grade 1 diagnostic data collection tool addition and

Grade 1 Diagnostic Data Collection Tool Addition and Subtraction Within 10 (Fluency) and Within 20

Montgomery County Public Schools, Maryland Curriculum 2.0 – Grade 1, July 2015

Progression of Learning Work with addition and subtraction within 10 and within 20 should be seen as a progression of learning that begins in Kindergarten and culminates in Grade 2. Schools need to have a vertical approach to planning for primary students so that they leave Grade 2 with a deep conceptual understanding of addition and subtraction, as well as automaticity with facts within 20. (Facts within 20 include facts with all single digit addends and facts with 10 as an addend.) Fluency is not to be achieved atthe expense of understanding; fluency involves efficiency, accuracy, and flexibility. What is efficient, however, varies from grade to grade based on the appropriateness of the methods within the progression of learning. This document explains fluency expectations for Grades 1 and 2.

Work with addition and subtraction within 10 and within 20 is crucial. It lays the foundation for understanding key properties of operations once students move to problems within 100, within 1000, and beyond. Decomposing numbers, composing a ten, decomposing a number leading to a ten, the meaning of the equal sign, the commutative property, understanding subtraction as a missing addend problem, and understanding addition and subtraction as inverse operations are all concepts first learned in Kindergarten and Grade 1. These are also concepts that allow students to make meaning of the Level 2 and Level 3 Methods for addition and subtraction within 20.

Methods, Modeling, and Automaticity Kindergarten students are introduced to Level 1 and 2 Methods within 10. Grade 1 students continue to refine Level 2 methods within 10. They also learn all of the Level 2 and Level 3 Methods within 20. As students are introduced to new methods, they are expected to incorporate them into the methods they have already learned, building their repertoire and their flexibility. This document explains different ways to model and reason with methods within 10 and within 20. Providing opportunities for reasoning is key to building conceptual understanding; Grade 1 students are responsible for modeling and reasoning with all of the methods within 10 and within 20 (except Doubles +2). Models, such as ten frames, double ten frames, fingers, expression, and equations, are intended to foster conceptual understanding of the methods so students can move on more abstract methods (such as from a Level 2 Count On Method to a Level 3 Make 10 Method). Through repeated opportunities modeling and reasoning, students develop mental strategies that will result in automaticity with facts.

Benchmarks and Proficiencies This document, in tandem with the Grade 2 document, delineates benchmarks for each Marking Period, as well as end-of-year proficiencies, for Grades 1 and 2 to help teachers, teams, and schools plan for building computational fluency with understanding. A key aspect of fluency is that it does not happen in a single grade, but requires attention to student understanding along the way. The benchmarks defined in Grades 1 and 2 should be seen as minimum requirements in order for students to reach the goal of knowing all facts within 20 from memory with conceptual understanding by the end of Grade 2. Students not meeting benchmarks will need targeted support and instruction with the methods in this document so they canachieve the end-of-year proficiencies. Other students will be at points beyond the benchmarks at any given time. For instance, some Grade 1 students, after repeated meaningful practice with addition and subtraction within 10, will know these facts from memory before Grade 2. These students should not be seen as above grade level; they are simply moving along the learning progression at a faster pace than other students. Students who know facts from memory at any point during Grades 1 and 2 are still expected to exhibit conceptual understanding. (See What about the Student Who’s Already Memorized?)

Recording Sheets Included are two versions of individual student recording sheets and one version of a group recording sheet. Teachers can choose to use any of the sheets. Some teachers will want to use the group recording sheet to collect data that may be transferred to an individual student’s recording sheet at a later time.

You will find the following sections in this document:

Benchmarks and Proficiencies by Marking Period Student Recording Sheet (Option 1) Student Recording Sheet – (Option 2) Group Recording Sheet – front and back Use of Fingers

Explanation of Level 1, 2, and 3 Methods: Addition and Subtraction within 10 Addition and Subtraction within 20

Clarifications Fluent Methods and Solving Word Problems Prerequisite Skills from Kindergarten Meaning of the Equal Sign Solving for the Unknown in Different Positions Mental Images, Mental Strategies, and From Memory What about the Student Who’s Already Memorized? Are Flashcards Appropriate? Are Timed Tests Appropriate? Grade 2 Teachers Need to Look to Grade 1 Standards Additional Resources

https://mymcps-instruction.mcpsmd.org/sites/ic/elementary/grade2/_layouts/MCPS.OLC.HOME/ResourceView.aspx?&ResourceId=45154&ViewPage=1

Addition and Subtraction within 10 and within 20 Grade 1 Benchmarks and End-of-Year Proficiencies


This table shows what students need to be able to do by the end of each marking period to be proficient. All statements refer to addition and related subtraction problems (e.g., 4 + 5, 9 – 4, 9 – 5). Students need to continue to show understandings and skills from previous marking periods and from Kindergarten.

