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First Grade

Operations and Algebraic Thinking

Shawna BakerJessica McKinleyNicole PlutaCari PotterAnna WilliamsClaire Williams

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+Task to develop

Operations and Algebraic Thinking

Valid Equalities

this lesson was patterned off of the lesson from

https://www.illustrativemathematics.org/illustrations/466)

+Put on your student hat!

Given the equality cards, and T-chart, determine if the equations are true or false.

1. Use strategy of your choice

2. With your group, place equation on true side or false side of T-chart.

3. Be prepared to explain your thinking in how you arrived at your decision of true or false.

4. Be prepared to agree, disagree, and/or add on to the thinking of your classmates.

+Put on your teacher hat!

While you are working on this task with your group, please make record of what students need to know and/or understand to be

able to “do” the task.

+Represent and solve problems involving addition and subtraction.

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

Students need to develop a concrete understanding of

numbers within 20 by gaining experience with the

three problem types of addition and subtraction: Result Unknown, Change

Unknown, and Start Unknown.

+What does it look like?

Student Example:17 children were in the class. 9 were boys and the rest were girls. How manygirls were in the class?

Student 1: I used red counters to build the 9 boys in the class. I counted on from 9 using white counters and stopped when I got to 17 to figure out how many boys were in the class. I then counted all the white counters I had set down. 17 = 9 + 8

Student 2: I first drew 9 circles to represent how many boys were in the class. I know there are 17 students in the class all together, so I drew more circles in a new group until I got to 17. After, I counted all the circles in my next set to get 8. 17 = 9 +8

+ProgressionIn order for first graders to successfully use addition and subtraction within 20 to solve word problems of all three problem types, they will need to know…

How to represent a problem in multiple ways using objects and manipulatives

How to take apart and combine numbers in a variety of ways

How to make sense of quantity and be able to compare numbers

How to flexibly and critically think about strategies that help to develop a conceptual knowledge of the standard algorithm

+Represent and solve problems involving addition and subtraction.

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

When adding more than two

addends, students need to understand that numbers can be grouped in multiple

ways. This is an informal introduction to the commutative

and associative property for addition.

+What does it look like?

Student example: There are 18 flowers planted beside the sidewalk. Jon planted 8 of them. Bailey planted 5 of them. Montez planted the rest of them. How many flowers did Montez plant? This student used a visual representation to count on. (insert picture)

A student who is proficient in this standard is able to consistently and independently solve multi-step word problems. They develop strategies for adding and subtracting whole numbers based on their prior work with small numbers. Students use a variety of mathematical tools to work through these problems such as objects, drawings, and equations.

There are cookies on the plate. There are 4 oatmeal raisin cookies, 5 chocolate chip cookies, and 6 gingerbread cookies. How many cookies are there total?

Student 1 Adding with a Ten Frame and Counters. I put 4 counters on the 10 Frame for the oatmeal raisin cookies. Then, I put 5 different color counters on the 10-Frame for the chocolate chip cookies. Then, I put another 6 color counters out for the gingerbread cookies. Only one of the gingerbread cookies fit, so I had 5 left over. One 10-Frame and five leftover makes 15 cookies.

Student 2 Look for ways to make 10

“I know that 4 and 6 equal 10, so the oatmeal raisin and gingerbread equals 10 cookies. Then, I add the 5 chocolate chip cookies and get 15 total cookies.”

= 10

10 + 5 = 15 cookies

Student 3 Number Line

“I counted on the number line. First, I counted 4, then I counted 5 more and landed on 9. Then, I counted 6 more and landed on 15. So there were 15 total cookies.”

+Progression

In order for first grade students to be able to solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, they must have conceptual understanding of…

Solving word problems within 10 using objects or drawings.

Decomposing and composing numbers up to 10.

Adding two addends.

Addition as a meaning of adding to/putting together.

Subtraction as a meaning of taking apart or taking from.

Using objects, drawings, or equations with a symbol for the unknown number to represent the problem.

+Misconceptions The equal sign Many students misunderstand the meaning

of the equal sign. Students often believe the answer is “coming up” when they need to understand that the equal sign means “the same as.” Children should see the equal sign written in multiple ways in order to develop understanding of this concept. Examples include: 5+7 = 12 and 12=5+7

Identifying key words or phrases to solve the problem Many students believe that a key word or phrase suggests the problem will involve the same operation each time. For example, students might assume that the word “left” means to subtract. It’s essential to provide problems in which key words represent different operations.

