Addi,on & Subtrac,on Building a Founda,on for Later Grades Delise Andrews Math Coordinator, Grades 3-5 Lincoln Public Schools dandrews@lps.org
Addi,on • On poster paper, together with several people at your table… Write a math story that could be solved by doing the addi:on problem: 8 + 7 = 15 • Be sure to write with large, bold print so that everyone in the room can read your problem. 2
Subtrac,on • On poster paper, together with several people at your table… Write a math story that could be solved by doing the subtrac:on problem: 15 – 8 = 7 • Be sure to write with large, bold print so that everyone in the room can read your problem. 3
Addi,on & Subtrac,on • Through numerous research studies, we know that young students can solve contextualized mathema:cs problems through reasoning and making sense of the rela:onships in the story. A., V. D., Lovin, L. H., Karp, K. S., & Bay-Williams, J. M. (2014) . Teaching student-centered mathematics . Upper Saddle River, NJ: Pearson. 4
Problem Sort Res Result lt U Unkn know own Cha Change Unkno nge Unknown wn Star St art U Unknown nknown K. Lucy has two apples. Julie Add to has five apples. How many Add t more apples does Julie have A. Lucy has two apples. Julie than Lucy? has five apples. How many fewer apples does Lucy H. Lucy has 3 fewer apples have than Julie? than Julie. Lucy has two D. Five apples are on the table. apples. How many apples Three are red and the rest does Julie have? are green. How many apples B . L u c y h a s are green? 3 f e t h w Take from a n e r J a u l i e p p Take from . l e s J u a l i e p p h l e s a s . f H o i v e w m d o a n e s y J. Grandma has five flowers. L a p u c p e l I. Three red apples and two y s h a v e ? How many can she put in green apples are on the her red vase and how many table. How many apples are G. Five apples were on the in her blue vase? on the table? table. I ate some apples. Then there were three E. Two bunnies sat on the apples. How many apples did grass. Three more bunnies I eat? Total Unknown Total Unknown Addend Unknown Addend U nknown Bot Both Addends h Addends U Unknown nknown hopped there. How many Take Apart bunnies are on the grass her/ Take Apart L. Two bunnies were sitting on now? the grass. Some more bunnies hopped there. Then g there were five bunnies. n t t i Together/ s i e r e e How many bunnies hopped w o r N s m . e F i v n i e e u n r e n a p b h e p l e T h Put Toget s m e T w e over to the first two? o s . t r e S a s r e . a b o n r e e l . t F . g h I a h e e t t e h d s . t w t p e e o n p n i m a p o h o u n a n y p l e b n a s . s o p p H o i e v e l e s w n n f i r e a u e t r e b r e w a b o n e l e t w e s Pu n o h e e n i w ? r n e b u t h y a n e ? m r f o w e H o b s s r a g e h t Difference Unknown Difference Unknown Big Bigger er U Unknown nknown Smaller Smaller U Unknown nknown O. Julie has three more apples t h e than Lucy. Lucy has two o n e r e w p e l s a p n m e T h e apples. How many apples S o s . C . p l e a p t w o a t e I a b l e . s . t p p l e e a does Julie have? h r e e t w e r r e n M. Julie has three more apples t h e e o re w e r s p p l e mpare y a m a n than Lucy. Julie has five w H o Compa r e ? e f o e b t a b l apples. How many apples h e t Co does Lucy have? 5
No,ce & Wonder • What did you no:ce about the types of addi:on and subtrac:on problems? • What do you wonder? 6
Classify Table Problems • Determine a classifica:on for the addi:on and subtrac:on problems you wrote earlier. • Clearly record the classifica:on on your chart paper. 7
No,ce & Wonder • What did you no:ce about the types of addi:on and subtrac:on problems we wrote? • What do you wonder? 8
Wri,ng Equa,ons • A situa:on equa:on represents a literal transla:on of the math story context. (represen:ng informa:on as it comes in the story) • A solu:on equa:on represents the mathema:cs required to find the solu:on to the problem. (the unknown quan:ty is isolated) Fuson, Karen C., Carroll, William M. and Landis, Judith(1996) 'Levels in Conceptualizing and Solving Addition and Subtraction Compare Word Problems', Cognition and Instruction, 14: 3, 345 — 371 9
