Provincial Grade 3 Assessment in Numeracy Fall 2009.

Provincial Grade 3 Assessment in Numeracy Fall 2009

Highlights of Revisions 4 Numeracy competencies Algebraic Reasoning Skills  Student predicts an element in a repeating pattern  Student understands that the equal symbol represents an equality of terms found on either side of the symbol 4 Numeracy competencies Algebraic Reasoning Skills  Student predicts an element in a repeating pattern  Student understands that the equal symbol represents an equality of terms found on either side of the symbol

Highlights Number Sense  Student understands that a given whole number may be represented in a variety of ways (to 100).  Student uses mental math strategies to determine answers to addition and subtraction questions to 18. Number Sense  Student understands that a given whole number may be represented in a variety of ways (to 100).  Student uses mental math strategies to determine answers to addition and subtraction questions to 18.

Purpose Improve student learning Communicate information about student achievement Improve student learning Communicate information about student achievement

Audiences for Assessment Info Parents School-based learning team Educational stakeholders Parents School-based learning team Educational stakeholders

Carousel Place different numbers onto chart paper around the room. Put your class into groups. Have the groups walk around the room and write representations of the number as many ways as they can. Tell them to use pictures, words, numbers, and manipulatives. Have them move onto the next number when you cue them with music. Place different numbers onto chart paper around the room. Put your class into groups. Have the groups walk around the room and write representations of the number as many ways as they can. Tell them to use pictures, words, numbers, and manipulatives. Have them move onto the next number when you cue them with music. 25 2 tens 5 ones Twenty-five

Number Sense Number sense involves deep and fundamental understandings of, and proficiency with, counting, numbers, and operations as well as an understanding of number systems and their structures.

Competency Student understands that a given whole number may be represented in a variety of ways (to 100). When students are representing numbers in a variety of ways, they demonstrate their understanding of the use of number, how a number compares to another number, how a number can be broken into parts, and place value knowledge. The ability to represent numbers in a variety of ways will help students with operations and mental math. Student understands that a given whole number may be represented in a variety of ways (to 100). When students are representing numbers in a variety of ways, they demonstrate their understanding of the use of number, how a number compares to another number, how a number can be broken into parts, and place value knowledge. The ability to represent numbers in a variety of ways will help students with operations and mental math.

Vocabulary Part-part-whole: The ability to conceptualize a number as being composed of other numbers. Concretely: Representing a situation or solving a problem using actual objects. Pictorially: Representing a situation or solving a problem using drawings or representations of actual objects. Symbolically: Representing a situation or solving a problem using an abstract representation. Most symbolic representations involves using numbers. Part-part-whole: The ability to conceptualize a number as being composed of other numbers. Concretely: Representing a situation or solving a problem using actual objects. Pictorially: Representing a situation or solving a problem using drawings or representations of actual objects. Symbolically: Representing a situation or solving a problem using an abstract representation. Most symbolic representations involves using numbers.

Representations Students need to be encouraged to represent numbers in a variety of ways (ie. Using manipulatives, words and pictures, number sentences, place value, money, ten frames, number line, connections to other strands and real life situations). Representations of numbers: Pictorial Part-part-whole using multiples of ten (ie. 85 = 80 + 5 or 40 + 40 + 5) Part-part-whole using non-multiples of ten (85 = 83 + 2) Regular place value (56 is 5 tens, 6 units) Irregular place value (56 is 4 tens, 16 units) Concrete with more than one type of manipulative (e.g., 36 can be represented with money, a model for tens, blocks, straws, etc.) Words Naming a number less than Naming a number greater than Odd or even Connections to real life situations Connections to other strands Story problem Another language Students need to be encouraged to represent numbers in a variety of ways (ie. Using manipulatives, words and pictures, number sentences, place value, money, ten frames, number line, connections to other strands and real life situations). Representations of numbers: Pictorial Part-part-whole using multiples of ten (ie. 85 = 80 + 5 or 40 + 40 + 5) Part-part-whole using non-multiples of ten (85 = 83 + 2) Regular place value (56 is 5 tens, 6 units) Irregular place value (56 is 4 tens, 16 units) Concrete with more than one type of manipulative (e.g., 36 can be represented with money, a model for tens, blocks, straws, etc.) Words Naming a number less than Naming a number greater than Odd or even Connections to real life situations Connections to other strands Story problem Another language

There are different, but equivalent, representations for a number. This is a key idea in mathematics. It is relevant for all number types and across multiple grades.