For example: Seth took the 8 stickers that he no longer wanted and gave them to Anna. Now Seth has 11 stickers left. How many stickers did Seth have to begin with?

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

1.OA.B.3. Apply properties of operations as strategies to add and subtract. (commutative and associative properties)

1.OA.B.4. Understand subtraction as an unknown-addend problem.

Students will build an understanding that addition

an subtraction are connected. They will build

understanding by representing contextual

situations in equations.

+What does it look like?

Students will work towards buildinga FLEXIBLE UNDERSTANDING of both the commutative property and associative property of addition as RELATIONAL.

Commutative Property: 1+3=4 is the same as 3+1=4 AND is also the same as 4=1+3 and 4=3+1 2+5=7 is the same as 5+2=7 AND is also the same as 7=2+5 and 7=5+2

Associative Property: 1+ 3+2+5=11 is also the same as 1+5+5=11The second and third of the two numbers can be added together to make 5 and then 5 and 5 will be added together to get 10. Add one more is 11.

+And…

Students will understand subtraction as an unknown-addend problem by using their flexible understanding of number combinations.10-6=____ (by finding the number that makes 10 when added to 6)

Hmm…What number do I

add to 6 to get 10? It’s 4! So then if I have 10 and

take 6 away, there will be 4 left. 4 and 6 is ten so I

can take either number away from 10 and get the

other one.

+Progression

In order for first grade students to be able to apply these properties of operations and understand addition and subtraction as relational, they must experience learning opportunities that emphasize:

Understand that addition is putting together and adding to, and understanding that subtraction is taking apart and taking from. (K.OA)

Represent addition and subtraction in contextual situations with objects, fingers, mental images, drawings, sounds (e.g., claps), verbal explanations, expressions, or equations. (K.OA.1)

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. (K.OA.2)

Decompose numbers less than or equal to 10 into pairs in more than one way (FLEXIBLY) 6 is the same as 3+3, 6 is the same as 4+2, and 6 is the same as 5+1 (K.OA.3)

For any number from 1-9, find the number that makes 10 when added to the given number. (K.OA,4)

+Possible Misconceptions

Equality is sameness not a solution. If students do not understand that the equal sign symbolizes sameness, they will not attend to the conceptual understanding of relational equality and then lack the understanding or misinterpret the meaning of the equal symbol.

Further, If students do not build that conceptual understanding of equality, Students might be confused about the meaning of the equals sign. In the problem 3+4=2+5 students may want to rename the equality as 7 instead of seeing these number combinations equalities. This is because they may see the equal sign as a symbol indicating that the answer should follow. Properties of operations are certain and undisputable. Students may

not understand that the properties of operations (associative or communicative) work consistently. Students must experience this concept through many examples to gain this understanding.

Commutative property works for subtraction. Students may believe that the commutative property applies to subtract so then, if 10-6=4 then 6-10=4. After students have discovered and applied the commutative property for addition, ask them to investigate whether this property works for subtraction. Have students share and discuss their reasoning and guide them to conclude that the commutative property does not apply to subtraction.

+Add and subtract within 20.

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

“Counting using the number line or the ten frame is a strategy I use when adding or subtracting.” –First

Grade Student

1.OA.C.6. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten; decomposing a number leading to a ten; using the relationship between addition and subtraction; and creating equivalent but easier or known sums.

+What does it look like?

•Students use many different strategies to add and subtract.  Strategies can include:-counting with fingers•Counting on from a given number•Using the number line•Using different colored counters to represent addition and subtraction problems•Acting out problems •Using models•Drawing pictures-tally marks

+Another example of what it looks like** Students can write numbers backwards and out of order.

+Misconceptions Students might miscount their manipulatives.

•Students might double hop or not make enough hops on the number line.

•Students might lose track of counters.

•Students might draw Sloppy and lose track of their work.

-Students might add too many or too little tally marks.

•Etc…

•Students might confuse the +, – and = sign.

Students might put the +,- and = sign in the wrong order example: +3=4+7

•Students might are more not look at the sign and assume addition. They may also see two numbers in a word problem and add them together without listening to the problem or key vocabulary.

