Grade: 1 Unit #4: Place Value, Comparison, Addition and Subtraction to 40 Time: 35 Days

Unit OverviewIn this unit, students extend the range of problem types and subtypes they can solve to within 40. Counting on should now be seen as a thinking strategy. The first addend is now seen as embedded in the total, showing the abstract shift of thinking of the cardinal meaning to the counting meaning of the first addend. Students will use the commutative property to shorten tasks (e.g., for 4 + 9, counting on from 9 instead of 4) while continuing to work with and becoming more proficient in solving problems using the problem solving situations (adding to, taking from, putting together, taking apart, comparison) using unknown in different places (start, change, results). Doubles facts are expanded to within 20 then expanded to doubles plus +1 and +2. Students become more fluid in computing numbers within 20 in preparation for adding and subtracting within 100.

Module 4 builds upon Module 2’s work with place value within 20, now focusing on the role of place value in the addition and subtraction of numbers to 40. Students study, organize, and manipulate numbers within 40. They compare quantities and begin using the symbols for greater than (>) and less than (<). Addition and subtraction of tens is another focus of this module as is the use of familiar strategies to add two-digit and single-digit numbers within 40. Near the end of the module, the focus moves to new ways to represent larger quantities and adding like place value units as students add two-digit numbers.

Using Level 2 and Level 3 strategies to extend addition and subtraction problem solving beyond 20, to problems within 40. As Grade 1 students are extending the range of problem types and subtypes they can solve, they are also extending the range of numbers they deal with and the sophistication of the methods they use to add and subtract within this larger range.1.OA.1

The advance from Level 1 methods to Level 2 methods can be clearly seen in the context of situations with unknown addends. These are the situations that can be represented by an addition equation with one unknown addend, e.g., 9 + □ = 13. Students can solve some unknown addend problems by trial and error or by knowing the relevant decomposition of the total. But a Level 2 counting on solution involves seeing the 9 as part of 13, and understanding that counting the 9 things can be “taken as done” if we begin the count from 9: thus the student may say, Niiiiine, ten, eleven, twelve, thirteen.

Students keep track of how many they counted on (here, 4) with fingers, mental images, or physical actions such as head bobs. Elongating the first counting word (“Niiiiine...”) is natural and indicates that the student differentiates between the first addend and the counts for the second addend. Counting on enables students to add and subtract easily within 20 because they do not have to use fingers to show totals of more than 10 which is difficult.

Understanding the concept of 10 is fundamental to children’s mathematical development because we live in a base ten number system. Students develop the structure to find a given 2-digit number and mentally produce 10 less or 10 more than that number without having to count. This will be an instrumental connection for the adding and subtracting of 2-digit numbers. The expectation within this unit is that students will utilize their knowledge of the place value system, properties of operations, and the connection between addition and subtraction to represent and solve problems within 100. Concrete models and illustrations are used, but students will also see that subtraction problems can be reformulated in terms of addition. Students will develop the understanding that when adding 2-digit numbers, one adds tens and tens, ones and ones, and that sometimes it is necessary to compose a ten. Students explain the strategies used and/or explain their thought processes by demonstrating individual numbers with base-10 organizers or manipulatives. Concrete objects, cards, ten frames or drawings afford connections with written numerical work and discussions and explanations in terms of tens and ones. In particular, showing composition of a ten with objects or drawings. When solving equations, students see a variety of addition and subtraction situations. Students should understand that “equality” means “the same quantity as.” This understanding will prevent the common pitfall that the equal sign means “to do something” or “the answer is.”

Important Note:If students progress from working with manipulatives to writing numerical expressions and equations, and they skip using pictorial thinking—students will then be more likely to use finger counting and rote memorization for work with addition and subtraction. So make sure that students are moving through the representations: manipulatives(concrete) to Pictures and then make the connection to the abstract equation or expression.Counting forward builds to the concept of addition while counting back leads to the concept of subtraction. However, counting is an inefficient strategy. Teachers need to provide instructional experiences so that students progress from the concrete level, to the pictorial level, then to the abstract level when learning mathematical concepts. (Concrete, Representational, Abstract CRA) Just knowing the basic facts is not enough. We need to help students develop the ability to quickly and accurately understand the relationships between numbers. They need to make sense of numbers as they find and make strategies for joining and separating quantities.

