Workshop goals Participants will…

– Represent and solve problems requiring addition (Activity 1).

– Discuss and reflect on pedagogy necessary to teach BIG idea ONE:

Discuss and identify representation models for addition and subtraction

(Activity 1 & 2)

Match and design a variety of addition and subtraction problems (Activity 3)

– Play a computation game and analyze possible fluency outcomes

(Activity 3)

– Discuss and reflect on pedagogy necessary to teach BIG idea TWO:

Analyze true/false equations illustrating understanding of place value and

properties (Activity 4)

Explain a variety of students’ addition and subtraction computation methods

(Activity 5)

Essential Understandings of Addition and Subtraction for Teaching Mathematics in PreKindergarten-Grade 2 BIG Idea ONE: Addition and subtraction are used to represent and

solve many different kinds of problems.

Essential Understanding 1a. Addition and subtraction of whole numbers are based on sequential counting with whole numbers.

Essential Understanding 1b. Subtraction has an inverse

relationship with addition.

Essential Understanding 1c. Many different problem situations can be represented by part-part-whole relationships and addition

or subtraction.

Essential Understanding 1d. Part-part-whole relationships can

be expressed by using number sentences like a + b = c or c- b = a. where a and b are the parts and c is the whole.

Essential Understanding 1e. The context of a problem situation

and its interpretation can lead to different representations.

BIG Idea TWO: The mathematical foundations for understanding

computational procedures for addition and subtraction of whole numbers are the properties of addition and place value.

Essential Understanding 2a. The commutative and associative

properties for addition of whole numbers allow computations to

be performed flexibly.

Essential Understanding 2b. Subtraction is not commutative or associative for whole numbers.

Essential Understanding 2c. Place-value concepts provide a

convenient way to compose and decompose numbers to facilitate addition and subtraction computations

Essential Understanding 2d. Properties of addition are central in

justifying the correctness of computational algorithms.

5 and 10 Frames

Open (Empty) Numberlines

The empty number line, or open number line as it is sometimes referred to, was originally proposed as a model for addition and

subtraction by researchers from the Netherlands in the 1980s. A number line with no numbers or markers, essentially the empty

number line is a visual representation for recording and sharing students’ thinking strategies during the process of mental

computation.

Below is an example of different ways a student might use the open numberline to solve 37 + 48

Story Problems (Activity 1) What models can be used to represent and solve addition and subtraction problems?

We will pose a story problem with a progression of number choices. For each number choice, consider one or more of the following as a way to represent

your solution to the problem: counters

5/10 frames Rekenrek

base 10 blocks 100 board open number line

Story Problem 1

Carl has ___ blocks. Rita gives him _____ more blocks. How many

blocks does Carl have now? (2, 3) (7, 4) (6, 7) (12, 10) (20, 23) (44, 38)

Story Problem 2 _____ penguins were standing on the iceberg. ____ jumped into the

water to swim. How many penguins are left on the iceberg? (5, 4) (15, 7) (30, 20) (50, 18) (86, 26) (83, 27)

We will pose a story problem with a progression of number choices. For each number choice, consider one or more of the following as a way to represent

your solution to the problem: counters

5/10 frames Rekenrek base 10 blocks

100 board open number line

Story Problem 3

Tarah has ___ melon balls. Kyrie has ___. How many more melon balls does Tarah have?

(5, 4) (15, 7) (30, 20) (50, 18) (86, 26) (83, 27)

Story Problem 4

Alicia counted ___ squirrels in the park on Tuesday. On Wednesday, she counted ___ squirrels. How many squirrels did she count on

Tuesday and Wednesday? (2, 3) (7, 4) (6, 7) (12, 10) (20, 23) (44, 38)

Examining Different Types of Problems (Activity 2) A. Take a look at the two problems below. Discuss with those at your table the similarities and differences between them.

