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Mathematics – Grade 1

Math_Curriculum Guide Gr 1_07.28.11_v1 Page | 1

Mathematics Curriculum Guide Grade 1

MILWAUKEE

PUBLIC SCHOOLS



Introduction: Mathematical Content Standards

In Grade 1, instructional time should focus on four critical areas: (1) developing understanding of addition, subtraction, and strategies for addition and subtraction within 20; (2) developing understanding of whole number relationships and place value, including grouping in tens and ones; (3) developing understanding of linear measurement and measuring lengths as iterating length units; and (4) reasoning about attributes of, and composing and decomposing geometric shapes. 1. Students develop strategies for adding and subtracting whole numbers based on their prior work with small numbers. They use a variety of models, including discrete objects and length-based models (e.g., cubes connected to form lengths), to model add-to, take-from, put-together, take-apart, and compare situations to develop meaning for the operations of addition and subtraction, and to develop strategies to solve arithmetic problems with these operations. Students understand connections between counting and addition and subtraction (e.g., adding two is the same as counting on two). They use properties of addition to add whole numbers and to create and use increasingly sophisticated strategies based on these properties (e.g., “making tens”) to solve addition and subtraction problems within 20. By comparing a variety of solution strategies, children build their understanding of the relationship between addition and subtraction. 2. Students develop, discuss, and use efficient, accurate, and generalizable methods to add within 100 and subtract multiples of 10. They compare whole numbers (at least to 100) to develop understanding of and solve problems involving their relative sizes. They think of whole numbers between 10 and 100 in terms of tens and ones (especially recognizing the numbers 11 to 19 as composed of a ten and some ones). Through activities that build number sense, they understand the order of the counting numbers and their relative magnitudes. 3. Students develop an understanding of the meaning and processes of measurement, including underlying concepts such as iterating (the mental activity of building up the length of an object with equal-sized units) and the transitivity principle for indirect measurement.1 4. Students compose and decompose plane or solid figures (e.g., put two triangles together to make a quadrilateral) and build understanding of part-whole relationships as well as the properties of the original and composite shapes. As they combine shapes, they recognize them from different perspectives and orientations, describe their geometric attributes, and determine how they are alike and different, to develop the background for measurement and for initial understandings of properties such as congruence and symmetry. 1 Students should apply the principle of transitivity of measurement to make indirect comparisons, but they need not use this technical term

Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning



The Assessment and Literacy Connections on this page are a general listing of research-based strategies which good teachers employ. The lists are not meant to be an “end all”. These are repeated for every grade level curriculum guide with the intent that our students experience some of the same strategies year to year, teacher to teacher. On the following pages, you will find specific usage of some of these strategies

Classroom Assessment for Learning Literacy Connections

Assessment for learning provides students with insight to improve achievement and helps teachers diagnose and respond to student needs. (Stiggins, Rick, Judith Arter, Jan Chappuis, Steve Chappuis. Classroom Assessment for Student Learning. Educational Testing Service, 2006.) Descriptive Feedback (oral/written) Effective use of questioning Exit Slips Milwaukee Math Partnership (MMP) CABS Portfolio items Student journals Student self-assessment Students analyze strong and weak work samples Use of Learning Intentions Use of rubrics with students Use of student-to-student, student-to-teacher and teacher-to-

student discourse Use of Success Criteria Formative Assessments

Measure of Academic Progress

Summative Assessments Unit Tests WKCE/WAA

Comprehensive literacy is the ability to use reading, writing, speaking, listening, viewing and technological skills and strategies to access and communicate information effectively inside and outside of the classroom and across content areas. (CLP, p.11) Literacy strategies can help students learn mathematics: Concept mapping Graphic organizers Journaling K-W-L Literature Partner talk RAFT Reciprocal teaching Talk moves Think Aloud strategy Think, Pair, Share strategy Three-minute pause Two column note taking Vocabulary strategies (e.g. Marzano’s 6 steps; Frayer model)



Operations & Algebraic Thinking 1.OA.

Represent and solve problems involving addition and

subtraction.

Understand and apply properties of operations and the

relationship between addition and subtraction.

Add and subtract within 20.

Work with addition and subtraction equations.

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

3. Apply properties of operations as strategies to add and subtract.2 Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.) 4. Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8. Add and subtract within 20.

5. Relate counting to addition and subtraction (e.g., by counting on 2 to add 2). 6. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

7. Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2. 8. Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ – 3, 6 + 6 = _.

1 See Glossary, Table 1. 2 Students need not use formal terms for these properties.

Operations & Algebraic Thinking



1.OA.

Essential/Enduring Understandings Assessment

Represent and solve problems involving addition and subtraction

within 20. Many different problem situations can be represented by part-part- whole relationships and addition or subtraction. Part-part whole relationships show how two numbers-the parts- are related to the whole.

Understand and apply properties of operations and the relationship between addition and subtraction. The Commutative and Associative properties for addition of whole numbers allow for computations to be performed flexibly.