Connection to Prior LearningIn previous units, students built understanding of tens and ones through conceptual understanding of place value with multiple representations. They began by building models of numbers using counting skills and building relationships between visual patterns or groups of tens. Students were able to tell one more or one less than any given number and relate counting to addition and subtraction. Two-digit numbers were then decomposed as multiples of ten with the decades, and multiples of ten with some ones, for other 2-digit numbers. Practice with fluent number combinations within 10 lays the foundation for extension of ten and some ones. These number combinations were first done with visuals then accompanied by written equations.

Major Cluster Standards

Extend the counting Sequence1.NBT.1 Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.Understand place value1.NBT.2 Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:a) 10 can be thought of as a bundle of ten ones- called a “ten”b) The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.c) The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90, refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).1.NBT.3 Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the

symbols >, =, <.

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 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 addition to 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 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.

Represent and solve problems involving addition and subtraction1.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 using objects, drawings, and equations e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Major Cluster Standards Unpacked

1.NBT.1 calls for students to rote count forward to 120 by Counting On from any number less than 120. Students should have ample experiences with the hundreds chart to see patterns between numbers, such as all of the numbers in a column on the hundreds chart have the same digit in the ones place, and all of the numbers in a row have the same digit in the tens place. This standard also calls for students to read, write and represent a number of objects with a written numeral (number form or standard form). These representations can include cubes, place value (base 10) blocks, pictorial representations or other concrete materials. They use objects, words, and/or symbols to express their understanding of numbers. As students are developing accurate counting strategies they are also building an understanding of how the numbers in the counting sequence are related—each number is one more (or one less) than the number before (or after).They extend their counting beyond 100 to count up to 120 by counting by 1s. Some students may begin to count in groups of 10 (while other students may use groups of 2s or 5s to count). Counting in groups of 10 as well as grouping objects into 10 groups of 10 will develop students understanding of place value concepts. After counting objects, students write the numeral or use numeral cards to represent the number. Given a numeral, students read the numeral, identify the quantity that each digit represents using numeral cards, and count out the given number of objects.

Arrow Cards

As first graders learn to understand that the position of each digit in a number impacts the quantity of the number, they become more aware of the order of the digits when they write numbers. For example, a student may write “17” and mean “71”. Through teacher demonstration, opportunities to “find mistakes”, and questioning by the teacher (“I am reading this and it says seventeen. Did you mean seventeen or seventy-one? How can you change the number so that it reads seventy-one?”), students become precise as they write numbers to 120.

Students should experience counting from different starting points (e.g., start at 83; count to 120). To extend students’ understanding of counting, they should be given opportunities to count backwards by ones and tens. They should also investigate patterns in the base 10 system.

Instructional StrategiesIn first grade, students build on their counting to 100 by ones and tens beginning with numbers other than 1 as they learned in Kindergarten. Students can start counting at any number less than 120 and continue to 120. It is important for students to connect different representations for the same quantity or number. Students use materials to count by ones and tens to a build models that represent a number, then they connect this model to the number word and its representation as a written numeral. Students learn to use numerals to represent numbers by relating their place-value notation to their models. They build on their experiences with numbers 0 to 20 in Kindergarten to create models for 21 to 120 with groupable and pregrouped materials. Students represent the quantities shown in the models by placing numerals in labeled hundreds, tens and ones columns. They eventually move to representing the numbers in standard form, where the group of hundreds, tens, then singles shown in the model matches the left-to-right order of digits in numbers. Listen as students orally count to 120 and focus on their transitions between decades and the century number. These transitions will be signaled by a 9 and require new rules to be used to generate the next set of numbers. Students need to listen to their rhythm and pattern as they orally count so they can develop a strong number word list.

Extend hundreds charts by attaching a blank hundreds charts and writing the numbers 101 to 120 in the spaces following the same pattern as in the hundreds chart. Students can use these charts to connect the number symbols with their count words for numbers 1 to 120.

Post the number words in the classroom to help students read and write them.

1.NBT.2a asks students to unitize a group of ten ones as a whole unit: a ten. This is the foundation of the place value system. So, rather than seeing a group of ten cubes as ten individual cubes, the student is now asked to see those ten cubes as a bundle- one bundle of ten.