Marking Period 1 Within 5 Know Facts with 0 from memory Continue to model and reason with Level 2 Methods (started

in K)Within 10 Model quantities to 10 using fingers with automaticity Know Facts with 0 from memory Continue to model and reason with Level 2 Methods (started

in K)

Marking Period 2 Within 5 Know +1/-1 Facts from memory Continue to model and reason with Level 2 Methods

Within 10 Know +1/-1 Facts from memory Continue to model and reason with Level 2 Methods

Within 20 Model and reason with Levels 2 and 3 Methods

Marking Period 3 Within 5 Continue to model and reason with Level 2 Methods

Within 10 Continue to model and reason with Level 2 Methods

Within 20 Continue to model, reason with, and refine Level 2 and Level 3

Methods

Marking Period 4 Within 5 Know all Facts within 5 from memory

Within 10 Continue to model and reason with Level 2 Methods using

fingers or mental strategies. The use of other models is not considered efficient for the end-of-year proficiency.

Use Count On/Greater Addend Method, when appropriateWithin 20 Continue to model, reason with, and refine Level 2 and

Level 3 Methods

Grade 1 students are responsible for modeling and reasoning with all Level 2 Methods within 10 and all Level 2 and Level 3 Methods within 20 (except Doubles +2, which are introduced in Grade 2). Students use models, such as fingers, ten frames, double ten frames, dot patterns, pictures, expressions, equations, and mental images and strategies to make sense of the methods, deepening their understanding so that they can meet the goal of efficiency, accuracy, and flexibility with addition and subtraction. For the end-of-year fluency within 10, students use only fingers or mental strategies with the facts in the table below in white; students should know all facts within 5 from memory.

The goal is computational fluency with conceptual understanding. Therefore, a student who knows the facts from memory, but who cannot reason with and explain a Level 2 or Level 3 Method has not met the expectations for conceptual understanding in Curriculum 2.0.

Addition and Subtraction Fluency within 10 Grade 1 Benchmarks and End-of-Year Proficiencies

+ 0 1 2 3 4 5 6 7 8 9 10

0 0 1 2 3 4 5 6 7 8 9 10

1 1 2 3 4 5 6 7 8 9 10 MP1 Benchmark MP2 Benchmark

2 2 3 4 5 6 7 8 9 10 Facts with 0 from memory

+1/–1 Facts from memory 3 3 4 5 6 7 8 9 10

4 4 5 6 7 8 9 10

5 5 6 7 8 9 10 End-of-Year Proficiencies

6 6 7 8 9 10 Other Facts within 5 from memory

Facts that do not need to be automatic* 7 7 8 9 10

8 8 9 10 This chart refers to addition facts and their related subtraction facts.

*These facts can be solved with fingers using Level 2 Methods or with mental strategies at the end of the year.

9 9 10

10 10

Recording Sheet, Grade 1 Addition and Subtraction Within 10 and 20 (Option 1) Student Name ___________________________________________________________


Level 1 Level 2 From Memory WW

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Facts with 0

+1/-1 Facts

Facts within 5

Count On

Count On/Greater Addend Count Up/Missing Addend Count Down**

Level 1 Levels 2 From Memory

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1100

Count On Facts with 0

Count On/Greater Addend* +1/-1 Facts

Count Up/Missing Addend Facts within 5

Count Down** 5 as Benchmark FactsDoubles +1 (Level 3) Make 10 Facts

All Facts within 10

Level 1 Level 2 Level 3 From Memory

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2200

Count On Doubles +1 +1/-1 Facts

Make 10 10 and Teen Numbers

Lead to 10 Doubles

Count On/Greater Addend Count Up/Missing Addend Count Down** Doubles +2 Doubles +1

Doubles +2

All Facts within 20

When students should first show proficiency: Kindergarten Grade 1 Grade 2 *Student starts with greater

addend when appropriate for End-of-Year Proficiency for addition within 10.

* *Count Down Methods are not required. Students can choose to use Count Up Methods for subtraction instead.

To see Grade 2 Benchmarks and Proficiencies, see the Grade 2 diagnostic tool.

MP1 MP2 MP4

Students must be able to model and reason with all Level 2 and Level 3 Methods within 10 and 20 (except Doubles +2).


Recording Sheet, Grade 1 Addition and Subtraction Within 10 and 20 (Option 2) Student Name _______________________________________________________


Within 5 Within 10 Level 1 Level 2 Memory Level 1 Level 2 From Memory

Count All

Count On

Count On/

Greater Addend

Count U

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Missing Addend

Count Down**

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Count On/

Greater

Addend*

Count U

p/

Missing Addend

Count Down**

Doubles +1

(Level 3)

Facts w

ith 0

+1/-‐1 Facts

Facts w

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5 as Benchmark

Make 10 Facts

All Facts

within 10

Within 20Level 1 Level 2 Level 3 From Memory

Count All

Count On

Count On/

Greater Addend

Count U

p/

Missing Addend

Count Down**

Doubles +1

Make 10

Lead to 10

Doubles +2

+1/-‐1 Facts

10 and Teens

Doubles

Doubles +1

Doubles +2

All Facts w/in 20

When students should first show proficiency:Kindergarten Grade 1 Grade 2

*Student starts with greater addend when appropriate for End-‐of-‐Year Proficiency for addition within 10.

** Count Down Methods are not required. Students can choose to use Count Up Methods for subtraction instead.

Students must be able to model and reason with all Level 2 and Level 3 Methods within 10 and 20 (except Doubles +2). MP1 MP2 MP4


Group Recording Sheet, Grade 1 Addition and Subtraction Within 10 and 20


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Students must be able to model and reason with all Level 2 Methods.

