Kentucky Education Cooperatives Conceptual Building Blocks Series Day 2 “All Kids, All Successful,...

Kentucky Education Cooperatives Conceptual Building Blocks Series

Day 2

“All Kids, All Successful, All the Time”

Addition and SubtractionDay 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 SubtractionStrategies

• 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 20Using:Five frames, ten frames, linking cubes, domino patterns, 20 framesK.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

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 EarsOne hand – to combine and partition within fiveTwo hands – to combine and partition within ten

Five and Ten FramesEmpty 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 tipsUse Velcro and structureUse foam ten frames with removable dotsUse Wiki Sticks to assist in building the symbols

Salute!

From Kentucky Center for Mathematics, Kentucky Numeracy Project Intervention Guide

www.kymath.org

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 toTake fromPut Together and Take ApartCompare(With unknown in all positions)

Addition and Subtraction SituationsProblem Type

Add To Result Unknown Change Unknown Start Unknown

Take From Result Unknown Change Unknown Start Unknown

Put together and take apart

Total Unknown Both Addends unknown

Addend Unknown

Compare Difference Unknown

Bigger Unknown Smaller 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, 2012. 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 onesDeveloping Number Knowledge, 2012. Wright, Robert J., Ellemor-Collins, David., and Tabor, Pam. Sage Publications.

Place Value

• Grouping itemsCups and countersSticks and bundlesDot stripsTen frames

K.NBT, 1.NBT.2,3, 4, 5,6, 2.NBT.1, 2, 3, 4

A Conceptual Understanding of Ten

=

Developing Number Knowledge, 2012. Wright, Robert J., Ellemor-Collins, David., and Tabor, Pam. Sage Publications.

Adding and Subtracting Tens to a Decade

= 3 tens or30

Developing Number Knowledge, 2012. 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, 2012. 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, 2012. 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, 2012. 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:

43 + 2865 - 27

1.NBT.4, .5, .6; 2. NBT.7 .8. 9; 3.NBT.2; 4. NBT.4

Student Work

Betty Chad Alice“40 + 20 is 60

43 43 and 8 + 3 is 11+ 28 + 28 and 60 + 10 is 6 1 711 70, and one

more is 71”

Student Work

Lisa’s Work Jason’s Work 65 65 – 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 34 + 28 = 52• Ignoring the zeros 2 13 303 – 154 259

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

38+ 49

Strategies based on Place Value

38 + 49 38 is 30 + 849 is 40 + 9

30 and 40 make 70 8 + 9 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:

56 + 32

134 + 643

A Jump Strategy

38 + 4949 + 10 is 59, plus 10 is 69, plus 10 is 79, then one more is 80, then 7 more is 87

+10 +10 +10 +1 +7______________________________________ 49 59 69 79 80 87

Your Turn

• Solve the following using a jump strategy

64 + 33

132 + 54

Partial Sums

38 136 267+ 49 + 553+ 841

70 600 100017 80 10087 9 8

689 1108

Partial Sums Expanded Notation

533 + 327

500 + 30 + 3300 + 20 + 7800 + 50 + 10 = 860

Your Turn

• Solve the following using a partial sums strategy

327 + 488

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 – 27

56 is SPLIT apart into 50 and 627 is SPLIT apart into 20 and 7

56 – 20 = 36 and 36 – 7 = 29

Your Turn

• Solve the following using a split strategy

435 - 227

Jump Strategy for Subtraction

45 – 27

-2 -5 -10 -10__________________________________ 18 20 25 35 45

Sources:• NCTM. 2011. Achieving Fluency: Special Education and Mathematics. Page 134,

136• Walle, Van de. 2006. Teaching Student Centered Mathematics K-3. Page 185• NCTM. 2011. Developing an Essential Understanding of Addition and Subtraction.

Your Turn

• Solve the following using a jump strategy:

56 - 38

Partial Differences

56 371 813- 23 - 285 - 139

30 100 700 3 -10 -20 33 -4 -6

86 674

NCTM. 2011. Developing an Essential Understanding of Addition and Subtraction. Pages 43-44

Compensation

• Addition by compensating 34 + 29 (add one to 29 to make it thirty; add, then subtract the one back off) 34 + 30 – 1 (64 - 1= 63)• Subtraction by compensating 53 – 28 (add two to 28 to make thirty, subtract, then take two back off) 53 – 30 + 2 = 23 + 2 is 25

Wright, Robert J., et. al. Developing Number Knowledge. 2012

Your Turn

• Solve the following problems using compensation:

45 + 39

75 - 38

Transforming

• Addition by Transforming: 58 + 27 (Add 2 to 58 and take 2 off 27; maintaining the quantity of the entire problem) 60 + 25 = 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:

68 + 25

77 - 39

Adding Ten

NCTM. 2011. Developing an Essential Understanding of Addition and Subtraction. Page 47.

Your Turn

• Use the adding ten strategy to solve the following:

53 - 27

“Same Change” method used in Subtraction to Avoid Regrouping

6000 6000 - 1 = 5999- 3642 3642 - 1 = 3641

2358

46 46 + 2 = 48- 28 28 + 2 = - 30

18

NCTM. 2011. 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. 7 + 8 = 14 C. 7 + 6 = 12

B. 8 + 6 = 13 D. 8 + 5 = 12

Taken from Error Patterns in Computation, Using Error Patterns to Help EachStudent Learn by Robert B. Ashlock. 2010

Identifying Error Patterns

A. 56 B. 18 C. 8 D. 42 + 6 + 30 + 16 + 56 17 48 15 98

E. 85 + 6 19

Taken from Error Patterns in Computation, Using Error Patterns to Help EachStudent Learn by Robert B. Ashlock. 2010

Identifying More Error Patterns

A. 32 B. 245 C. 524 D. 135 - 16 - 137 - 298 - 67 16 112 374 132

Taken from Error Patterns in Computation, Using Error Patterns to Help EachStudent 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