Use strategies to fluently add and subtract within 20.

Work with addition and subtraction equations – Part-part-whole

relationships can be expressed by using number sentences like a + b = c or c – b = a, where a and b are the parts and c is the whole.

Given a variety of problem solving situations, students use part-whole strategies to solve them. Example: Seth took the 8 stickers he no longer wanted and gave them to Anna. Now Seth has 11 stickers left. How many stickers did Seth have to begin with?

Classroom Assessment Based on Standards (CABS)

1.OA Standard 1 Grade 1 NOR: Bracelets, How Many of Each Marbles?

1.OA Standard 2 Grade 1 AR: How Many Cards, What’s Missing, Toy

Cars 1.OA.Standard 6

Grade 2 NOR: #9 1.OA. Standard 7

Grade 1 NOR: True False Number Sentences Grade 2 NOR: #8

1.OA Standard 8 Grade 1 NOR: Secret Number

Common Misconceptions/Challenges

Represent and solve problems involving addition and subtraction within 20 Students over rely on a key word or phrase in a problem and they use that word to decide on the operation needed to solve the task.

Understand and apply properties of operations and the relationship between addition and subtraction Students over generalize the properties in regards to the relationship between addition and subtraction. They need to explore how

numbers can be broken apart and put together for addition and then be challenged to verbalize what happens with subtraction situations. Students see explore commutative and associative property with subtraction in order to come to their own understanding of how these properties work in each situation.

Use strategies to fluently add and subtract within 20. Students’ strategies for solving addition and subtraction are limited to counting by ones. Students need to practice strategies building

on their understanding of place value e.g., anchoring to 5 and 10 and breaking apart numbers to apply properties of operations. Students’ multiple experiences with counting may hinder their understanding of counting on and counting back as connected to addition



and subtraction. To help them make these connections when students count on 3 from 4, they should write this as 4 + 3 = 7. When students count back (3) from 7, they should connect this to 7 – 3 = 4.

Work with addition and subtraction equations Students misunderstand the meaning of the equal sign. The equal sign means “is the same as” but most primary students believe the

equal sign means “the answer” and it usually sits to the right of the equal sign. Because students typically engage in number sentences set up with the answer to the right of the equal sign, children come to this over-generalization. First graders need to see equations written multiple ways, for example 5 + 7 = 12 and 12 = 5 + 7.

Instructional Practices

1. OA Cluster: Represent and solve problems involving addition and subtraction. 1. OA Standards: 1 Apply properties of operations and explore unknowns in all positions. Launch activities by discussion situations that naturally use one the formats when figuring out “how many” e.g., making sure all of the

crayons in an 8-pack box are accounted for or taking lunch count or attendance and noting absences. Explore contextual problems that are closely connected to students’ lives 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.

Examples of problem formats: Take-from Abel has 9 balls. He gave 3 to Susan. How many balls does Abel have now?

Compare 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?)

Summarize explorations by discussing the position of the unknown as it occurs in a problem situation. Students should be questioned about



where the answer to the problem is as it relates to the expression and what it means in the context of the situation. Placement of the unknown will determine the difficulty level of the problem situations.

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

1. OA Cluster: Represent and solve problems involving addition and subtraction. 1. OA Standards: 2 Word problems with three addends Launch working with 3 addends by having students discuss different number combinations that make a total. To further explore the concept of addition, students create word problems with three addends. Explore increasing estimation skills by

creating problems in which the sum is less than 5, 10 or 20. Use properties of operations and different strategies to find the sum of three whole numbers such as:

Counting on and counting on again (e.g., to add 3 + 2 + 4 a student writes 3 + 2 + 4 = ? and thinks, “3, 4, 5, that’s 2 more, 6, 7, 8, 9 that’s 4 more so 3 + 2 + 4 = 9.”

Making tens (e.g., 4 + 8 + 6 = 4 + 6 + 8 = 10 + 8 = 18) Using “plus 10, minus 1” to add 9 (e.g., 3 + 9 + 6 A student thinks, “9 is close to 10 so I am going to add 10 plus 3 plus 6 which

gives me 19. Since I added 1 too many, I need to take 1 away so the answer is 18.) Decomposing numbers between 10 and 20 into 1 ten plus some ones to facilitate adding the ones

Using doubles

Using near doubles (e.g.,5 + 6 + 3 = 5 + 5 + 1 + 3 = 10 + 4 =14)

Students will use different strategies to add the 6 and 8.



1. OA Cluster: Understand and apply properties of operations and the relationship between addition and subtraction. 1. OA Standards: 3-4 Properties of Operations Students should explore the important ideas of the following properties:

Identity property of addition (e.g., 6 = 6 + 0) Identity property of subtraction (e.g., 9 – 0 = 9) Commutative property of addition (e.g., 4 + 5 = 5 + 4) Associative property of addition (e.g., 3 + 9 + 1 = 3 + 10 = 13)

Students need several experiences exploring whether the commutative property works with subtraction. During summarizing discussions, reinforce the idea that taking 5 from 8 is not the same as taking 8 from 5. Students need to understand that you can take 8 from 5 but they will explore negative numbers later.