First Grade students are introduced to the idea that a bundle of ten ones is called “a ten”. This is known as unitizing. When First Grade students unitize a group of ten ones as a whole unit (“a ten”), they are able to count groups as though they were individual objects. For example, 4 trains of ten cubes each have a value of 10 and would be counted as 40 rather than as 4. This is a monumental shift in thinking, and can often be challenging for young children to consider a group of something as “one” when all previous experiences have been counting single objects. This is the foundation of the place value system and requires time and rich experiences with concrete manipulatives to develop.

40 + 2 = 42Make sure to reinforce the concept that 4 tens is the same as 40. Students should be asked to represent both ways. The use of arrow cards/hide zero cards will help solidify this thinking.

1.NBT.2b asks students to extend their work from Kindergarten when they composed and decomposed numbers from 11 to 19 into ten ones and some further ones. In Kindergarten, everything was thought of as individual units: ―ones. In First Grade, students are asked to unitize those ten individual ones as a whole unit: ―one ten. Students in first grade explore the idea that the teen numbers (11 to 19) can be expressed as one ten and some leftover ones. Ample experiences with ten frames will help develop this concept.

Example:For the number 12, do you have enough to make a ten? Would you have any leftover? If so, how many leftovers would you have?

Student 1I filled a ten frame to make one ten and had two counters left over. I had enough to make a ten with some leftover. The number 12 has 1 ten and 2 ones.

Student 2 counted out 12 place value cubes. I had enough to trade 10 cubes for a full ten frame. I now have 1 full ten frame and 2 cubes left over. So the number 12 has 1 ten and 2 ones.

1.NBT.2c builds on the work of 1.NBT.2b. Students should explore the idea that decade numbers (e.g. 10, 20, 30, 40) are groups of tens with no left over ones. Students can represent this with cubes or place value (base 10) rods. (Most first grade students view the ten stick (numeration rod) as ONE. It is recommended to make a ten with unfix cubes or other materials that students can group. Provide students with opportunities to count books, cubes, pennies, etc. Counting 30 or more objects supports grouping to keep track of the number of objects.)

Understanding the concept of 10 is fundamental to children’s mathematical development. Students need multiple opportunities counting 10 objects and ―bundling‖ them into one group of ten. They count between 10 and 20 objects and make a bundle of 10 with or without some left over (this will help students who find it difficult to write teen numbers). Finally, students count any number of objects up to 99, making bundles/groups of 10s with or without leftovers.

As students are representing the various amounts, it is important that an emphasis is placed on the language associated with the quantity. For example, 53 should be expressed in multiple ways such as 53 ones or 5 groups of ten with 3 ones leftover. When students read numbers, they read them in standard form as well as using place value concepts. For example, 53 should be read as ―fifty-three as well as five tens, 3 ones. Reading 10, 20, 30, 40, 50 as ―one ten, 2 tens, 3 tens, etc. helps students see the patterns in the number system.

A student’s ability to conserve number is an important aspect of this standard. It is not obvious to young children that 42 cubes is the same amount as 4 tens and 2 left-overs. It is also not obvious that 42 could also be composed of 2 groups of 10 and 22 leftovers. Therefore, first graders require ample time grouping proportional objects (e.g., cubes, beans, beads, ten-frames) to make groups of ten, rather than using pre-grouped materials (e.g., base ten blocks, pre-made bean sticks) that have to be “traded” or are non-proportional (e.g., money).Example: 42 cubes can be grouped many different ways and still remain a total of 42 cubes.

“We want children to construct the idea that all of these are the same and that the sameness is clearly evident by virtue of the groupings of ten. Groupings by tens is not just a rule that is followed but that any grouping by tens, including all or some of the singles, can help tell how many.” (Van de Walle & Lovin)

As children build this understanding of grouping, they move through several stages:Counting By Ones; Counting by Groups & Singles; and Counting by Tens and Ones.

Counting By Ones: At first, even though First Graders will have grouped objects into tens and left-overs, they rely on counting all of the individual cubes by ones to determine the final amount. It is seen as the only way to determine how many.Example:

Counting By Groups and Singles: While students are able to group objects into collections of ten and now tell how many groups of tens and left-overs there are, they still rely on counting by ones to determine the final amount. They are unable to use the groups and left-overs to determine how many.Example:

Counting by Tens & Ones: Students are able to group objects into ten and ones, tell how many groups and left-overs there are, and now use that information to tell how many. Ex: “I have 3 groups of ten and 4 left-overs. That means that there are 34 cubes in all.” Occasionally, as this stage is becoming fully developed, first graders rely on counting by ones to “really” know that there are 34, even though they may have just counted the total by groups and left-overs. Example:

Instructional Strategies:Essential skills for students to develop include making tens (composing) and breaking a number into tens and ones (decomposing). Composing numbers by tens is foundational for representing numbers with numerals by writing the number of tens and the number of leftover ones. Decomposing numbers by tens builds number sense and the awareness that the order of the digits is important. Composing and decomposing numbers involves number relationships and promotes flexibility with mental computation.

