin Lee Ann Pruske Hank Kepner Mathematics – Grade K Table 1 Addition and Subtraction Situations. Add to Take from Put Together / Take Apart Compare Result Unknown Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now? 2+3=? Change Unknown Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. How many bunnies hopped over to the first two? 2+?=5 Start Unknown Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before? ?+3=5 Five apples were on the table. I ate two apples. How many apples are on the table now? 5–2=? Five apples were on the table. I ate some apples. Then there were three apples. How many apples did I eat? 5–?=3 Total Unknown Three red apples and two green apples are on the table. How many apples are on the table? 3+2=? Addend Unknown Five apples are on the table. Three are red and the rest are green. How many apples are green? 3 + ? = 5, 5 – 3 = ? Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before? ?–2=3 Both Addends Unknown Grandma has five flowers. How many can she put in her red vase and how many in her blue vase? 5 = 0 + 5, 5 = 5 + 0 5 = 1 + 4, 5 = 4 + 1 5 = 2 + 3, 5 = 3 + 2 Difference Unknown (“How many more?” version): Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy? Bigger Unknown (Version with “more”): Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have? Smaller Unknown (Version with “more”): Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have? (“How many fewer?” version): Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have than Julie? 2 + ? = 5, 5 – 2 = ? (Version with “fewer”): Lucy has 3 fewer apples than Julie. Lucy has two apples. How many apples does Julie have? 2 + 3 = ?, 3 + 2 = ? (Version with “fewer”): Lucy has 3 fewer apples than Julie. Julie has five apples. How many apples does Lucy have? 5 – 3 = ?, ? + 3 = 5 DECEMBER HOMEWORK How will you implement this information into your work in the next month? Be prepared to share your successes and/or challenges at the January MTL meeting. Work with a teacher or teacher team Explore textbooks for story problem formats Try problem types with a group of students LEARNING INTENTIONS Understand strategies that can be used to solve various problem situations. Understand strategies that can be used to develop number fluency. SUCCESS CRITERIA You will be successful when you can identify and implement number strategies that students can use to develop number fluency and solve story problem situations. WHAT IS FLUENCY? What does fluency mean to you? Read the last paragraph on pg. 18 and 19. What is the “gist” of fluency as outlined in the Progressions for the Common Core State Standards in Mathematics? BUILDING FLUENCY AND INSTRUCTIONAL IMPLICATIONS “So the important press toward fluency should also allow students to fall back on earlier strategies when needed. By the end of the K-2 grade span, students have sufficient experience with addition and subtraction to know single-digit sums from memory… p. 19 0A Progressions document BUILDING FLUENCY AND INSTRUCTIONAL IMPLICATIONS “So the important press toward fluency should also allow students to fall back on earlier strategies when needed. By the end of the K-2 grade span, students have sufficient experience with addition and subtraction to know single-digit sums from memory… this is not a matter of instilling facts divorced from their meanings, but rather as an outcome of a multi-year process that heavily involves the interplay of practice and reasoning.” p. 19 0A Progressions document MOVING FROM CONCRETE TO ABSTRACT REPRESENTATIONS Direct modeling, Counting on and Numeric Reasoning Use the chart on pg. 36 and your problem type chart to make sense of the summary reading on pg 20-21. How are the problem types related to student strategies? HOW DOES OA GROW? OPERATIONS AND ALGEBRAIC THINKING K-2 K5 Grade 1 Grade 2 Responsible for 4 problem types Responsible for 8 problem types Responsible for 12 problem types Work within 5, then 10 Work within 20 Work within 100 Physically act out problems Count on Make 10 Decompose through 10 Doubles/Near doubles Concrete models to show all parts Semi-concrete models Matching with objects Tape diagrams Abstract model Drawings show quantities HOW DOES OA GROW? OPERATIONS AND ALGEBRAIC THINKING K-2 K5 Grade 1 Grade 2 Responsible for 4 problem types Responsible for 8 problem types Responsible for 12 problem types Work within 5, then 10 Work within 20 Work within 100 Physically act out problems Count on Make 10 Decompose through 10 Doubles/Near doubles Concrete models to show all parts Semi-concrete models Matching with objects Tape diagrams Abstract model Drawings show quantities LEARNING INTENTIONS Understand strategies that can be used to solve various problem situations. Understand strategies that can be used to develop number fluency. GRADE 1: CONTENT STANDARD 1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as • counting on • making ten • decomposing a number leading to a ten • using the relationship between addition & subtraction • creating equivalent but easier or known sums STRATEGIES THAT BUILD SINGLE DIGIT ADDITION FLUENCY Counting Make Use on. a ten. an easier “equivalent” problem. Use doubles Use fives Use a helping fact 8+6 Put 8 counters on your first frame & 6 counters on your second frame. Strategies: Make a ten. Use a double. Use fives. Use some other equivalent problem. COUNT ON: 8 + 6 Write an equation. 8+1+1+1+1+1+1=14. MAKE A TEN: 8 + 6 How could you make a ten? Move 2 counters to the top frame. Then you have 10 and 4 more counters. Write an equation. 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14 USE A DOUBLE: 8 + 6 What doubles might you use? USE A DOUBLE: 8 + 6 What doubles might you use? Reason 6 + 6 = 12; then add 2 more. Write an equation. 8+6=6+6+2= 12 + 2 = 14 USE FIVES: 8 + 6 Can you see some fives? Where? USE FIVES: 8 + 6 Can you see some fives? Where? Reason: 5 + 5 is 10; need to add 3 more and 1 more. Write an equation. 8+6=5+5+3+1 = 10 + 4 = 14 STANDARDS FOR MATH PRACTICE Which Standard for Math Practice were you using as you worked through these activities? 1. Make sense of problems and persevere in solving them. 3. Construct viable arguments and critique the reasoning of others. 6. Attend to precision. 7+9 9+8 6+7 Select a problem. Tell an addition story. Draw a strategy card for the group. Everyone uses ten frames and counters to reason through the strategy and writes an equation(s) that shows the reasoning. Share, compare, and discuss as a group. Repeat with another strategy card for the same problem or a new problem. Reflect: Which strategies seem to work best for each problem? STANDARDS FOR MATH PRACTICE Which Standard for Math Practice were you using as you worked through these activities? 1. Make sense of problems and persevere in solving them. 3. Construct viable arguments and critique the reasoning of others. 6. Attend to precision. WHY A 10 FRAME? How does modeling each strategy with the same manipulative model help develop student understanding? Develops number fluency and flexibility Anchors to 5 and 10 Place value fluency AS YOU THINK OF YOUR WORK… How does this information impact your work with teachers of all grades? … with Special Education , ELL, and older students who don’t have these skill sets. PROFESSIONAL PRACTICE EXPLORING STRATEGIES Number Strategies using Tens Frames: Work with a teacher or teacher team using tens frames strategies Model problem types with a group of students using tens frames Explore textbooks for number strategies using tens frames Video tape your work with teachers and or students working with ten frames and story problem types. LEARNING INTENTIONS Understand strategies that can be used to solve various problem situations. Understand strategies that can be used to develop number fluency. SUCCESS CRITERIA You will be successful when you can identify and implement number strategies that students can use to develop number fluency and solve story problem situations. FEEDBACK QUESTION: (1) Story problem types; (2) number strategies; (3) developmental learning progressions (from concrete to abstract); and (4) instructional models (like ten frames) are interrelated parts that must be cohesively connected for students to develop both fluency and number sense in the K-2 grades and the OA K-2 CCSS. Considering the 4 areas above, which areas do you need more practice to develop your skill in applying them with teachers and students?