Tab 5: Scaffolding to Increase Numerical Fluency … 5: Scaffolding to Increase Numerical Fluency: ... Handout 3-Doubles Plus One Ten-Frame 5-66 Speedy Tens 5-67 ... Numerical Fluency

Mathematics TEKS Refinement 2006 – K-5 Tarleton State University

Tab 5: Scaffolding to Increase Numerical Fluency: Table of Contents 5-i

Tab 5: Scaffolding to Increase Numerical Fluency Table of Contents

Master Materials List 5-ii

Strategies for Numerical Fluency 5-1

Spotting Numbers 5-13 Handout 1-Spotting Numbers Work Mat 5-17

Let's Frame It 5-18 Transparency 1/Handout 1 -Let's Frame It Work Mat 5-21

Give Me 10! 5-22 Handout 1-Give Me 10! Cards 5-25

Think Addition 5-36 Handout 1-Think Addition Cards 5-39 Handout 2-Think Addition Work Mat 5-41

Seeing Doubles 5-42 Handout 1-Seeing Doubles Cards 5-45 Handout 2-Seeing Doubles Poster 5-48

Half of Doubles 5-50 Handout 1-Half of Doubles Scenario Cards 5-52

Doubles Plus One 5-55 Handout 1-Doubles Plus One Dominos 5-58 Handout 2-Doubles Plus One Record Sheet 5-65 Handout 3-Doubles Plus One Ten-Frame 5-66

Speedy Tens 5-67 Handout 1-Speedy Tens Record Sheet 5-71

Supporting Materials for Compensation Activity Transparency 1-Double Ten-Frame 5-72 Transparency 2-Number Line 5-73


Tab 5: Scaffolding to Increase Numerical Fluency: Master Materials List 5-ii

Tab 5: Scaffolding to Increase Numerical Fluency Master Materials List

Card stock or construction paper Chart paper Counters Crayons (black and red) Dot plate arrangements of various numbers Double-sided counters Index cards Markers Notebook paper Overhead counters Paper Spinner with numbers 1-4 Spot Can Count by Eric Hill Tape

Spotting Numbers-Handout Let’s Frame It-Handout Give Me 10!-Handout Think Addition-Handouts Seeing Doubles-Handouts Half of Doubles-Handout Doubles Plus One-Handouts Speedy Tens-Handout Compensation-Transparencies Numerical Fluency Definition Poster and Question Transparency

The following materials are not in the notebook. They can be accessed on the MTR website until the K-5 MTR CDs are available. Slides 43-55, Numerical Fluency PowerPoint TEKS Refinements-small and enlarged printable versions Five-frame work mat (alternative handout) Procedures for the Strategy Activity Stations Math Fact Cards for Seeing Double Poster Math Fact Cards for Seeing Double Poster with Hints


Strategies For Numerical Fluency 5-1

Activity: Strategies For Numerical Fluency

TEKS: (K.1) Number, operation, and quantitative reasoning. The student

uses numbers to name quantities. The student is expected to: (A) use one-to-one correspondence and language such as more than,

same number as, or two less than to describe relative sizes of sets of concrete objects;

(B) use sets of concrete objects to represent quantities given in verbal or written form (through 20); and

(C) use numbers to describe how many objects are in a set (through 20) using verbal and symbolic descriptions.

(K.2) Number, operation, and quantitative reasoning. The student

describes order of events or objects. The student is expected to: (A) use language such as before or after to describe relative position in

a sequence of events or objects; and (B) name the ordinal positions in a sequence such as first, second, third,

etc. (K.4) Number, operation, and quantitative reasoning. The student

models addition (joining) and subtraction (separating). The student is expected to model and create addition and subtraction

problems in real situations with concrete objects. (1.1) Number, operation, and quantitative reasoning. The student uses

whole numbers to describe and compare quantities. The student is expected to: (A) compare and order whole numbers up to 99 (less than, greater than,

or equal to) using sets of concrete objects and pictorial models; (B) create sets of tens and ones using concrete objects to describe,

compare, and order whole numbers; (C) identify individual coins by name and value and describe

relationships among them; and (D) read and write numbers to 99 to describe sets of concrete objects. (1.3) Number, operation, and quantitative reasoning. The student

recognizes and solves problems in addition and subtraction situations. The student is expected to: (A) model and create addition and subtraction problem situations with

concrete objects and write corresponding number sentences; and (B) use concrete and pictorial models to apply basic addition and

subtraction facts (up to 9 + 9 = 18 and 18 – 9 = 9).



(1.5) Patterns, relationships, and algebraic thinking. The student recognizes patterns in numbers and operations.

The student is expected to: (D) use patterns to develop strategies to solve basic addition and basic

subtraction problems; and (E) identify patterns in related addition and subtraction sentences (fact

families for sums to 18) such as 2 + 3 = 5, 3 + 2 = 5, 5 – 2 = 3, and 5 – 3 = 2.

(2.1) Number, operation, and quantitative reasoning. The student

understands how place value is used to represent whole numbers. The student is expected to: (A) use concrete models of hundreds, tens, and ones to represent a

given whole number (up to 999) in various ways; (B) use place value to read, write, and describe the value of whole

numbers to 999; and (C) use place value to compare and order whole numbers to 999 and

record the comparisons using numbers and symbols (<, =, >). (2.3) Number, operation, and quantitative reasoning. The student adds

and subtracts whole numbers to solve problems. The student is expected to: (A) recall and apply basic addition and subtraction facts (to 18); (B) model addition and subtraction of two-digit numbers with objects,

pictures, words, and numbers; (C) select addition or subtraction to solve problems using two-digit

numbers, whether or not regrouping is necessary; (2.5) Patterns, relationships, and algebraic thinking. The student uses

patterns in numbers and operations. The student is expected to: (C) use patterns and relationships to develop strategies to remember

basic addition and subtraction facts. Determine patterns in related addition and subtraction number sentences (including fact families) such as 8 + 9 = 17, 9 + 8 = 17, 17 – 8 = 9, and 17 – 9 = 8.

(3.3) Number, operation, and quantitative reasoning. The student adds

and subtracts to solve meaningful problems involving whole numbers. The student is expected to: (A) model addition and subtraction using pictures, words, and numbers;

and (B) select addition or subtraction and use the operation to solve

problems involving whole numbers through 999. (4.3) Number, operation, and quantitative reasoning. The student adds

and subtracts to solve meaningful problems involving whole numbers and decimals.



