### Learning Story for Grade 4

#### Number and Computational Fluency

Grade 4 represents a significant shift in focus from whole number place value and addition/subtraction, to multiplication, division, and proportional reasoning. Addition and subtraction of whole numbers is still important in grade 4, but is not emphasized in the key concepts as students are expected to extend their previous strategies and understanding.

The grade 3 focus for multiplication and division is building an understanding of what the operations/processes are and how they can be represented concretely, pictorially and symbolically. As students move into grade 4, they focus more on learning multiplication facts to 100 and beginning to multiply and divide two and three-digit numbers by one-digit numbers.

In grade 4, students are also introduced to decimals for the first time. It is important to review the fraction of 1/10 using tools such as a ten frame or base ten blocks to bridge student understanding to the introduction of decimal numbers. In grade 4, students will further develop their understanding of fractions by comparing and ordering them, mainly through the use of benchmarks. Students will also use materials to begin to develop familiarity with the relationship between fractions and decimals.

### Key Concepts

#### Multiplication and Division Facts (Computational Fluency)

Introduction to computational strategies (patterns in multiplying by 5 and 10, doubles, halving, decomposition/distributive property), connection between multiplication and skip-counting

#### Multiplication and Division of Larger Numbers

Multiplication and division of two- and three-digit numbers by one-digit numbers without remainders

#### Fractions and Decimals (Concepts and Relationships)

Ordering and comparing fractions using benchmarks and common denominators, introduction to decimals as numbers and as another way of representing fractions; relationship between fractions and decimals (using models/concrete materials – not conversions)

#### Addition and Subtraction of Decimals to Hundredths

Extending mental math strategies from addition and subtraction of whole numbers to decimals (using materials or models at first like base-ten blocks, ten-frames, open number lines), then symbolically (making a whole instead of making 10, counting up, decomposition, etc.)

#### Key Number Concept 1: Multiplication and Division Facts (Computational Fluency)

##### Overview
Grade 4 is when students are introduced to computational strategies for multiplication and division. By this point, it is expected that students understand the concepts of multiplication and division, and that they can use this understanding to conceptualize the strategies. Some common computational strategies for multiplication facts include doubling, halving, and using decomposition (the distributive property) to simplify the multiplication. Students should work towards mastery of their 2s, 5s, and 10s facts first, as they can use these to derive the other facts. For division, students can use the think-multiplication strategy or decomposition.
##### Number Sense Foundations:

The following concepts and competencies are foundational in supporting the development of multiplication and division fluency in grade 4:

• Understanding the concepts of multiplication and division
• Fluency with addition and subtraction strategies
• Making reasonable estimates
##### Progression:
• Understanding of meaning and operations of multiplication and division from grade 3
• Understanding relationship between multiplication and skip-counting
• Recognizing and understanding patterns in x10
• Using x10 and halving or skip-counting to develop x5
• Understanding that x2 is the same as doubling
• Understanding x0 and x1, and that division by 0 is impossible (and why)
• Concepts of even and odd numbers (you cannot have an odd product by multiplying two even numbers)
• Using 0, 1, 2, 5, and 10 facts and/or models like arrays to derive the rest of the facts using decomposition (the distributive property)
• Students are expected to recall their 0, 1, 2, 5, and 10 facts by the end of grade 4. This should be a result of conceptual understanding combined with repeated practice and varied exposure (math workshops, games, drawing, etc.), not through forced memorization of facts.
##### Sample Week at a Glance:
Before this week of lessons, grade 4 students will have developed an understanding of and some fluency with their foundational multiplication (and corresponding division) facts (0, 1, 2, 5, 10).

Number talk: 6 x 7. How could you figure this out? (give students access to a whiteboard). Lead discussion toward using 5 x 7 to calculate 6 x 7. The following image may be helpful:

Ask students to develop all of their x 6 facts using this strategy, and challenge them to use it to derive x 4.

Closing circle – share and discuss strategies used and aha moments

Exploration: Complete a partially filled in multiplication chart

Investigation: What fraction of the multiplication chart is even? Odd? Why?

Closing Circle – How could what we have learned today help us check the reasonableness of our answers when we multiply? (e.g., I know 6 x 7 is even, so if I answer 49 by accident, I know it can’t be right!)

Number string:

5 x 6

6 x 6

3 x 6

2 x 6

4 x 6

Game: Play Strive to Derive from Math Fact Fluency, or Multiplication Tic-Tac-Toe from mathforlove.com. Note: encourage students to use the mental math strategies they have been developing rather than counting on their fingers.

