Distinguishing Between Shapes and Attributes

Transcription

1 Grade 1 Mathematics, Quarter 1, Unit 1.1 Distinguishing Between Shapes and Attributes Overview Number of Instructional Days: 5 (1 day = 45 minutes) Content to Be Learned Understand defining attributes (e.g., triangles are closed and three-sided). Understand non-defining attributes (e.g., color, orientation, overall size). Distinguish between defining and non-defining attributes. Build shapes to understand defining attributes. Draw shapes to understand defining attributes. Essential Questions What is this shape and how do you know? What attributes did you use to sort these shapes? Why did you choose those attributes to sort these objects? How can you sort these shapes in another way? Mathematical Practices to Be Integrated Construct viable arguments and critique the reasoning of others. Reason with shapes and their attributes. Make conjectures and build a logical progression of statements. Critique the reasoning of others. Look for and make use of structure. Discern a pattern or structure. How can you draw a (square, triangle, rectangle, hexagon, circle, trapezoid)? Show me. How can you build a (square, triangle, rectangle, hexagon, circle, trapezoid)? Show me. How do you know the shape you drew/built is a? 1

2 Grade 1 Mathematics, Quarter 1, Unit 1.1 Distinguishing Between Shapes and Attributes (5 days) Written Curriculum Common Core State Standards for Mathematical Content Geometry 1.G Reason with shapes and their attributes. 1.G.1 Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus nondefining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes. Common Core Standards for Mathematical Practice 3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. 2

3 Grade 1 Mathematics, Quarter 1, Unit 1.1 Distinguishing Between Shapes and Attributes (5 days) Clarifying the Standards Prior Learning Students named objects in the environment using names of shapes. They also named shapes correctly regardless of orientation. Students analyzed and compared two- and three-dimensional shapes describing their similarities and differences using informal language. Students built and drew shapes as well as composing simple shapes to form larger ones. Current Learning In this unit, students distinguish between defining (number of sides, number of vertices) and non-defining (color, size, orientation) attributes and use this formal language to describe shapes. They expand knowledge of two-dimensional shapes (square, circle, triangle, hexagon, rectangle, and trapezoid) using defining attributes and they draw and build shapes to distinguish defining attributes (draw/build closed shape with four equal sides). This is being taught at the developmental level. Later in the year (unit 2.3), students will compose two-dimensional and three-dimensional shapes to make composite shapes. Future Learning In second grade, students will use their knowledge of attributes to recognize and draw shapes having specified attributes (number of angles or faces). Additional Findings According to Principles and Standards for School Mathematics, Teachers must ensure that students see collections of triangles in different positions and with different sizes of angles and shapes that have a resemblance to triangles but are not triangles. (p. 98) See A Research Companion to Principles and Standards for School Mathematics for information on theories on geometric thinking learning and teaching geometry. (p. 152) 3

4 Grade 1 Mathematics, Quarter 1, Unit 1.1 Distinguishing Between Shapes and Attributes (5 days) 4

5 Grade 1 Mathematics, Quarter 1, Unit 1.2 Ordering Objects and Interpreting Data Overview Number of Instructional Days: 5 (1 day = 45 minutes) Content to Be Learned Order three objects by length. Interpret data with up to three categories. Answer questions about the total number of data points. Tell how many data points are in each category. Mathematical Practices to Be Integrated Model with mathematics. Apply mathematics known to everyday life. Use tools to identify important quantities in practical situations. Analyze and interpret data mathematically to draw conclusions. Attend to precision. Communicate precisely to others. Specify units of measure to clarify the quantity. Calculate/count accurately. Essential Questions How would you put these objects in order? Explain. What is the total number of data points in each category? What is the total number of data points in all? 5

6 Grade 1 Mathematics, Quarter 1, Unit 1.2 Ordering Objects and Interpreting Data (5 days) Written Curriculum Common Core State Standards for Mathematical Content Measurement and Data 1.MD Measure lengths indirectly and by iterating length units. 1.MD.1. Order three objects by length; compare the lengths of two objects indirectly by using a third object. Represent and interpret data. 1.MD.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. Common Core Standards for Mathematical Practice 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 6

