Strategic Math Series Professional Developer’s Guide developed by Susan P. Miller and Cecil D. Mercer The Learning Strategy Series 2006 The University of Kansas Center for Research on Learning Lawrence, Kansas University of Kansas Center for Research on Learning 2002

Overview of the Series Addition Facts 0-9 Subtraction Facts 0-9 Place Value: Discovering Ones & Tens Addition Facts 10-18 Subtraction Facts 10-18 Multiplication Facts 0-81 Division Facts 0-81 Addition with Regrouping Subtraction with Regrouping University of Kansas Center for Research on Learning 2006 University of Kansas Center for Research on Learning 2002

Purpose of the Series To provide students who are at risk for failure in school with: the skills needed to solve basic math problems. a foundation for solving more advanced problems in the future. University of Kansas Center for Research on Learning 2006

Rationales for the Series Students need: manipulative experiences to promote conceptual understanding of basic math facts and place value. conceptual understanding of math skills prior to rote memorization. strategies for solving math problems at the abstract level. University of Kansas Center for Research on Learning 2006

Rationales for the Series Students need: fluency-building practice to master and retain basic math facts and place value. strategies for solving word problems. successful experiences in math. University of Kansas Center for Research on Learning 2006

Guiding Principles The Concrete-Representational-Abstract (C-R-A) Teaching Method Concrete: use objects (e.g., pennies, cubes, sticks) Representational: use drawings (e.g., circles, boxes) and tallies Abstract: use numbers (e.g., 2, 7) University of Kansas Center for Research on Learning 2006

Guiding Principles Mastery Learning Facts learned to fluency Fluency assures retention Students then able to solve more complex problems University of Kansas Center for Research on Learning 2006

Guiding Principles Use of Mnemonics Enables students to engage in the problem-solving process Enables students to apply this process to everyday situations University of Kansas Center for Research on Learning 2006

Instructional Stages Stage 1 Pretest Stage 2 Teach Concrete Application Stage 3 Teach Representational Application Stage 4 Introduce the “DRAW” or “FIND” Strategy Stage 5 Teach Abstract Application Stage 6 Posttest Stage 7 Provide Practice to Fluency University of Kansas Center for Research on Learning 2006

The “DRAW” Strategy (used to teach all facts) D Discover the sign R Read the problem A Answer, or draw and check W Write the answer University of Kansas Center for Research on Learning 2006

The “FIND” Strategy (used to teach place value) F Find the columns I Insert the T N Name the columns D Determine the answer University of Kansas Center for Research on Learning 2006

Instructional Procedures Give an Advance Organizer Describe and Model Conduct Guided Practice Conduct Independent Practice Conduct Problem-Solving Practice Administer Timed Probe Administer Facts Review Probe Conduct Pig Game Practice Provide Feedback University of Kansas Center for Research on Learning 2006

Solving Addition Problems Concrete: manipulatives 2 _________ + 3 _________ _________ Representational: pictures or tallies 3 _________ Abstract: words 6 balls + 1 ball balls 3 birds sitting Sally has 4 books. + 5 birds sitting Irene has 3 books. birds sitting They have books altogether. University of Kansas Center for Research on Learning 2006

Solving Subtraction Problems Concrete: manipulatives 7 _________ - 4 _________ _________ Representational: pictures or tallies 5 _________ - 2 _________ Abstract: words 7 apples 3 baseball cards - 5 apples - 2 given away apples left in collection Jennifer had 4 pens. She lost 2 of them. She has pens left. University of Kansas Center for Research on Learning 2006

Solving Place Value Problems Concrete In 35 _________, there are ____ tens and ____ ones. The number 24 is another name for _________ and ___ _______. Representational 2 tens and 7 ones is another name for __________. Abstract 3 tens and 2 ones + 2 tens and 5 ones tens and ___ ones is another name for ____. Jerry has 5 tens. Linda has 3 tens. They have ___ tens altogether. Pop has 2 tens and 3 ones. Ray has 5 tens and 4 ones. How many tens and ones do they have altogether? University of Kansas Center for Research on Learning 2006

