### Learning Story for Grade 6

In Grade 6, there is significant new mathematical content added including an introductions to ratios and percentages and learning about factors and multiples. These new concepts all draw upon students’ developing proportional reasoning.

In Grade 6, students extend their understanding of fractions by learning about improper fractions and mixed numbers and moving fluently between these forms. They will represent these numbers in concrete, pictorial and symbolic forms and compare and order these types of numbers.

Building upon their understanding of multiplication and division from earlier grades, in Grade 6 students explore multiples and  factors of a number. Factors are used to identify whether a number is prime or composite. Through a process of factorization, students learn to identify factors of a number and a number’s prime factorization. They can then determine common factors between two or more numbers, including the greatest common factor (GCF). Students also explore the multiples of a number, including identifying common multiples between two or more numbers, and the least common multiple (LCM). Finally students also explore contextualized problems involving factors or multiples.

Students are introduced to ratios in Grade 6 and learn about part-to-part ratios and part-to-whole ratios as they see proportional relationships between quantities through concrete and pictorial representations. Percentages are also introduced formally in Grade 6 and students will learn to see percentages as connected to ratios, fractions and decimal numbers, using concrete materials and pictorial representations such as base ten blocks and hundred grids to represent percentages.

In Grade 6, students build on what they know about decimal numbers and place value as well as draw upon multiplication and division strategies with whole numbers that  they have learned in previous grades to multiply and divide decimal numbers.

### Key Concepts

#### Mixed Numbers

Representing mixed numbers and improper fractions concretely, pictorially, and symbolically; making connections between and within these forms of representations; understanding the concept of equivalence; using multiple strategies to compare and order mixed numbers and improper fractions; justify mathematical decisions (i.e., What strategy for comparing or ordering a set of fractions should I use? Which strategy makes the most sense or is the most efficient for this particular collection of numbers?)

#### Factors and Multiples

• Classify numbers as prime or composite.
• Apply divisibility rules to identify a factor of a number
• Determine the factors of a number (eg: arrays, factor pairs).
• Determine the prime factorization of a number (eg. factor tree)
• Determine the multiples of a number.
• Identify common factors and common multiples, including the GCF and LCM.
• Solve contextualized problems involving factors or multiples

#### Ratios & Percents

Introduction to ratio concepts to describe how values and quantities relate and compare. Provide concrete or pictorial representations of a given ratio and describe ratios as part-to-part  or part-to-whole ratios. Explore equivalent ratios. See percentage as a special ratio out of 100. Illustrate, record, express percents concretely and pictorially. Find missing parts (whole or percentage) using hundred grids or double number lines.

#### Multiplication and Division with Decimals

Drawing on knowledge of decimal place value and operations with whole numbers, students learn how to determine decimal products and quotients.  As with addition and subtraction of decimals, students see that for multiplication and division, they can use what they know about whole number multiplication and division as long as they pay close attention to place value.

#### Key Number Concept 1: Mixed Numbers

##### Overview

In Grade 6, students build on their understanding of proper fractions to represent and compare mixed numbers. Students build and represent mixed numbers concretely (e.g., pattern blocks, Cuisenaire rods, fraction strips), pictorially (e.g., rectangles on grid paper), and symbolically. Students should come to understand that a representation of any given fraction can often reveal equivalent fractions. For example, 1⅔ can be visualized as: one whole and two parts of one whole; eight thirds; one whole and eight twelfths; sixteen sixths; and so on. Flexibility is key; it is often helpful to visualize and describe a mixed number using an equivalent mixed number or improper fraction. Note that improper fractions present a natural decomposition: just like whole numbers to 100 can be decomposed into tens and ones, mixed numbers can be decomposed into wholes and parts. Students extend their use of strategies for comparing and ordering proper fractions to compare and order mixed numbers. These strategies include: using benchmarks (e.g., greater than or less than 1½ or “closeness” to 2); “thinking in images” (e.g., I can see 2⅛ and 2⅚); common numerators (e.g., 1¾ > 1⅗); common denominators (e.g., 7/6  < 8/6). Students move between these strategies; they develop proficiency in choosing an efficient strategy given the numbers at hand.