When students should first show proficiency: Kindergarten Grade 1 Grade 2

MP1 MP2 MP4



Group Recording Sheet, Grade 1 Addition and Subtraction Within 10 and 20


Within 20

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Grade 1 students must be able to model and reason with all Level 2 and Level 3 Methods (except Double +2).

When students should first show proficiency:Grade 1 MP2 Grade 2



Use of Fingers


Young children often count on their fingers or model quantities with their fingers. Teachers may need to communicate to parents that fingers are an important model for representing quantities and addition and subtraction for Kindergarteners and Grade 1 students. Throughout Kindergarten, students should be encouraged to use their fingers as one way to represent quantities. “It is important that students in Kindergarten develop rapid visual and kinesthetic recognition of numbers to 5 on their fingers” (Progressions, p. 8) as fingers become a strategy for keeping track of knowns and unknowns in addition and subtraction problems and lead to advanced methods for solving problems, especially in Grade 1. It is a Grade 1, Marking Period 1 expectation that students can quickly and accurately model quantities, 1-10, on their fingers. Additionally, using fingers (as well as 5 and 10 frames) establishes the importance of 5 and 10 as benchmark numbers (seeing 7 as 5 and 2 more).

There are different ways to model quantities with fingers depending on personal preference and on cultural background; all are acceptable. Some students will hold their hands palms down; others will hold their hands palms up. There are different ways to count out quantities, including:

Starting with 1 on the index finger, counting one more on subsequentfingers to 4 on the pinkie finger, with 5 counted on the thumb

Starting with 1 on the thumb, moving next to the index finger, endingwith 5 on the pinkie finger

Starting with 1 on the pinkie finger, counting one more on subsequent fingers, with 5 counted onthe thumb.

“Children … become able to use their fingers to add or to subtract using the direct modeling solution methods counting all or taking away. When counting all, they will count out and raise fingers for the first addend, then for the second addend, and then count all of the raised fingers. Some children learn at home or in a care center to put the addends on separate hands, while others continue on to the next fingers for the second addend. The former method makes it easier to see the addends, and the latter method makes it easier to see the total. Both methods can be modeled by the teacher. As children become more and more familiar with which group of fingers makes 4 or 5 or 7 fingers, they may not even have to count out the total because they can feel or see the total fingers. Similarly, children using the method of putting fingers on separate hands eventually can just raise the fingers for the addends without counting out the fingers. But they do need initially to count the total. Children who put addends on separate hands may have difficulty with problems with addends over 5 (e.g., 6 + 3) because one cannot put both such numbers on a separate hand. They can, however, continue raising fingers from 6 fingers.” (National Research Council, 2009, p. 158.)

Grade 1 students are expected to be able to show quantities to 10 on their fingers with automaticity by the end of Marking Period 1 in order to model and make sense of Level 2 and Level 3 Methods within 10 and 20 throughout the rest of the year.

When to wean from fingers? Sometimes students resist giving up using their fingers, even for easy problems, such as +1 or –1 problems. Throughout Grade 1, teachers should encourage students to solve problems such as these without fingers. As the year progresses, teachers should encourage students to solve progressively harder problems within 10 without fingers. Students can be encouraged to create mental images of their fingers or of other models, such as a ten frame or dot patterns – on dominoes or dot cubes. A teacher might prompt students to visualize: Make a mental picture of what 4 + 3 looks like on your fingers. At the end of the year, some Grade 1 students will still use fingers and Level 2 Methods when solving harder problems within 10 (e.g., 9 – 6, 3 + 5). Grade 1 students should be encouraged to use their fingers, as one representation, to make sense of and model Level 2 and 3 Methods for solving addition and subtraction within 20 throughout the year.

For problems within 10, Grade 2 students should move away from using fingers early in the year so that they can meet the Marking Period 2 Benchmark of knowing all facts within 10 from memory. Some Grade 2 students will rely on fingers to model problems within 20 using Level 2 and Level 3 Methods at the beginning of the school year. Teachers should encourage students to solve increasingly more difficult problems without fingers with the goal of students not needing fingers to solve even the most difficult problems within 20 by the end of Marking Period 3. (See the section Mental Images, Mental Strategies, and From Memory.)

Examples of Methods Within 10


Students begin working with Addition and Subtraction within 10 in Kindergarten and within 20 in Grade 1. Throughout Kindergarten to Grade 2, students are expected to reason with the methods described below. Students also move from basic Level 1 understandings of addition and subtraction to Level 2 Methods, and then to more sophisticated Level 3 Methods. The goal is computational fluency with conceptual understanding. Therefore, a student who might know their facts from memory, but who cannot reason with and explain a Level 2 or Level 3 Method has not met the expectations for conceptual understanding in Curriculum 2.0. Mathematical models, such as fingers, ten frames, double ten frames, dot patterns, expressions, and equations, help students make sense of the methods, deepening their understanding so that they can meet the goal of efficiency, accuracy, and flexibility with addition and subtraction. Teachers can use the terms method and strategy interchangeably; strategy may connect better to other contents, like reading.

