Mental Math Fact Learning Mental Computation Estimation Grade 2 Teacher’s Guide båÖäáëÜ=mêçÖê~ãë ml=_çñ=OMMM `Ü~êäçííÉíçïåI=mb===`N^=TkU qÉäW=EVMOF=PSU=QSMM c~ñW=EVMOF=PSU=QSOO ïïïKÖçîKéÉKÅ~LÉÇì 2008 Table of Contents Mental Math in the Elementary Mathematics Curriculum . . . . . . . . . . . . . 1 Definitions and Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Rationale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Introducing Thinking Strategies to Students . . . . . . . . . . . . . . . . . . . . . . . 8 Practice and Reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Response Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Timed Tests of Basic Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Parents and Guardians: Partners in Developing Mental Math Skills . . . . 13 Pre-operational Skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Patterned Set Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Part-Part-Whole Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Ten-Frame Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Other Number Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 One More/One Less and Two More/Two Less . . . . . . . . . . . . . . . 18 Next Number and Counting On and Back . . . . . . . . . . . . . . . . . . . 19 Fact Learning - Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Reviewing Addition Facts and Fact Learning Strategies . . . . . . . . . . . . . 23 Doubles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Plus 1 Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Plus 2, Plus 3 Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Using Five-Frames and Ten-Frames . . . . . . . . . . . . . . . . . . . . . . . 27 Model Numbers on a Five-Frame . . . . . . . . . . . . . . . . . . . . . 27 Visualize Combinations on a Five-Frame . . . . . . . . . . . . . . . 28 Random Numbers on a Five-Frame . . . . . . . . . . . . . . . . . . . 28 Ten-Frame Flash Cards . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Ten-Frame Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Empty Ten-Frame Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 New Strategies for Addition Fact Learning Introduced in Grade 2 . . . . . 30 Near Doubles (1-Apart Facts) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2-Apart Facts (Double In-Between Facts) . . . . . . . . . . . . . . . . . . . 30 Plus or Minus 0 (No Change Facts) . . . . . . . . . . . . . . . . . . . . . . . . 31 Relationships for Numbers 10-20 . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Developing the Concept With 2-Part Mats . . . . . . . . . . . . . . 32 Developing the Concept With Ten-Frames . . . . . . . . . . . . . . 33 Reinforcing the Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Make Ten Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Make Ten Flash Cards (example) . . . . . . . . . . . . . . . . . . . . . 35 Addition Facts With Sums to18 . . . . . . . . . . . . . . . . . . . . . . . 37 Fact Learning – Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Subtraction as “Think Addition” . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Ten-Frame Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Other Ways to Think About Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . 39 Up Through 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Back Down Through 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Subtraction Facts With Minuends to 18 . . . . . . . . . . . . . . . . . . . . . 41 Mental Computation - Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Addition Facts Extended to 2-Digit Numbers (New) . . . . . . . . . . . . 45 Front-End Addition (New) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Finding Compatibles (New) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Compensation (New) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Mental Computation - Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Using “Think Addition” in Subtraction (Extension) . . . . . . . . . . . . . 49 Estimation - Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Rounding in Addition and Subtraction (New) . . . . . . . . . . . . . . . . . 54 Appendixes Thinking Strategies in Mental Math . . . . . . . . . . . . . . . . . . . . . . . . 57 Scope and Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Mental Math in the Elementary Mathematics Curriculum Mental math in this guide refers to fact learning, mental computation, and computational estimation. The Atlantic Canada Mathematics Curriculum supports the acquisition of these skills through the development of thinking strategies across grade levels. Mental math refers to fact learning, mental computation, and computational estimation. The Atlantic Canada Mathematics Curriculum supports the acquisition of these skills through the development of thinking strategies across grade levels. Pre-Operational Skills Many children begin school with a limited understanding of number and number relationships. Counting skills, which are essential for ordering and comparing numbers, are an important component in the development of number sense. Counting on, counting back, concepts of more and less, and the ability to recognize patterned sets, all mark advances in children’s development of number ideas. Basic facts