The student is expected to: (A) use addition and subtraction to solve problems involving whole

numbers; and (5.3) Number, operation, and quantitative reasoning. The student

adds, subtracts, multiplies, and divides to solve meaningful problems. The student is expected to: (A) use addition and subtraction to solve problems involving whole

numbers and decimals; Overview: This section on Strategies for Numerical Fluency was developed to be

presented in a 1½ hour professional development session. The purpose of this professional development is help teachers understand how to guide students in developing mental images of addition and subtraction facts so that students will be able to compose and decompose numbers fluently; as well as, accurately. This section, in its entirety, needs to be presented to the K-2 teachers. Educators in grades 3-5 need to teach children through a diagnostic approach. Since these strategies will be new to many classrooms and the students in grades 3-5 may not have received instruction in prior grades, teachers in grades 3-5 should experience these strategies as well. The compensation piece of this section is vital to developing multi-step mental math strategies for children. This section is designed in a purposeful, sequential order with the intent that teachers can take the activities found in the appendix back to their classrooms. The Spotting Numbers and Let’s Frame It activities are to be used as whole group introductory lessons for not only introducing the strategies but also laying the foundation for using the ten-frame. From these strategies the compensation strategy is built, which is used with two-digit addition and subtraction. Compensation is used to find compatible numbers which help students compose and decompose numbers to solve more difficult problems mentally. It is important to note that these strategies will be meaningful once the students have a working knowledge of the Foundations of Numerical Fluency. Utilizing subitizing, fact families can be taught. Hands-on experiences will be given so that teachers can explore possible ways of implementing this procedure into the classroom. In identifying the TEKS for all sections of this professional development, the underlying processes and mathematical tools were not listed, but it is evident with all the problem solving that occurs that they are being addressed. It is the writers’ intention that the underlying processes and



mathematical tools are the framework of how you investigate the mathematical concepts for each grade level. The intention was to focus specifically on identifying the Number, Operation, and Quantitative Reasoning as well as the Patterns, Relationships, and Algebraic Thinking TEKS that directly affects numerical fluency.

Materials: Strategies for Numerical Fluency

Slides 43-55, Numerical Fluency PowerPoint TEKS Refinements-small and enlarged printable versions Numerical Fluency Definition Poster and Question Transparency

Spotting Numbers Handout 1-Spotting Numbers Work Mat, 1 per participant (page 5-17) Five-frame work mat Counters or black dots, 10 per participant Spot Can Count by Eric Hill Dot plate arrangements of various numbers

Let’s Frame It Transparency 1/Handout 1 -Let’s Frame It Work Mat, 1 handout per

participant and 1 transparency for trainer (page 5-21) Counters, unifix cubes, chips or beans, 10 per participant

Give Me 10! Handout 1-Give Me 10! Cards, 1 deck per group (pages 5-25 – 5-35) Paper

Think Addition Spinner with numbers 1-4 Handout 1-Think Addition Cards (pages 5-39 – 5-40) Handout 2-Think Addition Work Mat, 1 per participant (page 5-41) Counters, 10 per participant Notebook paper

Seeing Doubles Handout 1-Seeing Doubles Cards (pages 5-45 – 5-47) Handout 2-Seeing Doubles Poster (pages 5-48 – 5-49)

Half of Doubles Handout 1-Half of Doubles Scenario Cards, 1 set per group (pages 5-52 – 5-54)

Doubles Plus One Handout 1-Doubles Plus One Dominos Twice the Fun, 1 set per group

(pages 5-58 – 5-64) Handout 2-Doubles Plus One Record Sheet, 1 per participant (5-65) 1 red crayon and 1 black crayon, 1 set per participant Handout 3-Doubles Plus One Double Ten-Frame, 1 per participant (page 5-66)



Materials (cont.):

Speedy Tens Card stock or construction paper for Speedy Tens flap cards (1 set of 9 flap cards per group) Tape Handout 1-Speedy Tens Record Sheet (page 5-71)

Compensation Activity Transparency 1-Double Ten-Frame (page 5-72) Transparency 2-Number Line (page 5-73) Overhead counters Chart paper Markers

Fact Families (page 5-11) Paper Double-sided counters, 10 per participant Index cards

Grouping: Small and large group instruction depending on activity Time: 1½ hours Lesson: Place all manipulatives and materials on the tables before this portion of

the professional development begins.

Procedures Notes Slide 43 Strategies for Numerical FluencyStrategies for Numerical Fluency

Before participants arrive for Numerical Fluency Day 2, place the definition of numerical fluency and index cards on the table and the following question on the overhead: How could you use the concepts from yesterday to build a stronger numerical fluency foundation for students at any grade level? Have participants answer the question on the cards.

Slide 44 Goals & Purposes

• Increase teacher knowledge regarding the refinements of the TEKS relating to numerical fluency.

• Increase teacher knowledge of composing and decomposing numbers.

• Increase teacher knowledge of strategies to develop numerical fluency.

• Develop an understanding of the use of metacognition in problem solving.

When displaying the third bullet, draw attention to TEKS covered in this section, pages 5-1 – 5-3. Also, summarize the high points from the Overview, pages 5-3 – 5-4. Emphasizing the importance that the strategies are presented in a sequential order. If you are providing training for trainers, tell the participants not to turn the page. You want them to work the ensuing problem without seeing possible answers



Procedures Notes Slide

45 Solve the Following ProblemSolve the following problem mentally.

A group of teachers at a local school are involved in a walking contest. They are asked to wear a pedometer for eight weeks. The first week Janice walked 65,787 steps. The next three weeks she walked a total of 214,241 steps. On average how many steps did Janice walk per day during the four week period?

Write down the thought processes you used to solve the problem.

Below are two different strategies that could be used to find 65,787 + 214,241:

a. (65,787 + 213) + (214,241-213) = 66,000 + 214, 028 = 280,028 280,028 = 10,001 28

b. Working the problem as an estimation,

an example might be 66,000 + 214,000 = 280,000 280,000 = 10,000 28

Participants may have other strategies for solving this problem.

Slide 46 How Did You Solve the Problem?

• Share your strategies with your neighbor.• Share your strategies with the whole

group.

Walk around while participants are discussing their strategies, and choose several that are different. Once all of strategies you have chosen have been presented, ask the group to share other strategies.