Level 1 These methods involve the most basic representations and understandings of addition and subtraction. They are less efficient methods for solving and involve direct modeling of all of the quantities in the problem.

Count All - Addition: 3 + 6 Students apply a basic understanding of the cardinalities of the two addends. They model with objects, fingers, or pictures the quantities of the two addends. Then they count from 1, counting out the quantity in the first addend (1,2,3) and then counting out the quantity in the second addend (1,2,3,4,5,6). The student then recognizes the total as 9 (the number of fingers outstretched), or recounts the total from 1 if using objects or pictures. Students should be able to explain what the three quantities represent (the two parts are put together to make a total).

Count All - Subtraction: 9 – 3 The total of 9 is modeled with objects, fingers, or pictures. The take away quantity of 3 is counted out from the total. The remaining quantity is counted out from 1 to find the difference of 6.

Level 2 For Level 2 Methods, students apply a more advanced understanding of the cardinalities of the two addends, as well as an understanding that the addends are parts of the sum (or total).

Count On – Addition: 3 + 6 To solve this problem using a Count On method, understanding of the cardinality of the first addend is applied abstractly without being represented with objects, fingers, or pictures. Students then count on from the first addend, resulting in a more efficient method for solving. They may use fingers, mental images, or body movements to keep track of counting out the second addend: (think 3) … 4,5,6,7,8,9. The challenge is that when using fingers, for instance, the first finger is counted as 4, the second finger is counted as 5, etc. The student then must remember to stop at the sixth finger, recognizing the total to be 9.

Count On, Start with Greater Addend - Addition: 3 + 6 This applies to certain addition problems (like 3 + 6 or 1 + 8), but not to all (it may not be important for a problem like 4 + 5 to start with the greater addend). By starting with the greater addend, students apply understanding that the order of the addends does not matter (commutative property). This is the most efficient Level 2 method for addition, applying important understandings of Counting and Cardinality and properties of addition to problems within 10. Students may use fingers, mental images, or body movements to keep track of counting out the second addend. Students mentally change 3 + 6 to 6 + 3. The student starts at 6, and then counts on 3 to represent the smaller addend: (think 6) … 7,8,9.

Count Up (Missing Addend) – Subtraction: 9 – 3 Students apply understanding of the relationship between addition and subtraction as inverse operations and that subtraction can be understood as a missing addend problem. Students think of 9 – 3 as the missing addend problem 3 + ___ = 9. Students count up from the first addend, keeping track of when the total has been reached. Students understand that the quantity counted up is the missing addend, or difference (e.g., 6 outstretched fingers). (Think 3) … 4,5,6,7,8,9. Students may use fingers, mental images, or body movements to keep track. You will notice that using fingers to solve this problem is the same as the method for solving 3 + 6, above.



Count Down: 9 – 3 Students apply understanding of the cardinality of the total, 9. Students count back 3 to represent the take away quantity: (think 9) … 8,7,6. The student recognizes 6 as the difference. Students may use fingers, mental images, or body movements to keep track. Counting back can be challenging and confusing for many students; often, they will mistakenly include the 9 when counting back: 9,8,7, resulting in an incorrect difference. They may choose to use a Level 2 Count Up method of understanding subtraction as a missing addend problem, instead.

Level 3 Within 10, the only methods that would be considered Level 3, using a known fact to derive an unknown fact, is a Doubles +1 or Doubles + 2 Method. (See Methods Within 20 for more information.)

From Memory The following problem types within 10 are the easiest for students to memorize.

+1/–1 Facts Students have practice in Kindergarten and the beginning of Grade 1 relating addition and subtraction to the count sequence. One of the foundational understandings of Counting and Cardinality is that each successive number word in the count sequence refers to a quantity that is one more and that each previous number word in the count sequence refers to a quantity that is one less. Students with this understanding should be able to solve +1/–1 facts within 10 mentally early in Grade 1. Visualizing a number line or number path can also help students think about the number that is one more or one less.

Facts with 5 as a Benchmark Students in Kindergarten and Grade 1 can easily model the number 5 with fingers on one hand. They can feel the quantity 5 when all 5 of their fingers are outstretched. Both with fingers and when one row is filled on the ten frame, 5 becomes a powerful benchmark number. As students learn to model the quantities 6 to 9 on their fingers, they think of those quantities in relation to 5: 6 is 5 and 1 more; 8 is 5 and 3 more. The structure of the ten frame also encourages students to think of 5 as a benchmark. This reasoning allows students to think of and visualize decompositions of the numbers 6 to 9 as 5 and some more (e.g., 6 = 5 + 1; 9 = 5 + 4; 7 – 2 = 5; 8 – 5 = 3).

Make 10 Students start composing and decomposing 10 in Kindergarten. Models such as ten fingers and ten frames help students to reason about different pairs that make 10. When students can model quantities to 10 with their fingers with automaticity, they can begin to feel and visualize how many more fingers they need to make 10. Likewise, when their experiences with 10 frames allow them to visualize quantities on the ten frame, as well as reason about the missing quantity that will make 10, this will lead them to knowing the decomposition of 10 from memory at the beginning of Grade 2. When students are determining what number added to 6 makes 10, they can model 6 on their fingers and reason that they need 4 more to make 10. Or, they can model 6 on a ten frame, and reason that there are 4 empty spaces. The more students work with their fingers and ten frames in Kindergarten and Grade 1, the more they will be able to internalize the models, visualize quantities, and learn the pairs that make 10 from memory. Students practice making 10 using the online resource, NCTM Illuminations Ten Frame (Activity 3 – Fill).