are mathematical operations for which some students may not be conceptually prepared. Basic facts are mathematical operations for which some students may not be conceptually prepared. As a minimum, the following skills should be in place before children are expected to acquire basic facts. • Students can immediately name the number that comes after a given number from 0-9, or before a given number from 2-10. • When shown a familiar arrangement of dots # 10 on ten frames, dice, or dot cards, students can quickly identify the number without counting. Mental Math – Grade 2 1 • For numbers # 10 students can quickly name the number that is one-more, one-less; two-more, two-less. (the concept of less tends to be more problematic for children and is related to strategies for the subtraction facts) Mental mathematics must be a consistent part of instruction in computation from primary through the elementary and middle grades. 2 Mental Math – Grade 2 Curriculum Outcomes Thinking Strategies Grade 1 B7- use mental strategies to find sums P. 28 to 18 and differences from 18 or • Doubles Facts for addition and less subtraction facts B8- memorize simple addition and/or P. 36 subtraction facts from among those • Using patterns to learn the facts for which the total is 10 or less • Commutative property (3+2 = 2+3) C5- use number patterns to help solve addition and subtraction sentences Grade 2 B5- develop and apply strategies to P. 22 learn addition and subtraction facts • Doubles plus 1 B11- estimate the sum or difference • Make 10 (“bridging to 10”) of two 2-digit numbers • Two-apart facts; double in-between • Subtraction as “think addition” • Compensation Fact learning is a mental • Balancing for a constant difference exercise with an oral and/or visual prompt; the focus is P. 30 (Estimation) oral, rather than paper-and • Rounding both numbers to the pencil; drills should be short nearest 10 with immediate feedback over an • Round one number up and one extended period of time. number down • Front-end estimation Grade 3 B11/12- mentally add and subtract P. 34 two-digit and one-digit • Make 10 numbers, and rounded • Compatible numbers (“partner” numbers. numbers) B9- continue to estimate in addition and • Front-end addition subtraction situations • Back up through ten (“counting on”) B10- begin to estimate in multiplication • Compensation and division situations • Balancing for a constant difference C3 - use and recognize the patterns in a multiplication table P. 28 • Commutative property for multiplication (3x2 = 2x3) • Division as “think multiplication” • Helping facts Mental Math – Grade 2 3 Curriculum Outcomes Thinking Strategies Grade 4 B9 - demonstrate a knowledge of the P. 32 multiplication facts to 9 x 9 • Doubles B14 - estimate the product or quotient of • Clock-facts for 5’s 2- or 3-digit numbers and single • Patterns for 9’s digit numbers • Helping facts B15 - mentally solve appropriate addition and subtraction P. 36 (Estimation) computations • Rounding B16 - mentally multiply 2-digit numbers • Front-end by 10 or 100 • Clustering of Compatibles C2 - apply the pattern identified when multiplying by increasing powers P. 38 of 10 • Compatibles for division P. 40 • Front-end addition • Compensation • Up through 100 (counting on) • Back down through 100 (counting back) • Compatible numbers • Place-value-change strategy for mentally multiplying by 10, 100 4 Mental Math – Grade 2 Curriculum Outcomes Thinking Strategies Grade 5 B10- estimate sums and differences P. 40 to 41 (Estimation) involving decimals to thousandths • Rounding one up, one down B11- estimate products and quotients of • Looking for compatibles that make two whole numbers approximately 10, 100, 1000 B12- estimate products and quotients of • Front-end decimal numbers by single-digit whole numbers P. 44 B15- multiply whole numbers by 0.1, • Place-value-change strategy for 0.01, and 0.001 mentally mentally multiplying by 10, 100, 1000 C2- recognize and explain the pattern • “Halve-double” strategy for in dividing by 10, 100, 1000 and in multiplication multiplying by 0.1, 0.01 and 0.001 • Front-end multiplication B13- perform appropriate mental • Compensation multiplications with facility P. 46 to 50 By grade 5, students should • Place-value-change strategy for possess a variety of strategies mentally dividing by 10, 100, 1000 to compute mentally. It is • Place-value-change strategy for important to recognize that these mentally multiplying by 0.1, 0.01, strategies develop and improve 0.001 over the years with regular practice. Grade 6 B9- estimate products and quotients P. 40 (Estimation) involving whole numbers only, • Rounding one up, one down for whole numbers and decimals, and multiplication decimals only • Front-end method for multiplication B10- divide numbers by 0.1, 0.01, and and division 0.001 mentally C2- use patterns to explore division by P. 42 and 50 0.1, 0.01, and 0.001 • Place-value-change strategy for B11- calculate sums and differences in mentally dividing by 0.1, 0.01, 0.001 relevant contexts using the most appropriate method P. 44 • Compensation in multiplication • Front-end Students should perform mental computations with facility using strategies outlined in the Mental Math Guides. Mental Math – Grade 2 5