When determining the answer to a subtraction problem, 12 – 5. Explore recording the situation to match student thinking e.g., “If I have 5, how many more do I need to make 12?” In this case, using the format of an unknown e.g., 5 + ? = 12. This format will match the context of the situation and provide flexibility of thinking.

1. OA Cluster: Add and subtract within 20. 1. OA Standards: 5-6 Counting Strategies When working with students to add and subtract within 20, launch explorations by flashing dot patterns and asking kids to identify “How

many did they see?”, “How did they see it?” Explore counting strategies to add and subtract numbers to 20. Deepen student understanding by exploring and discussing patterns and

relationships in addition facts and relating addition and subtraction. Summarizing these relationships will work to build a foundation for fluency with addition and subtraction facts. Note: Fluency is defined as the knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them

flexibly, accurately, and efficiently. The use of objects, diagrams, or interactive whiteboards and various strategies will help students develop fluency.

1.OA Cluster: Work with addition and subtraction equations. 1.OA Standards: 7-8 Equal Sign Launch explorations so students develop an understanding of the meaning of the equal sign. Explore by using true/false number sentences

and having students justify why the problem is true or false. Summarize by listening to students discuss the quantity on one side of the equal sign must be the same quantity Explore equations by posing mathematical expressions in true or false formats. Summarize explorations by having students justify their

answers, make conjectures (e.g., if you add a number and then subtract that same number, you always get zero), and make estimations. Summarize lessons by taking opportunities to highlight the “equal sign’ means ‘the same as”. Use the language “equal to” and “the same as”

as well as “not equal to” and “not the same as” to help students grasp the meaning of the equal sign. Students should understand that “equality” means “the same quantity as”.



Summarize classroom explorations by discussing situations that help students:

Express their understanding of the meaning of the equal sign Accept sentences other than a + b = c as true (a = a, c = a + b, a = a + 0, a + b = b + a) Know that the equal sign represents a relationship between two equal quantities Compare expressions without calculating

Once students have a solid foundation of the key skills listed above, they deepen their understanding by exploring how to rewrite true/false

statements using the symbols, < and >.

Examples of true and false statements:

7 = 8 – 1

8 = 8

1 + 1 + 3 =7

4 + 3 = 3 + 4

6 – 1 = 1 – 6

12 + 2 – 2 = 12

9 + 3 = 10

5 + 3 = 10 – 2

3 + 4 + 5 = 3 + 5 + 4

3 + 4 + 5 = 7 + 5

13 = 10 + 4

10 + 9 + 1 = 19 Students should explore problems with the unknown in different positions as well as quantities on the other side of the equal sign. Having

students create word problems for given equations will help them make sense of the equation and develop strategic thinking.

Examples of student thinking when considering the unknown: 8 + ? = 11 “8 and some number is the same as 11. 8 and 2 is 10 and 1 more makes 11. So the answer is 3.”

5 = – 3 “This equation means I had some cookies and I ate 3 of them. Now I have 5. How many cookies

did I have to start with? Since I have 5 left and I ate 3, I know I started with 8 because I count on from 5. . . 6,

7, 8.”

Differentiation



Represent and solve problems involving addition and subtraction within 20 Students need to move through a progression of representations to learn a concept. They start with a concrete model (level 1 – direct model), move to a pictorial or representational model (level 2 - counting on or back), then an abstract model (level 3 – convert to an easier problem). Students start with level 1 situations, but will individually make progress as developmentally appropriate.

Understand and apply properties of operations and the relationship between addition and subtraction. After students have discovered and applied the commutative property for addition, ask them to investigate whether this property

works for subtraction. Students should study number expressions and the patterns they make for both addition and subtraction. To move to a generalization, some students need to work with larger numbers while others will have to have smaller numbers and models to manipulate. Have students share and discuss their reasoning and guide them to conclude that the commutative property does not apply to subtraction

Use strategies to fluently add and subtract within 20. Students need much practice developing strategies based on decomposing and recomposing single-digit numbers 5 + 7 = 2 + (3+7) as well as decomposing and composing numbers by units of tens and ones e.g., 5+7 = 2 + (3+7) = 10 + 2 = 12.

Work with addition and subtraction equations

Students work on 5 and 10 frames, rekenreks, bead strings, number lines, or 100’s charts to clarify their understanding of the base-ten system and make connections to the story formats. Students should use the strategies as a model for thinking.