1. Carl has 5 blocks. Rita gives him 8 more blocks. How many blocks does Carl have now?

2. Alicia counted 5 squirrels in the park on Tuesday. On Wednesday, she

counted 8 squirrels. How many squirrels did she count on Tuesday and

Wednesday?

Similarities and differences:

B. Compare and contrast these three problems. What makes them the same

or different? 1. Connie has 5 marbles. How many marbles does she need to have 13

marbles now?

2. Connie has 5 marbles. Juan gives her 8 more marbles. How many

marbles does she have now?

3. Connie had some marbles. Juan gave her 5 more marbles. Now she

has 13 marbles. How many marbles did Connie start with?

Similarities and differences:

A. 1. Carl has 5 blocks. Rita gives him 8 more blocks. How many blocks

does Carl have now?

2. Alicia counted 5 squirrels in the park on Tuesday. On Wednesday, she counted 8 squirrels. How many squirrels did she count on Tuesday and Wednesday?

Problem 1 is a joining problem – the story explicitly takes two sets and

combines them into one set that is counted. The result of the problem is

unknown. Problem 2 is a part-part-whole problem – in this story the sets

are counted together but not physically combined. The result of the problem

is also the unknown.

Both are addition problems, but the joining problem has “action” in it,

indicating that the sets should be combined. When students are first learning

about addition, they solve problems by directly modeling the story problem.

The action helps them develop the concept of addition as the combining of

sets. Part-part-whole problems do not contain action, which makes them

initially harder for children.

B. 1. Connie has 5 marbles. How many marbles does she need to have 13

marbles now?

2. Connie has 5 marbles. Juan gives her 8 more marbles. How many

marbles does she have now?

3. Connie had some marbles. Juan gave her 5 more marbles. Now she

has 13 marbles. How many marbles did Connie start with? The three problems are all join problems. The main difference between these

three tasks is the location of the unknown within the story problem. The unknown can be the result of the problem (as in #2), the change from the

first addend to the result (as in #1), or it can be the starting number in the problem (as in #3).

1. Connie has 5 marbles. How many marbles does she need to have 13 marbles now?

5 + ☐ = 13 (Join, Change Unknown: JCU)

2. Connie has 5 marbles. Juan gave her 8 more marbles. How many marbles does she have now?

5 + 8 = ☐ (Join, Result Unknown: JRU)

3. Connie had some marbles. Juan gave her 5 more marbles. Now she

has 13 marbles. How many marbles did Connie start with? ☐ + 5 = 13 (Join, Start Unknown: JSU)

C. Take a look at these two problems we solved earlier. Discuss with those at your table the similarities and differences between them.

1. 13 penguins were standing on the iceberg. 5 jumped into the water to

swim. How many penguins are left on the iceberg?

2. Tarah has 13 melon balls. Kyrie has 5. How many more melon balls

does Tarah have?

Similarities and differences:

D. Compare and contrast these three problems. What makes them the same

or different?

1. Connie had some marbles. She gave 5 to Juan. Now she has 8

marbles left. How many marbles did Connie have to begin with?

2. Connie had 13 marbles. She gave 5 to Juan. How many marbles does

she have left?

3. Connie had 13 marbles. She gave some to Juan. Now she has 5

marbles left. How many marbles did she give to Juan?

Similarities and differences:

C. 1. 13 penguins were standing on the iceberg. 5 jumped into the water to

swim. How many penguins are left on the iceberg?

2. Tarah has 13 melon balls. Kyrie has 5. How many more melon balls does Tarah have?

Problem 1 is a separate problem – the action is to remove one set from a larger set. Problem 2 is a compare problem – in this story the sets are

compared by subtraction. Both are subtraction problems, but the separate problem has “action” in it,

indicating that a set should be removed. The notion of set removal is not as evident in compare problems, which makes them initially more difficult.

D.

1. Connie had some marbles. She gave 5 to Juan. Now she has 8 marbles left. How many marbles did Connie have to begin with?

