Learning Story for Grade 3
Number and Computational Fluency
For grade 3, there are new number concepts and operations introduced and students move their thinking to more abstract ways with larger numbers and fractions and using all four number operations.
In grade 3, students move from thinking about numbers to 100 in concrete and visual ways in grade 2 to thinking about larger numbers to 1000 where they begin to think more abstractly about quantities. As they move to grade 4, they extend this thinking to numbers to 10 000. In grade 3, students build on their understanding of addition and subtraction facts to 20 and adding and subtracting two-digit numbers to develop their computational fluency with facts and use a variety of strategies to add and subtract three-digit numbers. This learning about addition and subtraction continues into grade 4 with larger numbers.
Fraction concepts are introduced formally for the first time in our BC curriculum in grade 3. The intent of this introduction is to have students understand what a fraction is (a type of number) and how it can be represented (concretely, pictorially, and symbolically). The focus is on fractions between 0-1 such as one-half, one-quarter, two-thirds etc. It is important to introduce the fraction of 1/10 using tools such as a ten frame to build their understanding towards the introduction of decimal numbers in grade 4. In grade 4, students will further develop their understanding of fractions by comparing and ordering fractions and then in grade 5, investigating equivalent fractions.
Multiplication and division are introduced formally for the first time in our BC curriculum in grade 3. The grade 3 focus is building an understanding of what the operations/processes of multiplication and division are and how they can be represented concretely, pictorially and symbolically. Grade 3 students may build or draw groups of objects to solve a multiplication question such as 3×5= or use tiles to build arrays to think of and visualize groups. For division questions such as 12÷4= students solve using concrete materials or drawing pictures using both sharing and grouping strategies. Students apply understanding of skip counting/multiples to think about multiplication and division as well as apply previous understanding of repeated addition and subtraction. Grade 3 students learn how multiplication and division are connected. The use of Cognitively Guided Instruction CGI problem types and math stories support students problem-solving strategies and making meaning of multiplication and division. As students move into grade 4, they will focus more on learning multiplication facts to 100 and beginning to multiply and divide two and three-digit numbers by one-digit numbers.
Key Concepts
Place value
Addition and subtraction
Students develop their computational fluency with addition and subtraction facts to/within 20, using mental math strategies such as making ten and using known facts to solve unknown facts.
Students apply their understanding of place value to add and subtract three-digit numbers using strategies such as decomposing and compensating.
Multiplication and division
Fraction concepts
Introduction of fractions as a type of number representing parts and wholes. Representing fractions such ½, ¼ and 1/10 in concrete, pictorial and symbolic forms.
Key Number Concept 1: Place Value
Overview
Number Sense Foundations:
The following concepts and competencies are foundational in supporting understanding of place value in grade 3:
- Counting by 10s, 100s, forward and back within 1000
- Understanding of grouping by 10s and two-digit numbers being composed of groups of tens and ones
- Reading and writing two-digit numbers
- Ordering and comparing two-digit numbers on a number line with benchmark numbers such as 50
- Connecting number words, symbolic number and quantities in concrete forms
- Concrete, pictorial and symbolic representations of two-digit numbers
- Make reasonable estimates
Progression:
- Place value at grade three builds on students’ understanding of two-digit numbers in grade 2
- Understanding that 100 is made up of ten groups of ten (connect to multiplication, use base ten blocks to represent this) and that the digit 1 in 100 represents one hundred based on its place in the number
- Understanding that 500 is five hundreds or five groups of one hundred
- Build different century numbers (200, 400, 800 etc) using concrete materials and record the number in symbolic form
- Add two-digit numbers/quantities to hundreds such as 300 and 47 and how to build and record this using concrete and symbolic forms (347)
- Invite students to represent different three-digit numbers with concrete materials and record in symbolic form
- Understanding that the digit to the left is ten times more than the digit to the right in a three-digit number
- Introduce standard form (347) and expanded form (300 + 40 + 7) for recording numbers
- Play with composing and decomposing three digit numbers in different ways (347 can be 300+40+7 or 200+100+40+7, 100+100+100+20+20+5+2, etc)
- Practice reading and writing three-digit numbers, paying attention to place value and zero as a place holder
- With a small collection of three-digit numbers, have students compare and order numbers along a number line, providing benchmark numbers of 0, 500 and 1000
- Make reasonable estimates for quantities up to 1000 using referents or other visual strategies
- Mentally adding 10s or 100s to any three-digit number such as 347 + 50
- Apply understanding of place value when adding and subtracting three-digit numbers and using strategies such as decomposing by place value or compensating by using complementary numbers
Sample Week at a Glance:
Read excerpts from Animals by the Numbers by Steve Jenkins inviting conversation about how numbers help us understand our world
Pose a problem inspired by the book or invite students to choose an animal and create a mini-project highlighting numbers about that animal (size, population, distance traveled, amount of food, etc)
Closing circle – share and discuss numbers used and compare animal’s numbers with a partner
Same but Different routine: comparing 428 and 824
Math Workshop
-exploring place value relationships through base ten blocks
-Three-Digit Face Off game
-Counting Collections between 100-300
-Teacher led small group instruction: reading and writing two or three-digit numbers
Closing Circle – students sharing what they did, what they learned and where they want to go next with their learning about place value
Class discussion having students share what they know about place value.