Doubles Facts See Methods Within 20.

https://mymcps-instruction.mcpsmd.org/sites/ic/_layouts/mcps.olc.home/resourceview.aspx?ResourceID=2561&ViewPage=1



Level 1 - These methods involve the most basic representations and understandings of addition and subtraction. They are less efficient methods for solving.

Count All – Addition: 6 + 9 Students apply a basic understanding of the cardinalities of the two addends. They model with objects, fingers, or pictures the quantities of the two addends. They count from 1, counting out the quantity in the first addend (1,2,3,4,5,6) and then counting out the quantity in the second addend (1,2,3,4,5,6,7,8,9). They then count the total in the entire model. Students should be able to explain what the three quantities represent (the two parts are put together to make a total). When students draw pictures, they can model quantities using the structure of a ten frame. This helps them to use 5 and 10 as benchmarks. In this example, after drawing the quantities of the two addends, they might reason about shifting one dot so that there are 3 groups of 5, or 15. This structure eventually encourages thinking about Level 3 Methods.

Count All - Subtraction: 15 – 6 The total of 15 is modeled with objects or pictures. The take away quantity of 6 is counted out from the total and often marked with Xes. The remaining quantity is counted out from 1 to find the difference of 9.

Level 2 – For Level 2 Methods, students apply a more advanced understanding of the cardinalities of the two addends, as well as an understanding that the addends are parts of the sum (or total).

Count On – Addition: 6 + 9 To solve this problem using a Count On Method, understanding of the cardinality of the first addend is applied abstractly without being represented with objects, fingers, or pictures. Students then count on from the first addend, resulting in a more efficient method for solving. They may use fingers, mental images, or body movements to keep track of counting out the second addend: (think 6) … 7,8,9,10,11,12,13,14,15.

Count On, Start with Greater Addend: 6 + 9 By starting with the greater addend, students apply understanding that the order of the addends does not matter (commutative property). This is the most efficient Level 2 method for addition, applying important understandings of Counting and Cardinality and properties of addition to problems within 20. Students may use fingers, mental images, or body movements to keep track of counting out thesecond addend. Students mentally change 6 + 9 to 9 + 6. The student starts at 9, and then counts on 6 to represent the smaller addend: (think 9) … 10,11,12,13,14,15.

Count Up (Missing Addend) – Subtraction: 15 – 6 Students apply understanding of the relationship between addition and subtraction as inverse operations and that subtraction can be understood as a missing addend problem. Student thinks of 15 – 6 as the missing addend problem 6 + ___ = 15. Students count up from the first addend, keeping track of when the total has been reached. Students understand that the quantity counted up is the missing addend, or difference (e.g., 9 outstretched fingers). (Think 6) … 7,8,9,10,11,12, 13,14,15. You will notice that using fingers to solve this problem is the same as the method for solving 6 + 9, above.

Count Down – Subtraction: 15 – 6 Students apply understanding of the cardinality of the total, 15. Students count back 6 to represent the take away quantity: (think 15) … 14,13,12,11,10,9. The student recognizes 9 as the difference. Counting back can be challenging and confusing for many students; they may choose to use a Level 2 Count Up method, understanding subtraction as a missing addend problem, instead.



Level 3 – Level 3 Methods involve using known facts to derive unknown facts. These methods rely heavily on students’ flexibility with numbers and on important properties of addition and subtraction. Level 3 Methods also lay the foundation for addition and subtraction beyond 20 for concepts such as composing and decomposing 10, using multiples of 10 as benchmarks, relating addition and subtraction as inverse operations (including understanding subtraction as an unknown addend problem and adding up to find a difference), and compensation strategies.

If students know all of their facts within 20 from memory, they are only proficient if they can understand, reason with, and model Level 2 and, most importantly, Level 3 strategies for addition and subtraction within 20. See the section, What about the Student Who’s Already Memorized? for more information.

Using Doubles Facts Doubles facts are typically some of the easiest for students to memorize, and they become the basis for some Level 3 Methods, such as Doubles +1 and Doubles +2. Repeated work with visuals and mental images in Grades 1 and 2 fosters learning these facts from memory. Dot patterns with and without ten frames, fingers, and other images can help students memorize doubles facts. Doubles facts with addition also need to be applied to subtraction. Students can think of this as halving (I know 9 is half of 18, so 18 – 9 = 9). Or they can simply relate

subtraction to the addition fact (9 + 9 = 18, so 18 – 9 = 9.).

Doubles +1 Students use doubles facts to derive other facts. A student should be able to visualize and then to reason that a problem like 6 + 5 can be thought of as one more than 5 + 5. Students can then think abstractly, relying more on equations they know: 7 + 8 = 7 + 7 + 1; or even abbreviating the thinking: 7 + 8 = 14 + 1.

