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Modeling Addition and Subtraction Structures

I am finishing out the school year with my first and second graders by trying to give them a solid foundation with the different structures for addition and subtraction problems. I’ve been using a marvelous  free resource called Thinking Blocks . They offer a variety of free iPad apps, but I’ve been using the web version on my interactive whiteboard because I wanted to bring in some concrete learning.

This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.

The Thinking Blocks program allows you to choose the type of problem to work with, and we began with part/whole problems with two parts. I made part/whole mats for the kiddos to use (on the backside is a part/whole mat with three parts) because I wanted them to be able to act out the models. My first graders used cubes while my second graders used base-10 blocks to find their solutions.

The problems cover all the structures the students need to be familiar with. All too often, we only expose them to join problems where the parts are known and they are asked to find the sum. But they also need to be able to solve missing part problems. Thinking Blocks mixes up the problem types, which I really like.

For each problem, we went through a series of steps that I hope will become a habit for the students as they work on problems independently. First, we read the problem straight through. When we finished reading, I asked them to tell me what the story was about with one word (pizzas, shoes, pages, etc.). This helps to focus them on what we’re counting in the story. Next, I asked them to summarize the story. This was really tough! I was looking for them to just retell the story without numbers or details. For example: Maria had some marbles. She gave some to her brother and she kept some.

Next, the Thinking Blocks program asks students to place the labels in the model. I love that this is the first step. Too often, students focus only on the numbers and lose the meaning of the problem. We re-read each sentence and discussed if it was a part or the whole and a student placed the labels in the model. After the labels are placed, the program offers a check feature.

Now it’s time to add numbers to the model. Again, we re-read each sentence to make sure we got the numbers in the correct place on the model. A question mark represents the missing number. Once the numbers are placed, the program offers a check feature and also re-sizes the blocks. That allows for some great discussions about what the size of the block tells us about the missing number.

After we had the completed model in Thinking Blocks, I had the students show the problem on their part/whole mat and use the manipulatives to find the missing number (sometimes the whole and sometimes a part).

After working with 2- and 3-part part/whole problems, we tackled comparison problems. This time, the kiddos used two colors of cubes to build trains to match the model. We used blue cubes for the difference and yellow cubes for the two quantities being compared. Again, sometimes the difference is known and one of the quantities being compared is missing, and sometimes the difference is missing.

We are working to the point when we will mix all the problem types. We’ve had such great discussions, and I’m excited to see their mathematical understanding grow each day!

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10 Comments

This is my focus for this summer to make students understand the structure of word problem. The pictures you shared are going to be very helpful. The website I am sharing actually gives a table for the different types of word problems. Thanks!

Thanks for sharing! The link Nuzhat is sharing is from the Common Core standards. You can see it here .

My first graders are about to be taught comparison with our new district math program, which is heavily into passive, picture driven lessons, ugghhh. I love how the difference is shown in blue! In the past, for comparison, I represented the two groups being compared with respective different colors. This model you shared will allow for a visual to see the difference and will be readily incorporated into the part, part, whole idea. When I teach word problems, I constantly ask my first graders to name what they are counting…numbers are the adjectives, what they are counting are the nouns….so important for them to keep that focus!

Love the reminder about having students name what they are counting. So important!

Oh wow, thanks for the link to the Thinking Blocks activities! This will be sooooo helpful with my second graders. :)))

Hi Donna, I love thinking blocks too and I love using bar models with my kids. I’m a math interventionist grades 1-6. My question is: How do you sustain all the great strategies you’re using with your kids, like the work with Thinking Blocks, if they’re not also seeing it in their classroom? I see my kids for 30 minutes every other day. Go Math, the program my district uses, presents multiple strategies (sometimes within 1 lesson). Sometimes the strategies I feel are the most crucial for my population (like bar models or number lines) just become one of myriad others, and it confuses my kids more than helps them. How do you make that work for your kids?

I also love Thinking Blocks! Another wonderful site I use as Math Coach is http://www.gregtangmath.com word problem generator. We can select the problem type (or choose random), choose what we want to be the unknown, and the number range. You can do one at a time or print out 10. Plus when you click on hint there is the tape diagram model that works for all add/sub problem types. Love it!

I’m not aware of a way to choose a specific problem.

