# 1.1.2A Addition & Subtraction

1
Subject:
Math
Strand:
Number & Operation
Standard 1.1.2

Use a variety of models and strategies to solve addition and subtraction problems in real-world and mathematical contexts.

Benchmark: 1.1.2.1 Represent Addition & Subtraction

Use words, pictures, objects, length-based models (connecting cubes), numerals and number lines to model and solve addition and subtraction problems in part-part-total, adding to, taking away from and comparing situations.

Benchmark: 1.1.2.2 Compose & Decompose

Compose and decompose numbers up to 12 with an emphasis on making ten.

For example: Given 3 blocks, 7 more blocks are needed to make 10.

Benchmark: 1.1.2.3 Relationship Between Counting & Addition/Subtraction

Recognize the relationship between counting and addition and subtraction. Skip count by 2s, 5s, and 10s.

## Overview

Big Ideas and Essential Understandings

Standard 1.1.2 Essential Understandings

Solving addition and subtraction problems embedded in a context helps first graders develop an understanding of what it means to add or subtract. Using objects to model and solve combining and separating situations also helps first graders make sense of what it means to add or subtract. The context of a problem also helps first graders make sense of the problem and choose a solution strategy.

First graders solve addition and subtraction problems using strategies including counting everything, counting on, counting back, making a ten, using a double or double plus one. As children solve problems they construct strategies that become more efficient. When given the problem, Lucy has 3 cookies and Dan has 8 cookies, how many more cookies does Dan have? Students may begin with direct modeling using materials to solve. As children's counting strategies become more efficient, they will be able to make sense of the problem, represent the situation, and plan a solution. By comparing a variety of solution strategies, children build their understanding of the relationship between addition and subtraction.

All Standard Benchmarks

1.1.2.1
Use words, pictures, objects, length-based models (connecting cubes), numerals and number lines to model and solve addition and subtraction problems in part-part-total, adding to, taking away from and comparing situations.
1.1.2.2
Compose and decompose numbers up to 12 with an emphasis on making ten. For example: Given 3 blocks, 7 more blocks are needed to make 10.
1.1.2.3
Recognize the relationship between counting and addition and subtraction. Skip count by 2s, 5s, and 10s.

Benchmark Cluster

1.1.2.1
Use words, pictures, objects, length-based models (connecting cubes), numerals and number lines to model and solve addition and subtraction problems in part-part-total, adding to, taking away from and comparing situations.

1.1.2.2
Compose and decompose numbers up to 12 with an emphasis on making ten.

1.1.2.3
Recognize the relationship between counting and addition and subtraction. Skip count by 2s, 5s, and 10s.

What students should know and be able to do [at a mastery level] related to these benchmarks:

• solve addition and subtraction problems using a variety of thinking strategies
• solve addition and subtraction problems in part-part-total, adding to, taking away from and comparing situations
• use number lines and empty number lines to represent their thinking when combining/separating numbers in addition and subtraction
• explain how they solved adding to, part-part-whole, taking away from, and comparing problems
• combine and partition numbers to 12 without counting by ones. For example, recognize that 6 is 5 and 1, 3 and 3, 4 and 2, etc.
• composing and decomposing is continued to all numbers to 12
• use doubles and doubles plus 1 to find ways to break numbers apart and put them together. For example 7 can be seen as double 3 plus one or 4 and 3.
• skip count by 2s, 5s and 10s
• recognize the relationship between skip counting and addition and subtraction

Work from previous grades that supports this new learning includes:

Accurately count a collection of objects and represent the count with a numeral

• Compose and decompose numbers up to ten with pictures and objects
• Count forward and backward to at least 20
• Find sums and differences of numbers between 0 and 10 using pictures and objects.

Correlations

NCTM Standards

Understand meanings of operations and how they relate to one another

Pre-K-2 Expectations:

• understand various meanings of addition and subtraction of whole numbers and the relationship between the two operations;
• understand the effects of adding and subtracting whole numbers;
• understand situations that entail multiplication and division, such as equal groupings of objects and sharing equally.

Compute fluently and make reasonable estimates

Pre-K - Grade 2 Expectations:

• develop and use strategies for whole-number computations, with a focus on addition and subtraction;
• develop fluency with basic number combinations for addition and subtraction;
• use a variety of methods and tools to compute, including objects, mental computation, estimation, paper and pencil, and calculators.

Common Core State Standards

Represent and solve problems involving addition and subtraction.

1.OA.1. Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.1

1.OA.2. Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Understand and apply properties of operations and the relationship between addition and subtraction.

1.OA.3. Apply properties of operations as strategies to add and subtract.2 Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)

1.OA.4. Understand subtraction as an unknown-addend problem. For example, subtract 10 - 8 by finding the number that makes 10 when added to 8. Add and subtract within 20.

Add and subtract within 20.

1.OA.5. Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

1.OA.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).

Use place value understanding and properties of operations to add and subtract.

1.NBT.4. Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

1.NBT.5. Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

1.NBT.6. Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

## Misconceptions

Student Misconceptions

Student Misconceptions and Common Errors

Students may think....

• the commutative property applies in subtraction.
• they can start at either number when counting back to subtract in the same way they could count up when adding.
• the only way to combine quantities is to count each quantity by ones and then count the combined groups. For example, when combining a group of six and a group of three, they first count 1, 2, 3, 4, 5, 6 and then 1, 2, 3. The groups are then combined and counted 1, 2, 3, 4, 5, 6, 7, 8, 9.