Doubles +2 With practice, a student should be able to reason and then to visualize that the addends in a problem like 6 + 8 are one more and one less than the addends in 7 + 7. In this early compensation strategy, a quantity of 1 is shifted from the greater addend to the lesser addend. A student could also reason that 6 + 8 is 6 + 6 + 2.

Make 10 Students have been making 10 since Kindergarten, reasoning about the structure of the ten frame and thinking about their 10 fingers to determine what number, when added to another number, makes 10. Grade 2 students are expected to know their Make 10 facts from memory by the end of Marking Period 1. Students combine the reasoning for making 10 and decomposing a number to solve problems like 6 + 8. Students can start with either of the addends (in this case 8), decompose the second addend (6 is 2 and 4) to be able to make 10, and then use their knowledge of teen numbers to quickly reason that 10 and 4 is 14. Students could solve 6 + 8 using a Make 10 strategy, or using a Doubles +2 strategy. The more strategies students have at their disposal, the more likely they are to think flexibly about numbers, leading to success with addition and subtraction beyond 20 and to automaticity with basic facts. The ten frame images come from the online resource, NCTM Illuminations Ten Frame (Activity 4 – Add). Another resource that prompts students to break apart addends to make a ten and then to compose a teen number is Greg Tang: Break Apart (Addition – Make 10, Easy). With repeated practice and reasoning, students will be able to use a Make 10 strategy mentally.


https://mymcps-instruction.mcpsmd.org/sites/ic/elementary/grade2/_layouts/MCPS.OLC.Home/ResourceView.aspx?viewpage=1&ResourceId=45178




Lead to 10 Students use 10 as a benchmark when subtracting with a Lead to 10 Method. Students should think with automaticity of a teen number as composed of a ten and some ones. So 15 is 10 and 5. To determine 15 – 7, a student can visualize 15 on a double ten frame, and reason that 15 – 5 = 10, and 2 less is 8, so 15 – 7 = 8. Alternately, a student can think more abstractly of subtracting 5 and then subtracting 2 for a difference of 8.

The resource, Greg Tang: Break Apart (Subtraction – Make 10, Easy), also uses a Lead to 10 Method, prompting students to use the same type of reasoning.

10 and Teen Numbers Since Kindergarten, students have thought of teen numbers, like 15, as ten ones and 5 more ones, or as a ten and 5 more ones. Early in Grade 2, students should have automaticity with decompositions using this understanding: 15 = 10 + 5. These facts should be easy to memorize for students with conceptual understanding. These facts and decompositions are essential for the more sophisticated Level 3 Methods, such as the Make 10 and Lead to 10 strategies.


Clarifications


Do students need to use a fluent method when solving word problems? In Grade 1, students solve word problems within 20; they do not need to use a fluent method. Some models students use to represent and solve word problems (e.g., pictures, part-part-whole diagrams) are not efficient, however these models are essential for students to make sense of the different problem situations and allow them to concentrate on problem solving. (See Grade 1 Word Problem Data Collection Tool.)

Prerequisite Skills from Kindergarten The following skills are instructed in Kindergarten and are essential to provide students with conceptual understanding and models as they continue with methods for solving addition and subtraction in Grades 1 and 2. See Kindergarten: Operations and Algebraic Thinking Diagnostic Tool for more information. Fingers: Automaticity with modeling quantities to 5 (By the end of Marking Period 1, Grade 1 students

model quantities to 10 with automaticity.) Making 10 – Knowing the number when added to another number that makes 10; using fingers and the

structure of the ten frame to reason about: 7 + ___ = 10 Decomposing number to 10 in more than one way: 6 = 1 + 5 = 2 + 4 = 3 + 3 Composing and Decomposing Teen Numbers with 10 ones: 19 is 10 ones and 9 more ones; 19 = 10 + 9 Conceptual Subsidizing – “Students come to quickly recognize the cardinalities of small groups without

having to count the objects; this is called perceptual subitizing. Perceptual subitizing develops intoconceptual subitizing – recognizing that a collection of objects is composed of two subcollections andquickly combining their cardinalities to find the cardinality of the collection (e.g., seeing a set as two subsetsof cardinality 2 and saying “four”). Use of conceptual subitizing in adding and subtracting small numbersprogresses to supporting steps of more advanced methods … that culminate in single-digit fluency.”(Progressions, p. 4)

Meaning of the Equal Sign In Kindergarten, the equal sign can be too abstract for students to understand; some students may have the misconception that the equal sign simply designates that the sum or difference should be given. In Grade 1, students solidify the important understanding that the equal sign denotes equality or balance.

1.1.B.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.

While students are making sense of addition and subtraction within 10 and 20, teachers should take every opportunity to discuss the meaning of the equal sign. When students are working with different addend pairs for the same sum, teachers and students can model and reason with equations, like 7 + 2 = 4 + 5:

What does the equal sign mean in this equation? 7 and 2 is 9 and 4 and 5 is 9. The equal sign shows that both sides are 9. The resource, Computation Balance, is a great way to explore the ideas of balance and equality.