Invite students to choose different materials to help them think about place value in new ways. Include base ten blocks, Cuisenaire Rods, ten frames, grid paper, loose parts, numerals, etc. Invite students to represent numbers in different ways.
Closing circle – have students share their findings/what they did with a partner and how materials helped them think about numbers in new ways.
Counting Collections between 100-500 (include estimation of quantity)
Math Workshop
-Posing problems or creating math stories based on counting collection from beginning of lesson
-Place the Digits math game
-half-sheet of five addition and subtraction questions with three-digit numbers (solve each in at least two ways)
-Teacher led small group instruction: provide cards with three-digit numbers and ask students to compare and order, explaining their thinking
Closing Circle – students sharing what they did, what they learned and where they want to go next with their learning about place value
How many ways can you make 458? Record on whiteboard.
CGI problems involving addition and subtraction with three-digit numbers such as: There were 247 tennis balls in the gym storage room. After our class finished playing, we added another box of tennis balls to the storage room and now they are 412 tennis balls. How many tennis balls did our class add to the storage room?
Closing Circle – student sharing and comparing strategies
Suggestions for Assessment
By the end of grade 3, students will be able to think about three-digit numbers flexibly and fluently. This would look like being able to represent and compose and decompose numbers in different ways using different forms (concrete, pictorial and symbolic) and order and compare three-digit numbers and explain their thinking and justify their choices.
By the end of grade 3, students will be able to apply their understanding of place value to strategies for adding and subtracting three-digit numbers. For example, they might decompose numbers by place value first and add like numbers and then recompose.
Suggested Links and Resources
Same but Different routine: https://www.samebutdifferentmath.com/
Counting Collections: https://www.stenhouse.com/content/choral-counting-counting-collections
About Teaching Mathematics: A K-8 Resource by Marilyn Burns (place value menu tasks and games)
CGI Math resources: https://blogs.sd38.bc.ca/sd38mathandscience/cgi-math/
Key Number Concept 2: Addition and Subtraction
Overview
The concepts of addition and subtraction involve two components in grade 3: computational fluency with facts to 20 and adding and subtracting three-digit numbers.
In grade 3, students develop their computational fluency with addition and subtraction facts to/within 20, using mental math strategies such as making ten and using known facts to solve unknown facts.
In grade 3, students build on their understanding and strategies for adding and subtracting two-digit numbers from grade 2 to develop a variety of strategies to add and subtract three-digit numbers. Three-digit addition and subtraction questions are initially presented horizontally so students focus on the whole numbers and not the “digits” and make sense of the quantities through estimating the sums and differences. Number talks as a whole class or in small groups is an essential practice for students to learn to explain their strategies and hear different strategies from their classmates. Many of these strategies can be done mentally and do not require students writing their computations on paper. Base ten blocks provide concrete support at grade 3 for the process of regrouping numbers when adding and subtracting three-digit numbers. Students apply their understanding using strategies such as decomposing and compensating. Students practice taking apart (example: decomposing using friendly numbers and compensating) and combining numbers in various ways which can also be considered regrouping (example 256+564 = 200+500+50+60+6+4=820. For subtraction strategies, students learn to use number lines to compare numbers and to add on to find the difference (example 432-175 can be thought of as 175+25 to make 200 then add 200 to make 400 and add 32 to make 432 then compose those add ons to find the difference of 257). Students will have two or more strategies to solve three-digit addition and subtraction questions and may have a preference for strategies that make sense to them. Students apply their understanding of addition and subtraction to 1000 through contextual problems such as Cognitively Guided Instruction (CGI) problem types.