The beginning concepts of place value are developed in Grade 1 with the understanding of ones and tens. The major concept is that putting ten ones together makes a ten and that there is a way to write that down so the same number is always understood. Students move from counting by ones, to creating groups and ones, to tens and ones. It is essential at this grade for students to see and use multiple representations of making tens using base-ten blocks, bundles of tens and ones, and ten-frames. Making the connections among the representations, the numerals and the words are very important. Students need to connect these different representations for the numbers 0 to 99.

Groups of ones (single objects) Groups of 2 tens and 3 ones (2 ten-rods & 3 singles) Place Value Table, Write the Number, Read and Say the Number.Students need to move through a progression of representations to learn a concept. They start with a concrete model, move to a pictorial or representational model, then an abstract model (CRA). For example, ask students to place a handful of small objects in one region and a handful in another region. Next have them draw a picture of the objects in each region. They can draw a likeness of the objects or use a symbol for the objects in their drawing. Now they count the physical objects or the objects in their drawings in each region and use numerals to represent the two counts. They also say and write the number word. Now students can compare the two numbers using an inequality symbol or an equal sign.

In addition, when learning about forming groups of 10, First Grade students learn that a numeral can stand for many different amounts, depending on its position or place in a number. This is an important realization as young children begin to work through reversals of digits, particularly in the teen numbers.Example: Comparing 19 to 911991Teacher: Are these numbers the same or different?Students: Different!Teacher: Why do you think so?Students: Even though they both have a one and a nine, the top one is nineteen. The bottom one is ninety‐one.Teacher: Is that true some of the time, or all of the time? How do you know?

1.OA.1 builds on the work in Kindergarten by having students use a variety of mathematical representations (e.g., objects, drawings, and equations) during their work. The unknown symbols should include boxes or pictures, and not letters.Teachers should be cognizant of the three types of problems (See Table)There are three types of addition and subtraction problems: Result Unknown, Change Unknown, and Start Unknown.Use informal language (and, minus/subtract, the same as) to describe joining situations (putting together) and separating situations (breaking apart).Use the addition symbol (+) to represent joining situations, the subtraction symbol (-) to represent separating situations, and the equal sign (=) to represent a relationship regarding quantity between one side of the equation and the other.A helpful strategy is for students to recognize sets of objects in common patterned arrangements (0-6) to tell how many without counting (subtizing).Contextual problems that are closely connected to students’ lives should be used to develop fluency with addition and subtraction. Table 1 describes the four different addition and subtraction situations and their relationship to the position of the unknown. Students use objects or drawings to represent the different situations.

Take From example: Abel has 9 balls. He gave 3 to Susan. How many balls does Abel have now?

Compare example: Abel has 9 balls. Susan has 3 balls. How many more balls does Abel have than Susan? A student will use 9 objects to represent Abel’s 9 balls and 3 objects to represent Susan’s 3 balls. Then they will compare the 2 sets of objects.

Note that even though the modeling of the two problems above is different, the equation, 9 - 3 = ?, can represent both situations yet the compare example can also be represented by 3 + ? = 9 (How many more do I need to make 9?)It is important to attend to the difficulty level of the problem situations in relation to the position of the unknown.

Result Unknown, Total Unknown, and Both Addends Unknown problems are the least complex for students. The next level of difficulty includes Change Unknown, Addend Unknown, and Difference Unknown The most difficult are Start Unknown and versions of Bigger and Smaller Unknown (compare problems).

More Examples: Result Unknown Change Unknown Start UnknownThere are 9 students on the playground. Then 8 more students showed up. How many students are there now? 9 + 8 = ______There are 9 students on the playground. Some more students showed up. There are now 17 students. How many students came? 9 + _____ = 17Here are some students on the playground. Then 8 more students came. There are now 17 students. How many students were on the playground at the beginning?_____ + 8 = 17Please see Table for additional examples. The level of difficulty for these problems can be differentiated by using smaller numbers (up to 10) or larger numbers (up to 20).

