 ### Learning Story for Grade 5

In Grade 5 students are building on their understanding of the key concepts from Grade 4. There is not anything new beyond extending place value. For example, with place value, addition, and subtraction, students are extending their understanding of these concepts from 10 000 to 1 000 000. With decimals, students are extending their knowledge of tenths and hundredths into thousandths. This includes addition and subtraction of decimal numbers. In Grade 6, students will extend their place value further: up to billions.

Grade 5 students continue to develop their understanding of multiplication and division from Grade 4 and the relationship between these two operations. Fact fluency is emerging and students are beginning to recall many facts to 100 (2s, 3s, 4s, 5s, 10s). Additionally, students begin to work with remainders and interpreting what they mean in different contexts in this grade. They will extend these concepts to decimal numbers in Grade 6.

Fraction concepts are introduced in Grade 3 and decimal concepts are introduced in Grade 4. Students build on previous understandings and use them to compare and order fractions using benchmark fractions such as 0, ½, and 1 in Grade 4. The concept of equivalent fractions help students to compare and order fractions in Grade 5. Students extend these understandings to mixed numbers and improper fractions in Grade 6. For decimals, students extend from a knowledge of tenths and hundredths in Grade 4 to thousandths in Grade 5. Grade 6 focusses on decimal computations.

### Key Concepts

#### Place value

Building on students’ understanding of place value (to 10 000), students build and represent numbers in different forms to 1 000 000 using 100 000s, 10 000s, 1000s, 100s, 10, and 1s. Students compare and order numbers along a number line, using benchmarks such as 500 000 and 100 000.

Students extend 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 estimate sums and differences and to add and subtract numbers from thousandths to one million using strategies such as decomposing and compensating.

#### Multiplication and division

Review of multiplication and division concepts (i.e., the various meanings of these operations); relationship to addition and subtraction; connection to skip counting/multiples; representing the process of each operation concretely, pictorially and symbolically.

Computational fluency with multiplication and division facts to/within 100 is emerging, using mental math strategies such as skip counting, using known facts to solve unknown facts, doubling & halving, annexing, and distributive property.

Multiplication and division to three digits (whole numbers), including division with remainders. Interpreting remainders in a variety of contexts.

#### Fractions and decimals

Reviewing fractions as a type of number representing parts and wholes. Representing fractions in concrete, pictorial and symbolic forms, such as using area models, set models, and number line models. Using equivalent fractions to order and compare fractions.

Review decimal concepts (tenths, hundredths) and extend to thousandths using visual representations (grid paper, number lines, base 10 blocks). Relate decimals to benchmark fractions (tenths, hundredths, thousandths).

#### Key Number Concept 1: Place Value

##### Overview

Building on students’ understanding of place value (to 10 000) in Grade 4, students build and represent numbers in different forms to 1 000 000 using 100 000s, 10 000s, 1000s, 100s, 10, and 1s (concrete, pictorial and symbolic) in Grade 5. Students are able to read and write numbers up to 1 000 000 and decimal place value extends from hundredths to thousandths and can recognize that decimals help with exactness and precision (building, measuring of speed etc). At this stage students should reflect a strong understanding of the relationship between the position and the value of digits in a number. Exploring different place value structures (binary coding, Mayan quinary and vigesimal systems, Exploding Dots) can help students better understand the base 10 system that we use.

A common question asked of students is  “What are the different ways you could show 100 000?”  In grade 5 we expect students will represent numbers using both standard and expanded forms, as well as continuing to work with concrete and pictorial models (base 10 blocks, place value charts). Students need opportunities to visualize and play with large numbers and look for examples in their world where they use larger numbers. “What is something that might cost 1 million dollars?” “How many grains of sand might be in a handful?” “When is 1 000 000 not a lot?” Students can compare and order numbers on a number line using benchmarks such as 500 000 and 100 000.  Students are able to count forward and backwards from a given point “What is one less than 100 000?” “What is one more than 999 999 and how do you know?” Estimating large numbers allows students to determine the reasonableness of their answers.  Students will have an easier time with addition, subtraction, multiplication, and division of numbers with a deep  understanding of place value.

*The use of a comma is not a world-wide standard. In continental Europe the groups of digits are separated by points and what we call a decimal point is replaced by a comma.

##### Number Sense Foundations:

Foundational, supporting concepts and related competencies that are needed to develop this grade level concept

The following concepts and competencies are foundational in supporting understanding of place value in grade 5:

• Exploration of the meaning of “base 10”
• Skip counting by 1 000, 10 000, 100 000 forward and backward from any starting point including points that do not fall within a count from 0. For example: count by 1 000 beginning at 5.
• Making, representing and identifying the value of a number using a range of materials and models
• Reading and writing numbers between hundredths (0.00) and ten thousand (10 000)
• Understanding of the connection between the place and the value of digits within a number
• Strategies for ordering whole and decimal numbers on a number line
• Renaming, partitioning and regrouping of numbers. (e.g.,1260 has 126 tens)
• Strategies for calculation that reflect place value (e.g., 25 by 1000 is 25 thousands, area models, etc.)
• Strategies for estimating (benchmarking, sampling, comparison) the magnitude of large numbers and quantities- how big is…?
• Connecting larger numbers and quantities to daily life
##### Progression:
• Connect place value concepts from previous learning:
• 10 000 is made up of 10 groups of 1000
• Understanding of Base 10 relationships
• Understanding that 10 000 is made up of 10 groups of 1000 and is read one hundred thousand
• Understands that hundredths is a number split into a hundred equal parts and connects to money and thousandths is split into a 1000 equal parts