Wri,ng Equa,ons • Joanna had some cookies. She gave 3 cookies to Eric. Now she has 5 cookies. How many cookies did Joanna have to begin with? • Situa:on equa:on: ☐ – 3 = 5 • Solu:on equa:on: 5 + 3 = ☐ 10
Wri,ng Equa,ons • Your turn! Write a situa:on equa:on and a solu:on equa:on for each story card. – A situa:on equa:on represents a literal transla:on of the math story context. (represent informa:on as it comes in the story) – A solu:on equa:on represents the mathema:cs required to find the solu:on to the problem. (isolate the unknown quan:ty) 11
No,ce & Wonder • Look at the equa:ons for all of the different problem types. What do you no:ce? • What do you wonder? 12
Modeling Addi,on & Subtrac,on • Use the place value blocks at your table to model these problems: – Sandra had 8 pennies. George gave her 4 more. How many pennies does Sandra have altogether? – Sandra has 8 pennies and 4 nickels. How many coins does she have? A., V. D., Lovin, L. H., Karp, K. S., & Bay-Williams, J. M. (2014) . Teaching student-centered mathematics . Upper Saddle River, NJ: Pearson. 13
Modeling Addi,on & Subtrac,on • Compare the way you modeled the problems. – How were your models similar? – How were they different? 14
Modeling Addi,on & Subtrac,on • Use the place value blocks at your table to model these problems: – Sandra had 12 pennies. She gave 4 pennies to George. How many pennies does Sandra have now? – George has 12 coins. Four of his coins are nickels, and the rest are pennies. How many pennies does George have? – George has 12 pennies and Sandra has 8 pennies. How many more pennies does George have than Sandra? A., V. D., Lovin, L. H., Karp, K. S., & Bay-Williams, J. M. (2014) . Teaching student-centered mathematics . Upper Saddle River, NJ: Pearson. 15
Modeling Addi,on & Subtrac,on • Compare the way you modeled the problems. – How were your models similar? – How were they different? 16
Structure & Problem Difficulty • Consider the following set of problems… – Maggie had 7 bracelets. She bought 8 more bracelets. How many bracelets does Maggie have now? – Maggie had 7 bracelets. She bought some more bracelets. She now has 15 bracelets. How many bracelets did she buy? – Maggie had some bracelets. She bought 8 more bracelets. She now has 15 bracelets. How many bracelets did Maggie start with? A., V. D., Lovin, L. H., Karp, K. S., & Bay-Williams, J. M. (2014) . Teaching student-centered mathematics . Upper Saddle River, NJ: Pearson. 17
SeTng the Stage for Mul,plica,on & Division Operations and Algebraic Thinking 2.OA Represent and solve problems involving addition and subtraction. 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 . 1 Add and subtract within 20. 2. Fluently add and subtract within 20 using mental strategies. 2 By end of Grade 2, know from memory all sums of two one-digit numbers. Work with equal groups of objects to gain foundations for multiplication. 3. Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends. 4. Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends. COMMON CORE FOR STATE STANDARDS Mathematics Common core state standards for mathematics. (2010). Washington, D.C.: National Governors Association Center for Best Practices (NGA Center). 18 a. b.
SeTng the Stage for Mul,plica,on & Division • Children engage in coun:ng equal groups and fair sharing (even with remainders) from an early age. K-2 students are well equipped to reason about such situa:ons given appropriate quan::es. • We can leverage this informal understanding to set the stage for the more complex mul:plica:on and division situa:ons students will encounter in later grades. A., V. D., Lovin, L. H., Karp, K. S., & Bay-Williams, J. M. (2014) . Teaching student-centered mathematics . Upper Saddle River, NJ: Pearson. 19
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