Number of the Day 25 Write in Number Words: ________________________ Show the number in pictures: Write the number that: Comes before:___ Comes after:___ Is 10 more:___ Is 10 less:___ Show the number two ways using base ten blocks:

Assessment Strategies and Observation Tips Using the number of the day is a valuable way to develop and assess representations of numbers. 25 2 tens 5 ones 20 +5 5 x 5

Number of the Day Students represent the number in a variety of ways. Students generate ways of combining numbers and operations to make that number. To help develop students’ flexibility with numbers and operations. To develop understanding of number composition and part-whole relationships. Relate a number of mathematical concepts with number of the day. Students represent the number in a variety of ways. Students generate ways of combining numbers and operations to make that number. To help develop students’ flexibility with numbers and operations. To develop understanding of number composition and part-whole relationships. Relate a number of mathematical concepts with number of the day.

Pictures 125 Words Place Value Equations 49 94 Compare

Competency Student uses mental math strategies to determine answers to addition and subtraction questions to 18. The use of mental math strategies and part-part-whole thinking will lead to long term understanding of basic facts. Student uses mental math strategies to determine answers to addition and subtraction questions to 18. The use of mental math strategies and part-part-whole thinking will lead to long term understanding of basic facts.

What are math strategies? Mental Math – Computations done by students “in their head” either in whole or in part. Strategy – A method or system of steps used to solve problems. Students must learn to make use of relationships between numbers and Operations to develop flexible math strategies. Mental Math – Computations done by students “in their head” either in whole or in part. Strategy – A method or system of steps used to solve problems. Students must learn to make use of relationships between numbers and Operations to develop flexible math strategies.

Teaching Strategies The MB curriculum has identified some strategies such as making ten, starting from known doubles and doubling. However, students will use different approaches to mental calculations. The development of a variety of strategies is greatly enhanced by the sharing and discussion of solutions. The student should be given the freedom to adapt, combine, even invent strategies by providing students with good problems to solve. The aim of mental math is help students achieve fluency so they can solve more difficult calculations. The MB curriculum has identified some strategies such as making ten, starting from known doubles and doubling. However, students will use different approaches to mental calculations. The development of a variety of strategies is greatly enhanced by the sharing and discussion of solutions. The student should be given the freedom to adapt, combine, even invent strategies by providing students with good problems to solve. The aim of mental math is help students achieve fluency so they can solve more difficult calculations.

6 + 5 What strategy would you use if you forgot the answer to 6 + 5? Turn to the person next to you and explain your strategy. What strategy would you use if you forgot the answer to 6 + 5? Turn to the person next to you and explain your strategy.

6 + 5 Building on a known double Doubles plus one Doubles minus one Building on a known fact Making ten Building on a known double Doubles plus one Doubles minus one Building on a known fact Making ten

Strategies What do you see? What do hear? What do you see? What do hear?

Venn Diagram Have students sort a set of math facts using a Venn Diagram. Observe how students arrange the set and if they apply the strategy correctly. Counting back Starting from a known double Using addition to subtract Subtract Zero

Strategy Cards Write the question on the front of the card then laminate the card. Student can use white board markers to record on the back of the flash card their answer and the strategy that they used to solve the problem.

Strategy Cards 8 + 5 = I use a known fact. I know that 8 + 4 is 12. 8 + 5 is one more than 8 + 4 so 8 + 5 would be one more which would be 13.

