Counting and Comparing Numbers to 120

Transcription

1 Grade 1 Mathematics, Quarter 4, Unit 4.1 Counting and Comparing Numbers to 120 Overview Number of Instructional Days: 20 (1 day = minutes) Content to be Learned Count to 120, starting at any number less than 120. Read and write numerals to 120. Represent a number of objects with a written numeral. Subtract multiples of 10 in the range from multiples of 10 in the range Relate the strategy to a written method and explain the reasoning used. Mathematical Practices to Be Integrated Look for and make use of structure. Detect a structure or pattern (i.e., in a counting sequence). Look for and express regularity in repeated reasoning. Look for general methods and shortcuts. Continually evaluate the reasonableness of intermediate results. Essential Questions Starting at, how far can you count? (Show me, if needed.) What pattern do you notice? What is this number? (Teacher shows numeral up to 120.) How many do you see? ( ) Write that number. What strategy would you use to solve: (Teacher may pose a word problem or equation. For example, There are 90 students in the gym. Forty students leave. How many students are left in the gym? or =.) How can you explain the strategy used to solve the problem? How does this written method explain your thinking? 49

2 Grade 1 Mathematics, Quarter 4, Unit 4.1 Counting and Comparing Numbers to 120 (20 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. Use place value understanding and properties of operations to add and subtract. 1.NBT.6 Subtract multiples of 10 in the range from multiples of 10 in the range (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. Common Core Standards for Mathematical Practice 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. 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x 2 + x + 1), and (x 1)(x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. 50

3 Grade 1 Mathematics, Quarter 4, Unit 4.1 Counting and Comparing Numbers to 120 (20 days) Clarifying the Standards Prior Learning In kindergarten, students composed and decomposed numbers into 10 ones and some further ones. They understood that these numbers are composed of 10 ones and 1, 2, 3, 4, 5, 6, 7, 8, or 9 ones. They counted to 100 by ones and tens, beginning at any given number in the known sequence. They wrote and represented numbers Current Learning Skills addressed in this unit are critical areas of instruction for grade 1. Earlier in the year, students read, write, and represent numerals 0 100, and the concept of equality is taught at the reinforcement level. They understand the numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to 1, 2, 3, 4, 5, 6, 7, 8, or 9 tens (and 0 ones). They also mentally find 10 more or 10 less of a two-digit number without having to count. They add within 100, adding a two-digit number and a multiple of 10. In this unit, skills taught at the developmental level are as follows: Students read and write numbers and use the symbols for greater than and less than to compare numbers up to 120. Students subtract multiples of 10 in the range from multiples of 10 in the same range using models and strategies. They relate the strategy to a written method and explain the reasoning used. Future Learning In grade 2, students will read and write numerals up to 1,000. They will mentally add 10 or 100 to a given number and mentally subtract 10 or 100 from a given number Additional Findings The number words continue to require attention in first grade because of their irregularities. (K 5 Number and Operations in Base 10 Learning Progressions, p. 6) K, Counting and Cardinality; K 5 Operations and Algebraic Thinking Learning Progressions states, Students understand that the last number name said in counting tells the number of objects counted. This connection will continue in grade 1 with the more advanced counting-on methods. (pp. 4 and 5) First graders also engage in mental calculation, such as mentally finding 10 more or 10 less than a given two-digit number without having to count by ones. They may explain their reasoning by saying that they have one more or one less 10 than before. Drawings and layered cards can afford connections with place value and be used in explanations. (K 5 Number and Operations in Base 10 Learning Progressions, p. 7) Differences of multiples of 10 such as can be viewed as 7 tens minus 4 tens and represented with concrete models such as objects bundled in tens or drawings. Children use the relationship between subtraction and addition when they view as an unknown addend addition problem, 70 + = 80, and reason that 1 ten must be added to 70 to make 80, so = 10. (K 5 Number and Operations in Base 10 Learning Progressions, p. 7) According to Principles and Standards for School Mathematics, Teachers should also encourage students to share the strategies they develop in class discussions. Students can develop and refine strategies as they hear other students descriptions of their thinking about number combinations. (p. 84) 51