Slide47 TEKS

• Each group will be assigned a grade level.• Identify the TEKS in your grade level (K-5)

that students must master in order to have success in solving this 5th grade problem.

• Are there any refinements that need to be identified? Add refined TEKS needed to teach this concept to the TEKS Refinement Wall.

Go back through strategies and connect TEKS to them. Some of the TEKS that should be identified at different grade levels are:

Kindergarten: K.1 (A),(B),(C); K.4

First grade: 1.1 (A),(B),(D); 1.3 (A),(B); 1.5 (D),(E)

Second grade: 2.1 (A),(B),(C); 2.3 (A),(B),(C); 2.5 (C)

Third grade: 3.1 (A); 3.3 (A); 3.5 (A)

Fourth Grade: 4.1 (A); 4.3 (A)

Fifth Grade: 5.1 (A), (B); 5.3 (A); 5.4

Have participants post the refined TEKS on the wall under their particular grade level. (See link in materials list for TEKS Refinements.)



Procedures Notes Let’s take a step back and look at some foundations in the early grades that lead to success in solving this problem.

Slide 48 Models

Concrete (such as) Counters

Double-sided countersThematic counters

Rods Base Ten Blocks

Semi-Concrete/Pictorial (such as) Ten-Frame Templates Drawing Pictures Number Lines

In the TEKS, it stresses concrete models. Manipulatives are the concrete models. This is a suggested list of manipulatives, but it is by no means comprehensive or are we saying that you have to purchase these tools. Math TEKS Connections (MTC) will focus on the use of tools for math instruction.

Slide49 Bridging

•Begin with stated problems that require children to think.

•Have children use manipulatives to develop a visualization of the problem.

•Have students write about their work.

•As the teacher, lead the students to abstraction.

Many times, teachers have been heard saying that the use of manipulatives does not work. These teachers see problems when students do not give up the use of manipulatives and do not move to the abstract. One reason for this is the lack of bridging from the concrete to the abstract. Students must be taught how to make this move. Teachers MUST model the conceptual understanding through manipulatives and at the same time write the procedures abstractly. For students to build numerical fluency, they need to discuss how they solve problems and record their steps. To illustrate the idea of bridging and its importance, draw a picture of concrete land and a picture of abstract land. Separate the two lands with a vast river. Ask participants: “What is the quickest way to cross a river?” They naturally will respond “by a bridge.” Continue the illustration by stressing that if teachers do not provide that bridge, students may just drown. This illustration was used in previous Texas Education Association work and hopefully will remind teachers who have participated in that training about the importance of manipulatives. For those not having gone through the training, this may be a new way of thinking about the use of manipulatives.



Procedures Notes Teachers can use manipulatives much like a boat. They may not give students the oars (all the pieces) to control the boat. Students may float along and be way down stream before they can put all the pieces together (illustrating several grades), or their boat may have so many holes that they sink. Teachers need to make sure students have all the pieces to connect the bridge to both lands and that the foundation of that bridge is strong.

Slide 50 Building on Unitizing

• Spotting Numbers• Let’s Frame It

Use the activities, Spotting Numbers (pages 5-13 – 5-17) and then Let’s Frame It (pages 5-18 – 5-21) to build on unitizing.

Slide51

Strategies for Addition and Subtraction of Whole Numbers

A. Give Me 10!B. Think AdditionC. Seeing DoublesD. Half of Doubles E. Doubles plus oneF. Speedy Tens

Prepare the materials for each of the following activities and put them in large plastic bags labeled alphabetically.

A. Give Me 10! (pages 5-22 – 5-35) B. Think Addition (pages 5-36 – 5-41) C. Seeing Doubles (pages 5-42 – 5-49) D. Half of Doubles (pages 5-50 – 5-54) E. Doubles Plus One (pages 5-55 – 5-66) F. Speedy Tens (pages 5-67 – 5-71)

Give participants 5 minutes to look at each of the baggies. The station activities are designed to give participants an idea of what each strategy may look like in their classrooms. The activities are taken from a 2-3 week long strategy investigation on how to introduce and teach each strategy in depth. These six activities are at various levels from concrete to symbolic. For further in-depth knowledge of each strategy, refer to the lessons for students in the appendix. There is also an example of one strategy investigation lesson in the appendix.



Procedures Notes Slide 52 Strategies of Compensation

• 17 +12 = 29

• 29 – 12 = 17

When you have finished the Speedy Tens Activity, go back to slides 50 and 51 to review the strategies that have been addressed before looking at slide 52. Say: You’ve learned about seven strategies for building numerical fluency. Some of these activities may be review, but we are building on prior knowledge. If participants are unfamiliar with these strategies, the introduction of the ten-frame will allow participants to proceed having the same background knowledge.

1. Make 5 (Spotting Numbers) 2. Make 10 (Let’s Frame It and Give Me 10!) 3. Think Addition 4. Seeing Doubles 5. Half of Doubles 6. Doubles Plus One 7. Speedy Tens

In these problems, the numbers used worked well with those strategies. But what if the numbers don’t work so well together? Another strategy for building numerical fluency is compensation. Ask: How would you define compensation? Give participants a chance to discuss their understanding of the word. (According to Yahoo online dictionary, compensation means to offset or counterbalance.) Based on that definition, let’s look at the use of compensation with addition. On slide 52, but only the addition problem: 17 +12 = 29. Have a participant come to the front and record answer on chart paper. Ask: How can we change this problem and still get the answer of 29? Sample responses include 18 + 11 = 29, 19 + 10 = 29. After about 5 responses have been shared and recorded on the chart paper, ask the



Procedures Notes participants to identify what patterns or relationships they see? Hopefully they will see that when you increase one addend, the other addend must be decreased. Ask: If I wanted to change 17 +12 = to work with compatible numbers, which of these (point to the equations they created) would I choose? You want the participants to understand that 19 + 10 = 29 or 20 + 9 = 29 would be the equations that take one of the addends to the nearest ten. That is the beauty of compensation. It allows you to take either addend to the nearest 10 so that mental math can easily be done. Demonstrate this using several ten-frame transparencies. Make several copies of Transparency 1-Double Ten-Frame (page 5-72) and cut out the number of ten-frames that you need. An example is found below. Set up one full ten-frame and then another with only 7 counters (representing 17). Below that grouping set up another full ten-frame and one with only two counters (representing 12). Model the compensation strategy by moving the two counters from the second addend to the ten- frame that has the 7 counters, thus making 19 + 10 = 29.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

To extend the compensation strategy to subtraction, click on the mouse and 29 – 12 = 17 will appear on Slide 52.