Strategy Cards The cards should be large enough to allow for students to record their strategy. As students work with the cards, they discover that there are number of successful strategies to choose from. The connections between these strategies will strengthen their understanding of the operations. A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6 – Volume Five The cards should be large enough to allow for students to record their strategy. As students work with the cards, they discover that there are number of successful strategies to choose from. The connections between these strategies will strengthen their understanding of the operations. A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6 – Volume Five

Question Show all the ways you can use your basic facts strategies to make 12. From this example, what can you determine this child knows. Interview the student for further evidence of their knowledge. Show all the ways you can use your basic facts strategies to make 12. From this example, what can you determine this child knows. Interview the student for further evidence of their knowledge.

Games Have students play games that have them determining math facts, observe and listen to what strategies students are using. Individually play a game with a student and ask them what strategies they are using to find the answers. Have students play games that have them determining math facts, observe and listen to what strategies students are using. Individually play a game with a student and ask them what strategies they are using to find the answers.

Operation Cards Each player needs a game board. The game board is the numbers 1 to 18. The object of the game is to be the first player to cross off all the numbers on his or her game board. The cards are placed facedown in a pile. Each player takes a turn and turns over the top two cards. The player can make either an addition or subtraction problem with the cards. The player decides on the operation to use, and marks off the answer to the problem on their game board. Note: A player may not be able to cross off a number on every turn so therefore he or she looses a turn. When all the cards in the pile have been used, a player shuffles the cards and they are used again. Each player needs a game board. The game board is the numbers 1 to 18. The object of the game is to be the first player to cross off all the numbers on his or her game board. The cards are placed facedown in a pile. Each player takes a turn and turns over the top two cards. The player can make either an addition or subtraction problem with the cards. The player decides on the operation to use, and marks off the answer to the problem on their game board. Note: A player may not be able to cross off a number on every turn so therefore he or she looses a turn. When all the cards in the pile have been used, a player shuffles the cards and they are used again. 123 456 789 101112 131415 161718

Strategy Players take turns rolling two regular dice. The object of the game is to be the first student to cover all of his/her numbers At each turn a player may add, subtract, multiply, or divide the number to cover a number on his/her gameboard. If all possible answers from a roll have been covered, the player looses that turn. Players take turns rolling two regular dice. The object of the game is to be the first student to cover all of his/her numbers At each turn a player may add, subtract, multiply, or divide the number to cover a number on his/her gameboard. If all possible answers from a roll have been covered, the player looses that turn.

Strategy Player 1 07 18 29 310 411 512 60 Player 2 07 18 29 310 411 512 60

Algebraic Reasoning Skills Algebraic reasoning involves analyzing, representing, and generalizing patterns and regularities in all aspects of mathematics. Conceptual understanding of algebra contributes to the development of number sense. Algebraic reasoning involves analyzing, representing, and generalizing patterns and regularities in all aspects of mathematics. Conceptual understanding of algebra contributes to the development of number sense.

Competency Student predicts: An element in a repeating pattern The ability to work with patterns is the basis for algebraic reasoning. Algebraic reasoning allows students to see the pattern and order in mathematics. Students who develop the ability to identify, reproduce, extend and create patterns are able to make generalizations and see relationships among numbers. Student predicts: An element in a repeating pattern The ability to work with patterns is the basis for algebraic reasoning. Algebraic reasoning allows students to see the pattern and order in mathematics. Students who develop the ability to identify, reproduce, extend and create patterns are able to make generalizations and see relationships among numbers.

Vocabulary Repeating pattern: A pattern that has a repeated sequence or arrangement about which predictions can be made. Core: The shortest string of elements that repeats in repeating pattern (e.g., in the pattern ABCABCABC…, ABC is the core). Increasing pattern: A pattern in which one or more elements of the sequence or arrangement increases. Term: An individual element in a pattern. Element: One of the objects or numbers belonging in a pattern. Repeating pattern: A pattern that has a repeated sequence or arrangement about which predictions can be made. Core: The shortest string of elements that repeats in repeating pattern (e.g., in the pattern ABCABCABC…, ABC is the core). Increasing pattern: A pattern in which one or more elements of the sequence or arrangement increases. Term: An individual element in a pattern. Element: One of the objects or numbers belonging in a pattern.

Group Work: Turn to your shoulder partner and explain how you would describe this pattern.

