Kentucky Education Cooperatives Conceptual Building Blocks Series Day 2 “All Kids, All Successful, All the Time”

Addition and Subtraction Day 2 Addition and Subtraction Strategy Progressions Story Problems Place Value Addition and Subtraction Strategies Identifying Error Patterns Center Activities

Alphabet Addition and Subtraction Add D and E Subtract C from F Add P and J

Addition and Subtraction Strategies Count all (two collections) Count on Count back/count down to/count up from Developing Essential Understanding of Addition and Subtraction Pre-K – Grade 2, NCTM publication, page 78, Early Numeracy Research Project (Clarke, 2001)

Addition and Subtraction Strategies Basic strategies Derived strategies Extending and applying addition and subtraction using basic, derived and intuitive strategies Developing Essential Understanding of Addition and Subtraction Pre-K – Grade 2, NCTM publication, page 78, Early Numeracy Research Project (Clarke, 2001)

Combinations and Partitions Activities Build fluency to 5 Build fluency to 10 Build Part-Part-Whole Understanding Combinations and Partitions of 20 Using: Five frames, ten frames, linking cubes, domino patterns, 20 frames K.OA, K.NBT, 1.OA, 2.OA.3

Begin with Five!

Move to ten! Think of a way to make 7

Begin building on and off from a number

Part-Part-Whole and Whole-part-part Use of five, ten, and twenty frames with no empty boxes. Show all partitions of the whole. Use of five, ten, and twenty frames with some empty boxes. Show all combinations of the whole.

Use of Frames

Doubles on a Bead Rack Objectives – Help students visualize doubles (e.g., 4+4; 6+6) – Help students use doubles in computation The visualization is key: 1+1=2 2+2=4 3+3=6 4+4=8

Check Up Bunny Ears One hand – to combine and partition within five Two hands – to combine and partition within ten Five and Ten Frames Empty Use counters

Check Up Ask the students to give the combinations and partitions of numerals to five What goes with three to make five? If I have two, how many more do I need to make five? What two numbers combine to make five? If I have two pencils, how many more do I need to have four pencils?

For our struggling students Some students have difficulty in thinking about partitioning quantity. Use cubes that will fit on their finger tips Use Velcro and structure Use foam ten frames with removable dots Use Wiki Sticks to assist in building the symbols

Salute! From Kentucky Center for Mathematics, Kentucky Numeracy Project Intervention Guide

Understanding Addition and Subtraction Situations The standards stress the importance of students being able to use addition and subtraction in all situations. The Four main problem types are: Add to Take from Put Together and Take Apart Compare (With unknown in all positions)

Addition and Subtraction Situations Problem Type Add ToResult UnknownChange UnknownStart Unknown Take FromResult UnknownChange UnknownStart Unknown Put together and take apart Total UnknownBoth Addends unknown Addend Unknown CompareDifference Unknown Bigger UnknownSmaller Unknown

We normally see problems that ask students to “join” or “separate” to find the unknown part. We tend to ask students to “put together” for addition and “take from” for subtraction. These definitions are limited and if these are the only exposure students have, they will have difficulty when the situation calls for something other than “put together” or “take away ”. Take for example, the following problem: Bob has 3 nickels and Bill has 7 nickels. How many more nickels does Bill have than Bob?

Problem Solving Mat

Story Problems Use of quantity in context assists students in attaching meaning to quantity, as well as, the actions of the operations Incorporate number talks along with the story problems Develop problem strings along with the story to assist students to develop efficient strategies K.OA.2, 1.OA.1, 2, 2.OA.1

Story Mats

Frog Story Mat

Gathering for Winter Whole Group Activity

Place Value Conceptual place value develops as students are involved in mental math strategies. Positional place value should not be taught in a procedural memorized manner. It gets in the way of a students development of a conceptual understanding. Developing Number Knowledge, Wright, Robert J., Ellemor-Collins, David., and Tabor, Pam. Sage Publications.

Unitizing and Place Value Unitizing in place value is the understanding that ten ones is the same one ten. Represents a HUGE shift in understanding 56 is 5 tens and 6 ones and is also: 4 tens and 16 ones 3 tens and 26 ones 2 tens and 36 ones 1 ten and 46 ones 56 ones Developing Number Knowledge, Wright, Robert J., Ellemor-Collins, David., and Tabor, Pam. Sage Publications.

Place Value Grouping items Cups and counters Sticks and bundles Dot strips Ten frames K.NBT, 1.NBT.2,3, 4, 5,6, 2.NBT.1, 2, 3, 4

A Conceptual Understanding of Ten = Developing Number Knowledge, Wright, Robert J., Ellemor-Collins, David., and Tabor, Pam. Sage Publications.

Adding and Subtracting Tens to a Decade = 3 tens or 30 Developing Number Knowledge, Wright, Robert J., Ellemor-Collins, David., and Tabor, Pam. Sage Publications.