Procedures Notes Using the same process as before list about 5 or 6 possible solutions where 17 is always the answer. 28 – 11 = 17, 27 – 10 = 17, 30 – 13 = 17, 31 – 14 = 17, etc. When you have about 5 answers, ask the participants to identify what patterns or relationships they see. You want participants to identify that when using compensation on a subtraction problem, you either subtract or add a constant to the minuend and subtrahend. Ask: Why do you think this happens? Demonstrate on Transparency 2-Number Line (page 5-73) that subtraction is the distance between two numbers. No matter how the minuend and subtrahend are changed, the distance between the two numbers must stay the same. So, when you add to the minuend and make it greater (subtract from it to make it smaller), you have to add to the subtrahend so you are taking more away (taking less away) to keep the distance the same. This example provides a great opportunity for a later connection to measurement.

Slide 53 Fact Families

Using subitizing to teach fact families.

Have participants use no more than two colors to compose the number 9. Then go through these steps with participants: Write down all the possible addition problems Look for patterns. Go back and take each one individually and look at relationships (commutative property). Ask participants the following questions:

How can you relate the equations to subtraction? What do all these number sentences have in common? Can this process help children develop



Procedures Notes numerical fluency? How could you use the dot cards to teach multiplication facts? For example, create dot cards for all possible multiplicative (repeated addition/ array method) combinations of 12 and look at those relationships. Think back to subitizing slide 33.

How can these arrangements be used to teach multiplication?

Slide 54

BREAK

Take a 10-minute break. As participants are breaking, ask them to think about how they could teach these strategies in their classrooms. Have them jot down on an index card, three things that they learned from this section that will help scaffold children to solve addition and subtraction problems mentally. Discuss how these strategies set the foundation of multiplication and division. Discuss how these strategies develop numerical fluency.

Slide 55

Once this slide appears, participants will have 12 seconds before you begin the next session.


Spotting Numbers 5-13

Activity: Spotting Numbers

TEKS: (K.1) Number, operation, and quantitative reasoning. The student uses

numbers to name quantities. The student is expected to: (A) Use one-to-one correspondence and language such as more than,

same number as, or two less than to describe relative sizes of sets of concrete objects,

(B) Use sets of concrete objects to represent quantities given in verbal or written form (through 20),

(C) Use numbers to describe how many objects are in a set (through 20) using verbal and symbolic descriptions.

(K.4) Number, operation, and quantitative reasoning. The student

models addition (joining) and subtraction (separating). The student is expected to model and create addition and subtraction

problems in real situations with concrete objects. (1.3) Number, operation, and quantitative reasoning. The student

recognizes and solves problems in addition and subtraction situations. The student is expected to: (A) Model and create addition and subtraction problem situations with

concrete objects and write corresponding number sentences, (B) Use concrete and pictorial models to apply basic addition and

subtraction facts (up to 9+9=18 and 18-9=9). (1.5) Patterns, relationships, and algebraic thinking. The student

recognizes patterns in numbers and operations. The student is expected to: (D) Use patterns to develop strategies to solve basic addition and

basic subtraction problems. Overview: Anchoring numbers to five will help students discover patterns and

relationships from a given number to other numbers. These relationships are especially important for understanding various combinations of numbers. Through the practice of discovering these relationships, students will be able to develop mental computation skills with larger numbers. The five-frame is a common model for this type of relationship. Its 1X5 array allows students to place counters or dots to represent a given number. Then students are encouraged to explain how they see their number. After several “free arrangements,” the students will be directed to specifically place the counters or dots starting from left to right. This will set the pattern for using the five-frame.



Materials: Handout 1-Spotting Numbers work mat, 1 per participant (page 5-17) Five-frame work mat (alternative handout) Counters or black dots, 10 per participant Spot Can Count by Eric Hill Dot plate arrangements of various numbers

Grouping: Whole group Time: 15 minutes Lesson:

Procedures Notes 1. Each participant will need Handout 1-

Spotting Numbers work mat (page 5-17). Ask: How many spaces are on the work mat? Explain to participants that because there are five spaces on the work mat, it is called a five-frame. The five-frame will be used to help show numbers in many different ways.

See the materials list for a link to a generic five-frame work mat.

2. Give each participant a set of ten black counters. Explain to participants that only one counter is permitted in each space.

3. To model this activity, hold up a number three dot plate arrangement. Ask participants to show the number on their five-frame. Ask: What can you tell us about the number three from looking at your mat? Some responses may be: It has spaces at the end or it has two dots then spaces and then one dot. After hearing from several participants, try other numbers from 0-5. Remember to always process with participants after asking them to show their number.

Participants may place their counters on the five-frame in any manner. Lead them to observe how their frame differs from other participants. There are no wrong answers. Focus attention on how many more counters are needed to make 5.

4. Next, try numbers that are between 5 and 10. The rule of one counter in each space

For 1st grade teachers, suggest that they may want to ask



Procedures Notes still applies. This time, however, demonstrate how the five-frame will be used like we read. Fill in the frame from left to right leaving no spaces. Hold up a number 8 dot plate arrangement. Ask participants to show the number on their five-frame. Participants will fill in from left to right and then place 3 counters below the five-frame. Ask: Why did you have to place some counters below the five-frame? Responses may include: All the counters can’t fit, or there was no more room. Focus attention on these larger numbers as 5 and some more: “8 is 5 and 3 more”. Try other numbers between 5 and 10. Remember to process after each number.

students to write the corresponding number sentence to match the five-frame.

5. Ask participants to clear their five-frame. Introduce the story Spot Can Count by Eric Hill. Give a brief synopsis that Spot is learning to count, and he will count several different animals throughout the book. As the story is read, participants will be asked to show on their five-frame how many animals Spot counted on the page. Remind participants how to use the five-frame (from left to right and only one counter or dot in each space).

6. Read Spot Can Count, stopping after each page and asking participants to show the number of animals Spot counted. Also ask one or more of these questions: What can you tell me about the number ___? How many more counters or dots do you need to fill up our five-frame? How many more counters or dots is the number ___ than five?