## Vignette

• In the Classroom

First graders in Mrs. B's class are using the Empty Number Line (ENL) to represent their strategies for the combination of eight and four. The strategies used include counting from one (Kahlid), counting-on (Elvis), and the grouping strategy of making ten (Malia).

Mrs. B:  Here are eight red marbles.  Here are four blue marbles . If we put all the marbles together, how many would there be? Mrs. B. displays eight red marbles and four blue marbles.

Kahlid:  I said one, two, three, four, five, six, seven, eight.  Then I went nine, ten, eleven, twelve.  There are twelve.

Mrs. B:  Kahlid, why did you count to eight?

Kahlid:  I started with the red marbles. I know there are eight so I counted to eight 1,2, 3, 4, 5, 6, 7, 8.

Mrs. B.:  Why did you continue to count after you reached eight? You said there were
eight red marbles.

Kahlid:  After the red marbles I counted more for the blue marbles..  I had to count
four more after eight  9, 10, 11, 12. Eight red and four blue marbles make
twelve marbles.

Mrs. B Draws on the ENL to represent Kahlid's strategy.

Kahlid counted the red marbles by one, 1, 2, 3, 4, 5, 6, 7, 8 and
then continued counting by one to add the blue marbles  9, 10, 11, 12.

Mrs. B. :  Did anyone else use Kahlid's count by one strategy?

Several students indicated they had used the count by one strategy.

Mrs. B. added a label to the ENL used for Kahlid's strategy.

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Kahlid:  I said one, two, three, four, five, six, seven, eight.  Then I went nine, ten, eleven, twelve.  There are twelve.

Mrs. B:  Kahlid, why did you count to eight?

Kahlid:  I started with the red marbles. I know there are eight so I counted to eight 1,2, 3, 4, 5, 6, 7, 8.

Mrs. B.:  Why did you continue to count after you reached eight? You said there were
eight red marbles.

Kahlid:  After the red marbles I counted more for the blue marbles..  I had to count
four more after eight  9, 10, 11, 12. Eight red and four blue marbles make
twelve marbles.

Mrs. B Draws on the ENL to represent Kahlid's strategy.

Kahlid counted the red marbles by one, 1, 2, 3, 4, 5, 6, 7, 8 and
then continued counting by one to add the blue marbles  9, 10, 11, 12.

Mrs. B. :  Did anyone else use Kahlid's count by one strategy?

Several students indicated they had used the count by one strategy.

Mrs. B. added a label to the ENL used for Kahlid's strategy.

Count by One Strategy

Mrs. B.:  Did anyone use a different strategy?  Several hands went up.

Mrs. B.:  Elvis, how did you combine the marbles?

Elvis:  I got eleven for my answer. I said eight, nine, ten, eleven.

Mrs. B:  How many did you start with Elvis?  Draws ENL.

Elvis:  Eight.

Mrs. B:  Why did you start with eight?

Elvis:  There are eight marbles..the red ones.

Mrs. B. Records an 8 on the ENL to show Elvis's thinking.

Mrs. B:  What did you do next Elvis?

Elvis:  There are blue marbles. So I did four more.

Mrs. B:  Elvis, would you show us how you thought about four more on the empty number line.

Elvis starts at eight and records the four hops, one for each of the blue marbles.

Mrs. B:  Elvis I see you started at eight, then made four hops. Say the numbers out aloud as you use your ENL to find how many marbles there are in all.

Elvis:  I started at eight. Then (pointing to each hop in turn) there's nine, ten, eleven, twelve.  Oh! The answer is really twelve!

Mrs. B:  Elvis used the Counting on Strategy. He started at eight and counted on four more 8, 9, 10, 11, 12. Did anyone else use this strategy?

A few children indicated they had used this strategy. One student said she had started at 4 and then counted eight more and the answer was still twelve!

Mrs. B. labeled the ENL used to show Elvis's strategy.

Counting On Strategy

Mrs. B:  Did anyone combine eight red marbles and four blue marbles
in a different way?

One hand went up.

Mrs. B:  Malia would you please share your strategy.

Malia:  I just knew it was twelve. It's easy. I made ten and then I thought ten and two is twelve. I know it is  twelve.

Mrs. B:  Tell us more about making a ten, Malia. How did you think about making ten?

Malia:  Well there are eight red marbles. Eight and two make ten.

Mrs. B:  I see the eight is for the red marbles. Where did the two come from? I don't see a group of two marbles.

Malia:  The two came from the blue marbles.  I broke four apart--two and two.  I used two of them with the eight to make ten.

Mrs. B:  Do you see what Malia did?  She used two of the blue marbles to make a group of ten. Eight red marbles and two blue marbles make ten marbles.

Mrs. B. uses the marbles to demonstrate Sara's thinking.

Malia:  There are still two blue marbles left after I made the ten so I did ten and two more. That makes twelve.

Mrs. B draws an ENL and Malia records her thinking.

Mrs. B:  Malia used the Make 10 Strategy.  She stared at 8, went 2 more to make 10 and then went 2 more to make 12. Eight red marbles and four blue marbles make twelve marbles.

Mrs. B. labeled the ENL used to show Sara's strategy.

Make a Ten Strategy

Students are now going to solve other combining problems using one or more of the strategies already identified. They will also represent a partner's strategy on an ENL.