##### Number Sense Foundations:

The following concepts and competencies are foundational in supporting understanding of mixed numbers:

• Representing proper fractions concretely, pictorially, and symbolically; these include set, length, and area models
• Understanding of fractions as numbers in and of themselves; these numbers express a relationship between the number and size of parts of wholes
• Comparing and ordering proper fractions using multiple–and flexible–strategies
##### Progression:
• Understand that, in a proper fraction, the denominator represents the number of same-sized parts of a whole and the numerator represents the number of those parts
• Represent mixed numbers concretely, pictorially, and symbolically; symbolic representations should record what students build concretely or show pictorially
• Understand that a mixed number is composed of both a whole number and proper fraction, each having the same whole
• Express mixed numbers as equivalent improper fractions (and vice versa)
• Represent mixed numbers in different ways in order to illustrate equivalence
• Before inviting students to compare and order mixed numbers, students should be able to apply and explain some strategies for comparing and ordering proper fractions
• Gradually develop and apply strategies for comparing mixed numbers, progressing from problems in which estimation or visualization is sufficient to problems in which more precision (i.e., determining common numerators or denominators) is helpful
• Order mixed numbers; order may differ from compare in that a single order problem may involve applying multiple compare strategies
##### Sample Week at a Glance:

Prior to this week, students have engaged in problems and tasks designed to activate and deepen their understanding of fraction concepts, limited to proper fractions. These learning experiences have been particularly focused on the concept of equivalence. Students have represented proper fractions concretely, pictorially, and symbolically and have made connections between these representations.

Pose a “Which One Doesn’t Belong?” prompt in which one quadrant contains a mixed number.

“Cover Me” Assign the yellow hexagon a value of 1. Have students build one and one-half. Using the same type of pattern block (i.e., colour, shape), find different ways to show 1½, then 2⅓. Assign the pink double hexagon a value of 1. Repeat the process to show 1¼.

Gallery Walk. As a whole class, select and sequence some responses for students to analyze. For example one and one-half is one yellow hexagon as well as one red trapezoid or three red trapezoids (a/k/a “three-halves”).

Display a Number Talk or Fraction Talk image that involves mixed numbers.

“What’s My Name?” Assign a value of 1 to the purple Cuisenaire rods. If purple is named “One,” what are the names of the white, red, and light green rods ? (Note that these are proper fractions.) What are the names of the yellow, dark green, black , brown, blue, and orange rods? (Note that these are mixed numbers or improper fractions.)

Whole class discussion. Select and sequence students to share different names for each Cuisenaire rod. For example, dark green is “1½,” “three-halves,” “one and two-quarters,” “six-quarters,” and “1 2/4.” Make connections between concrete, pictorial, and symbolic representations.

Pose a “Would You Rather?” prompt designed to have students think about different-sized wholes.

Have students choose a different colour rod and assign it a value of 1. Repeat the “What’s My Name?” exploration.

Gallery Walk. As a whole class, select and sequence some responses for students to analyze.

“Ways to Make a Number” Have students Record as many different ways as they can think of to make 36. For example, students might make 36 using place value (base ten blocks), factors (groups of, arrays, areas), and operations. This primes students to engage in a similar task involving a mixed number.

“Ways to Make a Number” Have students record as many different ways as they can think of to make 1 1⁄3. Nudge students to think about different visual representations and their corresponding symbolic representations.

Select and sequence students to share different ways in which they made 1⅓.

Pose an “Same but Different” prompt designed to have students compare two mixed numbers (or improper fractions).

Have students write about their solutions in their math journals or curricular competency portfolios. Have students emphasize how they reasoned about and persevered to solve the problem.

Next week, students will begin to compare and order proper fractions, then mixed numbers and improper fractions. These learning experiences will reinforce their understanding of representing and “naming” this category of numbers since thinking about equivalence is one approach that students may take.

##### Suggestions for Assessment

By the end of Grade 6, students will be able to move flexibly between–and within–concrete, pictorial, and symbolic representations of mixed numbers and improper fractions. Students will be able to compare and order mixed numbers and improper fractions using multiple strategies; they will justify the strategies that they choose based on the numbers at hand.

#### Key Number Concept 2: Factors and Multiples

##### Overview

Students have been developing their understanding of multiplication and division since Grade 3. Throughout these grades, they have been exposed to factors as the numbers that are multiplied to get a product, though they likely have not learned the vocabulary until this grade. Students can explore factors in many ways. For example, they can create arrays of a number to determine different ways it can be represented:

As students explore arrays of different numbers, they’ll discover that some numbers only have one pair of factors, 1 and the number itself. These are prime numbers. All other whole numbers are composite.