2. Connie had 13 marbles. She gave 5 to Juan. How many marbles does

she have left?

3. Connie had 13 marbles. She gave some to Juan. Now she has 5

marbles left. How many marbles did she give to Juan?

The three problems are all separate problems. The main difference between these three tasks is the location of the unknown within the story problem. The unknown can be the result of the problem (as in #2), the change from

the first addend to the result (as in #3), or it can be the starting number in the problem (as in #1).

1. Connie had some marbles. She gave 5 to Juan. Now she has 8

marbles left. How many marbles did Connie have to begin with?

☐ - 5 =8 (Separate, Start Unknown (SSU)

2. Connie had 13 marbles. She gave 5 to Juan. How many marbles does she have left?

13 – 5 = ☐ (Separate, Result Unknown: SRU)

3. Connie had 13 marbles. She gave some to Juan. Now she has 5 marbles left. How many marbles did she give to Juan?

13 - ☐ = 5 (Separate, Change Unknown: SCU)

Ways to Differentiate Problems

Problem Type

• Joining Stories (also called “Add To”)

• Separating Stories (also called “Take From”)

• Part, Part Whole Stories (also called “Put Together, Take Apart”)

• Comparing Stories

Location of the unknown

• Result-unknown

• Change-unknown

• Start-unknown

Number Choices

Total the Tiles (Activity 3)

Materials Needed: 3 dice, 18 number tiles (2 sets each of 1, 2, 3, 4, 5, 6, 7,

8, 9), recording sheet for each player

Object: To get the LOWEST score after 3 rounds Play:

• The 18 number tiles are placed face up. Players take turns. • The player rolls all three dice, then removes any tile(s) that match(es)

the total roll of the dice. • The player continues rolling until tiles cannot be removed.

• The score for that round is the total of the unused tiles. • The 18 tiles are placed again face up and the next player takes his or

her turn.

• At the end of 3 rounds the player with the lowest score wins.

Score Sheet

Round Player 1

Score

Player 2

Score

Player 3

Score

Player 4

Score

1

2

3

Total

Score Sheet

Round Player 1 Score

Player 2 Score

Player 3 Score

Player 4 Score

1

2

3

Total

What is Fluency?

• Basic Number Computation Fluency is the ability to recall the

combinations of numbers efficiently, accurately, and effortlessly.

• Computational Fluency refers to having efficient and accurate methods

for computing. Students exhibit computational fluency when they

demonstrate flexibility in the computational methods they choose,

understand and can explain these methods, and produce accurate

answers efficiently, A student cannot be fluent without conceptual

understanding and flexible thinking.

Development of Fluency

This diagram shows how fluency is developed. Please note that fluency is

NOT developed by memorizing flash card sums and differences; rather,

students first spend lots of time understanding the operations of addition and

subtraction, using a variety of addition and subtraction strategies, and then

practicing those strategies that will allow them to approach problems flexibly.

Do they need to memorize the addition and subtraction facts? YES by the

end of second grade. However, conceptual understanding and flexible

thinking come first!

Fluency: Simply Fast and Accurate? I Think Not!

By NCTM President Linda M. Gojak NCTM Summing Up, November 1, 2012

As mathematics educators at all levels consider effective implementation and instruction related to state or

Common Core standards, a frequently asked question is, “What does it mean to be fluent in mathematics?” The

answer, more often than not, is, “Fast and accurate.” Building fluency should involve more than speed and accuracy. It must reach beyond procedures and

computation.

Principles and Standards for School Mathematics states, “Computational fluency refers to having efficient and accurate methods for computing.

Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently. The computational

methods that a student uses should be based on mathematical ideas that the student understands well, including the structure of the base-ten number

system, properties of multiplication and division, and number relationships” (p. 152). What a wonderful description of fluency! It reminds us that a student cannot be fluent without conceptual understanding and flexible

thinking.