## Resources

Instructional Notes

Teacher Notes

• Students may need support in further development of previously studied concepts and skills.
• Using mental images of numbers aids in the ability to recall basic facts. Visualizing a quantity without counting is called subitizing. Students who are able to subitize can tell you the number of dots after seeing an image for just few seconds; they know the number without counting each dot. Some materials to help develop subitizing are shown below.

• Thinking about relationships between numbers and operations facilitates the learning of basic facts with understanding.
• Addition terminology: add, plus, combine. sum, join, put together
• Subtraction terminology: separate, take away, minus, difference, compare
• Number Talks are a great way to keep strategies visible in the classroom.  During a Number Talk, students are given three to five problems related to a specific strategy or problem.

Adding one is the focus of this set of problems.

 3 + 1 9 + 1 1 + 7 1 + 5

Subtracting two is the focus of this set of problems.

 4 - 2 8 - 2 7 - 2 10 - 2
• Students count back or use a comparison model to solve subtraction tasks and represent their thinking with pictures, words, and manipulatives.
• Using an empty number line also helps first graders represent their thinking.

For example, in the problem 5 + 6 a student might start at 5, add 5 (using a double) and then add 1 more to get to 11.

• Students need to use a variety of strategies when solving problems involving basic facts. These strategies include:
• doubles (4 + 4; 7 + 7; etc.)
• doubles plus 1 (7 + 8   think 7 + 7 + 1)
• doubles plus 2 (5 + 7   think 5 + 5 + 2 OR double the number in between: 6 + 6, because you're taking one off the 7 - leaving a 6, and adding that one to the five, creating a 6, resulting in 6 + 6)
• counting on - when adding 1, 2, or 3  (5 + 3, start with the larger number, 5, and count on three -- 6, 7, 8
• counting back - when subtracting 1, 2, or 3
• fact families:  7 + 5 = 12, 5 + 7 = 12, 12 - 7 = 5, 12 - 5 = 7
• adding 9 (4 + 9  think 4 + 10 - 1 = 14 - 1 = 13)
• adding 8  (4 + 8 think 4 + 10 - 2 = 14 - 2 = 12)
• make a ten
• relating addition to subtraction and subtraction to addition
(11 - 5 = 6 because 5 + 6 = 11 and 13 - 6 = 7 because 6 + 7 = 13)
• While the idea of teaching "key words" might be appealing it does not support student comprehension of word problems and can lead to incorrect solutions. Another approach might be "story problem theater." When students listen to a story problem and "act out" what is happening it is easier to understand what mathematics might be involved.
• Real-world contexts help first graders make sense of the problem and begins the process of choosing a solution strategy.
• First grade students should experience the following types of combining and separating problems (based on the work of Carpenter, 1999, p.12) The equations describe the problem structure and may not be the equation used to find the solution.

Combining and separating problems involve an action and are easily demonstrated by first graders. Result unknown is the most common and the easiest to solve. It is important that students experience problems involving change unknown and start unknown. Of these, start unknown is the most challenging.

Combining

 Result Unknown There are five dogs in a yard. Three more dogs walked into the yard. How many dogs are in the yard? 5 + 3 = ___ Change Unknown There are five dogs in the yard.  Some more dogs walked into the yard. There are now eight dogs in the yard. How many dogs came into the yard? 5 + ___ = 8 Start Unknown There were some dogs in the yard.  Three dogs came into the yard. There are now eight dogs in the yard.            How many dogs were in the yard to start with? ___ + 3 = 8

Separating

 Result Unknown There were eight dogs in the yard. Five dogs ran out of the yard. How many dogs are still in the yard? 8 - 5 = __ Change Unknown There were eight dogs in the yard. Some ran away.   Three dogs are still in the yard. How many dogs ran away? 8 - ___ = 3 Start Unknown There were some dogs in the yard. Five dogs ran away. Now there are three dogs in the yard. How many dogs were in the yard to start with? ___ - 5 = 3

Part-Part-Whole problems do not involve an action. They cannot be acted out or demonstrated. The equations describe the problem structure and may not be the equation used to find the solution. The location of the unknown in part-part-whole problems can change as well.

Part-Part-Whole

 Whole Unknown There are five big dogs and three little dogs in the yard. How many dogs are in the yard? 5 +  3 = __ Part Unknown There are eight dogs in the yard. Five dogs are big and the rest are little.  How many little dogs are in the yard? 8 = 5 +  ___

Comparing problems do not involve any action. Nothing is being gained or lost. Two amounts are being compared in order to find the difference or the difference is given with one amount and students are asked to find the other amount. Students may choose to add or subtract when finding an answer.

Comparing

 Difference Unknown There are five big dog and three little dogs in the yard. How many more big dogs than little dogs are in the yard? Compare Quantity Unknown There are three little dogs in the yard. There are two more big dogs than little dogs in the yard. How many big dogs are in the yar Referent  Unknown There are five big dogs in the yard. There are two more big dogs than little dogs in the yard. How many little dogs are in the yard?
• Modeling word problems is critical as students develop an understanding of

operations and the relationships that exist between and among those operations. The bar model is an effective tool that can be used to represent the relationships in a variety of problem solving situations.

For more information on the bar model, read Modeling Word Problems in the Mathematics Best Practice tab in the Resources section of the Minnesota Mathematics Framework.

Questioning

Good questions, and good listening, will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."

Getting Started
What do you need to find out?
What do you know now? How can you get the information? Where can you begin?
What terms do you understand/not understand?
What similar problems have you solved that would help?