Table 1 Common addition and subtraction situations1

Result Unknown Change Unknown Start Unknown

Add to

Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now?2 + 3 = ?

Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. How many bunnies hopped over to the first two?2 + ? = 5

Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before?? + 3 = 5

Take fromFive apples were on the table. I ate two apples. How many apples are on the table now?5 – 2 = ?

Five apples were on the table. I ate some apples. Then there were three apples. How many apples did I eat?5 – ? = 3

Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before? ? – 2 = 3

Total Unknown Addend Unknown Both Addends Unknown2

Put Together/Take Apart3

Three red apples and two green apples are on the table. How many apples are on the table?3 + 2 = ?

Five apples are on the table. Three are red and the rest are green. How many apples are green?3 + ? = 5, 5 – 3 = ?

Grandma has five flowers. How many can she put in her red vase and how many in her blue vase?5 = 0 + 5, 5 = 5 + 05 = 1 + 4, 5 = 4 + 15 = 2 + 3, 5 = 3 + 2

Difference Unknown Bigger Unknown Smaller Unknown

Compare4

(“How many more?” version):Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy?

(“How many fewer?” version):Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have than Julie?2 + ? = 5, 5 – 2 = ?

(Version with “more”):Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have?

(Version with “fewer”):Lucy has 3 fewer apples than Julie. Lucy has two apples. How many apples does Julie have?2 + 3 = ?, 3 + 2 = ?

(Version with “more”):Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have?

(Version with “fewer”):Lucy has 3 fewer apples than Julie. Julie has five apples. How many apples does Lucy have?5 – 3 = ?, ? + 3 = 5

2These take apart situations can be used to show all the decompositions of a given number. The associated equations, which have the total on the left of the equal sign, help children understand that the = sign does not always mean makes or results in but always does mean is the same number as.3Either addend can be unknown, so there are three variations of these problem situations. Both Addends Unknown is a productive extension of this basic situation, especially for small numbers less than or equal to 10.4For the Bigger Unknown or Smaller Unknown situations, one version directs the correct operation (the version using more for the bigger unknown and using less for the smaller unknown). The other versions are more difficult.

NOTE: Although First Grade students should have experiences solving and discussing all 12 problem types located in Table 1, they are not expected to master all types by the end of First Grade due to the high language and conceptual demands of some of the problem types. Please see Table 1

First Graders also extend the sophistication of the methods they used in Kindergarten (counting) to add and subtract within this larger range. Now, First Grade students use the methods of counting on, making ten, and doubles +/- 1 or +/- 2 to solve problems.

Example: Nine bunnies were sitting on the grass. Some more bunnies hopped there. Now, there are 13 bunnies on the grass. How many bunnies hopped over there?

Counting On Method

Student: Niiinnneee…. holding a finger for each next number counted 10, 11, 12, 13. Holding up her four fingers, 4! 4 bunnies hopped over there.”

Example: 8 red apples and 6 green apples are on the tree. How many apples are on the tree?Making Tens Method

Student: I broke up 6 into 2 and 4. Then, I took the 2 and added it to the 8. That’s 10. Then I add the 4 to the 10. That’s 14. So there are 14 apples on the tree.

Example: 13 apples are on the table. 6 of them are red and the rest are green. How many apples are green?Doubles +/- 1 or 2 Student: I know that 6 and 6 is 12. So, 6 and 7 is 13. There are 7

green apples.In order for students to read and use equations to represent their thinking, they need extensive experiences with addition and subtraction situations in order to connect the experiences with symbols (+, -, =) and equations (5 = 3 + 2). In Kindergarten, students demonstrated the understanding of how objects can be joined (addition) and separated (subtraction) by representing addition and subtraction situations using objects, pictures and words. In First Grade, students extend this understanding of addition and subtraction situations to use the addition symbol (+) to represent joining situations, the subtraction symbol (-) to represent separating situations, and the equal sign (=) to represent a relationship regarding quantity between one side of the equation and the other.