“This is a repeating pattern.” “The core (the part that repeats) of the pattern is circle, triangle, square.” “Every third shape is a square.” “If I start at a circle and I add three elements (shapes, …) I notice that I land on a circle” “If I start at a triangle and I add three elements (shapes, …) I notice that I land on a triangle.” “If I start at a square and I add three elements (shapes, …) I notice that I land on a square.” “This is a repeating pattern.” “The core (the part that repeats) of the pattern is circle, triangle, square.” “Every third shape is a square.” “If I start at a circle and I add three elements (shapes, …) I notice that I land on a circle” “If I start at a triangle and I add three elements (shapes, …) I notice that I land on a triangle.” “If I start at a square and I add three elements (shapes, …) I notice that I land on a square.” 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Group Work: Turn to your shoulder partner and explain how you would describe this pattern.

Assessment Strategies and Observation Tips When presenting repeating patterns, always repeat the core at least three times. To encourage students to make connections with numbers, present the numerical term position with the pattern. Guide students to try out their strategies with different patterns to see if it is a strategy that can be generalized or one that is specific to a particular pattern. When presenting repeating patterns, always repeat the core at least three times. To encourage students to make connections with numbers, present the numerical term position with the pattern. Guide students to try out their strategies with different patterns to see if it is a strategy that can be generalized or one that is specific to a particular pattern. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Think/Pair/Share How would you find the 20 th shape. “I think the 16 th shape plus four” “The core has 3 elements so I counted by 3s until I got to 21 and then I took one element away.” “I know the 21 st shape is a square and go back one” How would you find the 20 th shape. “I think the 16 th shape plus four” “The core has 3 elements so I counted by 3s until I got to 21 and then I took one element away.” “I know the 21 st shape is a square and go back one” 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Assessment Tool In math journal: continue this pattern for 5 more terms circle the core of the pattern what would the 25 th term would be? Explain and show how you know. In math journal: continue this pattern for 5 more terms circle the core of the pattern what would the 25 th term would be? Explain and show how you know. 1 2 3 4 5 6 7 8 9 10 11 12 13

Assessment Question If this pattern continues, what will the 50th bead be? Explain or show your thinking.

Look For the use of an efficient strategy (using the ‘fiveness’ of the core and skip counting, grouping into tens and counting, etc.) the correct use of mathematical language related to patterns the use of an efficient strategy (using the ‘fiveness’ of the core and skip counting, grouping into tens and counting, etc.) the correct use of mathematical language related to patterns

Learning Experience Prepare a large number line with the numbers 1 to 30. Place the number line on the floor or on a table. Use pattern bocks to make a repeating pattern with a two element core. Place each pattern block above a number on the number line. Example: 1 2 3 4 5 6 7 8 9 10 11 12 13 14... 30

Questions to Ask “Can you describe the pattern?” “What part of the pattern repeats? What is the pattern core?” “What will the next shape be? How do you know?” “If this pattern continues, what shape will be above the number 10? How do you know?” “What shape will be above the number 15? How do you know?” “If we read all of the numbers that have a hexagon above them, what do we know about these numbers?” (example: They are even numbers. They are counting by 2s.) “If we continue the pattern up to the number 20, how many squares will there be altogether?” “Can you describe the pattern?” “What part of the pattern repeats? What is the pattern core?” “What will the next shape be? How do you know?” “If this pattern continues, what shape will be above the number 10? How do you know?” “What shape will be above the number 15? How do you know?” “If we read all of the numbers that have a hexagon above them, what do we know about these numbers?” (example: They are even numbers. They are counting by 2s.) “If we continue the pattern up to the number 20, how many squares will there be altogether?”

Competency Student understands that the equal symbol represents an equality of the terms found on either side of the symbol. Understanding the equal symbol (=) means the quantity on the left is the same as the quantity on the right. Student understands that the equal symbol represents an equality of the terms found on either side of the symbol. Understanding the equal symbol (=) means the quantity on the left is the same as the quantity on the right.