Please note: The expectation is that students at grade 3 will have flexibility and fluency with multiple strategies. Students may have learned the “traditional algorithm” from outside of school sources and this can be acknowledged when students share this method during a number talk but they are expected to have other strategies to add and subtract than this.
Number Sense Foundations:
The following concepts and competencies are foundational in supporting understanding of addition and subtraction in grade 3:
- Skip-counting by 10s, 100s, forward and back within 100
- Understanding making and bridging 10, then 100
- Use flexible strategies for adding (example: doubles plus/minus __/near doubles) and subtracting (example: If I have 13 and you have 9, how many more do I have?)
- Reading and writing two-digit numbers
- Using flexible strategies for adding and subtracting (counting on, finding the difference, etc.)
- Understanding the commutative property of addition (example: 14+23 = 23+14)
- Connecting number words, symbolic number and quantities in concrete forms
- Conceptualize numbers as whole numbers and units (what is ten more than 44?)
- Decomposing and combining sets of tens and ones when given multi-digit numbers (example: 28+35=20+30+5+8=50+13)
- Concrete, pictorial and symbolic representations of two-digit and three-digit numbers
- Make reasonable estimates
Progression:
Addition and Subtraction Facts to 20
- Addition and subtraction at grade three builds on students’ understanding of addition and subtraction to 20 in grade 2 (introduced to computational strategies)
- Use a variety of strategies for facts to 20 (identifying related doubles, bridging over 10, decomposing, compensating)
- Ongoing practice with mental math strategies and being able to explain two or more strategies for facts such as 8+9 and 15-7 both orally and in with pictures, numbers and words
Addition and Subtraction to 1000
- Flexibly use different strategies (looking for multiples of 10, decomposing by place value) when adding and subtracting with two digit numbers
- Build numbers using concrete materials such as base ten blocks and compose and decompose considering place value (for numbers to 1000)
- Add three-digit numbers using different strategies (start with questions written horizontally to allow students to decompose by place value for more flexible thinking), with students explaining and sharing their strategies such as decomposing and compensating using concrete materials such as base ten blocks or visual tools such as an open number line
- Subtract three-digit numbers using different strategies (finding the difference, compensating) using concrete materials such as base ten blocks or visual tools such as open number line
- Demonstrate various ways to solve addition and subtraction problems using concrete, pictorial and symbolic forms.
- Use appropriate place value language value when explaining their thinking and strategies.
Sample Week at a Glance
Read excerpts from Two of Everything by Lily Toy Hong inviting conversation about how different numbers would change the outcome
Pose a problem inspired by the book and students can choose it to solve or create their own
Closing circle – share and discuss numbers used and compare with a partner
**Teach the game, Snap to It-Addition and play as a whole class (adapt to three-digits). This can be done at another point of the day**
Number Talk routine (example: what is 8+7? What is 38+7? 438+57?
Math Workshop
-Three Card Mixer math game
-Snap to It (Box Cars game)
-half-sheet of five addition and subtraction questions with two-digit and three-digit numbers (solve each in at least two ways)
-Teacher led small group instruction: mini number talk with focus on addition
Closing Circle – students sharing what they did, what they learned and where they want to go next with their learning about addition
Ways to make 100, ways to make 1000
Invite students to choose different materials to tell a number story about addition. Include base ten blocks, Cuisenaire Rods, ten frames, loose parts, numerals, etc. Invite students to represent their thinking in different ways.
Closing circle – have students share their findings/what they did with a partner and how materials helped them think about numbers and addition in new ways.
Open Question: The answer is 312. What might the question be?
Math Workshop
-Creating math stories or posing problems based on the number 312
-First to 20 math game (Bay-Williams)
-half-sheet of five addition questions with three-digit numbers (solve each in at least two ways)
-Teacher led small group instruction: provide cards with three-digit numbers and students practice addition strategies showing their thinking (could be verbally, on whiteboards, etc.)