7 Grade 1 Mathematics, Quarter 1, Unit 1.2 Ordering Objects and Interpreting Data (5 days) Clarifying the Standards Prior Learning Students used the attribute of length to describe objects. They compared two objects by direct comparison of length to find which object is taller/shorter, longer/shorter, etc. Current Learning In this unit, students order three objects by length. They interpret data with up to three categories. Students ask and answer questions about the total number of data points. This is taught at the developmental level. Later in the year (Unit 2.2), students compare the length of two objects indirectly by using a third object. Students organize and represent data with up to three categories (Unit 3.2). They 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. Future Learning In second grade, students will use and select appropriate tools to measure objects using standard units. Students will generate data by measuring line plots. They will draw a picture graph and a bar graph to represent data with up to four categories. They will solve word problems using information presented in a bar graph. Additional Findings Representing measurements and discrete data in picture and bar graphs involves counting and comparisons that provide another meaningful connection to number relationships (Curriculum Focal Points, p. 18). Students in grade 1 can ask and answer questions about categorical data based on a representation of the data (Mathematical Progressions, K 3, Categorical Data; Grades 2 5, Measurement Data p.5 6). Students struggle to see the data as a whole versus individual categories. Teachers need to help students make the conceptual leap to move from seeing data as disaggregate pieces to a comprehensive whole as referenced from A Research Companion to Principles and Standards for School Mathematics, p

8 Grade 1 Mathematics, Quarter 1, Unit 1.2 Ordering Objects and Interpreting Data (5 days) 8

9 Grade 1 Mathematics, Quarter 1, Unit 1.3 Introduction to Place Value and Comparing to 30 Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Count to 120, starting at any number less than 120. Read and write numerals to 30. Represent number of objects with written numerals. Understand that the two digits of a two-digit number are made up of tens and ones. Understand that 10 is a bundle of 10 ones called a ten. Understand that the numbers are composed of a ten and ones. Understand that the decade numbers (10, 20, 30, 40 ) refer to 1, 2, 3, 4 tens and zero ones. Compare two 2-digit numbers using various strategies including place value. Essential Questions Starting at, how far can you count? What patterns do you notice? What is this number (shown by this amount of objects)? How many (objects) do you see? How do you write this number? Mathematical Practices to Be Integrated Make sense of problems and persevere in solving them. Continually ask themselves, Does this make sense? Use concrete objects or pictures to help conceptualize and solve problems. Understand the approaches of others. Look for and make use of structure. Recognize the structure of two-digit numbers (understand two-digit numbers as composed of tens and ones). Detect the pattern in a counting sequence. Detect patterns in word names and their numeral representations. How can you use tens and ones to describe the number? (11 19 and 10, 20, 30?) Which number is more? Which number is fewer? How do you know? 9

10 Grade 1 Mathematics, Quarter 1, Unit 1.3 Introduction to Place Value and Comparing to 30 (15 days) Written Curriculum Common Core State Standards for Mathematical Content Number and Operations in Base Ten 1.NBT Extend the counting sequence. 1.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 value. 1.NBT.2 1.NBT.3 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). Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <. Common Core Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see