Vocabulary Equality: A mathematical statement indicating that two quantities (or expressions) are in balance; Two expressions that are equivalent (e.g., 2 + 1 + 5 = 4 + 4) Equation: A mathematical sentence stating that two expressions are equal. An equation contains an equals sign (=). Expressions: A mathematical representation containing numbers, variable, and/or operation symbols; an expression does not include a relational symbol (=,=). Equality: A mathematical statement indicating that two quantities (or expressions) are in balance; Two expressions that are equivalent (e.g., 2 + 1 + 5 = 4 + 4) Equation: A mathematical sentence stating that two expressions are equal. An equation contains an equals sign (=). Expressions: A mathematical representation containing numbers, variable, and/or operation symbols; an expression does not include a relational symbol (=,=).

Equality and Inequality Why are they equal? 16 + 18 = 18 + 16 13 + 9 = 15 + 7 Why are they equal? 16 + 18 = 18 + 16 13 + 9 = 15 + 7

Equalities and Inequalities Essential Questions: 1.How do you know the sets are equal? 2.How do you know the set are not equal? 3.How is a number sentence like a balance scale? Essential Questions: 1.How do you know the sets are equal? 2.How do you know the set are not equal? 3.How is a number sentence like a balance scale?

Balance Scale Students need to be provided with opportunities to describe equality and inequality, both concretely and pictorially. Place 12 cubes onto a one side and ask students how to make the sets equal. You may have to start with small sets of numbers and then onward to larger quantities.

Using the Balance Scale Students need to be provided with opportunities to describe equality and inequality, both concretely and pictorially. When using the balance scale to illustrate equalities and inequalities the objects, (bingo chips, stars unfix cubes, teddy bears) must always be uniform in size, shape and mass. The color of the objects can vary and will not be a factor in determining the equality. http://illuminations.nctm.org/ActivityDetail.aspx?id=26 Students need to be provided with opportunities to describe equality and inequality, both concretely and pictorially. When using the balance scale to illustrate equalities and inequalities the objects, (bingo chips, stars unfix cubes, teddy bears) must always be uniform in size, shape and mass. The color of the objects can vary and will not be a factor in determining the equality. http://illuminations.nctm.org/ActivityDetail.aspx?id=26 2 + 1 3 + 2

2 cylinders + 1 triangle ≠ 1 triangle + 3 cylinders Balancing the Scale - Possible responses: There is one more cylinder on the right side. To make the scale balance, one cylinder must be taken away on the right side. There is 1 less cylinder on the left hand side. To make the scale balance, we need to add a cylinder on the left side.

True/False Number Sentences Asking students to choose whether each number sentence is true or false can encourage them to examine their assumptions about the equal sign. a. 3 + 5 = 8 b.8 = 3 + 5 c.8 = 8 d.3 + 5 = 3 + 5 e.3 + 5 = 5 + 3 f.3 + 5 = 4 + 4 Think Mathematically Integrating Arithmetic and Algebra in Elementary School Asking students to choose whether each number sentence is true or false can encourage them to examine their assumptions about the equal sign. a. 3 + 5 = 8 b.8 = 3 + 5 c.8 = 8 d.3 + 5 = 3 + 5 e.3 + 5 = 5 + 3 f.3 + 5 = 4 + 4 Think Mathematically Integrating Arithmetic and Algebra in Elementary School

Learning Experience Equal or Not Equal – Group the class into small groups. Each group needs a balance scale and a set of number sentences/equations. Have each group use the balance scale and a set of cube or counters to determine whether the number sentences are equal or not equal. 2 + 9 11 10 4 + 6 12 + 4 13 + 2 9 9 7 5 + 3 6 + 8 5 + 9 Equal or Not Equal – Group the class into small groups. Each group needs a balance scale and a set of number sentences/equations. Have each group use the balance scale and a set of cube or counters to determine whether the number sentences are equal or not equal. 2 + 9 11 10 4 + 6 12 + 4 13 + 2 9 9 7 5 + 3 6 + 8 5 + 9

Question? There are 25 cubes on one side of the balance. I place 15 onto the other side of the balance. How do I balance the scale?