Closing Circle – students sharing what they did, what they learned and where they want to go next with their learning about addition
Open Question by Marian Small: What is a good way to add 490+490 in your head? or
Which estimate makes the most sense to you for the sum of 370+370?
CGI problems involving addition and subtraction with three-digit numbers such as: When we started playing the video game we had 648 points. Then we lost some points and ended with 362. How many points did we lose? Students are encouraged to show their thinking in multiple ways including adding on to find the difference.
Closing Circle – student sharing and comparing strategies
Suggestions for Assessment
Addition and Subtraction to 20
By the end of grade 3, most students will be able to recall addition and subtraction facts to 20 and can demonstrate fluency with addition and subtraction to 20. Students can use strategies such as finding related doubles, bridging over 10, decomposing, and the commutative property and can share their thinking in multiple ways.
Addition and Subtraction to 1000
By the end of grade 3, students will be able to apply their understanding of place value to strategies for adding and subtracting three-digit numbers. Students will be able to add and subtract three-digit numbers flexibly and fluently using both mental math strategies and recording with pencil and paper. Students will be able to use two or more strategies, use different forms (concrete, pictorial and symbolic) and justify their thinking. Strategies most students will be able to use for adding three-digit numbers include decomposing by place value, compensating and using benchmark numbers to add on using an open number line. For subtracting three-digit numbers, students will use both removal/take-away and finding the difference strategies and use strategies such as decomposing by place value, compensating and finding the adding on using an open number line to find the difference. All of these strategies focus on connecting place value understanding and number sense. Observe if students use mental math where appropriate as this demonstrates their growing number sense.
Suggested Links and Resources
Math Fact Fluency by Gina Kling and Jennifer Bay-Williams
Open Questions for Rich Math Lessons by Marian Small
Two of Everything by Lily Toy Hong (picture book)
Key Number Concept 3: Multiplication and Division
Overview
Multiplication and division are introduced in grade 3. The grade 3 focus is building an understanding of what the operations/processes of multiplication and division are and how they can be represented concretely, pictorially and symbolically. Students apply understanding of skip counting/multiples to think about multiplication and division as well as apply previous understanding of repeated addition and subtraction, for example bridging students’ understanding of repeated addition (2+2+2) to “three groups of 2” (3×2). Grade 3 students may build or draw groups of objects to solve a multiplication question such as 3×5= or use tiles to build arrays to think of and visualize groups. Arrays contribute to students’ understanding of multiplication as equal groups. For division questions such as 12÷4= students solve using concrete materials or drawing pictures using both sharing and grouping strategies. For example, 12÷4= can be thought of as how many groups of 4 make up 12 or how can I share out 12 into 4 groups. Grade 3 students learn how multiplication and division are connected. The use of Cognitively Guided Instruction CGI problem types and math stories support students problem-solving strategies and making meaning of multiplication and division.
Number Sense Foundations:
The following concepts and competencies are foundational in supporting understanding of multiplication and division in grade 3:
- Understanding of equal groups
- Grouping quantities and skip counting by 2s, 5s, and 10s (such as in counting collections)
Progression:
- Creating equal groups with concrete materials and then with visuals/pictures
- Skip counting, counting by multiples (number line, hundred chart, choral counting)
- Solve sharing and grouping questions with concrete materials (within 100)
- Introduce multiplication and division symbols and recording symbolic notation in equations
- Notice patterns in some multiplication facts (even numbers, ones place ending in 5 or 0 for 5x questions, etc)
- Solve multiplication and division fact questions within 100 using different strategies (concrete materials, drawing, tallies, open number line, hundred chart)
Sample Week at a Glance
Read Amanda Bean’s Amazing Dream, pausing throughout to have students notice and describe the arrays and equal groups they notice
Invite students to create their own illustrations that include arrays or equal groups and label them with multiplication equations and descriptive sentences.
Closing circle: Select a few students to share their illustrations and explain how arrays/equal groups help them think about multiplication.
Number Talk Image:
Choose an array image and project on screen and invite students to discuss: “Where do you see multiplication?”