Literacy Connections Academic Vocabulary



Literacy Strategies: Use the Think-Aloud strategy to help students identify the

situation and the operation that is working. Comparing and sorting true/false number sentences

add compare equal equation putting together solve strategy subtract symbol take from unknown whole number word problem

Literature Suggestions: Leedy, L. Mission: Addition. Holiday house, 1999. Leedy, L. Subtraction Action. Holiday house, 2002 Murphy, S. Animals on Board. HarperCollins, 2002. Ribke, S. Pet Store Subtraction. Children’s Press, 2007

Resources

A variety of objects for modeling and solving addition and subtraction problems Dot Card and Ten Frame Activities (pp. 9-11, 21-24, 26-30, 32-37) Numeracy Project, Winnipeg School Division, 2005-2006 ORC # 3992 From the National Council of Teachers of Mathematics: Balancing equations In this lesson, students imitate the action of a pan balance and record the modeled subtraction facts in equation form. ORC # 3978 From the National Council of Teachers of Mathematics: How many left? This lesson encourages the students to explore unknown-addend problems using the set model and the game Guess How Many?

Table 1. Common addition and subtraction situations.6



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 from

Five 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 Unknown1

Put Together / Take Apart

2

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 + 0 5 = 1 + 4, 5 = 4 + 1 5 = 2 + 3, 5 = 3 + 2

Difference Unknown Bigger Unknown Smaller Unknown

Compare3

(“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



Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning

Number & Operations in Base Ten 1.NBT

Extend the counting sequence.

Understand place value.

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

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.

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).

3. Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

4. Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten. 5. Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used. 6. Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and



subtraction; relate the strategy to a written method and explain the reasoning used.



Number & Operations in Base Ten 1.NBT


Extend the counting sequence to 120 recognizing that the

number system is based on a pattern anchored in five and tens and the numbers represent a quantity.

Understand place value and connect the counting words to a unitary value of tens and ones and use that understanding to compare 2-digit numbers

Use place value understanding and properties of operations to

add and subtract.

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.

Given a 2-digit number between 10 and 100, students will be able to

represent that number two different ways using pictures, words, or numbers.

Given two 2-digit numbers students will use place value strategies

to demonstrate understanding of addition and subtraction. Classroom Assessment Based on Standards (CABS)

1.NBT Standard 1 Grade 1 NOR Number Sense

1.NBT Standard 2 Grade 1 NOR Estimation Grade 2 NOR #1, #2, #3,

1.NBT Standard 3 Grade 1 NOR More, Less or Equal

1.NBT Standard 4 Grade 1 AR How Many Cards? Grade 2 NOR #6

1.NBT Standard 6 Grade 1 NOR Counting Objects


Extend the counting sequence Students need to begin to develop the understanding of what it means for one number to be greater than another. Rote counting or

writing numbers does not indicate student understand of quantity. When comparing quantities student must connect to the names: < Less Than, > Greater Than, and = Equal To. Students need to begin to understand that both inequality symbols (<, >) can create true statements about any two numbers where one is greater/smaller than the other, (15 < 28 and 28 >15).



Understand place value

Students don’t make a connection to the number they see and the quantity it represents. Students need experiences estimating and counting large amounts of objects. Opportunities to estimate and count will move students to unitize items in groups of tens in order to keep track of counting.

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

Students count by ones when adding 2-digit numbers. They don’t consider the digits as a quantity and ultimately don’t understand how the magnitude of number grows especially when adding multiples of 10. When adding 10 to any number students need to write it down (vertically and horizontally), track it on a 100’s chart, build it with cubes, and model it with base-ten drawings. Students also need to have conversations about what happens when adding or subtracting 10 from any number.


1.NBT Cluster: Extend the counting sequence. 1.NBT Standard: 1 Counting Launch lessons by giving students a number and asking them “What number one/ten before? What number comes one/ten after?” “How

far is this number from 10/20/30…?” Students model out their thinking on the 100’s chart. Explore using partially filled in 100s charts. Students complete the chart by filling in random numbers on the chart and justifying why the

number fits in the box or spot on the chart. Explore larger numbers by extending reading and writing numerals beyond 20 to 120. After counting objects, students write the numeral or

use numeral cards to represent the number. 1.NBT Cluster: Understand place value. 1.NBT Standards: 2 Understanding place value Launch counting experiences 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 explore patterns in the base 10 system. As students are representing the various amounts and sharing them during summaries of lessons, emphasize language associated with the

quantity of the number. Example:

Express 53 using base-ten drawings, words, pictures, numbers, tally marks, in multiple ways such as 53 ones or 5 groups of ten with 3 ones leftover.



Students read the numbers 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.

1.NBT.Cluster: Understand place value. 1.NBT.Standard 3. Launch lessons by presenting students with a picture, asking them how many they think are in the picture, or presenting them with a

number and students describe a situation that matched the number (e.g., 54 might be the number of students in two classrooms.) Students explore comparing numbers by using models (mini tens frames) that represent two sets of numbers. Explorations should include

opportunities for students to estimate an answer “to get them in the ball park”. Summarizing discussions should affirm student strategies when they (1) first attend to the number of tens, then to the number of ones, (2)

use pieces of numbers to adjust to an equivalent problem or make an easier problem to add or subtract. Students may also use pictures, number lines, and spoken or written words to compare two numbers.