11 Grade 1 Mathematics, Quarter 1, Unit 1.3 Introduction to Place Value and Comparing to 30 (15 days) equals the well remembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Clarifying the Standards Prior Learning In kindergarten, students counted to 100 by 1s and 10s. They began at any given number within the known sequence. Students also wrote numbers 0 20 and represented a number of objects with a written numeral Students began to build the foundation for place value by composing and decomposing numbers from into ten ones and some further ones by using objects or drawings or equations. Students compared the number of objects in one group to the number of objects in another group. Students also compared two numbers between 1 and 10 presented as a written numeral. Current Learning This is a critical area of learning in Grade 1. In first grade, students generalize their understanding by counting to 120 starting from any number less than 120. In this unit, students read and write numerals and represent a number of objects with a written numeral up to 30. By the end of the year, students represent, read, write, and compare numbers to 120. This is taught at the reinforcement level for numbers 0 20 and the developmental level for numbers Place value is a critical area of instruction in first grade. Students understand that the two digits of a 2- digit number represent bundles of tens and some ones. This is taught at the developmental level. They understand that 10 is a bundle of ten ones called a ten. Students understand the teen numbers (11 19) are composed of a ten and additional ones, and the decades (10, 20, 30, etc.) refer to a number of tens and no additional ones. This is taught at the reinforcement level. Students compare two 2-digit numbers. This is taught at the developmental level. At this time of year, students need to develop this understanding by using bundles, such as snap cubes and linking chains, that are easy to count and separate into ones. Students are at the discrete stage of development and solid place-value rods may promote confusion. As students gain experience with countable bundles, they may be able to transition to place-value rods and other more abstract visuals and models. Later in the year, students record the results of these comparisons using <, >, and =, and increase the number range to 120. Students will use their understanding of place value to compute sums within 100. (1.NBT.4) Future Learning In second grade, students will learn that 100 is a bundle of ten 10s. The numbers (100, 200, 300, etc.) refer to the number of hundreds (and 0 tens and 0 ones). They will count within 1,000. Students will skip count by 5s, 10s, and 100s. Students will read and write numbers to 1,000 using base-ten numerals, number names, and expanded form. They will compare 2- and 3-digit numbers and use symbols <, >, and = to record results. 11

12 Grade 1 Mathematics, Quarter 1, Unit 1.3 Introduction to Place Value and Comparing to 30 (15 days) Additional Findings Learning Progression K 5, Number and Operations in Base Ten (pp. 5 6): See the number-bond diagram, 5 and 10 frames, place value cards, numeral list, and visual supports as teacher resources to highlight the base-ten structure of numbers. The number words continue to require attention at first grade because of their irregularities. The decade words, twenty, thirty, forty, etc., must be understood as indicating 2 tens, 3 tens, 4 tens, etc. Many decade number words sound much like teen number works. For example, fourteen and forty sound very similar (p. 6) Principles and Standards for School Mathematics, students should recognize that the word ten may represent a single entity (1 ten), and, at the same time, ten separate units (10 ones) and that these representations are interchangeable. (p. 81) A Research Companion to Principles and Standards for School Mathematics, As with the teens words, the English number words between twenty and one-hundred complicate the teaching and learning processes for multidigit addition and subtraction. (p. 78) Children view numbers as single digits side by side: 827 is functionally eight two seven and not 8 groups of one hundred, 2 groups of ten, and 7 single ones. (p. 79) 12

13 Grade 1 Mathematics, Quarter 1, Unit 1.4 Understanding, Representing, and Solving Addition Within 20 Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Add within 20 to solve word problems (adding to, putting together). Relate counting to addition. Add within 20 using counting on, making 10, and decomposing strategies. Understand the meaning of equals sign (=) involving addition. Apply the commutative property as a strategy to solve addition problems. Represent an addition problem with an equation using symbols to represent the unknown number. Essential Questions How would you solve this word problem? How would you represent this word problem with an equation? How would you represent this problem using a model or drawing? Mathematical Practices to Be Integrated Reason abstractly and quantitatively. Represent a situation symbolically using objects, drawings, and equations. Given a symbolic representation, identify or create a situation. Look for and make use of structure. When solving and representing addition problems, understand that the sum is composed of several objects (addends). Detect the structure of the commutative property of addition (3 + 7 = 10, = 10). Recognize the structure of the distributive property to use the make 10 strategy to add (8 + 6 = = = 14). Use decomposing numbers by place value as a strategy for solving addition problems involving two-digit numbers. When counting on to add these two numbers, how do you decide which number to start with? What does = mean in this equation? How can counting help you find the total amount? 13