• Read and represent numbers to 1 000 000
• Understanding that 1 000 000 is made up of 1000 groups of 1000 and is read one million
• Use concrete, pictorial and symbolic representations (base 10 materials, expanded form)

• Read and represent numbers to thousandths
• Understanding that thousandths is a number split into a 1000 equal parts
• Use concrete, pictorial and symbolic representations (base 10 materials, expanded form, fractions)

• Apply strategies that employ concepts of place value
• In computation (arrays and area models, place value based algorithms)
• Recognising where place value can streamline counting, computation, ordering, etc. of numbers
##### Sample Week at a Glance:

Before the lessons included in this sample week students will have used place value charts and number lines to represent and compare numbers from 0.01 to 10 000.  They will have learned the routines for Same but Different, Number Strings  and Clothesline Math using familiar numbers, quantities and operations.  They will be familiar with building whole numbers using base 10 materials and have some experience connecting fractions to decimals (0.1=1/10).  In order to do Friday’s lesson, students will need to understand the relationship between meters/ millimeters and litres/millilitresThe focus for this sample week is understanding place value relationships.

Activate(whole class discussion):

Same but Different

0.01 and 0.001

0.001 to 1000

Explore(small groups):

Have students build some decimal numbers and whole numbers with Base 10 materials and then write the standard and expanded forms.  Provide them with Place value mats and  the Consolidation questions to discuss as they work.

Consolidate(whole class/individual):

Discuss:

• How is using the base 10 materials the same and different when representing whole numbers vs decimals?
• What do each of the representations help us understand about the numbers?

Exit Slip/Math Journal: Use pictures, numbers and words to explain what you know about one of the questions.

Activate(whole class):

Clothesline Math

Give 3 students one tent card each for benchmark numbers.  For example: 0.1, 0.01, 0.001 OR 1, 0.01, 2.  Have each student place their tent on the clothesline and explain why they have chosen where to put it.  Ask other students to agree/disagree.

Explore(small groups):

Give each group a set of tent cards to order on clotheslines or number lines. For example: 0.5, 1.2, 0.08, 0.055, 0.277, 0.009, 1/100, 1/1000, 1/10

Consolidate(whole class):

Gallery Walk:

• How does place value help us order numbers?
• Based on observations from the explore, select students to share helpful strategies they used when ordering the numbers.

Exit Slip/Math Journal: Use pictures, words and numbers to explain something you heard from someone else today about ordering numbers that is important/helpful.

Before

Share a video such as Let’s Count the Moons and discuss how in many Indigenous languages the words we use to count with depend on what it is we are counting. Practice saying the first three or four number names in hən̓q̓əmin̓əm̓ or other local Indigenous language.

During

Go outside for a counting walk (or within your school). What can we count? Encourage students to find different groups of items to count (ie. trees, leaves, homes, rocks, street signs, etc) and practice counting in both English and hən̓q̓əmin̓əm̓ or other local Indigenous language

After

Closing circle: Invite students to share what they counted and how they counted them. Were some items more difficult to count than others? For example, birds flying or something far away. What counting strategies did you use if you couldn’t touch the items?

Activate(whole class):

Number String

10mm= __cm

100mm=__cm

1000mm=__cm

__cm=1m

__mm=1m

__mm=0.5m

0.234m=__mm

Explore:

Give students a variety of decimal numbers represented to thousandths (0.003, 0.012, 0.345, 0.510, 0.700, 1.276) and place value materials (base 10, blocks, number line, grid paper, place value chart). Have them explore representing the numbers in as many ways as they can.  Then have them order the numbers using their representations to justify the order.

Consolidate(Whole Class):

Gallery walk

Based on observations from the explore, select students to share which representations were most helpful and how they used them.

Activate:

Measurement Clothesline: benchmark tents 1m, 1cm, 1mm

Have students place and discuss as on Tuesday

Explore(small groups):

Step 1:Provide students with several measurements in different units (m, cm, mm) and have them order them on number/clotheslines.

Step 2:  Have students select some object of various sizes and estimate where they would place them on the number line.  Then measure each object to the millimeter and adjust if needed.

Consolidate(individual reflection):

Math Journal: Use pictures, words and numbers to explain what someone should know about decimal place value to work with numbers successfully.

An interesting follow-up or cross curricular accompaniment to this week would be to have students research an alternative number system and compare it to the system we use.  For example Indigenous Nations from the southwest coast of British Columbia in the Halkomelem Language group use a base 10 system that is reflected in how the words for numbers are constructed. How might this help with understanding how the system works? What is the same/different from how English number words reflect the base 10 system.  The Mayans and the Babylonians used non-base 10 systems, which could also make for an interesting study.