You could screen shot the problems and post on an interactive white board or in a pwp. Wouldn’t have the utility of being in the program Thinking Blocks but it would be a work around.

Loved this post. So excited to try these methods and the virtual site. Thank you.

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Adding and subtracting rational numbers to solve problems.

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• Grade Levels 7th Grade
• Related Academic Standards CC.2.2.7.B.3 Model and solve real-world and mathematical problems by using and connecting numerical, algebraic, and/or graphical representations. CC.2.1.7.E.1 Apply and extend previous understandings of operations with fractions to operations with rational numbers.
• Assessment Anchors M07.A-N.1 Apply and extend previous understandings of operations to add, subtract, multiply, and divide rational numbers. M07.B-E.2 Solve real-world and mathematical problems using numerical and algebraic expressions, equations, and inequalities.
• Eligible Content M07.A-N.1.1.1 Apply properties of operations to add and subtract rational numbers, including real-world contexts. M07.B-E.2.1.1 Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate. Example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50 an hour (or 1.1 × $25 = $27.50).
• Competencies

Students will compute and solve problems using rational numbers. They will:

• add and subtract rational numbers.
• solve real-world problems by adding and subtracting rational numbers.

Essential Questions

• How is mathematics used to quantify, compare, represent, and model numbers? 
• How are relationships represented mathematically?
• How can expressions, equations and inequalities, be used to quantify, solve, and/or analyze mathematical situations?
• What makes a tool and/or strategy appropriate for a given task?
• How can recognizing repetition or regularity assist in solving problems more efficiently?
• Rational Number: A number expressible in the form a / b, where a and b are integers, and b ≠ 0.
• Repeating Decimal: The decimal form of a rational number in which the decimal digits repeat in an infinite pattern.

60–90 minutes

Prerequisite Skills

• Lesson 2 Exit Ticket ( M-7-5-2_Exit Ticket and KEY.docx )
• Lesson 2 Small-Group Practice worksheet ( M-7-5-2_Small Group Practice and KEY.docx )
• Lesson 2 Expansion Worksheet ( M-7-5-2_Expansion and KEY.docx )
• Lesson 2 Computations Worksheet ( M-7-5-2_Computations and KEY.docx )
• Lesson 2 Word-Problem Examples ( M-7-5-2_Word Problem Examples and KEY.docx )

Related Unit and Lesson Plans

• Computing and Problem Solving with Rational Numbers
• Adding and Subtracting Rational Numbers on a Number Line
• Multiplying and Dividing Rational Numbers to Solve Problems

Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

• http://ca.ixl.com/math/grade-7/add-and-subtract-rational-numbers

IXL’s Grade 7 Add and Subtract Rational Numbers will give students additional practice with addition and subtraction of rational numbers.

• http://ca.ixl.com/math/grade-8/add-and-subtract-rational-numbers-word-problems

IXL’s Grade 8 Add and Subtract Rational Numbers: Word Problems will give students additional practice with solving word problems that involve rational numbers.

Formative Assessment

• The modeling activity can be used to assess students’ prior knowledge and understanding regarding addition of rational numbers with unlike denominators.
• Activity 1 can be used to assess each student’s ability to create a word problem involving the addition and/or subtraction of rational numbers while also understanding the solution process.
• Use the exit ticket to quickly evaluate student mastery.

Suggested Instructional Supports

Instructional Procedures

As students come into class, have them evaluate the following expressions using a number line.

• 0.75 + 2.95      (3.7)

Walk around the classroom as students are working through the example problems. Briefly discuss the answers and make sure students are comfortable modeling addition and subtraction of rational numbers on a number line before moving on.

“In Lesson 1 of this unit, we learned how to model addition and subtraction of rational numbers on a number line. Today, we are going to focus on performing these computations without the use of a number line. We will then use these skills to solve some real-life problems.”

Computations: Adding and Subtracting Rational Numbers

Before presenting some real-world problems, give students the opportunity to practice adding and subtracting rational numbers without the help of a number line. If necessary, go over the following examples together as a class.

the other is not. Often, when computing with fractions, it is best to write all numbers in fraction form.”

common denominator. The lowest common denominator in this case would be 5.”

numerators as indicated. The denominator will stay as is.”

it is, but we may want to rewrite the fraction as a mixed number to get a better idea of the value.”