Instructional Strategies (1.OA. 1 & 2):Provide opportunities for students to participate in shared problem-solving activities to solve word problems. Collaborate in small groups to develop problem-solving strategies using a variety of models such as drawings, words, and equations with symbols for the unknown numbers to find the solutions. Additionally students need the opportunity to explain, write and reflect on their problem-solving strategies. The situations for the addition and subtraction story problems should involve sums and differences less than or equal to 20 using the numbers 0 to 20. They need to align with the 12 situations found in Table 1 of the Common Core State Standards (CCSS) for Mathematics.

Students need the opportunity of writing and solving story problems involving three addends with a sum that is less than or equal to 20. For example, each student writes or draws a problem in which three whole things are being combined. The students exchange their problems with other students, solving them individually and then discussing their models and solution strategies. Now both students work together to solve each problem using a different strategy.

Literature is a wonderful way to incorporate problem-solving in a context that young students can understand. Many literature books that include mathematical ideas and concepts have been written in recent years. For Grade 1, the incorporation of books that contain a problem situation involving addition and subtraction with numbers 0 to 20 should be included in the curriculum. Use the situations found in Table 1 of the CCSS for guidance in selecting appropriate books. As the teacher reads the story, students use a variety of manipulatives, drawings, or equations to model and find the solution to problems from the story.

Common Misconceptions:Many children misunderstand the meaning of the equal sign. The equal sign means ―is the same as‖ but most primary students believe the equal sign tells you that the ―answer is coming up‖ to the right of the equal sign. This misconception is over-generalized by only seeing examples of number sentences with an operation to the left of the equal sign and the answer on the right.First graders need to see equations written multiple ways, for example 5 + 7 = 12 & 12 = 5 + 7.

A second misconception that many students have is that it is valid to assume that a key word or phrase in a problem suggests the same operation will be used every time. For example, they might assume that the word left always means that subtraction must be used to find a solution. Providing problems in which key words like this are used to represent different operations is essential. For example, the use of the word left in this problem does not indicate subtraction as a solution method: Jose took the 8 stickers he no longer wanted and gave them to Anna. Now Jose has 11 stickers left. How many stickers did Jose have to begin with?Students need to analyze word problems and avoid using key words to solve them.

1.NBT.4 calls for students to use concrete models, drawings and place value strategies to add and subtract within 100. (Students should not be exposed to the standard algorithm.)

First Grade students use concrete materials, models, drawings and place value strategies to add within 100. They do so by being flexible with numbers as they use the base-ten system to solve problems. The standard algorithm of carrying or borrowing is neither an expectation nor a focus in First Grade. Students use strategies for addition and subtraction in Grades K-3. By the end of Third Grade students use a range of algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction to fluently add and subtract within 1000. Students are expected to fluently add and subtract multi-digit whole numbers using the standard algorithm by the end of Grade 4.

Example: 24 red apples and 8 green apples are on the table. How many apples are on the table?

Student A:I used ten frames. I put 24 chips on 3 ten frames. Then, I counted out 8 more chips. 6 of them filled up the third ten frame. That meant I had 2 left over. 3 tens and 2 left over. That’s 32. So, there are 32 apples on the table.

Student B:I used an open number line. I started at 24. I knew that I needed 6 more jumps to get to 30. So, I broke apart 8 into 6 and 2. I took 6 jumps to land on 30 and then 2 more. I landed on 32. So, there are 32 apples on the table.

Student C:I turned 8 into 10 by adding 2 because it’s easier to add.So, 24 and ten more is 34.But, since I added 2 extra, I had to take them off again.34 minus 2 is 32. There are 32 apples on the table.

Example: 63 apples are in the basket. Mary put 20 more apples in the basket. How many apples are in the basket?

Student A:I used ten frames. I picked out 6 filled ten frames. That’s 60. I got the ten frame with 3 on it. That’s 63. Then, I picked one more filled ten frame for part of the 20 that Mary put in. That made 73. Then, I got one more filled ten frame to make the rest of the 20 apples from Mary. That’s 83. So, there are 83 apples in the basket.

Student B:I used a hundreds chart. I started at 63 and jumped down one row to 73. That means I moved 10 spaces. Then, I jumped down one more row (that’s another 10 spaces) and landed on 83. So, there are 83 apples in the basket.

Student C:I knew that 10 more than 63 is 73. And 10 more than 73 is 83. So, there are 83 apples in the basket.