Students should also have repeated experience with problems such as: 7 + 2 = ___ + 5. Students are only able to determine the unknown in this problem if there is an understanding of equality. Determining whether an equation is true or false also involves understanding that the equal sign denotes equality: 5 + 6 = 15 – 6 is false, because the two sides are not equal.

Solving for the Unknown in Different Positions Starting in Grade 1, students should be asked to solve problems with unknowns in different positions. The Grade 1 standard states:

1.1.B.8 Determine the unknown whole number in an addition or subtraction equation relating to three whole numbers.

By varying the unknown, students solidify understandings of part-part-whole relationships and the understanding of addition and subtraction as inverse operations. Students should be asked to reason about these relationships, not taught procedures for solving. For instance, for a problem such as 12 = ___ + 4, a student should be asked what is known and unknown (e.g., I know the total is 12 and I’m trying to figure out what number to add to 4 to get 12.). Some students might add up from 4 to solve; others might think of the subtraction problem 12 – 4. Both methods allow the student to make sense of the part-part-whole relationship in a way that is meaningful.Ten frames, double ten frames, and part-part-whole diagrams can help students make sense of the known and unknown quantities in a problem.


https://mymcps-instruction.mcpsmd.org/sites/ic/elementary/kg/_layouts/MCPS.OLC.Home/ResourceView.aspx?viewpage=1&ResourceId=40158


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As students solve word problems, they confront unknowns in all positions. The more exposure they have working with part-part-whole relationships with computation, the more those understandings will translate in problem solving contexts.

Mental Images, Mental Strategies, and From Memory The Grade 2 standard states:

1.2.B.2 Fluently add and subtract within 20 using mental strategies.

By end of Grade 2, know from memory all sums of two one-digit numbers.

Models such as fingers, ten frames, and dot patterns help students visualize and conceptualize important understandings and methods. Throughout the progression of learning from Kindergarten to Grade 2, students move to more abstract models. One way to begin this transition is to ask students to visualize accessible models. A teacher can prompt students:

Picture how you would solve 4 + 7 with your fingers. How can visualizing a double ten frame help you solve 15 – 7?

These mental images are one form of mental strategy. Other mental strategies come from students’ repeated practice with composing and decomposing numbers and recording their thinking. A student might write the following to solve 15 – 7:

Once they are ready for a mental strategy, they can keep the following decompositions in their heads: 15 = 10 + 5 and 7 = 2 + 5. These decompositions allow them to mentally reason that to subtract 7, they can first subtract 5 to get to 10, and then subtract 2 more to get to 8. It is through these increasingly more sophisticated abstractions that students will be able to truly move to knowing facts from memory.

What about the Student Who’s Already Memorized? Students entering Grades 1 or 2 knowing all of their facts within 20 from memory are only proficient if they can understand, reason with, and model Level 2 and Level 3 strategies for addition and subtraction. This is what it means for students to have computational fluency with conceptual understanding. Sometimes, questioning students who have already memorized the facts can be challenging. Questions such as, How did you solve the problem? or How do you know? are often met with responses such as, I just know it. There are other ways to get at student understanding while respecting the fact that the student has automaticity with the facts. A student can be asked to explain a particular method:

How could you use the Make 10 Method to solve 6 + 9? or How does the Make 10 Method relate to 6 + 9? How could you use the Lead to 10 Method to solve 15 – 9? How could you use a doubles fact to solve 7 + 8? or What doubles fact relates to 7 + 8?

Students can also be asked to explain a particular solution method presented to them:

What problem does this solution represent? Explain why the problem was solved this way.

Are Flashcards Appropriate? Flashcards should be used with intention. A student should not receive a set of flashcards to practice facts rotely when the child does not know most of the facts included in the set. With flashcard and with timed tests, if a child practices over and over again, guessing at facts, the erroneous answers can be imprinted in the child’s mind, making it more difficult to undue the error than if the student never performed such activities.

“Practice that follows substantial first experiences that support understanding and emphasize ‘thinking strategies’ has been shown to improve student achievement with single-digit calculations. This approach allows computation and understanding to develop together and facilitate each other. Explaining how procedures work and examining their benefits, as part of instruction, support retention and yield higher levels of performance.” Adding It Up, page 193.

Flashcards can be used to practice strategies. Students can sort cards by a solution method, such as Doubles Facts, Make 10 Facts, and Leading to 10 Facts. Then, students are not guessing at answers, but applying strategies that will become more efficient for the student over time. A focus on strategies will be much more beneficial to

Clarifications


students in the long run. Alternate flashcards, such as Sums to 20 in Curriculum 2.0, provide models with double ten frames and emphasis on relating addition and subtraction as inverse operations.

Are Timed Tests Appropriate? In Curriculum 2.0, timed tests are used sparingly, and teachers are cautioned not to place undue emphasis on timed tests for basic facts. The emphasis should be on teaching solution strategies. “Teachers who use timed tests believe that the tests help children learn basic facts. This makes no instructional sense. Children who perform well under time pressure display their skills. Children who have difficulty with skills, or who work more slowly, run the risk of reinforcing wrong learning under pressure. In addition, children can become fearful and negative toward their math learning.” (Burns, 2000, p. 157) There is also a punitive nature to timed tests:

“Timed tests: Cannot promote reasoned approaches to fact mastery Will produce few long-lasting results Reward few Punish many Should generally be avoided” (Van de Walle, 2006, p. 119)

There are some timed tests included in Curriculum 2.0, however, not in Grade 1. Students are generally given more time than with traditional mad minute scenarios. On the Grade 2, Marking Period 4 Formative Task to assess fluency, students are given over 8 seconds per answer. If need be, assess students who are particularly sensitive to timing without announcing the time, observing whether there are a few problems they spend an inordinate amount of time on. The point of knowing facts fluently or from memory is not so students can perform on a timed test; it is to facilitate the application of fact fluency in real world situations and when solving multi-digit problems.