Example 2:      − 4.64 + 9.85

• −4.64 + 9.85     “Think about the number line. Based on the signs of each addend, do

you suspect our final answer here will be positive or negative?” (Positive, the absolute value of 9.85 is larger than the absolute value of −4.64.)

• Think:            9.85 – 4.64    
•                             

Distribute the Lesson 2 Computations Worksheet ( M-7-5-2_Computations and KEY.docx ). Instruct students to complete the worksheet individually. Walk around the room as students work to be sure they are on task and performing the computations accurately. Following the worksheet, provide time for students to discuss any problems they encountered, questions they have, or revelations they discovered. First, ask students to describe the computation process used to find each sum or difference. Then confirm their understanding by restating the correct process.

Problem Solving with Rational Numbers

Now it is time for students to apply their understanding of computation to solving real-world problems. Discuss the following examples together as a class.

• “The amount by which his savings have decreased is equal to the difference of 1018.20 and 920.45, written as 1018.20 – 920.45 or 97.75. Thus, Steven’s savings decreased by $97.75.”

Distribute Lesson 2 Word-Problem Examples ( M-7-5-2_Word Problem Examples and KEY.docx ). Have students discuss the solution process for each example problem in a manner similar to the process demonstrated above. Confirm the correct ideas students express. Then say: “Look through the problems you just received. Think of how the example word problems can be solved. Do you need to add or subtract the rational numbers? How will you go about doing this for fractions with unlike denominators, or for mixed numbers?”

Activity 1: Write-Pair-Share

Ask the whole class to think of some real-world contexts that involve the addition or subtraction of rational numbers. Students should make a list of at least five real-world contexts and provide one word problem. Ask students to share their ideas with a partner. Give students about 5 minutes to share contexts and word problems. During this time, each partner may ask questions of the other partner. Then, the whole class can reconvene. One member from each partner group will share the list of real-world contexts and word problems with the class. The teacher may wish to post the real-world contexts and word problems in a file on the class Web page or use them as a classroom display. These student examples would then serve as a reference tool.

Have students complete Lesson 2 Exit Ticket ( M-7-5-2_Exit Ticket and KEY.docx ) at the close of the lesson to evaluate students’ level of understanding.

Use the suggestions in the Routine section to review lesson concepts throughout the school year. Use the small-group suggestions for any students who might benefit from additional instruction. Use the Expansion section to challenge students who are ready to move beyond the requirements of the standard.

• Routine: Throughout the school year, encourage students to be on the lookout for real-world situations that involve the addition or subtraction of rational numbers. Students can present the problems to the teacher, who will facilitate class participation in solving the rational number problem.
• Small Groups: Students who need additional practice can be pulled into small groups to work on the Lesson 2 Small-Group Practice worksheet ( M-7-5-3_Small Group Practice and KEY.docx ). Students can work on the matching together or work individually and compare answers when done.
• Expansion: Students who are prepared for a greater challenge can be given the Lesson 2 Expansion Worksheet ( M-7-5-2_Expansion and KEY.docx ). The worksheet includes more difficult numeric expressions involving rational numbers.

Related Instructional Videos

Insert template.

1.1.2A Addition & Subtraction

Standard 1.1.2.

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.

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.

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

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.

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.

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.
• 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. 

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.

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.

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.

Subtracting two is the focus of this set of problems.

• 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)
• 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.

Separating  

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

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.

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)

NCTM Illuminations

• Comparing Connecting Cubes

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.

• Do It With Dominoes

In this unit, students explore four models of addition (counting, number line, sets, and balanced equations) using dominoes. They also learn about the order (commutative) property, the relation between addition and subtraction, and the result of adding 0. Students also write story problems in which the operation of addition is required and begin to memorize the addition facts. In this unit, students also investigate properties of addition and represent addition in pictures.Because familiarity with the many models of addition is important to students' success in problem solving, four models of addition are presented. Together they provide a structure for developing a rich conceptual schema for addition.

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.

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: Build 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: Build 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: Build 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: Wrappoing 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.

• How Many More Fish     

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

• Number: Counting and Cardinality
• Make Five! (with fingers)
• More, Less, Same with Fingers
• 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.
• Number: Adding and Subtracting
• Mathematics K-6 Resources
• 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.