4 Grade 1 Mathematics, Quarter 4, Unit 4.1 Counting and Comparing Numbers to 120 (20 days) 52

5 Grade 1 Mathematics, Quarter 4, Unit 4.2 Understanding, Representing, and Solving Addition and Subtraction Problems Within 20 Overview Number of Instructional Days: 20 (1 day = minutes) Content to be Learned Use addition and subtraction within 20 to solve word problems with unknown in all positions. (See CCSS glossary, Table 1, p. 88.) Solve word problems adding three whole numbers whose sum is less than or equal to 20 using equations with a symbol for the unknown number. Add within 20, demonstrating fluency for subtraction within 10 using strategies such as counting on, making 10, decomposing a number leading to a 10, using the relationship between addition and subtraction, and creating equivalent but easier or known sums. Mathematical Practices to Be Integrated Make sense of problems and persevere in solving them. Understand the problem and look at how to begin to solve the problem. Plan a strategy to find a solution instead of jumping right into the problem. Identify correspondences between different approaches. Construct viable arguments and critique the reasoning of others. Justify and communicate conclusions. Construct arguments using concrete objects, drawings, and actions. Recognize and use counterexamples. Essential Questions How would you solve this word problem? Can you solve it another way? (See CCSS glossary, Table 1, p. 88 to pose questions for start unknown in each category). How would you solve this word problem involving addition of three numbers with a symbol for the unknown? (e.g., There is fruit in a bowl. There are 4 apples, 6 bananas, and 5 pears. How many pieces of fruit are there in all? Write an equation to show your answer = and = ) What do you notice about these numbers? How could you use what you notice to solve this problem? (e.g., To add , the second two numbers can be added to make a 10, so = = 12) What strategy did you use to solve this problem? (Add or subtract these numbers within 20.) What is? (Fluently add and subtract within 10. e.g., 2 + 3; 6 + 4; 8 1) How can you solve these problems? (8 +? = 11; 5 =? 3; =?) 53

6 Grade 1 Mathematics, Quarter 4, Unit 4.2 Understanding, Representing, and Solving Addition and Subtraction Problems Within 20 (20 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. 1.OA.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. Add and subtract within 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). 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. 54

7 Grade 1 Mathematics, Quarter 4, Unit 4.2 Understanding, Representing, and Solving Addition and Subtraction Problems Within 20 (20 days) 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. Clarifying the Standards Prior Learning In kindergarten, students represented addition and subtraction word problems with pictures, sounds and mental images, or equations. They also decomposed numbers 10 and less into pairs more than one way. Students used pictures or objects and recorded the decomposition by a drawing or equation. When given any number 1 9, they found the number that made 10 by using objects or drawings. Students fluently added and subtracted within five. Current Learning This is a critical area of instruction for grade 1. Earlier in grade 1, students add within 20. They use subtraction to solve word problems and understand subtraction as an unknown addend problem. Students use strategies such as counting on, making 10, doubles, near doubles, fact families, etc. They understand the meaning of the equal sign involving addition and subtraction. These skills are reinforced in this unit. Students solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. (See CCSS glossary, Table 1, p. 88.) Comparing is taught at the reinforcement level, and all other strategies are reinforced in this unit. In addition, students determine the unknown whole number in an addition or subtraction equation relating three whole numbers. These skills are taught at the reinforcement level. Students add within 20, demonstrating fluency for addition within 10. This is also at the reinforcement level. In this unit, students use addition and subtraction within 20 to solve word problems involving unknowns in all positions. They solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20 using equations with a symbol for the unknown number to represent the problem. (See CCSS glossary, Table 1, p. 88.) Students demonstrate fluency for subtraction within 10. This is at the developmental level. An equation must include a symbol for the unknown, which can be a question mark, line, box, etc. 55

8 Grade 1 Mathematics, Quarter 4, Unit 4.2 Understanding, Representing, and Solving Addition and Subtraction Problems Within 20 (20 days) Future Learning In grade 2, students will use addition and subtraction within 100 in all situations to solve one- and twostep word problems with unknowns in all positions using drawings and equations. (See CCSS glossary, Table 1, p. 88.) Students will fluently add and subtract within 20 using mental strategies. They will know from memory all sums of two 1-digit numbers. Additional Findings The Learning Progressions recognize three levels of methods used for solving single-digit addition and subtraction problems. See K 5 Operations and Algebraic Thinking Learning Progressions appendix, p. 36 and pp , for a more detailed description. According to Principles and Standards for School Mathematics, Developing fluency requires a balance and connection between conceptual understanding and computational proficiency. On the one hand, computational methods that are over practiced without understanding are often forgotten or remembered incorrectly. On the other hand, understanding without fluency can inhibit the problem solving process. (p. 35) According to A Research Companion to Principles and Standards for School Mathematics, Intervention studies indicate that teaching counting on in a conceptual way makes all single-digit additions accessible to U.S. first graders, including learning disabled students and limited English proficient students. With practice, children can perform counting on rapidly and accurately enough to use this method in multidigit calculations of all kinds. Counting on is a powerful, general, and sufficiently rapid method for most purposes. (p. 75) See Figure 6.1 on page 73 for the Learning Progression for single-digit addition and subtraction. 56