When teachers are doing this activity with students, they may request students to record their answers on dry erase boards. This will help bridge the learning from concrete to symbolic.

7. After the story, ask participants: “What can Note: When students are able to



Procedures Notes you tell me about using the five-frame?” Responses should include you start from right to left, use only one counter in each space and there are five spaces on the five-frame.

quickly show a number and explain the number accurately referring to how many counters are needed to make five or how many more counters than five do we have, then students are ready for the ten-frame.

Resources: Hill, Eric (1999). Spot Can Count. New York: Penguin Group, Inc.

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Let’s Frame It 5-18

Activity: Let’s Frame It

TEKS: (1.3) Number, operation, and quantitative reasoning. The student



subtraction facts (up to 9 + 9 = 18 and 18 - 9 = 9). (1.5) Patterns, relationships, and algebraic thinking. The student

recognizes patterns in number and operations. The student is expected to: (D) Use patterns to develop strategies to solve basic addition and

basic subtraction problems. (2.3) Number, operation, and quantitative reasoning. The student

adds and subtracts whole numbers to solve problems. The student is expected to: (A) Recall and apply basic addition and subtraction facts (to 18). (2.5) Patterns, relationships, and algebraic thinking. The student uses

patterns in numbers and operations. The student is expected to: (C) Use patterns and relationships to develop strategies to remember

basic addition and subtraction facts. Determine patterns in related addition and subtraction number sentences (including fact families) such as 8+9=17, 9+8=17, 17-8=9, 17-9=8.

Overview: The ten-frame is a strategy to build a mental understanding of each

number. Prior to using a ten-frame, students must model the numbers and develop an understanding of the math facts that represent 0, 1, 2, 3, 4, & 5 using the five-frame.

Materials: Transparency 1/Handout 1 -Let’s Frame It work mat, 1 handout per

participant and 1 transparency for trainer (page 5-21) Counters, Unifix cubes, chips, or beans, 10 per participant

Grouping: Whole group Time: Only allow 5-7 minutes for each strategy.



Lesson: Procedures Notes 1. Begin by explaining how a ten-

frame board is set-up to be used. Use the ten-frame transparency and provide each participant a ten-frame handout for their own work. (Transparency 1/Handout 1 page 5-21) Stress the following four ideas. A. Each box must be filled in sequential order moving from left to right, beginning with the top row. Example: for the number 7, fill all 5 boxes of the top row and then the first two boxes in the second row. B. To model a math sentence, use a second ten-frame to show the second addend. For example, 7 + 4 would be modeled by showing 7 counters on the first ten-frame as mentioned above and 4 counters in a different color in the first four boxes on the second ten-frame. C. Instruct the participants to move the 3 last counters from the second ten-frame into the three empty boxes of the first ten-frame. D. Now the answer can be read easily—the first ten-frame is full which represents 10, and then count on to add the extra 1 left in the second ten-frame for a total of 11.

The ten-frame is used to form a bridge between numbers. Example: the number 7 can be tied to 5 and 2 more or 3 away from 10. Just as the dot plates help connect a pattern to a numeral, the use of ten-frames can be one of the most vital tools that a student uses to begin forming a mental picture in his/her mind of addition and/or subtraction math facts. Note: Important step in building to a mental understanding of each number: The children should know all the facts that represent each number before moving into math fact strategies so a solid foundation of each number is built. This step will also build to a strong mental picture of the number. Example: Number 6 means: 0 + 6, 1 + 5, 2 + 4, 3 + 3, 4 + 2, 5 + 1, and 6 + 0. This can be done by having the children make ten-frame books that represent each number.

2. Have participants practice making numbers using only

Be sure to walk around to observe that the participants are placing the objects on the



Procedures Notes one ten-frame to model numbers quickly. Call out random numbers for participants to demonstrate understanding of how to use the ten-frame.

boards in a left to right and top to bottom order. Remind them that it is just like teaching a child to read. Tell teachers, “After a few days, a fun new way to practice making numbers is to tell your students not to clear their boards between calling out the next number. See if your students can quickly figure out if they need to add more counters or to remove a few. Some of your students’ number development may require them to start counting from the number one each time while others will catch on quickly to the adding on or the removing of a few markers to make their number.”

3. Tell the participants that ten-frames are a basic tool for helping students learn other strategies for addition and subtraction. The next activities will address those strategies.

Resources: Burk, D., Snyder, A. & Symonds, P. (1999). Box it or bag it

mathematics. Salem, OR. Hope, J., Leutinger, L., Reys, B., & Reys, R. (1988). Mental math in

the primary grades. Parsippany, MJ: Pearson Learning Group. Van de Walle, J.A. (2007). Elementary and middle school

mathematics: Teaching developmentally (6th ed.) Boston, MA: Allyn and Bacon.

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Give Me 10! 5-22

Activity: Give Me 10!





recognizes patterns in number and operations. The student is expected to: (B) Find patterns in numbers, including odd and even, (D) Use patterns to develop strategies to solve basic addition and

basic subtraction problems. (1.12) Underlying processes and mathematical tools. The student

communicates about Grade 1 mathematics using informal language. The student is expected to: (A) Explain and record observations using objects, words, pictures,

numbers, and technology. (2.3) Number, operation, and quantitative reasoning. The student

adds and subtracts whole numbers to solve problems. The student is expected to: (A) Recall and apply basic addition and subtraction facts (to 18), (B) Model addition and subtraction of two digit numbers with objects,

pictures, words, and numbers. (2.5) Patterns, relationships, and algebraic thinking. The student uses



(2.13) Underlying processes and mathematical tools. The student

communicates about Grade 2 mathematics using informal language. The student is expected to: (A) Explain and record observations using objects, words, pictures,

numbers and technology.


Give Me 10! 5-23

Overview: Note to Trainer: Please clarify for participants - After students have had numerous opportunities of exploring the ten-frame and building number sentences representational of each number, students will engage in the following activity. During this activity, students will develop fluency of recognizing various combinations to make a set of ten.

Materials: Handout 1-Give Me 10! Cards, one deck per group (pages 5-25 – 5-35)

Paper Grouping: Small group Time: Only allow 5-7 minutes for each strategy. Lesson:

Procedures Notes 1. Shuffle Give Me 10! Cards and distribute so

that every participant receives seven cards. The remaining cards are placed in the middle of the group face down.