1.NBT.5 builds on students’ work with tens and ones and requires them to understand and apply the concept of 10 by mentally adding ten more and ten less than any number less than 100. This understanding leads to future place value concepts. It is critical for students to do this without counting. Prior use of models such as base ten blocks, ten frames, number lines, and 100 charts helps facilitate understanding. Continued experiences with ten frames will also help students see the pattern involved when adding or subtracting 10 and USE these patterns to solve such problems.First Graders build on their county by tens work in Kindergarten by mentally adding ten more and ten less than any number less than 100. First graders are not expected to compute differences of two-digit numbers other than multiples of ten. Ample experiences with ten frames and the number line provide students with opportunities to think about groups of ten, moving them beyond simply rote counting by tens on and off the decade. Such representations lead to solving such problems mentally.

Example: There are 74 birds in the park. 10 birds fly away. How many birds are in the park now?

Student AI thought about a number line. I started at 74. Then, because 10 birds flew away, I took a leap of 10. I landed on 64. So, there are 64 birds left in the park.

Student BI pictured 7 ten frames and 4 left over in my head. Since 10 birds flew away, I took one of the ten frames away. That left 6 ten frames and 4 left over. So, there are 64 birds left in the park.

Student CI know that 10 less than 74 is 64. So there are 64 birds in the park.

1.NBT.6 calls for students to use concrete models, drawings and place value strategies to subtract multiples of 10 from decade numbers (e.g., 30, 40, 50).This standard is foundational for future work in subtraction with more complex numbers. Students should have multiple experiences representing numbers that are multiples of 10 (e.g. 90) with models or drawings. Then they subtract multiples of 10 (e.g. 20) using these representations or strategies based on place value. These opportunities develop fluency of addition and subtraction facts and reinforce counting up and back by 10s.Examples:

70 - 30: Seven 10s take away three 10s is four 10s 80 - 50: 80, 70 (one 10), 60 (two 10s), 50 (three 10s), 40 (four 10s), 30 (five 10s) 60 - 40: I know that 4 + 2 is 6 so four 10s + two 10s is six 10s so 60 - 40 is 20

Example: There are 60 students in the gym. 30 students leave. How many students are still in the gym?

Student 1I used a hundreds chart and started at 60. I moved up 3 rows to land on 30. There are 30 students left.Student 2I used place value blocks or unifix cubes to build towers of 10. I started with 6 towered of 10 and removed 3. Had 3 towers left. 3 towers have a value of 30. There are 30 students left.Student 3Students mentally apply their knowledge of addition to solve this subtraction problem. I know that 30 plus 30 is 60, so 60 minus 30 equals 30. There are 30 students left.Student 4I used a number line. I started at 60 and moved back 3 jumps of 10 and landed on 30. There are 30 students left.Students may use interactive versions of models (base ten blocks,100s charts, number lines, etc.) to demonstrate and justify their thinking.

As illustrated above, many students will find the adding on strategy using a number line a more natural way of thinking to find the difference.

Student 5I used ten frames. I had 6 ten frames- that’s 60. I removed three ten frames because 30 students left the gym. There are 30 students left in the gym.

Student CI thought, “30 and what makes 60?”. I know 3 and 3 is 6. So, I thought that 30 and 30 makes 60. There are 30 students still in the gym.

Instructional StrategiesFor Standards 1.NBT.4-6 it is important to provide multiple and varied experiences that will help students develop a strong sense of numbers based on comprehension – not rules and procedures. Number sense is a blend of comprehension of numbers and operations and fluency with numbers and operations. Students gain computational fluency (using efficient and accurate methods for computing) when they are flexible and have many strategies from which to choose from, and as they come to understand the role and meaning of arithmetic operations in number systems.

Students should solve problems using concrete models and drawings to support and record their solutions. It is important for them to share the reasoning that supports their solution strategies with their classmates. Sets of laminated ten frames should be available to students as a concrete representation of their thinking.

Students will usually move to using base-ten concepts, properties of operations, and the relationship between addition and subtraction to

invent mental and written strategies for addition and subtraction. Help students share, explore, and record their invented strategies. Recording the expressions and equations in the strategies horizontally encourages students to think about the numbers and the quantities they represent. Encourage students to try the mental and written strategies created by their classmates. Students eventually need to choose efficient strategies to use to find accurate solutions.

Students should use and connect different representations when they solve a problem. They should start by building a concrete model to represent a problem. This will help them form a mental picture of the model. Now students move to using pictures and drawings to represent and solve the problem. If students skip the first step, building the concrete model, they might use finger counting to solve the problem. Finger counting is an inefficient strategy for adding within 100 and subtracting within multiples of 10 between10 and 90.