Solving Multiplication Problems Concrete: manipulatives 7 groups of 3 ________ ________ Representational: pictures or tallies 3 groups of 4 _________ _________ Abstract: words 3 plates My friend has 4 trains. of 7 oatmeal cookies Each train has 6 cars. oatmeal cookies There are cars in all. Cindy has 2 tapes. Each tape has 4 songs on it. How many songs are on the tapes? University of Kansas Center for Research on Learning 2006

Solving Division Problems Concrete: manipulatives 6 ___________ has how many groups of 2 _______? Solve: __________ : __________ = ____________ Total Objects/group Groups Representational: pictures or tallies How many groups of 4 _______ are in 16 _______? Total Objects/group Groups Abstract: words How many groups of 5 dogs are in a pack of 25 dogs? Solve: __________ : __________ = ____________ University of Kansas Center for Research on Learning 2006

“FAST DRAW” (used to teach word problems) F Find what you’re solving for A Ask yourself, “What are the parts of the problem?” S Set up the numbers T Tie down the sign D Discover the sign R Read the problem A Answer, or draw and check W Write the answer University of Kansas Center for Research on Learning 2006

Advanced Problem-Solving Practice Addition: Bob has 3 shirts. Tim has 6 shirts. How many shirts do they have altogether? Subtraction: Luis had 8 sheets of paper. Tom had 6 frogs. Luis gave his 8 sheets of paper to Sylvia. How many sheets of paper does Luis have left? University of Kansas Center for Research on Learning 2006

Advanced Problem-Solving Practice Place Value: Henry planted 12 flowers. Hank planted 27 more flowers. Phillip planted 2 trees. How many flowers were planted in all? Answer: _____ This number has ___ tens and ___ ones. Multiplication: Jan has 8 dolls. Each doll has 2 hats. Jan also has 3 pets. How many hats are there in all? University of Kansas Center for Research on Learning 2006

Advanced Problem-Solving Practice Division: Linda handed out 56 tickets. Ron wanted 10 tickets. Each student received 8 tickets. How many students received tickets? Students also create their own word problems. University of Kansas Center for Research on Learning 2006

Feedback Sequence 1. Score each student’s product for correct and incorrect responses. 2. Meet with student; plot score on the Progress Chart. 3. Specify incorrect responses and error patterns if they exist. 4. Demonstrate the task. 5. Ask student to practice application. 6. Close the feedback session with positive statement. University of Kansas Center for Research on Learning 2006

Field Test Results: Computation of Facts mean mean pretest posttest increase Addition 0-9 34% 93% 59% Subtraction 0-9 32% 96% 64% Place Value 20% 90% 70% Addition 10-18 49% 93% 44% Subtraction 10-18 20% 95% 75% Multiplication 48% 92% 44% Division 25% 96% 71% University of Kansas Center for Research on Learning 2006

Field Test Results: Rate Increases Following the Posttest Lesson mean mean beginning ending rate* rate* increase Addition 0-9 11 21 90% Subtraction 0-9 10 22 120% Place Value 12 34 183% Addition 10-18 14 24 71% Subtraction 10-18 8 14 75% Multiplication 7 16 128% Division 11 22 100% Note: Most students in the field tests needed additional fluency practice beyond the 21st or 22nd lesson. Continuation of Pig Games, Math Minutes, and/or Facts Reviews is appropriate during concrete and representational lessons of subsequent skills. * Digits correct per minute. University of Kansas Center for Research on Learning 2006

Materials to Obtain Manipulative devices (3 types) Chalkboard or overhead projector Timer or clock with a second hand All other materials contained in manuals. Pig dice accompany each manual. The color of the numbers on the dice match the color of the letters in the title of the manual. University of Kansas Center for Research on Learning 2006

Student Notebook Contents Contract Progress Chart Learning Sheets Math Minute(s) Facts Review(s) Pig Game Answer Key Relationship/Rules Sheet University of Kansas Center for Research on Learning 2006

Addition 0-9 and the Concrete Process 5 + 2 University of Kansas Center for Research on Learning 2006

Addition 0-9 and the Representational Process 3 + 2 4 + 3 5 + 1 University of Kansas Center for Research on Learning 2006