Focusing on efficiency rather than speed means valuing students’ ability to use strategic thinking to carry out a computation without being hindered by

many unnecessary or confusing steps in the solution process. Accuracy extends beyond just getting the correct answer. It involves considering the

meaning of an operation, recording work carefully, and asking oneself whether the solution is reasonable.

Fluency encompasses more than memorizing facts and procedures. In fact, I believe memorization is one of the least effective ways to reach fluency.

Anyone who has spent time teaching in the elementary grades realizes how many students are unsuccessful at rote memorization and how often they

revert to counting on their fingers. We would agree that third or fourth graders who are counting on their fingers certainly have not reached a level

of fluency, even though they may do it pretty quickly and accurately!

How do we help students progress from the early stages of counting to mathematical fluency? Let me give you a personal example. At the beginning of the school year, I gave a class of third-grade students a sheet with 10

addition facts. Under each fact was the word “explain,” followed by a line. I asked one of the students the sum of the first fact, 8 + 9, and she

immediately began to count on her fingers—certainly not the action of a student who is fluent with addition facts.

Before she reached the sum I asked her, “What do you know that would help

you find the sum of 8 and 9?” She thought for a brief time and replied, “Oh, it’s 17.” When I asked her how she had gotten that without counting, she

looked at me and said, “I just took 1 off the 8 and gave it to the 9. That made it 7 + 10. That’s easy—it’s 17.”

One might argue that child was not fluent. I believe, however, that she

demonstrated fluency and more. She was able to use her understanding of place value, addition, and the associative property to arrive at a correct response. She was efficient, accurate, and flexible in her thinking—all in a

matter of seconds. What made the difference between her fumbling first attempt and her successful second one? It was being provided with the

chance to stop and think about what she already knew and apply that understanding to 8 + 9.

Do we give students the opportunity to think about what they know and understand and use it in ways that make sense to them? Do we model

questions that students should be asking themselves as they strive to reach fluency in mathematics? As the student completed that assignment, she

didn’t need much more prompting. She continued to work on the rest of the facts efficiently and flexibly. She no longer needed to count on her fingers to

complete the assignment.

It is interesting to note that fluency isn’t mentioned in the high school Common Core Standards. The standards for grades K–8 refer to fluency in relation to mastery of basic facts and computational skills. As we think about

fluency, we should realize that it is more than procedural. Are there mathematical topics in which we want students’ thinking to be flexible,

efficient, and accurate beyond computation and procedures? Can a student reach fluency in areas of geometric thinking, algebraic thinking, statistical reasoning, or measurement? What does geometric fluency look like? What

are the characteristics of a student who is fluent in algebra? What areas of fluency in the K–12 curriculum reach beyond procedures and calculations but

are not mentioned in the standards?

Our students enter school with the misconception that the goal in math is to do it fast and get it right. Do we promote that thinking in our teaching

without realizing it? Do we praise students who get the right answer quickly? Do we become impatient with students who need a little more time to think? As we strive for a balance between conceptual understanding and procedural

skill with mathematical practices, we must remember that there is a very strong link between the two. Our planning, our instruction, and our

assessments must build on and value that connection. Fluency entails so much more than being fast and accurate!

True or False? (Activity 4) A. For activity four, we will look at the meaning of the equal signs and the models expressed by number sentences or number equations. Children in first grade were asked to answer this question, “Which one is false?” using

similar equations as the ones listed here. Most of the children simply responded that there was only ONE sentence that was true… the one that

was “written right”… 4 + 5 = 10. Obviously, that equation is one that is false! Discuss children’s misunderstanding. For more information on this misconception, see page 72 in Essential Understanding for Addition and

Subtraction.

8 = 8

4 + 5 = 10

7 + 1 = 8 + 2

9 = 4 + 5

11 – 6 = 10 – 5

The meaning of the equal sign is NOT “the answer is!” as some children believe. Rather it shows that both sides of the equal sign “have the same

value”.