##### Suggestions for Assessment

What to look for (Proficient):

• Place value language and representations when working with numbers (mental and written math)
• Expanded and standard form connections
• Use of number line to compare and order numbers
• Fluent counting with large numbers using appropriate place value language
• Connection to previous place value understandings. For example: 1 000 to 1 000 000 or 0.01 to 0.001
• Connection of decimals and fractions using place value

By the end of this grade students will be able to:

• Say and write numbers between 0.001 and 1 000 000
• Rename numbers using place value chart and expanded notation
• Explain and justify representations of numbers using base 10 language and materials
• Connect various representations of numbers between 0.001 and 1 000 000
• Order decimals, fractions and whole numbers using place value
• Justify ordering and other number choices using place value

#### Key Number Concept 2: Addition and Subtraction

##### Overview

In grade 5, students are adding and subtracting numbers from thousandths to one million. Being computationally fluent with addition and subtraction facts to 20 will help students when they are thinking about strategies for adding and subtracting larger numbers. For this reason, spending some time reviewing efficient strategies for recalling addition and subtraction facts is a useful endeavour and good place to start. These strategies could include bridging ten (e.g., 8 + 7 = 8 + 2 + 5 = 15) and using known facts (e.g., 8 + 8 = 16, so 8 + 7 = 16 – 1 = 15). These same strategies can be extended to larger numbers (e.g., 97 + 24 = 97 + 3 + 21 = 121; 97 + 20 = 117, so 97 + 24 = 117 + 4 = 121).

Students often become overly reliant on the traditional addition and subtraction algorithms to the detriment of their number sense. The problem with only using these algorithms is that students only ever get practice with single-digit computations. These algorithms do not consider the magnitude of the numbers, so you will find students needing to stack such simple examples as 99 + 99 instead of realizing that this is just 100 + 100 – 2. The alternative is to use a variety of strategies flexibly, such as these:

• For example, 278 + 47 = 278 + 40 + 7 or 1000 – 238 = 1000 – 200 – 30 – 8 or 1.9 + 3.3 = 1.9 + 3.0 + 0.3
• For example, 278+ 47 = 280 + 50 – 5 or 1000 – 238 = 999 – 238 + 1 or 1.9 + 3.3 = 2.0 + 3.3 – 0.1
• Adding to subtract. For example, for 1000 – 238, think 238 + 2 (to get 240) + 60 (to get 300) + 700 (to get 1000) yield the answer of 762 (which is 700 + 60 + 2).

There are other strategies too. Let students share their personal strategies for adding and subtracting numbers to support a culture of learning from one another. Mental Math routines such as Number Talks, Number Strings, and I Have You Need are great routines to support this kind of thinking. The traditional algorithms have a place here too. These algorithms are most useful when the numbers we are considering are too difficult for mental math (e.g., 234 857 + 78 669) and a calculator is not handy, or when an estimation is not good enough.

In addition to the above strategies, we want students to strategize about finding reasonable estimates for sums and differences. For example, rounding numbers to estimate. Estimating in this way helps students to build their number sense by considering the magnitude of the numbers they are working with rather than just single-digits. Students can use estimating to be able to ask themselves, “Is this answer reasonable?” when finding sums and differences, but the deeper reason for estimating is because in many life situations, an estimate is good enough.

Another consideration for this concept is clarity around the different meanings of adding and subtracting and how they are related to one another. Understanding addition as combining and subtraction not only as take-away but as comparison too (e.g., How much taller am I than you?) Having students work on addition and subtraction concurrently (e.g., doing mixed word problems) helps students to improve their operation sense around addition and subtraction.

##### Number Sense Foundations:

The following concepts and competencies are foundational in supporting understanding of addition and subtraction in Grade 5:

• Concrete, pictorial and symbolic representations of numbers
• Make reasonable computational estimates
• Developing mental math strategies
• Decomposing numbers by place value (e.g., 324 = 300 + 20 + 4)
• Computational fluency with addition and subtraction facts to 20
• Addition and subtraction of whole numbers to 10 000
• Decimal concepts to hundredths
• Addition and subtraction of decimals to hundredths
##### Progression:
• Review strategies for recalling addition and subtraction facts to 20
• Making 10 facts
• Bridging 10 (e.g., 8 + 4 = 8 + 2 + 2)
• Using a known fact (e.g., 10 + 8 = 18, so 9 + 8 = 18 – 1)
• Regrouping (e.g. 9 + 6 = 10 + 5)
• Inverse relationship (e.g. For 20 – 18 , think 18 + ⬚ = 20)
• Review the meanings of addition and subtraction.
• Subtraction as take-away and comparison
• Relationship between addition and subtraction
• Missing minuend
• Missing subtrahend
• Mixed problems involving addition and subtraction
• Adding and subtracting whole numbers (to 1 million)
• Review of place value concepts
• Estimating sums and differences
• Rounding strategies (front-end, compensation, etc.)
• Reasonableness of a calculation
• Personal strategies including mental math strategies
• Practice with efficient strategies
• Place value strategies
• Compensation strategies
• Problem-solving with numbers to 1 million
• Adding and subtracting decimal numbers (to thousandths)
• Review of decimal concepts
• Estimating sums and differences
• Rounding strategies (front-end, compensation, etc.)
• Reasonableness of a calculation
• Personal strategies including mental math strategies
• Practice with efficient strategies
• Place value strategies
• Compensation strategies
• Problem-solving with decimals to thousandths
##### Sample Week at a Glance

Students have been reviewing addition and subtraction facts to 20 and multi-digit sums and differences via instructional routines such as Number Talks since the beginning of the school year. Sums and differences with whole numbers to 1 million and decimal concepts to thousandths would be a focus for the first term. In the second term we extend these ideas to addition and subtraction with decimal numbers. The week below begins this unit, starting with addition.

