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Number & Operations

Number & Operations. 3–5 Module. Football Fun.

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Number & Operations

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  1. Number & Operations 3–5 Module

  2. Football Fun Awesome is the only way to describe Sunday’s battle between the Texans and the Panthers. At the end of the 20th quarter, the score was ½ to 3 with the Texans leading. The quarterback, number 157, threw 1,999 passes to his teammates during the first quarter. However, the Panthers kicked 3 field goals to take over the lead. All 87 players were on the field to defend their team’s record.

  3. Football Fun The excitement grew in the 4th quarter when the biggest player, Julius Peppers from the Panthers, weighing 98 pounds, stopped the Texans’ quarterback from throwing a pass with a crushing tackle on the 150 yard line. The Panthers won and continued their winning streak of 0 and 89.

  4. Football Fun • What is wrong with this story? • What understanding is missing? • What is number sense?

  5. Number Sense • Good intuition about numbers and their relationships • A “feel” for numbers • Approach numbers with flexibility • An awareness and understanding of what numbers are, their relationships, their magnitude, and the relative effect of operating on numbers

  6. Number Sense Students with number sense develop multiple meanings of numbers, know how operations work and how to apply them, and use numbers fluently (accurately, efficiently, flexibly)

  7. SCOS and Big Ideas • Big ideas go across grade levels and reflect important, fundamental understandings upon which SCOS objectives build • Use your Big Ideas handout throughout the workshop to make notes on how your grade level objectives fit

  8. Big Ideas in Number • Numbers can be classified in multiple ways to show relationships • Properties are the basis of classifications • Some representations show equivalent relationships • Part-whole relationships can reflect composing and decomposing

  9. Big Ideas in Number • Numbers in elementary mathematics are represented using a base-10 place value system • System - based on groupings of ten - allows us to represent all numbers with just 10 digits • Digits have different values depending on their positions (both whole numbers and decimals) • Students must understand both place value and face value • Composite groups (groups of more than one) can be counted multiple times and operated on as an entity

  10. Big Ideas in Number • Dealing with multiplicative reasoning requires a shift in thinking about numbers from numbers representing single units to composite units that are grouped • Composite groups (groups of more than one) can be counted multiple times and operated on as an entity • Instruction should focus on helping students identify, create, and count composite groups

  11. Big Ideas in Number • Operations (addition, subtraction, multiplication, division) are used to suggest distinct actions that are defined mathematically and are dependent on the context; performing those actions leads to consistent results for all numbers (whole and rational) • Operation relationships are an important part of number sense • Properties combined with operations are the foundation of arithmetic • Properties lead to mathematical generalizations

  12. Big Ideas in Number • Reasonable estimates reflect an understanding of both operations and number relationships • Context influences what an appropriate range (estimate) would be • Appropriate estimates reflect students’ sense making

  13. Big Ideas in Number • Fluency (accuracy, efficiency, flexibility) is reasoning about and using rational number operations with understanding • Mastery involves knowing strategies for retrieving basic facts and being able to apply them in other computations • Fluency is built upon number relationships, place value, properties, and operation understandings

  14. Early Computation • “Number sense” is a major goal of elementary mathematics • K-2 students learn the meanings of addition and subtraction as well as basic fact and problem solving strategies • Some students are still working toward fluency operating with one and two-digit numbers

  15. Developing Fluency • Understand relationship of numbers to 10 and part-part-whole relationships • Understand how addition and subtraction are related • Commutative property for addition 8 + 6 = ? 8 is 2 away from 10, 2 & 4 make 6 13 – 7 = ? “7 and what makes 13” 8 + 6 = ? 6 + 8 = 8 + 6

  16. Strategies for Addition Facts • One-More-Than and Two-More Than Facts • One addend is a 1 or 2 5 + 1, 5 + 2 • Facts with Zero • One addend is 0 4 + 0 • Doubles • Addends are the same 7 + 7

  17. Strategies for Addition Facts • Near-Doubles • One addend is one more than the other 5 + 6 (5 + 5 + 1) • Make-Ten Facts • One of the addends is 8 or 9 7 + 8 6 + 9 How could you make one addend ten?

  18. Developing Fluency Provide drill in the use and selection of those strategies once they have been developed • Practice – problem-based activities in which students develop flexible and useful strategies • Drill – repetitive non-problem-based activity; students have a strategy they understand and know how to use; helps to make the strategy automatic

  19. Addition Expectations Students should be able to write, interpret and solve problems that represent given problem situations using both horizontal and vertical equations for two, three and four-digit numbers by the end of third grade

  20. Multiple Solutions • Illustrate different ways to solve the problem • What thinking is used in each solution method? 239 +603

  21. Compare and Contrast Describe the strategies used to solve these 1 1 468 + 345 813 468 +345 700 100 13 813 48 +34 12 70 82 56 + 32 = 56 + 30 = 86 86 + 2 = 88

  22. Compare and Contrast Describe the misconceptions or incomplete understandings these solutions illustrate 382 + 445 7127 465 - 348 123 48 +34 712

  23. Multiplication • Multiplication is more than memorizing tables and practicing algorithms • Multiple perspectives bring a broader view & a more active approach to learning • Geometric Perspective • Numerical Perspective • Real-World Perspective

  24. Real-World Multiplication Things that come in groups: • What comes in 2s, 3s, and 4s? • Brainstorm things that come in groups Context determines the operations of multiplication and division

  25. Equal Group Problems Multiplication: • Situations when the number and size of groups is known Example: • The boys have 6 wagons. There are 4 books in each wagon. How many books do they have altogether?