Additional Instructional Resources:   

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.

"Vocabulary literally is the

 key tool for thinking."

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.

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  

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

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.

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.

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.

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 .

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.

• 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

In early addition and subtraction emergent 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. Emergent learners 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

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. 

Additional Resources

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 .

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:

• 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. 

Additional ELL Resources:

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

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.

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

Parents/Admin

Administrative/Peer 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.

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.

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

For Administrators

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.

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

Help Your Children Make Sense of Math

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?

Related Frameworks

• 1.1.2.1 Represent Addition & Subtraction
• 1.1.2.2 Compose & Decompose
• 1.1.2.3 Relationship Between Counting & Addition/Subtraction

To solve problems in context using addition and subtraction (Part 1)

Lesson details, key learning points.

• In this lesson, we will solve equations using addition and subtraction strategies.

This content is made available by Oak National Academy Limited and its partners and licensed under Oak’s terms & conditions (Collection 1), except where otherwise stated.

Starter quiz

3 questions.

4 Questions

Lesson appears in

Unit maths / addition and subtraction (applying strategies).

Addition Facts 20

Alien Addition

Jet Ski Addition

Monster Subtraction

Number Bonds 20

Koala Karts

Number Pairs 10

Kitten Match

Island Subtraction

Subtraction Facts 20

Jumping Chicks

Place Value Party

Treasure Quest Numbers

Monster Addition

Superhero Subtraction

Treasure Quest Addition

Galaxy Pals 20

Minus Mission

Bingo Number Pairs

Find the Bus Stop

Bridge Builder Addition

Dolphin Feed

Candy Cashier - Money

Math Surpass Compare

Galaxy Pals 100

Addition Blocks

Untamed Number Names

Monster Stroll

Number Trails Addition

Math Racer Addition

Galaxy Pals 200

Number Trails Pro

Bingo Hundreds

Skateboard Pups

Tug Team Addition

Two Digit Addition

Undercover Sums

Bingo 3 Numbers

Tandem Turtles Rounding

Space Jaunt Rounding

Bingo Rounding

Take the Cake Addition

Dolphin Dash

Addition to 100

Addition Snake

Pop Penguin

Amusement Park Addition

Venn Puzzle

Mystery Number

Do/Undo Addition

Subtraction to 100

Zogs and Monsters +

Puzzle Pics Money

Zogs and Monsters $

Math Surpass Prime

Integer Number Puzzles

Combining Integers

Double Cards

One to Twenty

All Aboard Addition

Monster Mischief

Secret Code Addition

Overlap Sums

Number Sequence

Math Man Addition

Missing Digits Addition

Adding Integers

Magic Triangle

Missing Digits Subtraction

Whole Number Puzzles

Sum Links 2

Place Value

Addition Regrouping

Regrouping 3 Digits

Subtraction Regrouping

Subtraction with Zero

Math Word Problems

Thinking Blocks Addition

Word Problem Practice

Triangle Pro

Undercover Pro

Pyramid Pro

Pyramid Double

Grid Junior

Grid Challenge

• Français

• No results found.
• Preparatory

Lesson Plan: One-step Equations: Addition and Subtraction

This lesson plan includes the objectives, prerequisites, and exclusions of the lesson teaching students how to form and solve equations in one step using either addition or subtraction and apply this knowledge to solve word problems.

Students will be able to

• understand that solving a one-step equation means finding the unknown value to make the statement true,
• solve a one-step equation by adding or subtracting a constant,
• solve geometric problems and word problems by forming and solving a one-step equation involving addition or subtraction.

Prerequisites

Students should already be familiar with

• forming linear expressions and equations,
• adding and subtracting positive and negative integers.

Students will not cover

• solving equations that require multiplication or division,
• solving two-step equations,
• solving nonlinear equations,
• solving equations with more than one unknown.

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Subtraction Activities for 1st Grade

Strengthen your child's subtraction skills with interactive educational resources for subtraction for 1st graders online. These learning resources include fun games and worksheets with eye-catching visuals and characters. Get started to help your 1st grader master this concept by engaging their critical thinking.