Lesson topic: Estimating decimal sums

Activate:
Number Talk with money. Costco’s marked down items end in 0.97. For example, shoes for \$32.97 and jackets for \$59.97. How much does it cost to purchase both (before tax)? How does rounding or estimating help?

Explore: • What strategy did you use when the number of decimal places was the same? Different?
• Did it matter whether the numbers were greater than one or not? Why?

Consolidate:

Discuss the strategies used in the explore activity (see question prompts above). Formalize efficient strategies, such as the different rounding techniques students used. Which methods are more precise? Which are faster? Why?

Practice:

Students play the above game with a partner. For those who like competition, the difference between the estimate and the true answer adds to one’s score. Lowest score wins! (Students use calculators to check answers.) You might need to impose time limits–we’re estimating after all!

Exit Ticket/Math Journal:

Give an example of when you would estimate the sum of two decimal numbers and explain how you would do this.

Lesson topic: Personal strategies for adding decimals

Activate:

Number Talk (4.25 + 8.9). Discuss the different strategies students share.

Explore:

Students are encouraged to explore the strategies that came up in the Number Talk or use other personal strategies to find 0.67 + 0.897 (or similar prompt). They do not need to use mental math. Provide base 10 materials too. How many different ways can you find the sum? Test your strategy with some other numbers. Check with a calculator or partner. Students who are used to the inquiry process will be happy to come up with their own numbers to add. Does the strategy always work? When is it best used and why?

Consolidation:

Students share their personal strategies. Discuss for which numbers the strategy works best. What are the limitations? Formalize and name efficient strategies (e.g., The “Harjot method” to give credit to student’s work) and encourage students to try some of these methods if they haven’t yet.

Practice:

Students “test out” some of the methods their peers used, and their own methods too, on a variety of decimal addition questions. The teacher can provide them or students used to inquiry can generate questions themselves to try. Students can check their solutions with one another or a calculator.

Exit Ticket/Math Journal:

Students show how to find 0.24 + 0.097 using as many different strategies as they can.

Lesson topic: Practice with efficient strategies for adding decimals

Activate:

I Have, You Need. Start with warmup targets like 10, 100, 1000. Then do a target of 1 using different decimals. For example, I have… 0.8, 0.25, 0.62, 0.05, 0.099, etc.

Explore:

Revisit strategies from the previous lesson. Explore the question: Which strategy is best for which types of numbers?

• When would you use friendly numbers and compensate? For example, 8.989 + 4.325
• When would you decompose numbers (e.g., by place value)? For example, 464.5 + 36.25
• When would you stack? For example, 235 295.214 + 45 922.09

Discuss when you would use each strategy and why. How do the base 10 materials help students to visualize and make sense of these strategies?

Practice: Provide opportunities for students to practice using efficient strategies, such as questions in a textbook, workbook, worksheet, online program, etc.

Exit Ticket/Math Journal:

Which strategy is your “go to” method and why? Which do you need more practice with? Illustrate with examples.

Activate:

Number Strings. Students use number relationships to find the sums. For example, 3.2 + 2, 3.2 + 1.9, 5.3 + 6, 5.3 + 5.8, 8.9 + 7.8
These examples encourage students to use “friendly numbers” and compensate.

Explore:

Have students write down a 4-digit decimal number (e.g., 23.54). Then reverse the digits while keeping the decimal place in the same spot (e.g., 45.32). Add the two numbers (e.g., 23.54 + 45.32). Repeat this with a variety of different 4-digit decimal numbers. What patterns do they notice? Students can make and test a conjecture. Does this always work or are there exceptions? Why or why not?

Have students explore other inquiries, too, such as… What if not 4-digits? What if the decimal is in a different spot? What if we subtract the numbers? Students can record this thinking in their Math Journals.

Consolidate:

Discuss student inquiries as a class. Share as many student inquiries as you have time for. What were their conjectures, procedures, and findings?

Exit Ticket/Math Journal:

Students reflect on what they learned from today’s inquiry.

Lesson topic: Assessment

Activate:

I Have, You Need. Similar to Wednesday’s lesson.

Explore:

Write a story problem that could be solved using 0.875 + 2.5. Students share their examples. Discuss how in all cases the idea of “combining” unifies these stories.

Consolidate:

Students do a short quiz on addition with decimals.

Based on Friday’s assessment, next week’s plan might begin with further review and practice of addition, look at different applications of addition, or move onto subtraction strategies. After subtraction strategies, students will explore mixed addition and subtraction problems.