Math Workshop
-multiplication equation cards and colour tiles (and repeat with other materials at another table) to build arrays
-Counting Collections to 100 asking students to focus on groupings such as 3 and 4 and connect to multiplication
-Teacher led small group instruction: teach the game Circles and Stars and have students share their understanding of equal groups
Closing Circle – students sharing what they did, what they learned and where they want to go next with their learning about multiplication
Table Groups create webs on charts: What do we know about multiplication? What do we wonder?
Invite students to use various materials to investigate their own wonders about multiplication. They may further investigate arrays or number patterns or another area of interest. Invite students to represent their thinking with materials such as cubes, tiles, drawing, painting, imprinting in clay, etc. Students may document their findings through photographs, mini-books or mini-posters.
Closing circle: Invite students to share their projects with a partner and then select a few projects to compare as a class. How did different materials help us show what we know about multiplication?
Number Talk: 4×12 (have students share different strategies for solving)
Math Workshop
-students choose multiplication equation cards and create a drawing of a context connecting to that equation
-half-sheet of paper with five multiplication facts to solve using materials, drawings or tallies
-Circles and Stars math game
-Teacher led small group instruction: Choral Counting choosing multiples based on students’ experience, using a small whiteboard to record and notice patterns
Closing Circle – students sharing what they did, what they learned and where they want to go next with their learning about multiplication
Choral Counting routine: counting by 4s, recording count on whiteboard in an array and having students notice patterns
Present a CGI problem to the class to solve using different strategies: Three friends went blackberry picking and each had a small basket. They picked 36 blackberries and shared them equally. How many blackberries did they each get?
Pause to have some students share their strategies. As some students continue to solve the problem, others may write their own problems to be used in future lessons.
Closing Circle – students sharing what strategies and materials are supporting their understanding of multiplication and generate a personal goal for practicing multiplication next week
Suggestions for Assessment
Suggested Links and Resources
Circles and Stars (Marilyn Burns math game) https://blogs.sd38.bc.ca/sd38mathandscience/wp-content/uploads/sites/14/2020/05/SD38_Circles_and_Stars.pdf
Number Talk Images http://ntimages.weebly.com/
Cognitively Guided Instruction (CGI) resources
Choral Counting and Counting Collections https://www.stenhouse.com/content/choral-counting-counting-collections
Amanda Bean’s Amazing Dream by Liza Woodruff (picture book)
Key Number Concept 4: Fraction Concepts
Overview
Grade 3 introduces concepts of fractions as students begin to develop fractional understanding across the intermediate grades. The intent of this introduction is to have students understand what a fraction is (a type of number) and how it can be represented (concretely, pictorially, and symbolically). Students understand that a fraction represents a part/s-whole relationship and that the parts are equal parts or fair shares of the whole. Some students may count the number of parts, such as in a composition of pattern blocks, and not attend to whether they are equal sizes or not. For example if there is a shape made of one hexagon, two trapezoids and six triangles, a student may think the hexagon is 1/9 of the whole as there are nine blocks in total and one hexagon when by area it is actually ⅓ of the whole. The focus in grade 3 is on fractions between 0-1 such as one-half, one-quarter, two-thirds etc. It is important to introduce the fraction of 1/10 using tools such as a ten frame to build their understanding towards the introduction of decimal numbers in grade 4. As students are learning about fractions, they often pay attention to the digit in the numerator and denominator without attending to the relationship. For example, students might think that ⅛ is greater than ½ as the number 8 is greater than 2. Visualizing fractions diagrams and using materials helps to build understanding of the relationships between parts and wholes and how if the number of parts in the denominator increases the size of the unit fraction or part decreases. Materials that support building fraction understanding include pattern blocks, Unifix Cubes, Cuisenaire rods and loose parts and ten frames. It is important that students do not only experience and see fractions as shapes/regions but also as sets and linear or length contexts. Building a strong foundation of what a fraction is (a quantity, a number, a relationship) in grade 3 is important as students move into grades 4 and 5 where they compare and order fractions and investigate equivalent fractions.