Summarize using comparative language such as more than, less than, greater than, most, greatest, least, same as, equal to and not equal to. 1.NBT Cluster: Use place value understanding and properties of operations to add and subtract. 1.NBT Standard: 4 Launch student explorations on operations by comparing two problem-solving situations with the same operations and asking students to

model them. Example Mary has 4 pumpkins and 6 apples. Sue has 6 pumpkins and 4 apples. Which one has more items?

Students explore and represent a problem situation using any combination of words, numbers, pictures, physical objects, or symbols. It is important for students to understand if they are adding a number that has 10s to a number with 10s, they will have more tens than they started with; the same applies to the ones.

Students should explore problems both in and out of context and presented in horizontal and vertical forms. As students are solving and sharing problems, it is important that teachers help them to summarize their learning using language associated with proper place value Teachers must help students reverbalize their place value work to help them explain and justify their mathematical thinking both verbally and in a written format. This standard focuses on developing addition - the intent is not to introduce traditional algorithms or rules.

Including estimating opportunities during the explore and summarize portion of the lesson helps identify an answer range for the solution

prior to finding the answer. Students will begin to focus on the meaning of the operation as well as attend to the actual quantities in the problem.

Examples: 43 + 36 Student counts the 10s (10, 20, 30…70 or 1, 2, 3…7 tens) and then the 1s.



28 + 34

Student thinks: 2 tens plus 3 tens is 5 tens or 50. S/he counts the ones and notices there is another 10 plus 2 more. 50 and 10 is 60 plus 2 more or 62.

45 + 18 Student thinks: Four 10s and one 10 are 5 tens or 50. Then 5 and 8 is 5 + 5 + 3 (or 8 + 2 + 3) or 13. 50 and 13 is 6 tens plus 3 more or 63.

29 +14

Student thinks: “29 is almost 30. I added one to 29 to get to 30. 30 and 14 is 44. Since I added one to 29, I have to subtract one so the answer is 43.”

1. NBT. Cluster: Use place value understanding and properties of operations to add and subtract. 1.NBT. Standard 5-6: Given a 2-digit number mentally find 10 more and 10 less and subtract quantities of 10.



Launch explorations by use mini 10’s frames to model counting by 10s and adjusting number forward and backward by 10s. Guide discussions by asking “What number would it take to be 10 more? – 10 less?

Explore the number relationships that emerge by highlighting pattern and connecting it to numbers on 100’s charts and summarize by reverbalizing student thinking and using different versions of 100s charts to confirm number patterns.

Examples: 10 more than 43 is 53 because 53 is one more 10 than 43 10 less than 43 is 33 because 33 is one 10 less than 43

Students explore subtracting with ten by using mini-ten frames, or checking their thinking on the 100s charts. Summarize discussions by displaying examples of students’ strategies on chart or on the overhead.

Examples of strategies using base-ten thinking for computation: 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


Literacy Strategies: Partner talk will help students explore numbers together.

When using larger quantities to count, students will devise methods with partners to help them keep track of groups and record their thinking.

add compare compose count less multiple number ones place value range tens strategy subtract



Differentiation

Literature Suggestions: Franko,B. Zero is the leaves on a tree. Tricycle Press, 2009 Merriam, E. 12 Ways to get to 11. Aladdin, 1996. Ryan, M. One Hundred is a Family. Hyperion, 1992. Wells, R. Can you count to a google? Albert Whitman, 2000.

symbol

Extend the counting sequence Use adding machine tape to write number strings to help students identify the system of numbers e.g., 7, 8, 9, 10 moving to 77, 78, 79, 80.

Ask how the numbers are similar and how they are different. This will help students internalize number patterns within and between decades. Students need to verbalize the pattern.

Understand place value Provide students with multiple opportunities exploring counting 10 objects and “bundling” them into one group of ten. Students can scoop

or grab handfuls of objects and make a bundle of 10 with or without some left over. Student could then compare their quantities with that of others in the class. Items to be counted could include: buttons, screws, crayons, paperclips, or other things that could contribute to a class collection jar.

Use place value understanding and properties of operations to add and subtract. 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. Students move from a closed number line (1-10) to an open number line to record computation strategies. Help students verbalize jumps of

10 and make connections to the 100s chart. Numeral cards may help students decompose the numbers into 10s and 1s. Students should be able to apply their place value skills to

decompose numbers. For example, 17 + 12 can be thought of 1 ten and 7 ones plus 1 ten and 2 ones.



Resources Groupable models Linking cubes Plastic chain links Pregrouped materials Base-ten blocks Mini-tens frames Rekkenrek Ten-frame Place-value mat with ten-frames

2-3 versions of hundreds charts filled and blank hundreds chart



Measurement & Data 1.M.D.

Measure lengths indirectly and by iterating length units.

Tell and write time.

Represent and interpret data.