##### Suggestions for Assessment

What to look for:

• Reasonable estimates of sums and differences
• Flexibly use a variety of strategies to add and subtract numbers
• Use of models (base 10 materials, open number lines) to represent their thinking
• Extending grade 4 knowledge of adding and subtracting decimals to one million and thousandths
• Use inquiry to explore sums and differences
• Understands when to add and when to subtract in a variety of contexts

By the end of this grade students will be able to:

• Reasonably estimate sums and difference from thousandths to one million
• Use an appropriate strategy to add and subtract numbers from thousandths to one million
• Connect visual representations or models to the strategies used to find sums and differences
• Explain and justify their thinking (e.g., steps to an algorithm or findings in an inquiry)
• Explain the different meanings of addition and subtraction
• Solve problems involving addition and subtraction of decimals

#### Key Number Concept 3: Multiplication and Division

##### Overview

Reviewing basic multiplication and division concepts sets a strong foundation for the deeper learning to come. This includes discussing the various meanings of these operations. Students first learn about multiplication as equal groups, repeated addition, and arrays. The idea of an array can be extended to include area of a rectangle which is taught at this grade level. Multiplication as a rate is another useful meaning and connects to the idea of “times” that students often use in a number of contexts (e.g., “I have three times as many as you do.”) Students can also use the relationship between multiplication and division to understand that, for example, 5 ✕ ___ = 35 can be solved by 35 ÷ 5.

The major meanings of division include the idea of fair share, which students tend to be more familiar with (i.e., partitive division), and equal groupings (i.e., quotative division). Partitive division is when the number of groups are known and we are trying to find out how many in each group. For example, sharing 8 items between 2 groups and asking how many in each group (i.e., 8 ÷ 2). Quotative division is the opposite: the number of items in each group is known and we are trying to find the number of groups. For example, Distributing 8 items 2 at a time until we are out and asking how many groups (i.e., 8 ÷ 2). Students often do not experience many examples of the latter and yet many division algorithms are built from this concept. Other meanings of division include repeated subtraction, the missing length of a rectangle in area problems, and rates.

Having learned about different computational strategies in Grade 4, students’ computational fluency with multiplication and division facts to/within 100 is emerging. Students share a variety of personal strategies to recall facts, from counting strategies such as skip counting on fingers and counting forward and backward from a known fact (e.g., 8 ✕ 6 = 8 ✕ 5 + 8), to more sophisticated strategies such as doubling & halving (e.g., 4 ✕ 8 = 2 ✕ 16), the distributive property (e.g., 8 ✕ 7 = 8 ✕ 5 + 8 ✕ 2), and factoring (e.g., 6 ✕ 8 = 6 ✕ 2 ✕ 2 ✕ 2). Students use these strategies flexibly. That is, they use the strategy they find most useful depending on the fact they are recalling.

Students extend the aforementioned computations to multiplication and division of three digits with whole numbers. This includes division with remainders and interpreting remainders in a variety of contexts, such as when to round up or down. Annexing (e.g., 3 ✕ 100 is 3 ✕ 1 with two 0s annexed to the end: 300) is a useful strategy for multiplying larger numbers (e.g. 125 ✕ 4 = 100 ✕ 4 + 20 ✕ 4 + 5 ✕ 4 = 400 + 80 + 20). Using area models to illustrate the distributive property helps students to use what they know to solve more complex problems as in the previous example. Division of multi-digit numbers can be developed by using the partitive definition and sharing base ten blocks, or the quotative definition and counting the number of equal groups needed to build a number. We will delve into these ideas below.

##### Number Sense Foundations:

The following concepts and competencies are foundational in supporting understanding of multiplication and division in Grade 5:

• Concrete, pictorial and symbolic representations of 3-digit numbers
• Decomposing 3-digit numbers by place value (e.g., 324 = 300 + 20 + 4)
• Computational fluency with addition and subtraction facts to 20
• Addition and subtraction of whole numbers to 10 000
• Multiplication as equal groups, repeated addition, and arrays
• Partitive and quotative division (modeling with concrete materials, drawing pictures)
• Skip counting forward and backward by 2, 3, 4, 5, 6, 7, 8, 9, and 10, starting from any number to 100
• Make reasonable computational estimates
##### Progression:
• Review the different meanings of multiplication and division.
• Multiplication as equal groups, repeated addition, and arrays
• Partitive and Quotative definitions of division
• Personal strategies for recalling multiplication and division facts to 100
• Connecting meanings of multiplication to strategies: skip counting, counting on/back from a known fact, commutative property (3 ✕ 2 = 2 ✕ 3), halving & doubling, distributive property, factoring
• Using the inverse relationship between multiplication and division. That is, using the related multiplication fact to recall a division fact.
• Multiplication of 2- and 3-digit numbers
• Annexing numbers multiplied by 10, 100, 1000
• Partition larger arrays into smaller arrays and summing its parts. Progress from concrete to pictorial to abstract. For example, using base 10 blocks to make an array, then grid paper, then open area models.
• 2- and 3-digit by 1-digit multiplication before multi-digit multiplication
• Developing algorithms incrementally and connect to previous models
• Area model (visual)
• Expanded form (abstract: 324 ✕ 4 = 1200 + 80 + 16)
• Efficient strategies (mental math)
• Division of 2- and 3- digit numbers by 1-digit numbers
• Division as sharing (partitive) using base 10 blocks. Progress from simple problems to those requiring regrouping to those with remainders.
• Division as equal groups (quotative) using base 10 blocks. Progress from simple problems to those requiring regrouping to those with remainders.
• Developing algorithms and connecting to quotative meaning of division
• Decomposing dividend (e.g., 324 ÷ 3 = 300 ÷ 3 + 24 ÷ 3)
• Repeated subtraction. How many times do you subtract the divisor from the dividend to get to zero? It’s faster to subtract in groups. (e.g., 324 ÷ 3… subtract 100 3s to get 24, then 8 3s to get 0, giving us 100 + 8 = 108). There are many ways to do this.
• Traditional algorithm is repeated subtraction done a particular way
##### Sample Week at a Glance