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• Add With 10 (20)
• Doubles And Near Doubles Addition Strategy (35)
• Make 10 Strategy (18)
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Subtract with Pictures

Subtract Using Pictures Game

Shine bright in the math world by learning how to subtract using pictures.

Subtract and Match the Number Game

Begin the exciting journey of becoming a math wizard by learning to subtract and match the number.

Use Pictures to Subtract Worksheet

Learners must use pictures to subtract to enhance their math skills.

Identify the Correct Subtraction Sentence Worksheet

Focus on core math skills with this fun worksheet by identifying the correct subtraction sentence.

Subtraction Strategies

Subtract using Part Part Whole Models Game

Kids must subtract using part whole models to practice subtraction.

Think Addition using Bar Models Game

Take the first step towards building your math castle by practicing think addition using bar models.

Create Number Sentences Worksheet

Boost your ability to create number sentences by printing this playful worksheet.

Relate Addition & Subtraction Sentences Worksheet

Be on your way to become a mathematician by practicing to relate addition & subtraction sentences.

2-Digit Subtraction

Subtract Multiples to 10 using Visuals Game

Enjoy the marvel of mathematics by exploring how to subtract multiples of 10 using visuals.

Identify the Difference of Multiples of 10 Game

Unearth the wisdom of mathematics by learning how to identify the difference of multiples of 10.

Find 1 More or 1 Less Worksheet

Put your skills to the test by practicing to find 1 more or 1 less.

Find 10 More or 10 Less Worksheet

Make math practice a joyride by solving problems to find 10 more or 10 less.

All Subtraction Resources

Solve Word Problems on Take From Scenarios Game

Enjoy the marvel of math-multiverse by learning to solve word problems on "Take From" scenarios.

Complete the Sentences Using the Bar Model Worksheet

Dive into this fun-filled printable worksheet by completing the given sentences using bar models.

Solve Difference Unknown Scenarios Game

Kids must solve difference unknown scenarios to practice subtraction.

Subtract 1-Digit Numbers from 2-Digit Numbers: Vertical Timed Practice Worksheet

This worksheet tests your speed in subtracting 1-digit numbers from 2-digit numbers in a 1-minute challenge.

Subtraction Scenario Game

Take a look at subtraction scenarios with this game.

Complete the Related Facts Game

Have your own math-themed party by learning how to complete the related facts.

Subtraction on 10-frames Worksheet

Use this printable worksheet to practice subtraction on 10-frames to strengthen your math skills.

Subtract Using Think Addition Strategy Worksheet

Help your child revise subtraction by solving to subtract using think addition strategy.

Take Away Scenario Game

Use your subtraction skills to solve 'Take Away' scenarios.

Subtract 1-Digit Numbers from 2-Digit Numbers: Horizontal Timed Practice Worksheet

Timed worksheet to boost proficiency in subtracting 1-digit numbers from 2-digit numbers swiftly.

Represent Subtraction Sentence on 10-frames Worksheet

Put your skills to the test by practicing to represent subtraction sentences on 10-frames.

Write the Related Facts Game

Shine bright in the math world by learning how to write related facts.

Think Addition to Subtract Strategy Worksheet

Reinforce math concepts by practicing think addition as a subtraction strategy.

Subtract Numbers within 100: Vertical Timed Practice Worksheet

Fast-paced subtraction worksheet challenging students to solve problems within 100 in a minute!

Word Problems on Subtraction Game

Add more arrows to your child’s math quiver by solving word problems on subtraction.

Complete the Related Subtraction Facts Game

Unearth the wisdom of mathematics by learning how to complete related subtraction facts.

Solve Using 10-frames Worksheet

Assess your math skills by solving using 10-frames in this worksheet.

Subtract to Find the Difference Worksheet

This downloadable worksheet is designed to practice subtraction to find the difference.

Subtract Multiples of 10 using Place Value Game

Shine bright in the math world by learning how to subtract multiples of 10 using place values.

Word Problems on Take Away Game

Kids must solve word problems on 'Take Away' to play.

Subtract Numbers within 100: Horizontal Timed Practice Worksheet

A quick-paced, timed worksheet focused on enhancing students' skills in subtracting numbers within 100.

Find 1 Less than a Number Worksheet

Solidify your math skills by practicing to find 1 less than a number.