While Working
How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if...?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any patterns or relationships that will help you solve this?
How does this relate to...?
Why did you...?
What assumptions are you making?

Reflecting about the Solution
How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
What did you try that did not work? Has the question been answered?
Can the explanation be made clearer?

Responding (helps clarify and extend their thinking)
Tell me more.
Can you explain it in a different way?
Is there another possibility or strategy that would work?
Is there a more efficient strategy?
Help me understand this part ...

(Adapted from They're Counting on Us, California Mathematics Council, 1995)

Instructional Resources

NCTM Illuminations

In this unit, students explore five models of subtraction (counting, sets, number line, balanced equations, and inverse of addition) using connecting cubes. The lessons focus on the comparative mode of subtraction. In them, children explore the relationship between addition and subtraction, write story problems in which comparison is required, and practice the subtraction facts.
The unit consists of lessons that build on and extend early understandings about counting, addition, and subtraction in the comparative mode. Familiarity with the many models of subtraction in each mode is important to children's success in problem solving. Together the five models provide a structure for developing a rich conceptual schema for subtraction.

• #### Let's Count to 20

In this unit, students make groups of 10 to 20 objects, connect number names to the groups, use place value concepts, compose and decompose numbers, and use numerals to record the size of a group. Visual, auditory, and kinesthetic activities are included in each lesson.This unit is most appropriate for students typically in the first grade.

• #### Ten Frame

Thinking about numbers using frames of 10 can be a helpful way to learn basic number facts. The four games that can be played with this applet help to develop counting and addition skills. (This applet works well when used in conjunction with the Five Frame applet.)

• #### Let's Count To 10

Lesson 1 - Building Numbers to Five
In this lesson, students make groups of zero to 5 objects, connect number names to the groups, compose and decompose numbers, and use numerals to record the size of a group. Visual, auditory, and kinesthetic activities are used to help students begin to acquire a sense of number.
Lesson 2 - Writing Numerals to Five
As students construct groups of a given size, recognize the number in the group, and record that number in numerals, they learn the number words through 5 in order (namely, to rote count), and develop the ability to count rationally.
Lesson 3 - Building Sets of Six
In this lesson, students construct sets of six, compare them with sets of a size up to six objects, and write the numeral 6. They also show a set of six on a "10" Frame and on a recording chart.
Lesson 4 - Building Sets of Seven
Students construct and identify sets of seven objects. They compare sets of up to seven items, and record a set of seven in chart form.
Lesson 5 - Building Sets of Eight
Students explore the number 8. They make and decompose sets of eight, write the numeral 8, and compare sets of up to eight objects.
Lesson 6 - Building Sets of Nine
Students construct sets of up to nine items, write the numeral 9, and record nine on a chart. They also play a game that requires identifying sets of up to nine objects.
Lesson 7 - Building Sets of Ten
Students explore sets of up to 10 items and practice writing the numbers 0 through 10. Students count back from 10, identify sets of up to 10 objects, and record 10 on a chart. They also construct and decompose sets of up to 10 items.
Lesson 8 - Wrapping Up the Unit
Students review this unit by creating, decomposing, and comparing sets of zero to 10 objects and by writing the cardinal number for each set.

This unit focuses on comparative subtraction. The students use fish-shaped crackers to explore five meanings for the operations of subtraction (counting, sets, number line, balance, and inverse of addition). Comparative subtraction extends the students early understandings about counting, addition, and subtraction in the take-away mode. In this unit, the students investigate properties of subtraction, represent subtraction in objects and pictures, and record subtraction in both vertical notation and equations. They create and solve problems involving comparative subtraction. Students answer such questions as "How many more?" and "How many less?" Missing addend activities provide students with an experience in algebraic thinking.

Other Instructional Activities

Composing and Decomposing Numbers

• Butterfly Wings
Use numbers to make another number.
• Speedy Pictures 1
• Flash fingers, dice or a ten frame to students and have them answer questions like: What did you see?  How did you see it?  How many are there altogether?  The flash time is adjustable.
• Composing Numbers
• Speedy pictures of fingers, dice, math rack, egg carton, and coins. You can select the picture and the flash time.
• Combining Numbers to 10  Adapted/Taken from:  Teaching Number Advancing Children's Skills & Strategies, by Wright, Martland, Stafford and Stanger, 2006
• Flash a ten frame with 5 blue counters in the upper row, 3 red and 2 blue in the lower row (Illustrating seven and three).
• How many blue counters?  How many red counters ? How many counters altogether?
• Similarly with other combinations to 10 (9 and 1, 1 and 9) and so on).

•  Partitioning 10  Adapted/Taken from:  Teaching Number Advancing Children's Skills & Strategies, by Wright, Martland, Stafford and Stanger, 2006.
• Flash an empty 10 frame.

How many squares altogether?
I am going to put on 8 red counters.  How many empty squares will there be?  Place on 8 red counters.  Watch to see if you were correct.  Flash the ten frame.  Were you correct?
How many squares altogether? How many counters?  How many empty squares?
Repeat for other addend pairs for ten --  7 and 3, 6 and 4, etc.