B. For these next tasks, please put away all paper and pencils and only think of the equations here mentally. For each equation decide if the equation is true or false, tell their partner, and then give a reason for why they as they

do. See if you can find multiple ways to justify your thinking.

54 + 10 = 11 +53

120 + 30 = 148 + 10

23 + 76 + 45 = 144

56 = 50 + 4 + 2

89 + 1 + 40 = 30 + 10 + 80

Solution Strategies Chart

Strategies Used In What Situations

Count all (two collections)

Addition situation

Count on (from one addend and eventually

from the higher number)

Addition situation

Count back Count down to Count up from (‘thinking addition” to find the

missing addend)

Subtraction situation

Basic strategies Doubles

Commutativity Adding 10 Making 10

Finding partners for tens Other known facts

Addition or subtraction problems

Derived strategies

Near doubles Add 9 Build to next 10

Fact families (from part, part, whole model)

Addition or subtraction

problems

Extending and applying strategies, i.e., basic

strategies, derived strategies, or intuitive strategies

Compensation Partial sums and differences Adding or subtracting tens and then

ones on an open number line Visualizing a 100s board to add and

subtract

Range of tasks (including

problems with multi-digit numbers),… can be

completed mentally.

(adapted from Clark , 2001, as cited on pg. 78 of Developing Essential Understandings of Addition & Subtraction)

Strategy Progression

Model Representation Progression

(for Addition and Subtraction)

Name of Model and

Grade Level Use

Model Diagram Description Sample Problems

1. Five Frame (Primarily in Kindergarten)

(Model can be filled with counters; the inside of the squares can also be colored or drawn to represent what happened )

Five squares horizontally. They may be filled with two-colored counters and the parts and the whole counted (addition) and/or they can begin with a filled five frame, some are taken away (subtraction, and the resulting number that are left can be found.

I had three candies. I received two more. How many do I have now? I drew one shape from a bag of red and blue shapes five times. I have two blue shapes and three red shapes. I began with five pennies. I spent three pennies. How many do I have now?

2. Ten Frame (Kindergarten and First Grade)

and

Ten squares placed horizontally in one long line AND ten squares placed as two groups of five. Both should be used to show the parts of 10. The long grid could be colored coded so that each five is easily seen. Again, the squares may be filled to show the parts of 10 (addition or subtraction) OR numbers can be added to show addends for 10 OR subtraction can be modeled as students take away some of the counters.

My ten frame is filled with red and yellow counters. There are 8 yellow counters. How many counters are red? Jonathan had 10 sticks of gum. He gave his friends 3 sticks. How many does he have left? Susan rolled a die that had 1, 2, or 3 pips. She rolled a 2 and a 3 and put that many counters on the grid. How many squares did she cover now? How many more does she need to completely cover the squares in the grid?

3 Part-Part-Whole Kindergarten, First Grade, and Second Grade

1st and 2nd Grade

Kindergartners use the part -part--whole model to make number partners for 5, 6, 7, 8, 0, and 10 objects. The place the whole number of counters on the part-part-whole mat, some that are on blue and some that are on red. 1st, and 2nd graders place the total number as a numeral in the top box and either counters in the two parts (early 1st grade) or numerals in the parts.

(Kindergarten) What are the partners for 8… 6….0?

Add to, result unknown Take from, result unknown Put together/take apart total

unknown, Put together/take apart both

addends unknown (1st Grade)

All situations, however the following do not need to be mastered -

o Add to , start unknown,

o Take from, start unknown,

o Compare, bigger unknown

o Compare, smaller unknown

(2nd Grade)

All situations need to be mastered.

4 Lasso

Separating -

This model illustrates joining OR separating stories. Typically, connecting cubes or unifix cubes would be used to model this process

(Kindergarten) Take from and Add to, result

unknown problems (1st Grade)

All take from and Add to problems.