Closing circle – have students share what they learned from playing the game, and if they noticed a connection to the number string at the beginning

Number talk: 9 x 6. How could you figure this out? Lead discussion toward using 10 x 6 or 9 x 5 to help.

Math Workshop

-provide a collection of symbolic multiplication cards that show facts that students could derive from x2, x5, x10 such as x3, x4, x6, x9 and have students represent each fact in terms of known facts. They can draw arrays, other diagrams, or write symbolically

-subtraction facts math game (maintaining computational fluency)

-multiplication facts math game (developing computational fluency)

-Teacher led small group instruction: ways to multiply using known facts OR practice building fluency with foundational facts

Closing Circle – students sharing what they did, what they learned and where they want to go next with their learning about multiplication facts

How many ways can you figure out 6 x 8? Record on whiteboard.

Start connecting multiplication facts and strategies with division. Play The Factor Game from Math Fact Fluency.

Closing Circle – why is it important to develop and use these strategies instead of just skip-counting?

Based on formative assessment information from this week, next week’s planning would include extending multiplication strategies to two-digit by one-digit multiplication, as well as more practice with single digit multiplication, and division within 100.
##### Suggestions for Assessment
By the end of grade 4, students will be able to think about multiplication and division facts flexibly and fluently. This would look like being able to use known facts to derive facts in different ways. By the end of grade 4, students will not need to use skip-counting as their primary strategy for multiplication and/or division.

#### Key Number Concept 2: Multiplication and Division of Larger Numbers

##### Overview

In grade 4, students extend their thinking of multiplication and division to larger numbers. Students can use strategies such as decomposition (the distributive property) to multiply using models such as arrays, and can use decomposition and the partial quotients method to divide (again, using models and pictures). In Grade 5, once area has been introduced, students can use the area model for multi-digit multiplication.

Please note: The expectation is that students in grade 4 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, but they are expected to have other strategies for multiplication and division.

##### Number Sense Foundations:

The following concepts and competencies are foundational in supporting the development of multiplication and division fluency in grade 4:

• Understanding the concepts of multiplication and division
• Multiply a one-digit number by 10.
• Fluency with addition and subtraction strategies
• Make reasonable estimates
##### Progression:
• Understanding of meaning and operations of multiplication and division from grade 3
• Fluency with multiplying a one-digit number by 10
• Multiplying a teen number by a one-digit number by decomposing teen number into 10 and some more
• Multiplying a one-digit number by a multiple of 10
• Multiplying a two digit number by a one-digit number by decomposing the two-digit number into tens and ones
• Introduction to division using partial quotients (with materials or diagrams – no formal recording at this point) – start with an estimate
• Extend multiplication and division strategies to three-digit by one-digit
##### Sample Week at a Glance

Before this week of lessons, grade 4 students will have developed an understanding of and some fluency with their foundational multiplication (and corresponding division) facts (0, 1, 2, 5, 10).

 Exploration:Introduce this as a slide show:first show the 1st ring of candy, then the first two, then all 3, each time asking students, “Now how many”? How do you know? Lead discussion towards how we can solve 18 x 3 by breaking it up into 10 x 3 + 8 x 3.Lesson and practice with multiplying teen numbers by one digit numbers using 3 possible mental math strategies: distributive property, doubles/halves, or decomposition. For example,16 x 5 = 10 x 5 + 6 x 5 OR16 x 5 = 16 x 10/2 = 160/2 OR16 x 5 = 4 x 4 x 5 = 4 x 20Students can start by building an array to help them visualize the situation.The Number Frames tool here is great as it creates a fillable array with adjustable dimensions: https://mathigon.org/polypad#number-framesClosing circle – share and discuss strategies used and aha moments

Number Talk: How could you figure out 12 x 5?

Lead students to the 3 strategies from Monday.

12 x 5 = 10 x 5 + 2 x 5 OR

12 x 5 = 12 x 10/2 = 120/2 OR

12 x 5 = 3 x 4 x 5 = 3 x 20

Continue practicing the three strategies from Monday.

Closing Circle – Which of these strategies is your favourite? Why? What are the pros/cons of each strategy?

Number string:

10 x 4

9 x 4

18 x 4

19 x 4

20 x 3

2 x 3

2 x 30

Lesson and practice multiplying multiples of 10 by 1-digit numbers

Closing circle – have students share what they learned from their practice and how that connects with their learning from Monday and Tuesday.