Adding and Subtracting Tens off the Decade and 23 and 20 “23… 33, 43, “43” Developing Number Knowledge, Wright, Robert J., Ellemor-Collins, David., and Tabor, Pam. Sage Publications.

Adding and Subtracting Tens and Ones + 23 and 22 20, “30, 40, 43, 44, 45” Developing Number Knowledge, Wright, Robert J., Ellemor-Collins, David., and Tabor, Pam. Sage Publications.

Using Bundles and Sticks 300 (3 groups of hundreds), 20 (2 groups of tens), and 4 (4 sticks) 324 Developing Number Knowledge, Wright, Robert J., Ellemor-Collins, David., and Tabor, Pam. Sage Publications.

+ What moves do you want to make? 7 5

Using Dots 300 (3 hundreds), 20 (2 tens) and 4 (4 ones) is 324

“Gretchen” Video

Mental Math Solve the following: NBT.4,.5,.6; 2. NBT ; 3.NBT.2; 4. NBT.4

Student Work Betty ChadAlice “ is and is and is , and one more is 71”

Student Work Lisa’s Work Jason’s Work – 27 = “37, 47, 57, - 27 is 30, then 3 more is 60, 42 5 more is 33+5 is 38”

Common Misconceptions when adding and subtracting “Subtract the smaller from the larger” is a rule that children apply regardless of minuend or subtrahend. 62 – 45 = 23 Not regrouping = 52 Ignoring the zeros –

Instructional materials that support children in using mental math strategies Use of composite units Bundles and Sticks Dot Strips Frames Dots on popsicle sticks Unifix Cubes – Towers Cups and counters 100 Bead Rack Base Ten Blocks

Thou Shall Not CARRY

Strategies based on Place Value is is and 40 make make 17 which is 10 and 7, so 70 and 10 is 80 plus 7 is 87

Your Turn Use the split strategy to solve the following:

A Jump Strategy is 59, plus 10 is 69, plus 10 is 79, then one more is 80, then 7 more is ______________________________________

Your Turn Solve the following using a jump strategy

Partial Sums

Partial Sums Expanded Notation = 860

Your Turn Solve the following using a partial sums strategy

Why do so many students struggle with subtraction?  We teach them to “take away” or borrow.  Subtraction is neither commutative nor associative  Our sequence of learning is wrong  Start with the little stuff first.

Place Value Strategy for Subtraction 56 – is SPLIT apart into 50 and 6 27 is SPLIT apart into 20 and 7 56 – 20 = 36 and 36 – 7 = 29

Your Turn Solve the following using a split strategy

Jump Strategy for Subtraction 45 – __________________________________ Sources: NCTM Achieving Fluency: Special Education and Mathematics. Page 134, 136 Walle, Van de Teaching Student Centered Mathematics K-3. Page 185 NCTM Developing an Essential Understanding of Addition and Subtraction.

Your Turn Solve the following using a jump strategy:

Partial Differences NCTM Developing an Essential Understanding of Addition and Subtraction. Pages 43-44

Compensation Addition by compensating (add one to 29 to make it thirty; add, then subtract the one back off) – 1 (64 - 1= 63) Subtraction by compensating 53 – 28 (add two to 28 to make thirty, subtract, then take two back off) 53 – = is 25 Wright, Robert J., et. al. Developing Number Knowledge. 2012

Your Turn Solve the following problems using compensation:

Transforming Addition by Transforming: (Add 2 to 58 and take 2 off 27; maintaining the quantity of the entire problem) = 85 Subtracting by Transforming: 56 – 29 (add one to both numerals – keeping the distance between the numerals the same) 57 – 30 = 27

Your Turn Solve the following using a transforming strategy:

Adding Ten NCTM Developing an Essential Understanding of Addition and Subtraction. Page 47.

Your Turn Use the adding ten strategy to solve the following:

“Same Change” method used in Subtraction to Avoid Regrouping = = = = NCTM Developing an Essential Understanding of Addition and Subtraction. Page 47

Your Turn Solve the following using the “same change” strategy: 5000 – 2657

Identifying Error Patterns A = 14C = 12 B = 13D = 12 Taken from Error Patterns in Computation, Using Error Patterns to Help Each Student Learn by Robert B. Ashlock. 2010

Identifying Error Patterns A.56 B. 18C. 8D E Taken from Error Patterns in Computation, Using Error Patterns to Help Each Student Learn by Robert B. Ashlock. 2010

Identifying More Error Patterns A.32B. 245C. 524D Taken from Error Patterns in Computation, Using Error Patterns to Help Each Student Learn by Robert B. Ashlock. 2010

Reflection Activity

Center Activities for Day 2 What Number Is…? Game of Chance 100 or Bust Clear the Board Make Ten Rummy Real Counting On Race to Twenty

Make and Take Day 2 Bead Strings Frame Cards

Day 2 Reflection and Post-Assessment Three things I learned today are…. Two things I will implement in my classroom are….. I’m still wondering about… Complete the post-assessment