Handout 1-Give Me 10! Cards (pages 5-25 – 5-35) should be used to make a deck of cards for each group. Each deck is composed of four cards of each ten-frame.

2. Each participant identifies all pairs that equal to 10 within their hand and lays them down on the table. (Example: 6 and 4, 7 and 3)

3. After everyone has laid down their “make 10” pairs, the group must agree that all pairs are correct. Once everyone is in agreement, continue the game.

4. The participant to the left of the dealer begins the game by asking another participant for a card that would make a pair equal to 10. (Example: participant is holding an 8 and would need to ask for a 2)

5. If the participant has the card that was asked for, it is given to the participant asking for it.

6. If the participant does not have the card that was asked for, the participant replies with “Go Fish”.

7. To “Go Fish,” the participant who asked for a


Give Me 10! 5-24

Procedures Notes card draws a card from the deck.

8. If the card that was drawn completes a pair to make 10, then the participant lays the pair down.

9. Then, the next participant proceeds following the same procedures.

10. Continue the game until everyone has had a chance to ask for a card.

11. Next, each participant records on a piece of paper the number sentences they were able to build throughout the game. Participants compare lists with someone at their table by identifying same number sentences and number sentences that show the commutative property. (Example: 6 and 4 and the partner had 4 and 6)

Resources: Nugent, G. (1995). Hands-On Math: Manipulative Activities for the 2-3

Classroom. Cypress, CA: Creative Teaching Press, Inc. Van de Walle, J.A. (2004). Elementary and middle school

mathematics: Teaching developmentally. (5th ed.) Boston, MA: Allyn and Bacon.


Handout 1-1 Give Me 10! 5-25






















Think Addition 5-36

Activity: Think Addition





recognizes patterns in numbers and operations. The student is expected to: (D) Use patterns to develop strategies to solve basic addition and basic

subtraction problems. (1.11) Underlying processes and mathematical tools. The student applies

Grade 1 mathematics to solve problems connected to everyday experiences and activities in and outside of school.

The student is expected to: (B) Solve problems with guidance that incorporates the processes of

understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness.

(2.3) Number, operation, and quantitative reasoning. The student adds

and subtracts whole numbers to solve problems. The student is expected to: (A) Recall and apply basic addition and subtraction facts (to18), (B) Model addition and subtraction of two digit numbers with objects,

pictures, words, and numbers. (2.5) Patterns, relationships, and algebraic thinking. The student uses



(2.12) Underlying processes and mathematical tools. The student adds

applies Grade 2 mathematics to solve problems connected to everyday experiences and activities in and outside of school.

The student is expected to: (B) Solve problems with guidance that incorporates the processes of

understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness.


Think Addition 5-37

Overview: A prerequisite for this strategy is for students to know the addition math facts and be able to use the ten-frame fluently. According to John Van de Walle (2004, p.165), “subtraction facts prove to be more difficult than addition.” Encouraging students to think, “What goes with this part to make the total?” is the foundation for the strategy Think Addition. This strategy requires students to use known addition facts to create the unknown quantity or part.

Materials: Spinner with numbers 1-4

Handout 1-Think Addition Cards (pages 5-39 – 5-40) Handout 2-Think Addition Work Mat, 1 per participant (page 5-41) Counters, 10 per participant Notebook paper

Grouping: Small group Time: Only allow 5-7 minutes for each strategy. Lesson:

Procedures Notes 1. Place the number cards face down and the

spinner in the middle of the group.

Make the number cards from Handout 1-Think Addition Cards pages 5-39 – 5-40).

2. Begin by spinning the spinner to get a number between 1 and 4, and then all participants build the number on their ten-frames.

Ten-frame master is Handout 2-Think Addition Work Mat (page 5-41).

3. Next, turn over one number card and begin “thinking addition”.

4. The participant who flipped the number card will ask the group, “_______ and what makes ________?” (# on spinner) (# on card) (Example: 3 and what makes 10 ? (# on spinner) (# on card)

5. Record the subtraction fact onto a piece of notebook paper.

6. Continue playing until everyone has a turn.


Think Addition 5-38

Procedures Notes 7. Have participants reflect on the following two

questions with their group. a. How does the Think Addition strategy

lay the groundwork for Fact Families?

b. What are some problems students may have using this strategy?

Resources: Van de Walle, J.A. (2004). Elementary and middle school

mathematics: Teaching developmentally (5th ed.) Boston, MA: Allyn and Bacon. Van de Walle, J.A. (2007). Elementary and middle school



Handout 1-1 Think Addition 5-39

Think Addition Cards

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Handout 1-2 Think Addition 5-40

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Seeing Doubles 5-42

Activity: Seeing Doubles





recognizes patterns in numbers and operations. The student is expected to: (B) Find patterns in numbers, including odd and even, (D) Use patterns to develop strategies to solve basic addition and basic

subtraction problems. (1.11) Underlying processes and mathematical tools. The student


The student is expected to: (A) Identify mathematics in everyday situations. (2.3) Number, operation, and quantitative reasoning. The student

adds and subtracts whole numbers to solve problems. The student is expected to: (A) Recall and apply basic addition and subtraction facts (to 18), (B) Model addition and subtraction of two digit numbers with objects,

pictures, words, and numbers. (2.5) Patterns, relationships, and algebraic thinking. The student

uses patterns in numbers and operations. The student is expected to: (C) Use patterns and relationships to develop strategies to remember


(2.12) Underlying processes and mathematical tools. The student


The student is expected to: (A) Identify the mathematics in everyday situations.


Seeing Doubles 5-43

Overview: After several days of practicing building double facts on ten-frames using problem situations, a mental picture of doubles math facts should begin to form. At this point, image pieces are used to help make a comparison between the double math fact and what it represents in meaningful situations. To reinforce this meaningful representation, participants will engage in a matching activity where the image is paired with the corresponding number sentence.

Materials: Handout 1-Seeing Doubles Cards (pages 5-45 – 5-47)

Handout 2-Seeing Doubles Poster (pages 5-48 – 5-49) Grouping: Small group Time: Only allow 5-7 minutes for each strategy. Lesson:

Procedures Notes 1. Place the set of Seeing Doubles image cards

in the center of the group where all participants can see them and be able to inspect them up close. Show only the pair of giggly eyes from the Seeing Doubles poster (Handout 2, pages 5-48 – 5-49). Ask each group to discuss why the pair of giggly eyes represents 1+1.