Before this week of lessons, grade 5 students will have developed an understanding of the different meanings of multiplication and division by way of real-life examples, representing multiplication and division expressions using concrete materials and visual models, and exploring/playing with these concepts in a variety of ways. This would be over several days.

They would also be doing daily Number Talks or Number Strings to discuss mental math strategies for recalling multiplication and division facts, which the classroom community participates in all year long. Using Number Strings with the idea of annexing 0s for products of 10, 100, and 1000 would be a prerequisite for the upcoming lessons.

Lesson topic: Array models (1- and 2-digit by 1-digit)

• Number String (6 ✕ 10, 12 ✕ 10, 6 ✕ 100, 60 ✕ 100, 60 ✕ 500)
• Small arrays (up to 10 ✕ 10). Build using unit cubes, base 10 blocks, or drawing on grid paper. Students share personal strategies for determining the number of little squares/cubes in each array, such as:
• Counting by 1s
• Skip Counting
• Multiplying

We want students to see that multiplication is the most efficient way!

• Do the same with larger arrays (more than 10 ✕ 10). Guiding question: How does decomposing or partitioning the array into smaller arrays help? Can you partition the array in other ways? How is place value helpful? (Connect partitioning by place value to annexing.) [insert picture]
• Formalize student strategies as your “lesson”:
• We can use multiplication to determine how many unit squares are in an array.
• We can break up a large array into smaller arrays which makes the multiplications simpler. Then find the sum of its parts.
• Using an array model to solve a multiplication question.
• Independent practice with the lesson ideas above. Provide students with grid paper and several questions (e.g., 23 ✕ 6). Students show how they can solve these questions using an array model.

Exit ticket: Use an array model to calculate 14 ✕ 7

Lesson topic: Area models (2- and 3-digit by 1-digit)

• Number String (8 ✕ 10, 24 ✕ 10, 80 ✕ 100, 40 ✕ 200, 30 ✕ 700). Make explicit connections between this exercise and the multiplication strategies used this week.
• Review multiplication with arrays. Students may need more practice with these. When students are ready, connect the array model to an area model (multi-digit by 1-digit numbers). How are they similar? How are they different? [insert picture]
• Key understanding: Area models allow students to work with larger numbers without having to draw all the unit squares, so it is more efficient than an area model. Grid paper is no longer necessary. Students visualize the unit squares as a reminder that multiplication is linked to arrays.
• Exploring area models. Students decompose (or partition) 2- and 3-digit by 1-digit area models to make finding the whole area easier. How many ways can you do this? Which way is the easiest? Why? Partitioning by place value facilitates an algorithmic approach which will help students understand the traditional multiplication algorithm better.
• Independent practice. Students play in small groups. Give students dice or playing cards. Allow them to roll 3 or 4 dice (or draw 3 or 4 cards, Ace to 9) and order them to make the greatest product: ⬜⬜ ✕ ⬜ or ⬜⬜⬜ ✕ ⬜. Students draw area models on mini white boards to calculate. Who got the greatest product? How do you know?

Exit ticket: Use an area model to calculate 123 ✕ 5

Lesson topic: Area models (multi-digit; e.g., 2-digit by 2-digit)

• Review multiplication with area models from last lesson. Ask students how they could partition the area model into multiple sections both vertically and horizontally (e.g., four quadrants)–if this has not already come up on its own! How could this way of partitioning help with multi-digit multiplication examples? Let students explore and share! Partitioning the area model by place value for both numbers has strong connections to the traditional multiplication algorithm. [insert picture]
• Math Workshop. Students are working on one or more of the following:
• Working on multiplication questions with the teacher
• Card/Dice game from yesterday with format: ⬜⬜ ✕ ⬜⬜
• Short worksheet with different multiplication questions
• Practice online (e.g., Mathletics)

Debrief today’s Math Workshop.

Lesson topic: From area models to expanded form of traditional algorithm

• Number String (7 ✕ 10, 60 ✕ 3, 40 ✕ 50, 500 ✕ 70, 800 ✕ 400). Again, connect this exercise to the multiplication strategies used this week.
• Show a similar image to the one below comparing multiplying with area models vs expanded form of the traditional algorithm. Then ask students to make sense of the different solutions. Why do they work? How are they connected? How do they differ? [insert picture]
• Students explore using expanded form with various multi-digit multiplication problems. Teacher facilitates discussion on how this method compares to previously taught models: array and area.
• Independent practice with today’s lesson ideas.