Subtraction 0-9 and the Concrete Process 5 - 3 University of Kansas Center for Research on Learning 2006

Subtraction 0-9 and the Representational Process 5 - 2 4 - 3 6 - 4 University of Kansas Center for Research on Learning 2006

Place Value and the Concrete Process Tens Ones 24 = 17 = 46 = Tens Ones Tens Ones University of Kansas Center for Research on Learning 2006

Place Value and the Representational Process 14 = Tens Ones Tens Ones Tens Ones University of Kansas Center for Research on Learning 2006

Addition 10-18 and the Concrete Process 3 + 8 University of Kansas Center for Research on Learning 2006

Addition 10-18 and the Representational Process 5 + 6 6 + 7 8 + 5 University of Kansas Center for Research on Learning 2006

Subtraction 10-18 and the Concrete Process 12 - 4 University of Kansas Center for Research on Learning 2006

Subtraction 10-18 and the Representational Process 12 - 6 11 - 3 16 - 8 University of Kansas Center for Research on Learning 2006

Multiplication and the Concrete Process 4 x 3 University of Kansas Center for Research on Learning 2006

Multiplication and the Representational Process 4 x 2 3 x 5 = _____ 6 x 3 University of Kansas Center for Research on Learning 2006

Division and the Concrete Process 6 : 3 = ____ 5 10 University of Kansas Center for Research on Learning 2006

Division and the Representational Process 8 : 2 = _____ 12 : 6 = _____ 6 : 2 = _____ University of Kansas Center for Research on Learning 2006

Subtraction Rules & Relationships In subtraction, when the bottom number in the ones column is bigger than the top number in the ones column, the ten is traded. To help you learn the subtraction trade rule, remember the BBB sentence: Bigger number on Bottom means Break down the ten and trade. Example: 14 -7 ? University of Kansas Center for Research on Learning 2006

Subtraction Rules & Relationships Addition and subtraction facts have the same numbers, but in different orders. e.g.: 8 + 7 = 15 15 - 8 = 7 Because addition and subtraction are related, you always can state a subtraction problem as an addition problem. e.g.: 14 - 6 = ? ? + 6 = 14 University of Kansas Center for Research on Learning 2006

Subtraction Rules & Relationships To check a subtraction answer, add your answer to the number that is being subtracted. If the sum of these two numbers equals the first number in the subtraction problem, your answer is correct. e.g.: 14 - 5 = 9 Check: 9 + 5 = 14 University of Kansas Center for Research on Learning 2006

Multiplication Rules and Relationships Any number times 0 equals 0. e.g.: 6 x 0 = 0 9 x 0 = 0 Any number times 1 equals the original number. e.g.: 7 x 1 = 7 9 x 1 = 9 University of Kansas Center for Research on Learning 2006

Multiplication Rules and Relationships 2 times any number equals the number added to itself e.g.: 2 x 8 = 16 8 + 8 = 16 Changing the order of the numbers in multiplication does not change the answer. e.g.: 6 x 3 = 18 3 x 6 = 18 University of Kansas Center for Research on Learning 2006

Division Rules and Relationships 0 divided by any number equals 0. 0 ÷ 6 = 0 0 ÷ 4 = 0 Any number divided by 1 equals the number. 7 ÷ 1 = 7 2 ÷ 1 = 2 University of Kansas Center for Research on Learning 2006

Division Rules and Relationships Any number divided by that number equals 1. 7 ÷ 7 = 1 2 ÷ 2 = 1 Multiplication and division facts have the same numbers, just in different orders. 8 x 7 = 56 56 ÷ 7 = 8 56 ÷ 8 = 7 University of Kansas Center for Research on Learning 2006

Division Rules and Relationships Because multiplication & division are related, you can always state a division problem as a multiplication problem. 35 ÷ 7 = ? ? x 7 = 35 ? To check a division answer, multiply your answer by the number that the total is being divided by. If the answer to this multiplication problem equals the total, your answer is correct. 35 ÷ 5 = 7 Check: 7 x 5 = 35 University of Kansas Center for Research on Learning 2006