Print Handout1-Seeing Doubles Cards (pages 5-45 – 5-47) and cut them apart. Each group will need one set of cards.

2. Next, each participant selects one image card, examines the image, and identifies its doubling attribute.

3. Each participant takes a turn describing his/her image card based on the doubling attribute. Then the participant uses Velcro to put the card beside the corresponding double math fact.

It is important to note that some participants may place the lizard with the 8+8 math fact. The lizard has 2 types of doubling attributes: 4 legs and 4 toes on each foot. There is value in allowing participants to discover the correct corresponding math fact. Through this discovery, participants will be justifying their reasoning, which is what students ultimately need to achieve.

4. Once all image cards have been identified, document all double images next to their


Seeing Doubles 5-44

Procedures Notes corresponding double math fact.

5. Participants will document by drawing the given image for each math fact on the concentration cards. (Example: 1+1= 1 pair of eyes) Remind participants that Handout 2 is a completed doubles poster.

If teachers use this activity in their classrooms, suggest that they use real objects instead of image cards.

Resources: Rightsel, P. S., & Thorton, C. A. (1985). 72 addition facts can be

mastered by mid-grade 1. Arithmetic Teacher, 33(3), 8-10. Van de Walle, J.A. (2004). Elementary and middle school



Handout 1-1 Seeing Doubles 5-45





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Half of Doubles 5-50

Activity: Half of Doubles



concrete objects and write corresponding number sentences. (B) Use concrete and pictorial models to apply basic addition and

subtraction facts (up to 9+9=18 and 18-9=9) (1.5) Patterns, relationships, and algebraic thinking. The student

recognizes patterns in number and operations The student is expected to: (B) Find patterns in numbers including odd and even (D) Use patterns to develop strategies to solve basic addition and basic

subtraction problems Overview: After several experiences working with ten-frames and using the strategy

of Seeing Doubles (pages 5-42 – 5-49), students will then begin looking at how to apply Seeing Doubles to the subtraction operation. Students will need to have numerous opportunities to model and explain how to show subtraction starting with the sum of a double fact on a ten-frame and explaining what 12 of the difference would be. Example: Doubles addition fact is 9+9=18. The subtraction fact is 18-9=9. Students should be able to model and explain that 12 of 18 is 9. The following activity will be introduced to students after students are able to demonstrate and explain Half of Doubles on a ten-frame successfully. This activity helps build fluency of addition and subtraction double math facts.

Materials: Handout 1-Half of Doubles Scenario Cards, 1 set per group (pages 5-52 –

5-54) Grouping: Small group Time: Only allow 5-7 minutes for each strategy. Lesson:

Procedures Notes 1. Shuffle Half of Doubles scenario cards and

distribute ALL cards to participants. Some participants may have more than one card.

Make scenario cards from Handout 1 (pages 5-52 – 5-54).


Half of Doubles 5-51

Procedures Notes 2. One participant reads the scenario and then

asks the question from the card.

3. The participant who has the answer, replies with the “I have” statement and then reads the scenario on that card.

4. The game proceeds until the answer of one of the scenarios is on the card of the participant who began the game.

5. After the game is concluded, have participants discuss the following questions with their group:

a. How does this activity reinforce doubles addition facts?

b. How does this activity build fluency with double subtraction facts?

c. What modifications would be needed for ELL and inclusion students?

Resources: Rightsel, P. S., & Thorton, C. A. (1985). 72 addition facts can be

mastered by mid-grade 1. Arithmetic Teacher, 33(3), 8-10. Van de Walle, J.A. (2004). Elementary and middle school



Handout 1-1 Half of Doubles 5-52

I have 5.

We planted 18 flowers in our school garden. The

weather was so hot 12 of our flowers died.

Who has how many flowers are still growing

in our garden?

I have 9.

Our neighbors have 8 dogs and cats. 12 of the animals

are dogs.

Who has how many cats my neighbors

have?

I have 4.

My mom bought a dozen eggs. She tripped coming into the house and only 6

eggs were not broken.

Who has how many eggs were broken?

I have 6.

Our car has 4 tires. My uncle drove over a nail in

the street and 12 of the tires went flat.

Who has how many tires are not flat?



I have 2.

My grandmother gave me $16. She asked me to

share 12 of the money with my brother.

Who has how much money I was suppose to

give to my brother?

I have 8.

A pie was cut into 6 pieces. I ate 12 of it.

Who has how many pieces I ate?

I have 3.

I went to summer camp for 2 weeks. It rained 12 of the

days I was there.

Who has how many days it rained at camp?

I have 7.

My father bought a pair of soccer shoes. 12 of the pair

of shoes did not have shoestrings.

Who has how many shoes did not have

shoestrings?



I have 1.

Our class filled 10 boxes with cans to recycle. Our

teacher took 12 of the boxes to get recycled.

Who has how many boxes did not get

recycled?


Doubles Plus One 5-55

Activity: Doubles Plus One





recognizes patterns in number and operations. The student is expected to: (D) Use patterns to develop strategies to solve basic addition and basic

subtraction facts. (2.3) Number, operation, and quantitative reasoning. The student

adds and subtracts whole numbers to solve problems. The student is expected to: (A) Recall and apply basic addition and subtraction facts (to18) (2.5) Patterns, relationships, and algebraic thinking. The student



Overview: Before students begin working with the Doubles +1 strategy, they must

know the sums to the doubles facts and have learned to model doubles math facts on two ten-frames. Double +1 strategy is then introduced and practiced using the two ten-frame work mats. Once the students are able to fluently model and explain how they are solving the doubles +1 math facts through pictorial and symbolic levels, teachers may then use this Doubles +1 domino game. This particular game will build fluency in recognizing doubles +1 math facts and reinforce the mental thinking that occurs to solve the math fact equation.

Materials: Handout 1-Doubles Plus One Dominos, 1 set per group (pages 5-58 – 5-

64)



Handout 2-Doubles Plus One Record Sheet, 1 per participant (page 5-65) 1 red crayon, 1 per participant 1 black crayon, 1 per participant Handout 3-Doubles Plus One Double Ten-Frame, 1 per participant (page 5-66)

Grouping: Small group Time: Only allow 5-7 minutes for each strategy. Lesson:

Procedures Notes 1. Gather materials: Doubles Plus One

dominos and record sheet, 1 red crayon, and 1 black crayon.

