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 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 = ? = 8 + 6

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

17. Strategies for Addition Facts
Near-Doubles
One addend is one more than the other ( )
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 =
= 86
= 88 48
+34 12 70 82

22. Compare and Contrast 382 + 445 465 48 7127 - 348 +34 123 712
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 Model for 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. Number Sort Sort the number cards into groups95
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 2 4
Students need both procedural and conceptual understandings related to fractions and decimals 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:Reasons for 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 Be careful about using only a few models
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

51. Models to Symbols Top Number Bottom Number how many what
The counting number tells how many shares or parts
Numerator -- Latin root meaning number
Bottom Number
Tells fractional part being counted
If a 4, counting fourths or if a 6, counting sixths, and so on…
Denominator -- Latin root meaning namer

52. Counting with Fractions
Count around the room
Count around the room by halves
Whole numbers stand
Count by fourths
Will more, fewer, or the same number of people be standing when we count by halves or fourths? Why?

53. Set Models
Whole is total set of objects and subsets of the whole are fractional parts
Number of objects not size is important
Examples: counters, people, M & Ms, discrete objects

54. Set Models Describe your table group with 3 – 4 fraction statements
One-half of the group is male
One-half of the group is wearing a hat

55. Set Models Sets with color tiles One-half of the set is red
Two-thirds of the set is blue
Three-sevenths of the set is yellow and one-seventh is green
One-third of the set is yellow, one-sixth is red, and one-half is green

56. Set Models
The idea of referring to a collection of objects as a single whole may be confusing
Three objects is one-fourth of a set of two-color counters
When partitioning sets, children frequently confuse the number of counters in a share with the name of the share

57. Comparing Fractions
Like denominators – When denominators are the same, the fraction with the bigger numerator is larger because there are more of the same-sized parts… Ex. 3/8 < 6/8
Like numerators – When numerators are the same, the fraction with the smaller denominator is larger because the size of the parts is larger… Ex. 2/5 > 2/8

58. Comparing Fractions
More or less than 0, 1/2, 1 – Fractions can be compared by determining whether they are more than or less than a benchmark number
Distance from 0, 1/2, 1 – Fractions can be compared by finding the distances from a benchmark number and then comparing the distances (Ex. 5/6 < 7/8 because 7/8 is closer to 1 since it is only 1/8 away)

59. Caution
Be cautious of teaching rules or algorithms for comparing two fractions
Rules require no thought about the size of the fractions
Students must develop number sense about fraction size

60. Caution
A fraction tells only about the relationship between the part and the whole
It does not say anything about the size of the whole or the size of the parts
Comparisons with any model can be made only if both fractions are parts of the same whole

61. Adding & Subtracting Fractions
1. cherry + orange =
2. lime + blueberry =
3. lemon + cherry =
4. blueberry + lemon =
5. orange + lime + blueberry =
6. One sweetie bar - orange =
7. One sweetie bar - blueberry =
Sweetie Bar

62. Adding & Subtracting Fractions
Explore unlike denominators with models
Use simple contextual tasks
Paul and his brother were eating the same kind of pizza. Paul had 3/4 of his pizza. His brother still had 7/8 of a pizza. How much pizza did the two boys have together?
Estimate answers and check for
reasonableness of solutions

63. Decimals In what real world situations do we use decimals?
Are there times when it is easier to use a decimal notation rather than a fraction notation?

64. 3 ¾ 3.75 Representations Compare these two numbers
How are the numbers alike and how are they different?
3.75
3 ¾

65. Connections
Decimals allow us to represent fractional quantities using our base ten number system
Tasks and multiple representations help students connect fractions and decimals and see how they represent the same concepts

66. Strategies Familiar fraction models can be used to explore decimals
2/4 = .50 = 1/2 = .5
Familiar fraction models can be used to explore decimals
Base ten system can be extended to include numbers less than one as well as large numbers
Models help make meaningful translations and connections between fractions & decimals

67. Extending Models 10 x 10 Square Ten-frame Two-color counters
Fraction circles
Base 10 blocks
Decimal circles

68. Decimals on the Meter Stick
If the meter is 1 whole, how would you represent 2.45 concretely using meter sticks, rods or longs, and unit cubes?
How can the model be used to show equivalence?
Is 0.3 the same amount as 0.30?
How can you tell? Are they the same length?

69. Connection to Measurement
If the length of the meter represents one unit, what is the decimal value of the decimeters, centimeters, and millimeters?
Represent each number using the meter stick and place value blocks

70. Important Ideas
Students can make sense of decimals through models – they need to develop a sense of the magnitude of the numbers
Models can help students focus on place value instead of face value (.18 < .2 even though 18 > 2)
Models need to be connected to notation symbols and to fractions as parts of wholes

71. Reflection
In grade level groups, identify which objectives fall under each big idea

72. DPI Mathematics Staff Everly Broadway, Leanne Barefoot Robin Barbour
Carmella Fair
Chief Consultant
Mary H. Russell
Johannah Maynor
Partners for Mathematics Learning is a Mathematics-Science Partnership Project funded by the NC Department of Public Instruction. Permission is granted for the use of these materials in professional development in North Carolina Partner school districts.
Partners
for Mathematics Learning

73. PML Consultants Amanda Baucom Julia Cazin Anna Corbett Gail Cotton
Ryan Dougherty
Tery Gunter
Kathy Harris
Joyce Hodges
Karen McCain
Vicki Moss
Kayonna Pitchford
Ron Powell
Susan Riddle
Judith Rucker
Shana Runge
Kitty Rutherford
Penny Shockley
Pat Sickles
Nancy Teague
Bob Vorbroker
Jan Wessell
Carol Williams
Stacy Wozny
Tim Hendrix, Advisor Anita Bowman, Evaluator
Freda Ballard, Technology Meghan Griffith, Administrative Assistant 73

74. Partners Writers Ana Floyd Jeane Joyner Rendy King Katherine Mawhinney
Gemma Mojica
Elizabeth Murray
Wendy Rich
Catherine Stein
Please give appropriate credit to the Partners for Mathematics Learning project when using these materials. Permission is granted for their use in professional development in North Carolina Partner school districts.
Jeane Joyner, Project Director
Partners
for Mathematics Learning