• Make Ten Fish  Adapted/taken from:  Teaching Number Advancing Children's Skills & Strategies, by Wright, Martland, Stafford and Stanger, 2006.
• Combining and partitioning Numbers 1-10

Materials: At least four sets of numeral cards from 1 to 10
Intended learning: To learn the partitions of ten.
Description: The activity is played the same as Fish. Instead of traditional playing cards, numeral cards 1-9 ( with or without 10 frames) are used. Five cards are dealt to each player. The remaining cards are placed in a stack in the center of the players. Players try to make matching pairs of cards adding to 10. Players take turns to ask another player for a particular card. If the player being asked does not have the required card they say Fish! The player seeking the card takes the top card off the stack. The game continues until one player has no more cards. Players see how many pairs they have.
Note:
The game provides examples of missing subtrahend tasks ( for example: I have six.
How many more will make ten?) This game can be modified for structuring 5 through 9 and 12.

• "Quick images" ( dot patterns, or ten-frames) can be flashed for 3 seconds. Students are then asked to describe how they saw the quantity on the card. Flashed or partially hidden materials encourage students to use imagery when solving problems.
• Count Around  Adapted/Taken from:  Teaching Number Advancing Children's Skills & Strategies, by Wright, Martland, Stafford and Stanger, 2006

Choose a number range. For example, 5 to 45, counting by fives. Children stand in a circle and count around, each child saying the next number in the sequence. Start the count at 5, the smallest number in the range. The child who says 45, the largest number in the range, sits down. Counting continues as the next child begins the count again at 5, the smallest number in the range. Try counting by ten and two when playing Count Around.

Cavanagh, M., Dacey, L., Findell, C., Greenes, C., Sheffield, L., & Small, M. (2001). Navigating through number and operations in prekindergarten-grade 2. Reston, VA: National Council of Teachers of Mathematics.

Confer, C. (2005). Teaching number sense: grade 1. Sausalito, CA:  Math Solutions.

Conklin, M. (2010). It makes sense! Using ten-frames to build number sense, grades k-2. Sausalito, CA: Math Solutions Press.

O'Connell, S., & SanGiovanni, J. (2011). Mastering the basic math facts in addition and subtraction, strategies, activities & interventions to move students beyond memorization.  Portsmouth, NH: Heinemann.

Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K-5. Sausalito. CA: Math Solutions.

Richardson, K. (1999). Developing number concepts addition and subtraction. White Plains, New York: Dale Seymour Publications.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction.  New York, NY: Teachers College Press.

Van de Walle, J., Karp, K., Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.) Boston, MA: Allyn & Bacon.

Van de Walle, J. & Lovin, L. (2006). Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.

New Vocabulary

"Vocabulary literally is the

key tool for thinking."

Ruby Payne

Mathematics vocabulary words describe mathematical relationships and concepts and cannot be understood by simply practicing definitions.  Students need to have experiences communicating ideas using these words to explain, support, and justify their thinking.

Learning vocabulary in the mathematics classroom is contingent upon the following:

Integration:   Connecting new vocabulary to prior knowledge and previously learned vocabulary.  The brain seeks connections and ways to make meaning which occurs when accessing prior knowledge.

Repetition:    Using the word or concept many times during the learning process and connecting the word or concept with its meaning.  The role of the teacher is to provide experiences that will guarantee connections are made between mathematical concepts, relationships, and corresponding vocabulary words.

Meaningful    Multiple and varied opportunities to use the words in context.  These

Use:              opportunities occur when students explain their thinking, ask clarifying questions, write about mathematics, and think aloud when solving problems.  Teachers should be constantly probing student thinking in order to determine if students are connecting mathematics concepts and relationships with appropriate mathematics vocabulary.

Strategies for vocabulary development

Students do not learn vocabulary words by memorizing and practicing definitions. The following strategies keep vocabulary visible and accessible during instruction.

Mathematics Word Bank:  Each unit of study should have word banks visible during instruction.  Words and corresponding definitions are added to the word bank as the need arises.  Students refer to word banks when communicating mathematical ideas which leads to greater understanding and application of words in context.

Labeled pictures and charts:  Diagrams that are labeled provide opportunities for students to anchor their thinking as they develop conceptual understanding and increase opportunities for student learning.

Frayer Model: The Frayer Model connects words, definitions, examples and non-examples.

Example/Non-example Charts: This graphic organizer allows students to reason about mathematical relationships as they develop conceptual understanding of mathematics vocabulary words.  Teachers should use these during the instructional process to engage student in thinking about the meaning of words.

Vocabulary Strips:  Vocabulary strips give students a way to organize critical information about mathematics vocabulary words.

 word definition illustration

Encouraging students to verbalize thinking by drawing, talking, and writing increases opportunities to use the mathematics vocabulary words in context.

Additional Resources for Vocabulary Development

Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.

Sammons, L. (2011).  Building mathematical comprehension: Using literacy strategies to make meaning.  Huntington Beach, CA: Shell Education.

Professional Learning Communities

Professional Learning Communities

Reflection - Critical Questions regarding the teaching and learning of these benchmarks.

What are the key ideas related to composing and decomposing numbers at the first grade level? How do student misconceptions interfere with mastery of these ideas?

What representations should a student be able to make for the number 9 or 12 if they understand composition and decomposition?

When checking for student understanding of composition/decomposition of numbers, what should teachers:

• listen for in student conversations?
• look for in student work?
• ask during classroom discussions?

How can teachers assess student learning related to composition/decomposition of numbers?

Examine student work related to a task involving composition/decomposition of numbers. How do you know a student is proficient? Using three pieces of student work, determine what student understanding is observed through the work.

