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Math Models

This comprehensive guide explores the distinction between strategies and algorithms in mathematics education for students from Pre-K through Grade 5. It details how students utilize various mathematical strategies for addition, subtraction, multiplication, and division, while also emphasizing the importance of standard algorithms by the end of Grade 4 and Grade 5. The guide covers key concepts in counting, cardinality, operations, algebraic thinking, measurement, geometry, and number operations, providing educators with insights to effectively teach these essential mathematical skills.

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Math Models

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  1. Math Models Parent Academy Presented by: Allison Curran http://websites.nylearns.org/acurran1

  2. Math Domains • Counting and Cardinality (P-K) • Operations and Algebraic Thinking (P-5) • Measurement and Data (P-5) • Geometry (P-5) • Number and Operations in Base Ten (K-5) • Number and Operations – Fractions (3-5)

  3. Math Models as Strategies Algorithms VS. Strategies: • The Standards distinguish strategies from algorithms • Students use strategies for addition &subtraction in K-3 • Fluently add & subtract whole numbers using standard algorithms by the end of Grade 4 • Use strategies for multiplication & division 3-4 but use standard algorithm by the end of Grade 5 • Computation Algorithm: A set of predefined steps applicable to a class of problems that gives the correct result in every case when the steps are carried out correctly • Computation Strategy: Purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at converting one problem into another

  4. Example of ProgressionsNumbers in Base 10 Kindergarten • Draw special attention to 10 • Children learn to view the whole numbers 11 through 19 as ten ones and some more ones • They decompose 10 into pairs such as 1/9, 2/8, 3/7 …

  5. Number BondsBundlesRekenrek5 and Ten Frames

  6. Grade 1 • View ten ones as a unit called a ten • Compose and decompose this unit flexibly • View the numbers 11-19 as composed of one ten and some ones • See two digit numbers as representing some tens • Students add and subtract using this understanding

  7. BundlesNumber BondsRekenrek

  8. Ten FramesNumber Bonds

  9. Number Bonds

  10. Number Lines

  11. Grade 2 • Students extend their base-ten understanding to hundreds • Add and subtract within 1,000 • Use composing and decomposing • Fluent with addition and subtraction within 100

  12. Number BondsBundlesNumber Lines

  13. Place Value Discs

  14. Grade 3 • Major focus is multiplication • Adding and subtracting fluently within 1,000

  15. Number BondsArraysDistributive Property

  16. Tape DiagramsNumber Bonds

  17. Grade 4 • Use standard algorithm to add and subtract • Multiply and divide with multi-digit numbers • Extend knowledge on base-ten system • Work with decimals – tenths & hundredths

  18. Tape DiagramVertical Number LineNumber Bonds

  19. Grade 5 • Decimals to the thousandths place • Reason using all four operations with decimals to the hundredths place • Standard multiplication algorithm with multi-digit numbers • Reason about dividing whole numbers with two-digit divisors

  20. Tape Diagram“Box Method”Standard Algorithm

  21. Multiple Representations

  22. Standards for Mathematical Practice • Make sense of problems and persevere in solving them • Reason abstractly and quantitatively • Construct viable arguments and critique the reasoning of others • Model with mathematics • Use appropriate tools strategically • Attend to precision • Look for and make use of structure • Look for & express regularity in repeated reasoning

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