Have students connect a 0-99 chart or a 1-100 chart to their invented strategy for finding 10 more and 10 less than a given number. Ask them to record their strategy and explain their reasoning.

Focus Standards for Mathematical PracticeMP.3 Construct viable arguments and critique the reasoning of others. Students describe and explain their strategies for adding within 40, and critique and adjust student samples to more efficiently solve addition problems.MP.5 Use appropriate tools strategically. After learning varied representations and strategies for adding and subtracting pairs of two-digit numbers, students choose their preferred methods for representing and solving problems efficiently. Students may represent their computations using arrow notation, number bonds, quick ten drawings, and linking cubes. As they share their strategies, students explain their choice of counting on, making ten, adding tens and then ones, or adding ones and then tens.MP.6 Attend to precision. Students recognize and distinguish between units, demonstrating an understanding of the difference between 3 tens and 3 ones. They use this understanding to compare numbers and to add like place value units.MP.7 Look for and make use of structure. Students are introduced to the place value chart, deepening their understanding of the structure within our number system. Throughout the module, students use this structure as they add and subtract within 40. They recognize the similarities between 2 tens + 2 tens = 4 tens and 2 + 2 = 4, and use their understanding of tens and ones to explain the connection.

Understandings-Students Will Understand that… Properties of addition and subtraction reflect the relationship of addition and subtraction, the parts of the whole within an equation Strategies can be used to decompose complex problems to make an easier problem (counting on, make a ten, near ten, doubles,

doubles +1.+2) Word problems can be represented using multiple modalities Problem solving structures reinforce part/part/whole and number combinations within 40

Essential Questions

Why is it important to know multiple strategies in solving addition/subtraction problems? What is the purpose of using properties in adding or subtracting numbers? How are problem solving strategies connected to number relationships?

Prerequisite Skills/Concepts: Students should already be able to… Decompose numbers less than or equal to 10 into pairs in

more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

For any number from 1 to 9, find the number that makes 10 when added to the given number,e.g., by using objects or drawings, and record the answer with a drawing or equation.

Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g. ,by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

Advanced Skills/Concepts: Some students will be able to… Add and subtract within 100 using concrete models or drawings

and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Knowledge: Students will know… Skills: Students will be able to …

Strategies to solve addition and subtraction problems within 40.

Each type of word problem situation (adding to, taking from, putting together, taking apart, comparing).

All three unknown problem types (results, change, start).

Use addition and subtraction within 40 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing involving results unknown using objects, drawings, and equations with a symbol for the unknown number. (1.OA.1)

Use addition and subtraction within 40 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing involving change unknown using objects, drawings, and equations with a symbol for the unknown number. (1.OA.1)

Use addition and subtraction within 40 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing involving start unknown using objects, drawings, and equations with a symbol for the unknown number. (1.OA.1)

Count to 120 from any number less than 120. (1.NBT.1) Read and write any number up to 120. (1.NBT.1) Represent a group of objects with a number up to 120. (1.NBT.1) Tell how many tens and how many ones are in a number. (1.NBT.2) Compare two-digit numbers using <, =, and >. (1.NBT.3) Use manipulatives and pictures to solve problems addition and

subtraction problems within 40. (1.NBT.4) Use concrete models, or drawings and strategies based on place

value, properties, and relationships between addition and subtraction to solve addition and subtraction problems within 40. (1.NBT.4)

Fine 10 more or 10 less mentally. (1.NBT.5) Subtract multiples of 10 from multiples of 10 under 100 using

concrete models, or drawings and strategies based on place value, properties, and relationships between addition and subtraction.

(1.NBT.6)

Transfer of Understanding-Students will apply…

Addition and subtraction fluency skills within 10 to solve a variety of word problem types within 40. Properties of operations to solve word problems using three whole numbers (part/part/whole) to combine to sums less than or equal

to 40.

Academic Vocabulary

>(greater than) < (less than) Place value Equal (=) Numerals Ones Tens

Unit Resources

Pinpoint: Grade 1 Unit #4

Connections to Subsequent Learning

Students will continue to build their understanding of arithmetic properties to solve single-digit addition and subtraction problems. These mathematical properties will combine with place value reasoning to extend to multi-digit computations.