What are the key ideas related to modeling and solving addition and subtraction problems at the first grade level? How do student misconceptions interfere with mastery of these ideas?

What are the key ideas related to skip counting and addition and subtraction?

What models should a student be able to make when solving addition and subtraction problems? What strategies would a first grader use when solving a problem? What do the strategies children use to solve problems tell teachers about student understanding?

When checking for student understanding when solving addition and subtraction problems, what should teachers

• listen for in student conversations?
• look for in student work?
• ask during classroom discussions?

How can teachers assess student learning related to solving problems involving addition and subtraction?

Examine student work related to a task involving modeling and solving addition and subtraction problems. What evidence do you need to say a student is proficient?  Using three pieces of student work, determine what student understanding is observed through the work.

How can teachers assess student learning related to these benchmarks?

How is this benchmark related to other benchmarks at the first grade level?

Professional Learning Community Resources

Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.

Caldwell, J., Karp, K., Bay-Williams, J., Rathmell, E., & Zbiek, R. (2011). Developing essential understanding of addition and subtraction for teaching mathematics in prekindergarten-grade 2. Reston, VA: National Council of Teachers of Mathematics

Carpenter, T., Fennema, E., Franke, M., Levi, L., & Empson, S. (1999). Children's mathematics cognitively guided instruction. Portsmouth, NH: Heinemann.

Chapin, S., & Johnson, A. (2006). Math matters, Understanding the math you teach grades K-8, (2nd, ed.). Sausalito, CA: Math Solutions Press.

Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions:  Using math talk to help students learn (Grades K-6). Sausalito, CA: Math Solutions.

Fuson, K., Clements, D., & Beckmann, S.. (2009). Focus in grade 1 teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.

Fosnot, Catherine., Dolk, Maarten. (2001). Young mathematicians at work, constructing number sense, addition, and subtraction. Portsmouth, NH: Heinemann.

Hyde, Arthur. (2006) Comprehending math adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.

Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA:  National Council of Teachers of Mathematics.

Murray, M. (2004) Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.

Sammons, L., (2011). Building mathematical comprehension-Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.

Wright, R., Stanger, G., Stafford, A., & Martland, J. (2006). Teaching number in the classroom with 4-8 year-olds. Paul Chapman Publishing.

References

Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.

Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA.: Corwin Press.

Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting english language learners in math class grades k-2. Sausalito, CA: Math Solutions Publications.

Burns, M. (Ed.). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.

Caldwell, J., Karp, K., Bay-Williams, J., Rathmell, E., & Zbiek, R. (2011). Developing essential understanding of addition and subtraction for teaching mathematics in prekindergarten-grade 2. Reston, VA: National Council of Teachers of Mathematics

Carpenter, T., Fennema, E., Franke, M., Levi, L., & Empson, S. (1999). Children's mathematics cognitively guided instruction. Portsmouth, NH: Heinemann.

Cavanagh, M., Dacey, L., Findell, C., Greenes, C., Sheffield, L., & Small, M. (2001). Navigating through number and operations in prekindergarten-grade 2. Reston, VA: National Council of Teachers of Mathematics.

Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.

Chapin, S. and Johnson, A. (2006). Math matters, understanding the math you teach, grades k-8. (2nd ed.). Sausalito, CA: Math Solutions Press.

Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K-6). Sausalito, CA: Math Solutions.

Confer, C.. (2005). Teaching number sense: Grade 1. Sausalito, CA: Math Solutions.

Dacey, L., & Salemi, R. (2007). Math for all: Differentiating instruction k-2. Sausalito, CA: Math Solutions.

Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.

Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.

Fosnot, Catherine., Dolk, Maarten. (2001). Young mathematicians at work: Constructing number sense, addition, and subtraction. Portsmouth, NH: Heinemann.

Fuson, K., Clements, D., & Beckmann, S (2009). Focus in grade 1 teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.

Hyde, Arthur. (2006). Comprehending math adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.

Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: helping children learn mathematics. Washington, DC.: National Academies Press.

Leinwand, S., (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH:  Heinemann.

Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.

Murray, M. & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann.

Murray, M. (2004). Teaching Mathematics Vocabulary in Context. Portsmouth, NH: Heinemann.

O'Connell, S., & SanGiovanni, J. (2011). Mastering the basic math facts in addition and subtraction, strategies, activities & interventions to move students beyond memorization.  Portsmouth, NH:  Heinemann.

Parrish, S. (2010). Number talks: helping children build mental math and computation strategies grades K-5. Sausalito. CA: Math Solutions.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.

Sammons, L. (2011). Building mathematical comprehension-using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.

Schielack, J., Charles, R., Clements, D., Duckett, P., Fennell, F., ... & Zbiek, R. M., (2006). Curriculum focal points for kindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: NCTM.

Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction.  New York, NY: Teachers College Press.

Van de Walle, J., Karp, K., Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.

Van de Walle, J., & Lovin, L.  (2006). Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.

West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.

Wright, R., Stanger, G., Stafford, A., & Martland, J. (2006). Teaching number in the classroom with 4-8 year-olds. Paul Chapman Publishing.

## Assessment

Performance Assessment

• ##### Additive Tasks Involving Two Screened Collections  1.1.2.1

adapted from Wright, R., Stanger, G., Stafford, A., & Martland, J. (2006.) Teaching number in the Classroom with 4-8 year-olds. Paul Chapman Publishing.