Find Missing Numbers in Subtraction Sentences Game

Find out the missing numbers in the given subtraction sentences.

Match the Difference of Multiples of 10 Game

Add more arrows to your child’s math quiver by matching the difference of multiples of 10.

Related Addition & Subtraction Equations Worksheet

Dive into this printable worksheet by practicing related addition & subtraction equations.

Add and Subtract 2-Digit and 1-Digit Numbers: Vertical Timed Practice Worksheet

A fast-paced worksheet to sharpen your 2-digit and 1-digit addition and subtraction skills under time pressure.

Represent Subtraction Game

Learn to solve math problems by representing subtraction.

Identify the Difference and Match Game

Add more arrows to your child’s math quiver by helping them identify the difference and match.

Select the Correct Subtraction Sentence Worksheet

Learners must select the correct subtraction sentence to enhance their math skills.

Subtract Using Number Bonds of 10 Worksheet

In this worksheet, learners will get to subtract using number bonds of 10.

Word Problems to Subtract Multiples of 10 Game

Help your child take flight by learning how to solve word problems to subtract multiples of 10.

Identify the Difference Game

Let your child see the world through math-colored shades with our 'Identify the Difference' game!

Add and Subtract 2-Digit and 1-Digit Numbers: Horizontal Timed Practice Worksheet

Fast-paced worksheet to practice adding and subtracting 2-digit and 1-digit numbers in 2 minutes!

Create Subtraction Sentences Worksheet

Be on your way to become a mathematician by practicing to create subtraction sentences.

Identify the Difference to Match Game

Apply your knowledge of subtraction to identify the difference to match.

Solve Word Problems to Find the Difference Game

Kids must solve word problems to find the difference to practice subtraction.

Form Math Equations Using Bar Model Worksheet

Use this printable worksheet to form math equations using bar models to strengthen your math skills.

Add and Subtract Two 2-Digit Numbers: Vertical Timed Practice Worksheet

Fast-paced worksheet to master addition and subtraction of 2-digit numbers in 3 minutes!

Your one stop solution for all grade learning needs.

• Inspiration

Singapore math support and resources

Using Ten Frames for Subtraction

Ten frames is a useful visualization tool when kids first learn about numbers and is often used to introduce addition at the kindergarten or pre-school level, but what about subtraction? How should we use properly ten frames for subtraction? Today, we will explain how we use ten frames to build up the number sense required for learning subtraction at early elementary levels.

If you are not familiar with what a ten frame is, please see our introduction to ten frames and number bonds here .

Subtraction within ten

The first stage of learning subtraction using ten frames is for numbers less than 10, for example 7 – 2 = 5.

The objective of this stage is to let kids get familiar with “seeing” numbers in relation to the two landmark numbers, i.e. 5 and 10. By showing quantities in two rows of five, we practice counting (subtracting) within 10 and build up subitizing skills. Here we also use number bond to see numbers as being “broken” into two parts:

The important thing when mapping from ten frames to the number bond is to see where to place the two numbers, i.e. 7 and 2, on the number bond circles.

Subtraction from Ten

The second stage in using ten frames for subtraction is subtracting from 10, for example, 10 – 3 = 7.

Here, we are trying to visualize how to take away from 10 to get the required result. Both forms of the number family should be used:

This is closely related to the concept of making ten , and is especially important for the next step.

Subtraction within 20

The final stage is subtraction within 20. This relies on the solid understanding in the previous two stages, for example, 12 – 5 = 7.

Here, we can think of the subtraction problem as a multi-step process.

First, we know that to make ten, we need to take 2 away from 12. So, out of the 5 we need to take away, we have accounted for 2.

Then, we ask: how many do we still need to take away? 5 – 2 = 3. Here, students rely on the foundation they built up in stage 1 (see above) for subtraction involving numbers less than 10.

Finally, we need to take 3 away from 10, which is exactly what we practiced in stage 2.

In number bond, we can visualize the problem as:

Ten frames and number bonds are great visualization tools that can be used to build up a strong foundation in number sense at early elementary levels. In this article, we discussed the use of ten frames to teach subtraction. In particular, ten frames are used in three stages to build up towards subtraction within 20. Once students mastered addition and subtraction within 20, extending to higher order numbers will be easy.

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