+Progression In order for first grade students to be able add and subtract within 20 they

must first be able to:

 •K.CC.4c: Understand that each successive number name refers to a quantity that is one larger.

•K.OA.1: Represent addition and subtraction with objects, fingers, mental images, drawings2, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

•K.OA.2: Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

•K.OA.3: Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

Progression

•Following first grade students will use their understanding of adding and subtracting within 20 to:

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

+Work with addition and subtraction equations

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

Equality is a relational term and in math it means that

two mathematical expressions separated by

an equal sign have the same value.

+What does it look like?

Examples of problem types:

7=4+3 10=101+4 5 4+3=6+15=5 8-4=3+1

+ProgressionFrom Kindergarten:

K.CC.C.6: Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group by using matching and counting strategies.K.CC.C.7: Compare two numbers between 1 and 10 presented as written numerals.

They also need to have the understanding that: A number represents a quantity Joining of two groups ( 4 cubes and 4 cubes is 7 cubes) Separating of groups (4 cubes taken from 7 cubes leaves 3 cubes left

over) The quantities/numbers on both sides of the equal sign name the same

number The equal sign is a relational sign; not an operational sign. It is a sign for

equality. 4+3=7 is the same as 3+4=7 (commutative property) The label “true” means that the quantities on both sides of the equation

are the same and “false” means that the quantities on both sides of the equations are not the same.

When students understand these concepts, they will then be able to determine the unknown whole number in an addition or subtraction equation, such as 4+ __ = 7 (1.OA.8)

+Misconceptions

Students see the equal sign as an operational sign that means that something needs to be done, or the answer comes next.

Student may become confused when the equation is represented differently. ( 7= 4+3 with the answer being on the left rather than on the right)

Students may not know the meaning of “true” and “false” or may use them incorrectly.

Students may add all numbers together, including both sides of the equal sign. Ex. 4+1=5+2 would be 12.

+Work with addition and subtraction equations

1.OA.D.8. Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.

Not only do students have to find two quantities that

are the same, but they have to determine what

makes those two quantities the same when there is an

amount missing.

+What does it look like?

Students should see the unknown number in multiple positions in an equation, such as:a + b = ___a + ___ = c___ + b = ca – b = ___a - ___ = c___ - b = cHere are some example problems:7 + __ = 108 = 4 + ___9 – 6 = ___10 - ___ = 8

Different strategies students can use to solve these problems would be a pan balance, their fingers, X’s and O’s, five/ten/twenty frame, number line, and a part-part-whole box.

+Progression of the StandardHow does this concept unfold and build?

First students are able to decompose numbers.

Understand the meaning of the (+), (-) and (=) signs.

Begin to solve addition or subtraction problems with the unknown on one side of the equal sign __ = a +b or a + b = __

Following the mastery of the unknown on one side, students begin to find the missing value by making each side of the equal sign balanced or the same value.

When the teacher reads an equation it is important for it to be read in a meaningful way. For example 4 = 13 – 9 “four is what is left after 9 is taken away from 13.

+MisconceptionsStudent Thinking

While working with equations such as 5 + 6 = ___ +3 students typically find 11 to be the unknown number because they are trying to find the answer as opposed to making the equation balanced.

When seeing equations written as 5 = ___ + 2, students think they are written backwards and add 5 and 2 together.

Most misconceptions are formed from a misunderstanding of the (+), (-), and (=) signs.

After watching this clip https://www.youtube.com/watch?v=DEvNhwn0DEc What is one way that the teacher could present the problem differently so that students are less likely to build the misconceptions mentioned above?

+High Quality Assessment Practices

+Formative Assessment

+References

Common Core State Standards Progressions Documents. K, Counting and Cardinality; K-5, Operations and Algebraic Thinking

Kansas City Department of Education. KATM Created Common Core Flip Books

Nevada Academic Content Standards for Mathematics

Singapore Math

Small, M. Uncomplicating Algebra to Meet Common Core Standards in Math, K-8

Van de Walle, Elementary and Middle School Mathematics Teaching Developmentally (8th ed.)

https://www.illustrativemathematics.org

https: //www.ReadTennessee.org/math

https://mdk12.org

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