1. Order three objects by length; compare the lengths of two objects indirectly by using a third object. 2. Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.

3. Tell and write time in hours and half-hours using analog and digital clocks.

4. Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.






Measure lengths indirectly and by iterating length units.

Tell and write time to the hour and half-hour.

Represent and interpret data in order to find an answer to a

question posed.

Given an object students identify an attribute of the object to be

measured, the tool they will use to measure with reasons why that tool will be appropriate. Finally students will provide a measurement for the item.

Given a time, students will record the correct time in both digital and analog and identify what would be happening during that time period.

Students will pose a question, collect data informally, display the data and use it to support their conclusions based on the data.

Classroom Assessment Based on Standards (CABS)

1.MD Standard 1 Grade K M – Longer or Shorter Grade 1 M – Measure, Compare and Order

1.MDStandard 2 Grade 1 M – Measure, Compare and Order Grade 2 M #2

1.MD Standard 3 Grade 1 M Telling Time Grade 2 M #6, 7, 8

1.MD Standard 4 Grade 1 Statistics and Probability – What’s Your

Favorite?, Looking at Graphs, Grab and Graph


Measure lengths indirectly and by iterating length units. Some students may view the measurement process as a procedural counting task. They might count the markings on a ruler rather than

the spaces between (the unit of measure). Students need numerous experiences measuring lengths with student-made tapes or rulers with numbers in the center of the spaces.

Students lay down materials leaving gaps between measurement tools or overlapping the tools. Students need much practice first laying out many of the same unit and then moving onto using one unit many times. The use of non-standard items to measure supports the idea of unit size and the relationship to distance being measured.

Tell and write time.



Students confuse the hour hand and the minute hand on an analog clock. They should begin practicing telling time by only using the hour hand. Make sure to connect segments of time to events occurring throughout the day.

Represent and interpret data. Students don’t understand the difference between posing a question and sharing a story. They need a lot of practice posing questions

and sorting out which ones will give them information. Posing “yes or no” and having students sort them would be helpful in providing them some information on types of questions.

Students will have limited experience interpreting data and will need a lot of teacher questioning and modeling to help them understand that data answers questions but can also pose new ones.


1. MD Cluster: Measure lengths indirectly and by iterating length units. 1. MD Standards: 1-2 Launch measurement lessons by asking students to discuss different attributes of objects in the room. Look at a desk and discuss all the

different parts that could be measured and why we would need to know the measurement. Discuss what measurement is and how it connects to using what they know about numbers and quantities. Quantifying measurement will feel different to students as they are not directly counting objects, but are counting units of an object as it is placed end to end.

Explore direct and indirect measurement. Focus discussion by summarizing that length is measured from one end point to another end

point. Continue explorations by physically aligning the objects and reinforcing the language taller, shorter, longer, and higher. Students identify and justify the length they are measuring during the summary of the lesson. Direct measurement is when objects are compared directly with each other. Indirect measurement is when a conclusion is reached about a measurement based on the comparison of two objects not directly connected to the third.

Examples for ordering: Order three students by their height Order pencils, crayons, and/or markers by length Build three towers (with cubes) and order them from shortest to tallest Three students each draw one line, then order the lines from longest to shortest

Example for comparing indirectly: Two students each make a dough “snake.” Given a tower of cubes, each student compares his/her snake to the tower. Then

students make statements such as, “My snake is longer than the cube tower and your snake is shorter than the cube tower. So, my snake is longer than your snake.”



1.MD. Standard 2. Express length as a whole number unit. Brainstorming measurement devices could launch a discussion about things that can be used to measure. Students use their counting skills while exploring measuring with non-standard units. During summarization portions of the lessons, monitor students‘explanations of unit measurements making sure they are using

language such as “about 6 paper clips”. While this standard limits measurement to whole numbers of length, in a natural environment, not all objects will measure to an exact whole unit.

Example: Ask students to use multiple units of the same object to measure the length of a pencil.

(How many paper clips will it take to measure how long the pencil is?)

1. MD Cluster: Tell and write time. 1. MD Standard: 3 Telling time Launch telling time explorations by recording events throughout the day. Use multiple representations of different clock faces to help

students get used to matching the hands of a clock. Monitor explorations and focus summary discussions around these ideas.

within a day, the hour hand goes around a clock twice (the hand moves only in one direction) when the hour hand points exactly to a number, the time is exactly on the hour time on the hour is written in the same manner as it appears on a digital clock the hour hand moves as time passes, so when it is half way between two numbers it is at the half hour there are 60 minutes in one hour; so halfway between an hour, 30 minutes have passed half hour is written with “30” after the colon

“It is 4 o’clock”

“It is halfway between 8 o’clock and 9 o’clock. It is 8:30.”