Exit ticket: Use expanded form to calculate 15 ✕ 8 and 24 ✕ 32

Lesson topic: Efficient strategies

• Give students one or two multiplication questions and ask them to solve them in as many different ways as they can, such as 99 ✕ 5 and 35 ✕ 18. Facilitate discussion about efficient strategies. For example, 99 ✕ 5 = 100 ✕ 5 – 5 and 35 ✕ 18 = 7 ✕ 5 ✕ 2 ✕ 9 = (7 ✕ 9) ✕ (5 ✕ 2). Not all students will be ready for such “short cuts” and this is okay. Some students will use the traditional algorithm. In this case, show students how this algorithm connects to area models and expanded form.
• Math Workshop. Students are working on one or more of the following:
• Working on multiplication questions with the teacher
• Card/Dice game with format: ⬜⬜ ✕ ⬜⬜
• Short worksheet with different multiplication questions
• Practice online (e.g., Mathletics)

Debrief today’s Math Workshop.

Based on formative assessment information from this week, next week’s plan might begin with further review and practice of 2- and 3-digit multiplication and related story problems. If students are proficient in their understanding, we will move on to dividing 2- and 3- digit numbers. The progression would be similar to multiplication: exploring and sharing personal strategies, formalizing the most efficient strategies, and developing these ideas using a progression that goes from concrete (e.g., base 10 blocks) to pictorial (e.g., area models) to abstract.

##### Suggestions for Assessment

What to look for:

• mental math strategies for multiplication and division facts and applicable multi-digit multiplication and division calculations
• representing multiplication using arrays, area models, and expanded form
• relationship between multiplication and division
• division using partitive (fair share) and quotative (equal groups) meanings

By the end of this grade students will be able to:

• Use mental math strategies to recall multiplication and division facts
• Model multiplication using arrays and area models
• Communicate the different meanings of multiplication and division
• Connecting the distributive property to multiplication & division and multiplication & division to one another.

#### Key Number Concept 4: Fractions and Decimals

##### Overview

Reviewing fraction concepts that were first introduced in Grade 3 helps students to build a strong foundation in fractions. Some key ideas include understanding the meaning of the top and bottom numbers in a fraction (i.e., going beyond simply naming the “numerator” and “denominator”). The bottom number represents the unit size and can be defined as the number of same size parts that make up the whole. The top number represents how many of these parts or units we count. It is also important for students to have a clear understanding of what the “whole” represents in a fraction. Another key idea is that a fraction is a number (not to numbers placed one on top the other), which is why it can be placed on a number line. We also want students to connect fractions with division. For example, ¾ means 3 ÷ 4. This will help students to have an easier time with mixed numbers and fractions greater than 1 in Grade 6.

Fraction representations help students to deepen their understanding of fraction concepts. These representations include area models, set models, and number line models. Students often get a lot of practice with area models and find them the easiest to understand, but they also need exposure to set models and fractions on a number line, as well. The number line in particular helps students to understand that a fraction is indeed a number. These models help students to visualize fractions so that they can order and compare them, which was the focus in Grade 4 for this topic. In Grade 5, students explore equivalent fractions. The models help students to see why the numerator and denominator are multiplied or divided by the same number to generate an equivalent fraction (e.g., finding common denominators).

Again, reviewing decimal concepts (tenths, hundredths) from Grade 4 helps students to extend the concept to thousandths. Visual representations (grid paper, number lines, base 10 blocks) are great tools to build understanding. In particular, we want students to see that each place value is ten times greater than the one preceding it. Representations are also useful in relating decimals to benchmark fractions (tenths, hundredths, thousandths) which is an important relationship to understand, and they allow us to understand equivalent decimals, as well. For example, 0.2 = 2/10 = 0.20 = 20/100 = 0.200 = 200/1000. Note that 0.2 and 2/10 are both read as “2 tenths” because they are equivalent. Encouraging students to name decimals and fractions in this way (as opposed to “point 2” and “2 over 10”) supports this understanding. We also want students to understand that the zero in the hundredths place of 0.20 can be removed; whereas, the zero in tenths place of 0.02 cannot be removed and still maintain equivalence.

##### Number Sense Foundations:

The following concepts and competencies are foundational in supporting understanding of fractions and decimals in Grade 5:

• Whole number place value concepts
• Number line models with whole numbers
• Multiplication and division facts
• Fraction concepts (see Grade 3 curriculum)
• Comparing and ordering fractions
• Decimal concepts (tenths and hundredths)
##### Progression:
• Review of fraction concepts
• Meaning of numerator and denominator and the “whole”
• Representing fractions using area models, set models, and number line models
• Comparing and ordering fractions using models
• Equivalent fractions
• Using concrete materials
• Using area, set, and number line models
• Using abstract strategies or algorithms
• Common denominators
• Comparing and ordering fractions using common denominators
• Review of decimal concepts (tenths and hundredths extended to thousandths)
• Relationship between benchmark fractions and decimals
• Representing using decimal grids and number lines
• Equivalent decimals
• Using concrete materials (e.g., base 10 blocks)
• Using decimal grids and number lines
• Annexing and removing zeros
• Comparing and ordering decimals and fractions
##### Sample Week at a Glance

Before this week of lessons, grade 5 students will have reviewed basic fraction concepts, including the meaning of the numerator and denominator and what constitutes a “whole”. Lots of time was spent representing fractions using area models, set models, and linear models (e.g., number line) and making connections between these representations. Up to this point, comparing and ordering fractions has been done using concrete materials or models. Doing this with the concept of equivalent fractions will be the focus of this week’s lessons.