Materials:  Collection of counters of one color (red) and a collection of another color (blue), one screen (piece) of cardboard or cloth

What to do and say: Briefly show and then screen (cover) eight red counters. Here are 8 red counters. Here are 2 blue counters. How many counters all together? ___
Follow the same procedure using collections such as
9 red, and 3 blue, ___
5 red and 3 blue, ___
7 red and 2 blue, ___

The purpose of these tasks is to see if the child can "count on."
Solution:  Student correctly counts on to determine the total number of objects.
Benchmark:  1.1.2.1

• #### Removed Items Task Involving a Screened Collection  1.1.2.1

Materials: A collection of counters of one color (red); two screens
What to do and say: Briefly display and the screen 7 red counters. Here are seven red counters. I am going to take two away. Remove 2 counters and screen them without the child seeing them. I took away two of the red counters. How many counters are left?
Similarly:7 remove 2___; 10 remove 3___; 12 remove 4___ ; 16 remove 2____
Keep the number of items removed in the range of 2 to 5.

Wright, R., Stanger, G., Stafford, A., & Martland, J. (2006). Teaching number in the classroom with 4-8 year-olds. Paul Chapman Publishing.

• Partitions of 5 and 10. Adapted from Wright, R., Stanger, G., Stafford, A., & Martland, J. (2006). Teaching number in the classroom with 4-8 year-olds.Paul Chapman Publishing.

Materials: None
What to do or say:
I will say a number and you say a number that goes with it to make five:
3, ___              4,___               2, ___              1___.
I will say a number and you say a number that goes with it to make ten:
9___,               5___,               8,___               3,___               0___
6___                7___                2___                4___                1___
Solution:  Student correctly gives the number needed to make 5/10.
Benchmark:  1.1.2.2

• Count by 10 to 120.
Count by 5 to 100.
Count by 2 to 50.
Solution:  Student correctly counts by 10, 5, and 2.
Benchmark:  1.1.2.3
• There are 10 pieces of fruit in your basket. Some are oranges and some are bananas. How many of each could there be?  Find as many combinations as you can.  Use pictures, numbers or words to show your thinking.

Solution:  Student correctly identifies combinations of ten oranges and bananas.
Benchmark  1.1.2.2

• Peter has 8 toy cars. He gives 5 toy cars to his friend Mike.

How many cars does Peter have now?
Solution:  Peter has 3 toy cars now.
Benchmark:  1.1.2.1

• Bisola has a bag of 3  marbles. Sami gives her 6 marbles.

How many does Bisola have now?
Solution:  Bisola has 9 marbles.
Benchmark:  1.1.2.1

## Differentiation

Struggling Learners

In early addition and subtraction struggling learners will need many experiences using materials they can see and touch. Numerals do not provide for counting, so students solve problems with a variety of concrete materials. Struggling students also need support in forward and backward number word sequences. Problems in addition and subtraction are selected in the range of counting sequences in which a student is most successful.

Concrete - Representational - Abstract Instructional Approach

(Adapted from The Access Center: Improving Access for All K-8 Students)

The Concrete-Representational-Abstract Instructional Approach (CRA) is a research-based instructional strategy that has proven effective in enhancing the mathematics performance of students who struggle with mathematics.

The CRA approach is based on three stages during the learning process:

Concrete     -     Representational     -     Abstract

The Concrete Stage is the doing stage. The concrete stage is the most critical in terms of developing conceptual understanding of mathematical skills and concepts.  At this stage, teachers use manipulatives to model mathematical concepts. The physical act of touching and moving manipulatives enables students to experience the mathematical concept at a concrete level.  Research shows that students who use concrete materials develop more precise and comprehensive mental representations, understand and apply mathematical concepts, and are more motivated and on-task.   Manipulatives must be selected based upon connections to the mathematical concept and the students' developmental level.

The Representational Stage is the drawing stage.  Mathematical concepts are represented using pictures or drawings of the manipulatives previously used at the Concrete Stage. Students move to this level after they have successfully used concrete materials to demonstrate conceptual understanding and solve problems.  They are moving from a concrete level of understanding toward an abstract level of understanding when drawing or using pictures to represent their thinking.  Students continue exploring the mathematical concept at this level while teachers are asking questions to elicit student thinking and understanding.

The Abstract Stage is the symbolic stage.  Teachers model mathematical concepts using numbers and mathematical symbols.  Operation symbols are used to represent addition, subtraction, multiplication and division.  Some students may not make a clean transfer to this level. They will work with some symbols and some pictures as they build abstract understanding.  Moving to the abstract level too quickly causes many student errors.   Practice at the abstract level will not lead to increased understanding unless students have a foundation based upon concrete and pictorial representations.

Bender, W. (2009). Differentiating math instruction-Strategies that work for k-8 classroom! Thousand Oaks, CA: Corwin Press.

Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades k-2.Sausalito, CA: Math Solutions.

Murray, M. & Jorgensen, J. (2007). The differentiated math classroom-A guide for teachers k-8. Portsmouth, NH: Heinemann.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction.  New York, NY: Teachers College Press.

Van de Walle, J., Karp, K., Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.) Boston, MA: Allyn & Bacon.

Van de Walle, J. & Lovin, L. (2006). Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.

English Language Learners

English Language Learners

It is important for teachers to be consistent with their own language when talking about operations with number. For example, take-away, minus, and subtract all can be confusing for ELL students when teachers use words interchangeably.