If you are implementing a program, such as mad or math minute, it is suggested that the timed component be deemphasized. It is also suggested that instruction, including reteaching of strategies, receive more time than administration of the drills; far too often, administering the drills several times a week takes an inordinate amount of time away from math instruction and does not help students who are already struggling with applying strategies and learning facts from memory. It is also suggested that 10 to 20 problems is more appropriate and far less daunting for many students than 40 or 50 problems. It is suggested that a mix of addition and subtraction problems be presented to students. However, if you have chosen to start with addition and move to subtraction, addition problems should have unknowns in different positions (e.g., 3 + __ = 8, +__ = 5 + 4, __ + 6 = 11). This way, students will benefit from solving unknown addend problems, making addition and subtraction more connected.

Grade 2 Teachers Need to Look to Grade 1 Indicators In Grade 2, the only indicator relating to addition and subtraction within 20 states:

1.2.B.2 Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

The issue is that this indicator may oversimplify the progression of learning that allows students to reach these goals. As is commonly the case in the Common Core State Standards, many of the important conceptual understandings for one grade can be found in the standards from previous years when they were introduced. The standards are also cumulative, meaning, in this case, that a Grade 2 student is responsible for the understandings explicit in the Grade 1 standards. Therefore, Grade 2 teachers need to look to the Grade 1 standards to truly appreciate the complexity of the understandings for adding and subtracting with 20:

Applicable Grade 1 Indicators 1.1.B.3 Apply properties of operations as strategies to add and subtract. 1.1.B.4 Understand subtraction as an unknown-addend problem. 1.1.B.5 Relate counting to addition and subtraction (e.g., by counting on 2 to add 2) 1.1.B.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use

strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

1.1.B.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.

https://mymcps-instruction.mcpsmd.org/sites/ic/elementary/_layouts/MCPS.OLC.Home/ResourceView.aspx?ResourceId=2347

Clarifications


1.1.B.8 Determine the unknown whole number in an addition or subtraction equation relating to three whole numbers.

In particular, Indicator 1.1.B.6 calls out the specific Level 3 Methods Grade 1 and Grade 2 students are responsible for understanding.

Likewise, there are relevant Kindergarten standards Grade 1 and Grade 2 teachers need to be aware of as students are expected to continue to apply, refine, and abstract these understandings:

Kindergarten Indicators 1.K.B.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or

drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1). 1.K.B.4 For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by

using objects or drawings, and record the answer with a drawing or equation. 1.K.C.1 Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using

objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

Additional Resources The following link includes additional resources to instruct methods for adding and subtracting within 10 and 20. They can be used for centers, as the focus of targeted instruction in small group, or for independent practice.

Howard County Wikispace Dot Card and Ten Frame Activities Illuminations Five Frame and Ten Frame Activities Computation Balance NC Dept. of Public Instruction: Building Conceptual Understanding and Fluency Through Games

References

Burns, M. (2000). About Teaching Mathematics: A K-8 Resource. Sausalito, CA: Math Solutions Publications.

Clements, D. (2004). Engaging Young Children in Mathematics: Standards for Early Childhood Mathematics Education. Mahwah, NJ: Lawrence Erlbaum Associates, Inc.

McCallum, B. (2012). Tools for the Common Core Standards. "General Questions About the Mathematics Standards."

National Research Council (2001). Adding It Up: Helping Children Learn Mathematics. Washington, DC: National Academy Press.

National Research Council (2009). Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity. Committee on Early Childhood Mathematics.Washington, DC: The National Academies Press.

Partnership for Assessment of Readiness for College and Careers (PARCC). September 2014. PARCC Model Content Frameworks: Mathematics Kindergarten through Grade 2.

Progressions for the Common Core State Standards in Mathematics: Counting and Cardinality; K–5 Operations and Algebraic Thinking (2011). The Common Core Standards Writing Team.

Reys, R. (2012). Helping Children Learn Mathematics. New York: John Wiley & Sons, Inc.

Van de Walle, J. (2006). Teaching Student-Centered Mathematics, Grade K - 3. New York: Pearson.







http://commoncoretools.me/2012/04/02/general-questions-about-the-standards/

http://www.nap.edu/catalog.php?record_id=9822

http://www.nap.edu/catalog.php?record_id=12519

http://parcconline.org/parcc-k-2-model-content-frameworks

http://parcconline.org/parcc-k-2-model-content-frameworks

http://commoncoretools.files.wordpress.com/2011/05/ccss_progression_cc_oa_k5_2011_05_302.pdf

grade 1 diagnostic data collection tool addition and

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