Lesson topic: Exploring fraction equivalence

• Clothesline Math. Estimating benchmarks on a 0 to 1 number line. For example, ½, ¼, ¾ , ⅓, ⅔ (or other simple fractions).
• Use concrete materials such as fraction circles, pattern blocks, or Cuisenaire Rods. Given which piece is the “whole”, name the other pieces. For example, with Cuisenaire rods, if the orange rod is 1, what number (fraction) is each of the other rods? If the yellow pattern block is 1, what are the red, blue, and green pattern blocks? Note that pattern blocks are easier than fraction circles and Cuisenaire rods are more challenging.
• Explore equivalence. Can you combine shapes of one colour to get the same shape of another colour? For example, for Cuisenaire rods, 2 red = 1 purple. This means that 2/5 = 4/10. For pattern blocks, 3 greens = 1 red. This means 3/6 = 1/2. Have students find as many equivalent fractions as they can and record their thinking.

Debrief lesson. Students share what they found while the teacher records the equivalent fractions in a visual manner.

Lesson topic: Equivalent fractions using models

• Matching game. Display fractions using different representations (area, set, and number line models) for several equivalent fractions, such as ½, 2/4, 3/6; ⅔, 4/6; 2/8, ¼, etc. Which representations show equivalent fractions? How do you know?
• Math Workshop. Students rotate through the following activities:
• Matching game (like above)
• Equivalent area models activity (e.g., game)
• Equivalent fractions on a number line with teacher
• Equivalent set models activity (e.g., worksheet)

Debrief today’s Math Workshop.

Lesson topic: Generating equivalent fractions

• Clothesline Math. Place these fractions on the number line: 1/2, 2/3, 2/4, 4/6. Make connections between equivalent fractions.
• Prompt: Find all the equivalent fractions that you can for 1/2. Facilitate a discussion on how these fractions are related and what students did to find equivalent forms. Encourage students to look for patterns between the numbers and to look for number relationships (e.g., multiplicative relationships). Formalize their key take-aways.
• Equivalent fraction games.
• Fraction Track (need device).
• Use playing cards (1-10). Draw three cards and use the template: ⬜/⬜ = ⬜/⬜. Place the three cards in any position. Then figure out what the third number needs to be. Choose another three cards if the ones drawn are too challenging.

Debrief games. What strategies did you use? What challenges arose?

Lesson topic: Finding common denominators

• Prompt: Which is greater? 3/5 or 7/10? How do you know? Class discussion of strategies. Teacher formalizes strategies of finding a common denominator (e.g., using materials, drawing a picture, multiplication/division)
• Math Workshop. Students are working on one or more of the following:
• Finding common denominators with the teacher.
• Matching game (need a device)
• Common denominator worksheet (includes visuals)
• Online practice (e.g., Mathletics)
• Fraction Track (need device)
• Card game from Wednesday: ⬜/⬜ = ⬜/⬜
• Building equivalent fractions with materials (e.g., fraction circles)

Debrief today’s Math Workshop.

Lesson topic: Comparing and ordering fractions

• Count around the circle. Use fractions such as ½, ⅓, ¼, and have students state an equivalent form if they can! For example, students count: ⅛, 2/8 (or ¼), ⅜, 4/8 (or ½), …
• Prompt: Order a set of fractions from least to greatest (e.g., ⅕, ⅔, ¾, 3/10). Materials are provided. Students work on the task and share their strategies as a class. Teacher helps students to connect the key ideas learned this week to the strategies used in today’s task.
• Practice using fraction cards. Provide fraction materials too. Students work in pairs or small groups. They draw 2 or more fraction cards and place the fraction cards in order (e.g., least to greatest). Students record their solutions to show the teacher.

Exit ticket. Students use the digits 1, 2, 3, 4, 5, and 6 each only once to create 3 fractions that they have to put in order from least to greatest. Students must show their thinking.

Based on formative assessment information from this week, next week’s plan might begin with further review and practice of equivalent fractions. For example, do Math Workshop with the activities from last week while checking in on students needing more support. When students are proficient in their understanding, we will move on to decimal concepts.

##### Suggestions for Assessment

What to look for:

• Reasonably estimating fraction benchmarks on a number line
• Representing fractions using area models, set models, and linear models
• Can use models to determine fraction equivalence
• Can compare and order fractions using models

By the end of this grade students will be able to:

• Represent fractions using an area model, set model, and linear model
• Determine whether or not two fractions are equivalent
• Generate a list of equivalent fractions
• Compare and order a set of fractions