• Connecting counting materials with words is helpful for ELL students. When adding or subtracting, have counters, dominoes, bead racks, etc. available for students to solve.
• Drawing pictures when solving problems is a powerful way to communicate mathematical thinking.
• Word banks need to be part of the student learning environment in every mathematics unit of study.
• Use vocabulary graphic organizers such as the Frayer model (see below) to emphasize vocabulary words count, first, second, third, etc.

• Sentence Frames

Math sentence frames provide support that English Language Learners need in order to fully participate in math discussions.  Sentence frames provide appropriate sentence structure models, increase the likelihood of responses using content vocabulary, help students to conceptualize words and build confidence in English Language Learners.

Sample sentence frames related to these benchmarks:

 I found the answer by __________________________________________.
 I can make 10 with _________ and ___________.
 I can make ______ with _______ and _________.
• When assessing the math skills of an ELL student it is important to determine if the student has difficulty with the math concept or with the language used to describe the concept and conceptual understanding.

Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications.

Extending the Learning

Students who are successful in early addition and subtraction should be encouraged to begin using non-count by one strategies to solve addition and subtraction problems. For example students who can combine and partition numbers to 10 and then 20 (without counting) are ready for strategies that require challenging problem solving. As always, it is important for instruction to be embedded in real-world problems that have meaning for students. Children can solve problems mentally as they move away from using materials.

Bender, W. (2009). Differentiating math instruction-Strategies that work for k-8 classrooms! Thousand Oaks, CA.: Corwin Press.

Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades k-2.Sausalito, CA: Math Solutions.

Murray, M. & Jorgensen, J. (2007).The differentiated math classroom-A guide for teachers k-8. Portsmouth, NH: Heinemann.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.

Classroom Observation

What should I look for in the mathematics classroom? (adapted from SciMathMN, 1997)

What are students doing?

• Working in groups to make conjectures and solve problems.
• Solving real-world problems, not just practicing a collection of isolated skills.
• Representing mathematical ideas using concrete materials, pictures and symbols. Students know how and when to use tools such as blocks, scales, calculators, and computers.
• Communicating mathematical ideas to one another through examples, demonstrations, models, drawing, and logical arguments.
• Recognizing and connecting mathematical ideas.
• Justifying their thinking and explaining different ways to solve a problem.

What are teachers doing?

• Making student thinking the cornerstone of the learning process. This involves helping students organize, record, represent, and communicate their thinking.
• Challenging students to think deeply about problems and encouraging a variety of approaches to a solution.
• Connecting new mathematical concepts to previously learned ideas.
• Providing a safe classroom environment where ideas are freely shared, discussed and analyzed.
• Selecting appropriate activities and materials to support the learning of every student.
• Working with other teachers to make connections between disciplines to show how math is related to other subjects.
• Using assessments to uncover student thinking in order to guide instruction and assess understanding.
 Students are . . . Teachers are . . . describing connections between addition and subtraction in a variety of contexts, such as number grids, number lines, counters, ten frames, numerals, connecting cubes, etc. highlighting the connections between addition and subtraction. talking about how they solved problems and listening to other students' solutions. choosing good problems that invite exploration of addition and subtraction solving problems and explaining strategies using questions that help students construct conceptual understanding. Examples of effective questions include: Tell me more about that. Can you show me? Why do you say that? Do you have a different way to solve this? What do you think about that student's answer? What else can you tell me? How do you know? Does that make sense to you? Why do you think that happened? Do you think this will happen every time? finding combinations of numbers, particularly combinations making 10. providing experiences that require students to find combinations. For example, I have 8 pencils and crayons.  How many of each could I have? counting by 2s, 5s and 10s and finding patterns in those sequences. using count around activities to help students learn the counting sequence for counting by 2s, 5s, and 10s. providing opportunities to look for patterns in these counting sequences.

For Mathematics Coaches

Chapin, S., and Johnson, A. (2006).  Math matters: Understanding the math you teach: Grades k-8. (2nd ed.).  Sausalito, CA: Math Solutions.

Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.

Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.

Sammons, L., (2011).  Building mathematical comprehension: Using literacy strategies to make meaning.  Huntington Beach, CA: Shell Education.

West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.

Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA:  Math Solutions.

Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.

Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH:  Heinemann.

Lester, F. (2010). Teaching and learning mathematics: Transforming research for school administrators. Reston, VA:  National Council of Teachers of Mathematics.

Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.

Parents

Parent Resources

Cavanagh, M. (2004). Math to Know: A mathematics handbook. Wilmington, MA:  Great Source Education Group, Inc.

Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA:  Great Source Education Group, Inc

Helping your child learn mathematics

Provides activities for children in preschool through grade 5

What should I look for in the mathematics program in my child's school? A Guide for Parents developed by SciMathMN

Ask the right questions

In helping children learn, one goal is to assist children in becoming critical and independent thinkers. You can help by asking questions that guide, without telling them what to do.

Good questions, and good listening, will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."

Getting Started
What do you need to find out?
What do you know now? How can you get the information? Where can you begin?
What terms do you understand/not understand?
What similar problems have you solved that would help?

While Working
How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if . . . ?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any patterns or relationships that will help you solve this?
How does this relate to ...?
Can you make a prediction?
Why did you...?
What assumptions are you making?

Reflecting about the Solution
How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
What did you try that did not work?
Has the question been answered?
Can the explanation be made clearer?

Responding (helps clarify and extend their thinking)
Tell me more.
Can you explain it in a different way?