  26. Equal Group Problems Division: • Situations when either the number or size of sets is unknown Two Types: • Partition Division • Measurement Division

  27. Equal Group Problems • Partition Division • Rick has 24 apples. He wants to share them equally with 4 of his friends. How many apples will each friend receive? • Measurement Division • Rick has 24 apples. He put them into bags containing 6 apples each. How many bags will Mark use? How might children solve these differently?

  28. Multiplication Explorartion Patterns on the Hundred’s Chart: • Looking for patterns helps students understand multiplication better than simply rote counting multiples (3, 6, 9, 12, 15, …) • Find interesting patterns of t multiples on the chart

  29. Geometric Modelfor Multiplication • Rectangular Arrays • What are all of the rectangles you can build with the factors of the number 24? • Build a rectangle: How would you name this rectangle? What is the multiplication fact you’ve modeled? What do you notice?

  30. Multiplicative Reasoning • Quantitative relationships center around equal-sized groups • Whether dealing with place value or operations, in order to reason multiplicatively, students must understand the idea that composite groups (groups of more than one) can be counted multiple times and operated on as an entity

  31. Solve Using Two Methods 5 x 63 = ? Try solving 4 x 27 using a new/different strategy

  32. Array Model How would 63 x 5 work as an array model? 60 3 300 15 5 What would 63 x 57 look like as an array?

  33. Array Model 60 3 Partial Products: 3000 150 420 + 21 3591 3000 150 50 420 21 7

  34. 63 X 57 441 3150 3591 Array Model 60 3 3150 3000 150 50 441 420 21 7

  35. Context is Key! For 32 ÷ 5 = • Participant A: Answer is 6 • Participant B: Answer is 6 r 2 • Participant C: Answer is 7 • Participant D: Answer is 6 2/5 • Participant E: Answer is 6 or 7 …What is the question????

  36. Multiplication Strategies • Using repeated addition • Skip-counting • Doubling • Using partial products • Using five-times and ten-times • Doubling and halving • Nifty nines • Factoring and grouping flexibly • Properties of mathematics (commutative, associative, distributive, identify)

  37. SCOS: Fractions & Decimal s Standard Course of Study Objectives for Fractions and Decimals • Note the progression of ideas • Check K-2 objectives • What are the specific objectives for your grade?

  38. 95 Number Sort • Sort the number cards into groups How did you sort the cards? How are the numbers alike? • Sort the number cards into groups again How is your sort different from the first sort?

  39. Classifying Numbers • Why would we classify numbers with students?

  40. 1 10 Numbers Less Than One Students need both procedural and conceptual understandings related to fractions and decimals 2 4 5 6 .68 3 5 .75 What makes fractions difficult for students? .09

  41. Fraction Concepts & Skills • Name fractional parts as wholes and sets • Understand different models of fractions • Representing fractions using standard notation, concrete and pictorial representations • Equivalence; relation to “whole” • Compare and order fractions • Compute with fractions • Applications; Solving problems

  42. Fractions:Reasonsfor Difficulties • Material is taught • Too abstractly • Too procedurally • Without meaningful contexts • Through rote memorization of procedures • With attention on algorithms and less attention on number sense and reasoning • Without connections • With limited models

  43. Area/Region Models • Region is cut into equivalent parts • Regions may/may not be congruent • Examples: pattern blocks, grid or dot paper, geoboards, circles, squares and other rectangles

  44. Unit Fractions • One of equal portions • Numerator is 1 • Denominator is number of equal portions

  45. Exploring Fourths • Red lines divide the figure into fourths • How many small squares in each fourth? • Are each of these sections fourths? • How do you know?

  46. Tangrams • Square divided into seven pieces • If the value of the entire square is 1, what is the value of each tangram piece? • If the value of the large triangle is 1, how does this change the value of each tangram piece?

  47. Caution • Be sure to identify the whole or whole unit • Different pieces can be identified as the whole

  48. Caution Examples • Be careful about using only a few models • It is important to model fractions with manipulatives and also to draw fraction representations • Include non-examples Non-examples

  49. Linear Models • Length is divided into smaller parts; lengths can be compared • Examples: fraction tiles, paper strips, Cuisenaire rods, number lines, rulers

  50. Linear Models • Number Lines • Useful tool for ordering fractions • Useful in helping students recognize that fractional parts can be subdivided (halves into fourths; fourths into eights) • Yard or meter sticks can be linear models

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