1. MD Cluster: Represent and interpret data. 1. MD Standard: 4 Data Students create object graphs and tally charts using data relevant to their lives (e.g., favorite ice cream, eye color, pets, etc.). Graphs may be constructed by groups of students as well as by individual students. Counting objects should be reinforced when collecting, representing, and interpreting data. Students describe the object graphs and tally charts they create. They should also ask and answer questions based on these charts or graphs that reinforce other mathematics concepts such as sorting and comparing. The data chosen or questions asked give students opportunities to reinforce their understanding of place value, identifying ten more and ten less, relating counting to addition and subtraction and using comparative language and symbols. Students may use an interactive whiteboard to place objects onto a graph. This gives them the opportunity to communicate and justify their thinking.

Differentiation

Measure lengths indirectly and by iterating length units. Measurement units share the attribute being measured. Students need to use as many copies of the length unit as necessary to match the

length being measured. For instance, use large footprints with the same size as length units. Place the footprints end to end, without gaps or overlaps, to measure the length of a room to the nearest whole footprint. Use language that reflects the approximate nature of measurement, such as the length of the room is about 19 footprints. Students need to also measure the lengths of curves and other distances that are not straight lines.

Have students use reasoning to compare measurements indirectly, for example to order the lengths of Objects A, B and C, examine then

compare the lengths of Object A and Object B and the lengths of Object B and Object C. The results of these two comparisons allow students to use reasoning to determine how the length of Object A compares to the length of Object C. For example, to order three objects by their



lengths, reason that if Object A is smaller than Object B and Object B is smaller than Object C, then Object A has to be smaller than Object C. The order of objects by their length from smallest to largest would be Object A - Object B - Object C.

Tell and write time to the hour and half-hour.

Make a human clock with students. Focus on hour hands and move the hands around. Match the human clock with an analog and digital clock. Summarize discussions by connecting the time on the clock to events during the day.

Represent and interpret data in order to find an answer to a question posed.

Provide students with questions to collect data around. Provide students with multiple ways to organize data and to make sense of it. Students may use counters, tally marks, or pictures. Summarize discussions by asking students what they notice and providing them an opportunity to discuss more than “which one has more” “which one has less.”


Literacy Strategies: Concept map will help students to develop a

compare length object whole number

Literature Suggestions:



Resources

Clothesline rope Yarn Toothpicks Straws Paper clips Connecting cubes Cuisenaire rods A variety of common two- and three-dimensional objects Strips of tagboard or cardboard ORC # 4329 From the National Council of Teachers of Mathematics, Illuminations: The Length of My Feet This lesson focuses students’ attention on the attributes of length and develops their knowledge of and skill in using nonstandard units of measurement. ORC # 1485 From the American Association for the Advancement of Science: Estimation and Measurement In this lesson students will use nonstandard units to estimate and measure distances.



Geometry 1.G

Reason with shapes and their attributes.

1. Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size) ; build and draw shapes to possess defining attributes. 2. Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.1 3. Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

1 Students do not need to learn formal names such as “right rectangular prism.”





Geometry 1.G


Reason with shapes and their attributes.

Given an assortment of triangle figures, students identify which ones are triangles and which ones are not. They provide justification about each triangle. Classroom Assessment Based on Standards (CABS)

1.G Standard 1 Grade K G Sorting Shapes Grade 1 G Comparing Shapes

1.G Standard 2 Grade 1 G Fill the shape Grade 1 G Faces

1.G Standard 3 Grade 1


Reason with shapes and their attributes. Students may think that a square that has been rotated so that the sides form 45-degree angles with the vertical diagonal is no longer a

square but a diamond. They need to have experiences with shapes in different orientations. For example, in the building-shapes strategy above, ask students to orient the smaller shapes in different ways.

Some students may think that the size of the equal shares is directly related to the number of equal shares. For example, they think that

fourths are larger than halves because there are four fourths in one whole and only two halves in one whole. Students need to focus on the change in the size of the fractional parts as recommended in the folding shapes strategy. The first activity in the unit Introduction to Fractions for Primary Students (referenced above) includes a link, Parts of a Whole, to an interactive manipulative. It allows students to divide a circle into the number of equal parts that they choose. Students can easily see the change in the size of the equal shares as they increase or decrease the number of parts.




1.G.1. Reason with shapes and their attributes. 1.G.1 Standard 1 Launch conversations about shapes by going on a shape hunt in the classroom and classifying the shapes that you see into catagories. Note: Attributes refer to any characteristic of a shape. Students use attribute language to describe a given two-dimensional shape: number of sides, number of vertices/points, straight sides, closed. A child might describe a triangle as “right side up” or “red.” These attributes are not defining because they are not relevant to whether a shape is a triangle or not. Students should articulate ideas such as, “A triangle is a triangle because it has three straight sides and is closed.” It is important that students are exposed to both regular and irregular shapes so that they can communicate defining attributes. Students should use attribute language to describe why these shapes are not triangles.

When summarizing lessons use appropriate language to describe a given three-dimensional shape: number of faces, number of

vertices/points, number of edges.

o Example: A cylinder would be described as a solid that has two circular faces connected by a curved surface (which is not considered a face). Students may say, “It looks like a can.”