Make dominos from Handout 1 (pages 5-58 – 5-64). Handout 2 (page 5-65) is the record sheet.

2. Place dominoes face down in middle of group.

3. Every participant selects 1 domino from the group.

4. If the domino is a double (i.e., 5 and 5, 4 and 4, etc.), then return the domino to the middle of the group and select another domino.

5. If the domino is a double + 1 (i.e., 5 and 6, 2 and 3, etc.,), hold your domino vertically with the greater number at the bottom.

This procedure is to ensure that students are able to identify the number that is less. When students record their dominos on the record sheet, the teacher will have an understanding of how the student is processing the strategy. If students are unable to identify the number that is less, difficulty with Doubles Plus One strategy will occur. After students demonstrate the understanding of identifying the number that is less, then dominos may be held in any position.

6. Have participants draw, on the record sheet, the dots to match the domino that they have selected.



Procedures Notes 7. Identify the number that is less on the

domino. Record that number on the top and bottom ten-frames using a black crayon.

Handout 3 (page 5-66) is a double ten-frame.

8. Record the plus 1 onto one of the ten-frames using a red crayon. Your double ten-frames should match your domino.

Recording the “plus 1” in a red color allows the students to create a mental picture for this strategy. Not designating which ten-frame the “plus 1” should go on is important. Students need to have an understanding it doesn’t matter which number has the “plus 1”. The strategy is still the same. Whether the plus 1 red dot is on the top ten-frame or the bottom ten-frame. The strategy still applies.

9. Record the number sentence to the record sheet.

The following is an example of what the recording sheet should look like when it is completed.

10. Continue selecting different dominos and

recording your answers until the record sheet is completed.

Resources: Van de Walle, J.A. (2004). Elementary and middle school

mathematics: Teaching developmentally (5th ed.) Boston, MA: Allyn and Bacon


Handout 1-1 Doubles Plus One 5-58














Handout 2 Doubles Plus One 5-65

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Speedy Tens 5-67

Activity: Speedy Tens


uses whole numbers to describe and compare quantities. The student is expected to: (B) Create sets of tens and ones using concrete objects to describe,

compare, and order whole numbers, (D) Read and write numbers to 99 to describe sets of concrete objects. (1.3) Number, operation, and quantitative reasoning. The student



subtraction facts (up to 9+9=18 and 18-9=9) (1.5) Patterns, relationships, and algebraic thinking. The student

recognizes patterns in number and operations. The student is expected to: (B) Find patterns in numbers including odd and even, (D) Use patterns to develop strategies to solve basic addition and basic

subtraction problems. (2.1) Number, operation, and quantitative reasoning. The student

understands how place value is used to represent whole numbers. The student is expected to: (A) Use concrete models of hundreds, tens, and ones to represent a

given whole number (up to 999) in various ways. (B) Use place value to read, write, and describe the value of whole

numbers to 999. (2.3) Number, operation, and quantitative reasoning. The student

adds and subtracts whole numbers to solve problems. The student is expected to: (A) Recall and apply basic addition and subtraction facts (to 18), (B) Model addition and subtraction of two-digit numbers with objects,

pictures, words, and numbers. (2.5) Patterns, relationships, and algebraic thinking. The student




Speedy Tens 5-68

Overview: After several experiences modeling the addition facts of ten plus another addend, students will practice their fluency of adding ten or a multiple of ten to a single digit on the ten-frame working towards mental recall of the facts.

Materials: Card stock or construction paper for Speedy Tens flap cards

Tape Handout 1-Speedy Tens Record Sheet (page 5-71)

Grouping: Pairs Time: Only allow 5-7 minutes for each strategy. Lesson:

Procedures Notes 1. For this lesson participants will use flap

cards which show one or more filled ten-frame(s) plus, a second ten-frame partially filled with 1 to 9 dots. The flap, when folded down, shows the total of the two ten-frames. Make flap cards to use in this activity.

To make flap cards: Cut out several card stock or construction paper cards (9" x 3") and a second set (4 -1/2 " x 3 -1/2 "). Tape the shorter card to the top right side of the larger card. Cut out and attach a filled ten-frame and a second ten-frame partially filled on the long card and the answer written on the flap that is folded down. The student holds the card with the flap up then folds it down to check their answer.


Speedy Tens 5-69

Procedures Notes

The ten-frames that were made for the Give Me 10 cards (Handout 1, pages 5-25 – 5-35) can also be used as the black-line masters for these flap cards. If trainers wish to make the flap cards smaller, a smaller version of the ten-frames can be found in Elementary and middle school mathematics: Teaching developmentally (5th ed.) (Van De Walle, 2004).

2. Divide into pairs. If there is an odd number of participants in a group, form one triad.

3. Distribute flap cards to pairs.

4. One partner will hold up flap card and ask the other partner to quickly say the total. Then raise the flap to check answer.

5. Both partners are to record the fast ten fact and the total onto Handout 1-Speedy Tens Record Sheet (page 5-71). .

6. When both partners have recorded their answers, trade cards with another partner.

7. Continue the process until record sheet is completed.

8. After everyone is finished with the activity, have participants share reflections with the group over the following questions:


Speedy Tens 5-70

Procedures Notes a. How do these flap cards help students

create a mental image of addition facts?

b. How can flap cards be used to help bridge learning from the pictorial to symbolic level?

Resources: Hope, J., Leutinger, L., Reys, B., & Reys, R. (1988). Mental math in the

primary grades. Parsippany, MJ: Pearson Learning Group. Van de Walle, J.A. (2004). Elementary and middle school mathematics:

Teaching developmentally (5th ed.) Boston, MA: Allyn and Bacon.


Handout 1 Speedy Tens 5-71

Speedy Tens Record Sheet

10-Frame Totals

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When you finish all flap cards, complete the following problems with your partner: 27+10= _______ 43+10+10+10= _________ When adding 10 to a given number, which place value position changes?_________ What causes only that one place value position to change? __________________________________________________________________________________________________________________________________________________________________________________________

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Documents

Tab 5: Scaffolding to Increase Numerical Fluency … 5: Scaffolding to Increase Numerical Fluency: ... Handout 3-Doubles Plus One Ten-Frame 5-66 Speedy Tens 5-67 ... Numerical Fluency