Primarily

Kindergarten

Joining -

5 Bead Counter (first grade)

This model is helpful for developing fluency with sums and differences up to 20. (My students use plastic beads on dowel sticks). The alternating colors are purposeful. It helps them see number in sets of five and ten. Students make one digit numbers on each of the bars. They move the beads to the far right of the bar (The bar is much longer than the picture shows). For example, if they make 7 on the top, they would slide five blue and 2 red to the right. If they make 8 on the bottom, they slide to the right five red and three blue. To find the sum, they “see” 10 as five on the top and the bottom bars and then they can see five extra (2 on the top and 3 on the bottom). This is just another way to use the “make it ten” strategy … the “most important” strategy!

Jennifer was having a birthday party. 8 of her friends were upstairs and 6 were downstairs. How many of her friends came to the party? Make 9 on the top bar. Make 7 on the bottom bar. How many are there altogether? Quick look - (Show 8 on the top; show 8 on the bottom)… How many? How do you know how many without counting?

6 Open Number Line (to 20) First Grade

This model provides practice for adding and subtracting on an open number line. Children can either think about sets of 5 or 10 as they “count on” to find the sum. It also extends the “make a 10” strategy as well as “counting back” to subtract. This simple number line is a precursor to the open number line that is used in 2nd -6th grade with whole and rational numbers.

Adding and subtracting problems can be used (to 20) to ask students to think in 5s and 10s.

7 Place Value Blocks (tens and ones) first grade and (hundreds, tens and ones) second grade

A familiar model that is used throughout the grades. The terms regrouping and trading are taught . This model helps students transition to the standard algorithm.

Any computation of adding, subtracting, and word problems within 100 (first grade) and within 1000 (second grade)

8 100s board (Kindergarten 1st and 2nd Grade)

I love to create a hundreds chart with strips of 10. I cut apart a long number line and form the ten strips. However it is done, this is a very important model. To add, the first addend is circled, and an arrow going down shows adding first by 10s and then an arrow going to the right indicates adding by ones. To subtract, the process is reversed.

For kindergarten, use for counting patterns and counting on activities.

Count on from 26… Count on from 53… Count on from 29…

For first and second grade, use mental math-

56 + 5; 62 - 5 45 + 32; 79 - 31 48 + 43; 80 - 52 one and two digit addition and

subtraction problems within 100.

8 + 9 = ?

“eight plus 2 more makes 10

plus 7 more.” OR “eight plus 2

more makes 10

plus 5 more and plus 2”

8 10 17

9 Open number line (to 100) 1st and 2nd grade

55 + 33

This model extends the previous number line to 20. Students must be able to count on and count back to use it proficiently. First they count on or count back by tens and then they count on or back by ones. In second grade, they may include 100s as well. In that case, you would first add or subtract by 100s, then 10s, then 1s.

Any computation with adding and subtracting word problems within 100 (first grade and 2nd grade) Any computations with adding and subtracting word problems within 1000 (2nd grade) Please note that an open number line is most applicable when counting on or back with numbers that don’t require many steps of counting on or back.

“55, 65, 75, 85, 86, 87, 88”

62 - 33

“62, 52, 42, 32, 31, 30, 29”

NAME _____________________

Computational Methods

Review each student’s computation method. Answer the questions below…

• Explain the method used.

• Is it correct? Why or why not?

• What properties or understanding (or misunderstanding) of place value would

justify (or NOT justify) the method used?

Method One

Method Two Method Three

606 438 – 172 = ?

- 359

353 438

- 100

338

- 70

268

- 2

266

Method Four

NAME _____________________

Method Five

400 – 165 = ?

Method Six

128

+258

300

70

+ 16

386

Method Seven

628 – 219 = ?

NAME _____________________

Method Eight

628 – 219 = ?

Method Nine

99 + 1 100

- 5 - 5

95 -1 94

Method Ten

Plan of action

Something that squares with your beliefs

Something going ‘round in your head

Three things you will try and when: 1.

2.

3.