Students should compare and contrast two-and three-dimensional figures at the same time in order to explore the defining attributes of

each type of shape.

Examples: List two things that are the same and two things that are different between a triangle and a cube. Given a circle and a sphere, students identify the sphere as being three-dimensional but both are round. Given a trapezoid, find another two-dimensional shape that has two things that are the same.

Students could explore using interactive whiteboards or computer environments to move shapes into different orientations and to enlarge or decrease the size of a shape still keeping the same shape. They can also move a point/vertex of a triangle and identify that the new shape is still a triangle. When they move one point/vertex of a rectangle they should recognize that the resulting shape is no longer a rectangle.

1.G.1. Reason with shapes and their attributes. 1.G.Standard 2. Launch conversations about shapes and their attributes by reading a literature book or looking at pieces of artwork and discussing the shapes. Note: Shape explorations support the development of describing, using and visualizing the effect of composing and decomposing shape. It is not



only relevant to geometry, but is related to children’s ability to compose and decompose numbers. Students explore with a variety of pattern blocks, plastic shapes, tangrams, or computer environments to make new shapes. The teacher can provide students with cutouts of shapes and ask them to combine them to make a particular shape.

Example:

What shapes can be made from four squares?

1.G.1. Reason with shapes and their attributes. 1.G.Standard 3

Launch lessons by asking students about things they share and how they are shared. Students need to explore with different sized circles and rectangles to recognize that when they cut something into two equal pieces,

each piece will equal one half of its original whole. Children should recognize that halves of two different wholes are not necessarily the same size.

Examples:

Student partitions a rectangular candy bar to share equally with one friend and thinks “I cut the rectangle into two equal parts. When I put the two parts back together, they equal the whole candy bar. One half of the candy bar is smaller than the whole candy bar.”

Student partitions an identical rectangular candy bar to share equally with 3 friends and thinks “I cut the rectangle into four equal parts. Each piece is one fourth of or one quarter of the whole candy bar. When I put the four parts back together, they equal the whole candy bar. I can compare the pieces (one half and one fourth) by placing them side-by-side. One fourth of the candy bar is smaller than one half of the candy bar.

Students partition a pizza to share equally with three friends. They recognize that they now have four equal pieces and each will receive a fourth or quarter of the whole pizza.



Summarize discussions emphasizing equality (equal shares, equal portion size)

Mathematics - Grade 1

Differentiation

1.G.1. Reason with shapes and their attributes. Students can easily form shapes on geoboards using colored rubber bands to represent the sides of a shape. Ask students to create a

shape with four sides on their geoboard, then copy the shape on dot paper. Students can share and describe their shapes as a class while the teacher records the different defining attributes mentioned by the students.

Pattern block pieces can be used to model defining attributes for shapes. Ask students to create their own rule for sorting pattern

blocks. Students take turns sharing their sorting rules with their classmates and showing examples that support their rule. The classmates then draw a new shape that fits this same rule after it is shared.

Students can use a variety of manipulatives and real-world objects to build larger shapes. The manipulatives can include paper shapes,

pattern blocks, color tiles, triangles cut from squares (isosceles right triangles), tangrams, canned food (right circular cylinders) and gift boxes (cubes or right rectangular prisms). Students can make three-dimensional shapes with clay or dough, slice into two pieces (not

necessarily congruent) and describe the two resulting shapes. For example, slicing a cylinder will result in two smaller cylinders.

Folding shapes made from paper enables students to physically feel the shape and form the equal shares. Ask students to fold circles and rectangles first into halves and then into fourths. They should observe and then discuss the change in the size of the parts.




Literacy Strategies: Venn Diagram – use a venn diagram to compare and contrast

shapes. Students’ play “guess my rule” to identify if a shape belongs or not

triangle three-sided shape two-dimensional rectangle square trapezoid half-circles quarter-circles cube right rectangular prism right circular cone right circular cylinder compose equal shares halves fourths quarters

Literature Suggestions: Burns, M. The Greedy Triangle. Scholastic. CA 1994 McMillan, B. Fire Engine Shapes. William Morrow. New York. 1988 Murphy, S. Circus Shapes. 1998 Rogers, P. The Shapes Game. Henry Holt. New York. 1989.

Resources

Paper shapes Pattern blocks Color tiles Isosceles right triangles cut from squares Tangrams Canned food (right circular cylinders) Gift boxes (cubes and right rectangular prisms)



ORC # 1481 From the Math Forum: Introduction to fractions for primary students http://mathforum.org/varnelle/knum1.html http://mathforum.org/varnelle/knum2.html http://mathforum.org/varnelle/knum5.html This four-lesson unit introduces young children to fractions. Students learn to recognize equal parts of a whole as halves, thirds and fourths. van Hiele Puzzle

mathematics curriculum guide grade 1 · pdf filemathematics curriculum guide grade 1 ......

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