Number Sense Foundations:
The following concepts and competencies are foundational in supporting understanding of fractions in grade 3:
- Informal use of fraction language and understanding (sharing halves of a sandwich)
- Understanding decomposing into equal parts (quantity or space)
Progression:
- Understanding that a fraction is a number representing a part-whole relationship
- Understanding that a quantity or area is decomposed into equal parts or fair shares when we think about fractions
- Visually comparing sizes of fractions using area, set and linear models (one-half is more than one-quarter)
- Specific math vocabulary: equal parts, whole, numerator, denominator
- Using concrete materials, represent fractions such as ½, ¼, ⅕, 1/10
- Using concrete materials, represent fractions with numerators greater than one such as ¾. ⅖, 8/10 (for 8/10, with a collection of 10 cubes, 8 are blue and 2 are yellow)
- The size of the fraction depends on the size of a whole (⅛ of a small pizza is smaller than ⅛ of a large pizza)
- If the wholes are the same size, the size of the parts decreases as the quantity of equal parts increases (¼ of a small pizza is smaller than ½ of a pizza)
- Decompose and compose fractions to extend understanding of parts-whole relationships (¾ is made up of ¼ and ¼ and ¼)
Sample Week at a Glance
In partners, invite students to record what they know about fractions and how they use them in their lives. Collect responses on a chart or whiteboard. Drawing on students’ sharing, highlight the important elements of a fraction; parts, whole, equal parts, symbolic notation.
Invite students to think about two tasks to think about 1/2:
1) Marian Small question: When is 1⁄2 a lot of something? When is it not? Use pictures, numbers and words to show your thinking.
2) Using concrete materials (pattern blocks, Unifix Cubes, base ten blocks, Cuisenaire rods) create many different ways to show ½.
Closing circle: Select a few students to share new learning they had about fractions and how contexts and materials supported their thinking.
Same but Different routine: comparing 1/2 and 2/10
Math Workshop
-matching game matching picture of fraction to symbolic notation
-Build Ten Tenths game using ten frame, counters and dice
-addition facts math game (developing computational fluency)
-Teacher led small group instruction: show ways to make one-half and explain thinking
Closing Circle – students sharing what they did, what they learned and where they want to go next with their learning about fractions
Fraction Talks: Choose an image and have students respond to “What fractions do you see?”
Offer a range of materials that provide opportunities for creating fractions using set, area and linear models and include tools such as ten frames. Invite students to choose a fraction and represent it in as many different ways as they can. Take photographs of fraction representations or do a gallery walk.
Closing circle: Choose a few fraction representations to compare and discuss.
Same but Different routine: comparing two fraction images
Math Workshop
-provide a collection of symbolic fraction cards or fraction dice and have student choose a card or roll the dice and represent that fraction in three different ways
-have students create their own images for the Same but Different routines to be used next week (photographs of materials, drawings, iPad design)
-subtraction facts math game (developing computational fluency)
-Teacher led small group instruction: show ways to make one-tenth and explain thinking
Closing Circle – students sharing what they did, what they learned and where they want to go next with their learning about fractions
Read The Lions’ Share by Matthew McElligott, providing each student with a square piece of paper to fold into fractions as the story is read. Discuss what the animals learned about fractions and equal parts in the story.
Invite students to choose to:
1) Continue to investigate fractions through paper folding, labeling their paper with fractions as they fold
2) Write their own math story about sharing something, considering fractions.
Closing circle: In partners, have students share what they learned and show each other their paper folding or stories. As a whole class, collect emerging questions the students have about fractions.
After this week of lessons, based on formative assessment information, the following week would likely include lessons on creating unit fractions (numerator of one) and fractions with numerators greater than 1 using a range of materials and pictorial representations. Some lessons will also focus on time for students to investigate their own questions about fractions from Friday’s closing circle.
Suggestions for Assessment
By the end of grade 3, students will be able to explain what a fraction is– a type of number that represents a part/s-whole relationship in which the parts are equal. A student might describe the fraction ⅔ as “This fraction has three parts that make the whole and 2 of them are the parts we are thinking about. So the 2 means the parts and that’s the numerator, the number on top and the the three means the number of parts that make the whole and that 3 on the bottom is called the denominator.” By the end of grade 3, students will be able to represent fractions such as ½, ¾, ⅔ or 1/10 using concrete, pictorial and symbolic forms. They might use materials such as pattern blocks, Cuisenaire rods or ten frames and loose parts to show these relationships through area, linear and set models.
Suggested Links and Resources
Same but Different https://www.samebutdifferentmath.com/fractions-ratios-percentage
Fraction Talks http://fractiontalks.com
The Lion’s Share by Matthew McElligott (picture book)