14 Grade 1 Mathematics, Quarter 1, Unit 1.4 Understanding, Representing, and Solving Addition Within 20 (15 days) Written Curriculum Common Core State Standards for Mathematical Content Operations and Algebraic Thinking 1.OA Represent and solve problems involving addition and subtraction. 1.OA.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. 2 2 See Glossary, Table 1. Understand and apply properties of operations and the relationship between addition and subtraction. 1.OA.3 Apply properties of operations as strategies to add and subtract. 3 Examples: If = 11 is known, then = 11 is also known. (Commutative property of addition.) To add , the second two numbers can be added to make a ten, so = = 12. (Associative property of addition.) 3 Students need not use formal terms for these properties. Add and subtract within OA.5 Relate counting to addition and subtraction (e.g., by counting on 2 to add 2). 1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., = = = 14); decomposing a number leading to a ten (e.g., 13 4 = = 10 1 = 9); using the relationship between addition and subtraction (e.g., knowing that = 12, one knows 12 8 = 4); and creating equivalent but easier or known sums (e.g., adding by creating the known equivalent = = 13). Work with addition and subtraction equations. 1.OA.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, = 2 + 5, =

15 Grade 1 Mathematics, Quarter 1, Unit 1.4 Understanding, Representing, and Solving Addition Within 20 (15 days) Common Core Standards for Mathematical Practice 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Clarifying the Standards Prior Learning In kindergarten, students solved addition word problems within 10 using objects, fingers, mental images, simple drawings, sounds, acting out, verbal explanations, or equations. Students used drawings or equations to decompose numbers less than or equal to 10 (5 = 2 + 3; 5 = 4 + 1). For any number 1 9, students found the number that made 10 when added to a given number by using objects or drawing and recorded answers with an equation or drawing. Students became fluent adding and subtracting within 5. Kindergarten students were exposed to equations and the equals sign symbol (=), and were encouraged to write equations, but writing equations was not required. Current Learning This is a critical area of learning in Grade 1. In this unit, students relate counting to addition to solve word problems within 20 using strategies involving adding to and putting together using objects, drawings, and equations with a symbol for the unknown number. Working with the commutative property of addition (8 + 3 = 11; = 11) encourages students to use counting on strategies when combining quantities. Students add within 20 by counting on, making a ten, and creating equivalent, but easier known sums. They develop understanding of the meaning of the equals sign (=). This is taught at the developmental level. 15

16 Grade 1 Mathematics, Quarter 1, Unit 1.4 Understanding, Representing, and Solving Addition Within 20 (15 days) Later this year, students apply their understanding of counting and decomposing a number leading to a ten to subtract within 20. Students use this understanding to solve word problems by taking from, taking apart, and comparing with unknowns in all positions, solving all problem types listed in CCSS glossary table 1 (p. 88). Students use strategies based the relationship between addition and subtraction to solve problems involving unknown addends. They will determine if equations involving addition and subtraction are true or false. Future Learning In second grade, students will develop understanding of odd and even within 20 and will fluently add within 20 using mental strategies. Students will add and subtract within 1,000 using concrete models, drawings, and strategies based on the properties of operations, place value, and the relationship between addition and subtraction; they will fluently add and subtract within 100 using these strategies. Third grade students will generalize these strategies to larger numbers, and in fourth grade students will be expected to fluently apply the standard algorithm for addition and subtraction. Additional Findings Kindergarten, Counting and Cardinality; K 5, Operations and Algebraic Thinking Learning Progressions gives more information on representing and solving subtypes for all unknowns in all three positions. Students will need level 2 and 3 strategies to extend addition problem solving beyond 10 (pp ). Linking equations to concrete materials, drawings, and other representations of problem situations afford deep and flexible understandings of these building blocks of algebra (p. 13). According to A Research Companion to Principles and Standards for School Mathematics, Teachers should not just focus on keywords in word problems, but rather build a complete mental model of the problem. (p. 68) Children move through an experiential progression of single-digit addition methods. Helping children progress through methods can lead all first graders to methods that are efficient enough to use for all later multidigit calculations (p. 73). A major step is that children notice that they can count on. Children in time learn to chunk (doubles, make 10). Ten frames enable children to see sums of 10. (p. 74) 16