Possible Solution 25 = 15 + 25 = 15 + 10 25 = 15 + 25 = 15 + 10

Possible Solution 25 - = 15 25 – 10 = 15 25 - = 15 25 – 10 = 15

Resource Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School by Thomas P. Carpenter, Megan Loef Franke, and Linda Levi Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School by Thomas P. Carpenter, Megan Loef Franke, and Linda Levi Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School

Assessment Strategies and Observation Tips What goes in the box: 8 + 4 = + 5 At your tables, read and be ready to discuss to the whole group how the child responded to the problem. What goes in the box: 8 + 4 = + 5 At your tables, read and be ready to discuss to the whole group how the child responded to the problem. ChildPage Lucy10-11 Randy11 Barb11 Ricard o 13 Gina13

Assessment Strategies and Observation Tips What goes in the box: 4 + 2 = + 4 We have to get students thinking that the equal symbol does not mean a command to carry out a calculation but a sign that denotes the relation between two equal quantities. Think Mathematically Integrating Arithmetic and Algebra in Elementary School What goes in the box: 4 + 2 = + 4 We have to get students thinking that the equal symbol does not mean a command to carry out a calculation but a sign that denotes the relation between two equal quantities. Think Mathematically Integrating Arithmetic and Algebra in Elementary School

Literature Equal Shmequal by Virgina Kroll The story begins with a group of animals attempting to make equal teams for a tug-o-war. The meaning of the word “equal” is discussed. The animals explore using equal numbers for the teams and later use a seesaw to look at equal weight. The story provides the basis for a discussion about using objects of the same size and mass when using a balance scale to explore equality and inequality. Show this using pictures and equations. Equal Shmequal by Virgina Kroll The story begins with a group of animals attempting to make equal teams for a tug-o-war. The meaning of the word “equal” is discussed. The animals explore using equal numbers for the teams and later use a seesaw to look at equal weight. The story provides the basis for a discussion about using objects of the same size and mass when using a balance scale to explore equality and inequality. Show this using pictures and equations.

Documentation of Evidence Conversations: over the shoulder, conference, peer, journals, book talks Observations: focused, around the room, individual/partner/group, checklists Products: journals, portfolios, projects, skill applications Adapter from A. Davies, Making Classroom Assessment Work, Second Edition © 2007 Connections Publishing, p. 52.)

Focused Observation Target one or two outcomes Identify a small number of students to observe Note what you ‘see and hear’ these students ‘do and say’ Collect observations over a period of time as students provide evidence in using criteria and meeting expectations Target one or two outcomes Identify a small number of students to observe Note what you ‘see and hear’ these students ‘do and say’ Collect observations over a period of time as students provide evidence in using criteria and meeting expectations

‘Get students working harder. School is not a place to watch old people get tired. The person working the hardest is growing the most dendrites! It’s about the learning’ Anne Davies

Assessment is a Human Process “ It is worth noting, right from the start, that assessment is a human process, conducted by and with human beings, and subject inevitable to the frailties of human judgment. However crisp and objective we might try to make it, and however neatly quantifiable may be our ‘ results, ’ assessment is closer to an art than a science. It is, after all, an exercise in human communication. ” (Ruth Sutton, Assessment: A framework for teachers, 1992) “ It is worth noting, right from the start, that assessment is a human process, conducted by and with human beings, and subject inevitable to the frailties of human judgment. However crisp and objective we might try to make it, and however neatly quantifiable may be our ‘ results, ’ assessment is closer to an art than a science. It is, after all, an exercise in human communication. ” (Ruth Sutton, Assessment: A framework for teachers, 1992)

Levels of Performance Needs ongoing help to reach Grade 3 entry level of performance. Approaching expectations to reach Grade 3 entry level of performance. Meeting expectations for Grade 3 entry level of performance. Needs ongoing help to reach Grade 3 entry level of performance. Approaching expectations to reach Grade 3 entry level of performance. Meeting expectations for Grade 3 entry level of performance.

Our Goal Mathematics education must prepare students to use mathematics confidently to solve problems.

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Provincial Grade 3 Assessment in Numeracy Fall 2009.

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