Read Amanda Bean’s Amazing Dream and choose at least one page for a number talk. Have students discuss their strategies. https://www.youtube.com/watch?v=g07vteeiz_o

Math Workshop:

• Practice the 3 strategies from Monday. Have teen number x 1-digit number cards (and some larger 2-digit by 1-digit) and students choose a card and try to find the solution using at least two strategies. Use individual whiteboards for this
• Multiplication/division facts fluency game/activity: Multiplication Tic-Tac-Toe
• Teacher led small group instruction: ways to multiply 2-digit numbers by 1-digit numbers using the three strategies from Monday

Closing Circle – students sharing what they did, what they learned and where they want to go next with their learning about multiplication facts

Number talk: How could you figure out 24 x 6?

Just right practice: students choose between 3 independent practice activities: Multiplying a 1-digit number by a multiple of 10

Multiplying a 1-digit number by a teen number

Multiplying a 1-digit number by any two digit number

Closing Circle – what strategies did you learn this week that you did not know before? How will this change the way you multiply going forward?

Based on formative assessment information from this week, next week’s planning would include extending multiplication strategies to two-digit by one-digit multiplication, as well as more practice with single digit multiplication, and division within 100.

##### Suggestions for Assessment

By the end of grade 4, students will be able to think about multiplication and division flexibly and fluently. This would look like being able to break apart (decompose) factors into numbers easier to work with. By the end of grade 4, students will not need to use skip-counting or a formal procedure they have memorized to multiply and divide. Rather, they will have developed fluency working with the strategies presented here. For assessment, focus on strategies and not just correct answers.

#### Key Number Concept 3: Fractions and Decimals (Concepts and Relationships)

##### Overview

It is important at this level that students do not come to see fractions and decimals as separate or different. Fractions and decimals are two different ways of representing a number. In grade 4, students are introduced to decimals with models, and have extensive exposure to benchmark fractions and decimals, as well as how to order and compare them. This forms the foundation for fraction and decimal number sense. Note: at this level, students are only expected to compare and order fractions by comparing to benchmarks or looking at common numerators/denominators. They are not expected to convert fractions with unlike denominators to fractions with like denominators, as they do not learn about equivalent fractions until grade 5.

##### Number Sense Foundations:

The following concepts and competencies are foundational in supporting the development of fraction and decimal concepts in grade 4:

• Place value – number relationship between places
• Using concrete/pictorial/symbolic representations of fractions
• Understanding tenths and different representations
• Understanding that a fraction is a number
• Fluency with the three representations of fractions (linear/number line, set, area/region) and how to move between them
##### Progression:
• Review of three representations of fractions (linear/number line, set, area/region) including tenths
• Ordering and comparing fractions including tenths
• using benchmarks
• using properties of fractions with common numerators or common denominators
• Introduce decimals as another way to represent tenths
• Introduce tenths on a ten frame and represent with fraction and decimal
• Introduce tenths and hundredths as fractions on a 100-grid and how to represent using decimals
• Connect number line model of fractions to decimals
• Review of place value and how to extend it to tenths and hundredths
• Use estimation to place decimals and fractions on a number line
##### Sample Week at a Glance

Before this week of lessons, grade 4 students will have developed an understanding of the different representations of fractions, including representations of tenths on ten-frames. This week, students are introduced to decimals for the first time.

Instructional Routine: What do you notice? What do you wonder?

Ask students what they notice/wonder about these two images and how they are alike and different. Introduce decimal notation as another way to show tenths.

Have students work in table groups and give each group several fractional representations of tenths (area model, number line, symbolic, ten-frames). Place 10 cards or papers at each table, each with one of the following decimals written on it: 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. Have students sort the fraction cards based on what decimal card they go with.

Closing circle – share and discuss new learning from the table group activity

Instructional Routine: Count around the circle by tenths. Students may notice that 10 tenths is a whole. This is a great discussion and is good to touch on but equivalent fractions are not in the curriculum until grade 5.

Give students several 100 grids and have them shade in a series of tenths expressed as fractions or decimals to help them see the relationship. Then project images of partially shaded grids (tenths only)  and see if students can write the corresponding fraction and decimal.

Closing Circle – Did you notice the small squares on the grid? What fraction of the grid do you think they are? How could we write that as a decimal?

Review and extend: Have students work in pairs to roughly place the following numbers on an open number line from 0 to 1.

⅓, 0.2, 0.5, ¾, 8/10, ½

Talk about what strategies they used and what was difficult.

Same activity as yesterday except with tenths and hundredths.

Closing circle – have students share their biggest new learning so far this week. Invite them to share how tenths and hundredths are related.

Number Talk: How are these numbers alike? How are they different?

10, 1, 0.1, 0.01.

What does the decimal point tell us?

During: Review of place value and extending concept to tenths and hundredths

Closing circle: As a group, express the decimals from the opening number talk as fractions.

Clothesline Numberline: give each student 2-3 fraction and decimal cards and have them place the cards on a class clothesline (or masking tape/string) number line.

Closing discussion: Discuss strategies students used and what they learned.

Based on formative assessment information from this week, next week’s planning would include introducing addition and subtraction of decimals.

##### Suggestions for Assessment

By the end of grade 4, students will be able to think about fractions and decimals flexibly and fluently. This would look like being able to represent fractions and decimals in multiple forms – symbolically and using an area or number line model for example. Students will be comfortable ordering and comparing fractions and decimals using benchmark numbers.

#### Key Number Concept 4: Addition and Subtraction of Decimals to Hundredths

##### Overview

In grade 4, students will begin to extend the mental math strategies they have developed for adding and subtracting whole numbers, to decimals.

Please note: The expectation is that students at grade 4 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, but they are expected to have other strategies to add and subtract decimals.

##### Number Sense Foundations:

The following concepts and competencies are foundational in supporting the development of decimal addition and subtraction in grade 4:

• Place value – number relationship between places
• Representing, ordering, and comparing decimals to tenths and hundredths
• Estimating sums
• Fluency with mental math addition and subtraction strategies (making/bridging 10, decomposing, near doubles, etc.)
##### Progression:
• Review different representations of decimals (linear/number line, area (hundreds grid), symbolic) including tenths and hundredths
• Ordering and comparing decimals using benchmarks
• Estimate the sum of two decimal numbers, tenths first, then hundredths, then combinations of the two including whole numbers
• Use mental math strategies to add decimals to hundredths (make a whole, open number line)
• Estimate the difference of two decimal numbers, tenths first, then hundredths, then combinations of the two including whole numbers
• Use mental math strategies to subtract decimals to hundredths (think addition, open number line)
##### Sample Week at a Glance

Before this week of lessons, grade 4 students will have developed an understanding of the different representations of decimals and how to order and compare them. This week, students are introduced to addition and subtraction of decimals for the first time.

Number Talk:

What should 3.9 + 5.12 be close to?

Students will likely notice that 3.9 is close to 4 and 5.12 is close to 5 so the sum is close to 9.

Have students work in partners. Give each pair of students several cards with decimal sums (like 2.5 + 6.31 for example) and whole number cards 1-15. Ask the group to put each sum card with the whole number that the sum will be closest to. Students can do a gallery walk at the end to see how other students organized their cards.

Closing circle – share and discuss new learning from the partner activity

Which one doesn’t belong?

Clothesline Number line: give each student 2-3 cards showing decimal addition expressions. Have students place the cards on a class clothesline (or masking tape/string) number line based on their estimates of the sums.

Closing discussion: Discuss strategies students used and what they learned.

Number Talk:

How could we add 3.9 + 5.12?

Give students some grid paper with several 10 x 10 grids so they can investigate. If they do not notice on their own, invite them to look at the relationship between the 9 tenths of 3.9 and the 1 tenth of 5.12. They might notice that 3.9 + 5.12 = 4 + 5.02 (make-a-whole strategy).

Have students work in partners. Give each pair of students several cards with decimal sums (like 2.8 + 6.31 for example) and grid paper. Have them estimate the sum first, and then represent the addends and sum pictorially. Finally, invite students to use mental math strategies to add, and to compare with the sums they found pictorially. How close is the sum to the estimate?

Closing circle – discuss how estimating first could be helpful in finding a calculation error.

Number Talk: I’m hungry! I want a sandwich, fries, and a drink for lunch. The sandwich costs \$5.75, the fries cost \$3, and the drink costs \$2.90. I have \$10. Do I have enough?

Math workshop:

Story problems with money

Comparing and ordering decimals and fractions (review)

Adding with decimals using mental math strategies

Teacher circulates to assess and offer support

Closing circle: Discuss one thing you learned this week about adding decimals.

Open with two more money addition story problems.

Closing circle:

Invite students to share the strategies they used to play the game.

Based on formative assessment information from this week, next week’s planning would include subtraction of decimals.

##### Suggestions for Assessment

By the end of grade 4, students will be able to add and subtract decimals (to hundredths) flexibly and fluently. This would look like being able to represent equations pictorially as well as being able to use efficient mental math strategies to find the sum or difference. Before adding or subtracting decimals, students should be able to answer the question, “What should this be close to?”.