Students can also discover and apply rules for divisibility to find factor pairs of a number. Using a factor rainbow is one way to organize the list of factors.

Using a factor tree, students can decompose a number into its prime factors. This is a process called prime factorization. Even though the elaboration in the curriculum uses exponents, that is optional for Grade 6.

When comparing the factors of two or more numbers, students can identify common factors. A Venn diagram is one way to organize the factors of two or more numbers:

Because 8 is the greatest of the common factors, it is called the greatest common factor (GCF) of 24 and 32.

Whereas finding factors involves dividing a number, finding multiples involves multiplying a number. A multiple of a number is found by multiplying that number by a whole number. Students have encountered multiples in earlier grades through skip-counting and patterns that start and increment by the same number.

When comparing the multiples of two or more numbers, students can identify common multiples.

Because 18 is the least of the common multiples, it is called the least common multiple (LCM) of 6 and 9.

Cuisenaire Rods are another tool that represent common multiples very well. For example, for the common multiples of 3 and 4:

Students can also explore and solve contextual problems involving factors and multiples. For example, if hot dog buns come in packages of 6 and wieners come in packages of 8, what is the least number of buns and wieners you can buy without having any left over?

Understanding factors and multiples will be extended in later grades by using them to determine square roots and cube roots of numbers, and factoring polynomials.

##### Number Sense Foundations:

The following concepts and competencies are foundational in supporting understanding of factors and multiples in grade 6:

• Concepts of multiplication and division and how they are related.
• Representing products using arrays.
• Understanding multiplication facts to 100 and related division facts.
• Strategies for multiplying and dividing whole numbers.
##### Progression:
• Explore arrays of numbers to determine factors of the numbers. Some students may be able to engage in this process symbolically without arrays.
• Recognize that some numbers only have two factors (1 and itself), and these are called prime numbers. Whole numbers with more than two factors are called composite numbers.
• Discover and apply divisibility rules to identify factors of numbers.
• Use a strategy (e.g., factor tree) to express the prime factorization of a number.
• Explore the factors of two or more numbers to identify common factors and the greatest common factor (GCF).
• Determine multiples of a number using a strategy such as skip counting.
• Explore the multiples of two or more numbers to identify common multiples and the least common multiple (LCM).
• Solve contextualized problems involving factors and multiples.
• Students can explore problems with factors earlier in the unit when learning about factors. It is good to revisit problems involving factors after learning about multiples so that they have to decide whether factors or multiples will be key to solving each problem.
##### Sample Week at a Glance

Prior to this week, students would have learned the term factor and explored different arrays of numbers to determine factors of the numbers. Some may have also used a symbolic strategy to find the factors. In this process they will have noticed that some numbers only have two factors, and these are called prime numbers. All other whole numbers are called composite numbers. They may also have explored some problems involving factors.

Focus: Prime Numbers up to 100

• Before: Prime numbers they already know.
• Brainstorm as a class the numbers which they know are prime, and how do they know they are prime.
• To prove a number is prime, there can only be two factors (i.e., one array in either direction).
• Even though 1 has one array, it is not a prime number. There are complex reasons why 1 is not prime, but at this grade level it is sufficient to say it is because it has only one factor, whereas prime numbers have two.
• The primes which are needed for the next activity are 2, 3, 5, 7 so those should emerge during brainstorming.
• Other primes less than 20 (11, 13, 17, 19) are likely to emerge as well. Larger primes can be more challenging to figure out as there may be factor that is challenging to find. For example 91 may seem like a prime number, but 91 = 7 x 13.
• During: Sieve of Eratosthenes
• Provide each pair of students with a hundred chart. Tell them to cross out 1 and put a circle around 2, 3, 5, and 7.
• Tell them to cross out all numbers which are multiples of 2, 3, 5, and 7.
• Tell them to circle all of the numbers which are not crossed off. These are the prime numbers less than 100.
• After:
• Have the class turn & share with another group to make sure everyone has the same numbers circled. Continue sharing and discussing until everyone agrees.
• Ask the class if they noticed any patterns. For example:
• “Except for 2, all of the prime numbers end in an odd number, which makes sense because all even numbers are composite except for 2.”
• “Many primes come in pairs, like 17 and 19, 71 and 73.”
• Ask the class if there were any surprises. For example:
• “I was surprised a bunch of numbers weren’t prime, like 87.”
• “I thought there would be more of a pattern, but each row looks so different.”
• Explain to the class that prime numbers are really important for encryption (i.e., password security) because it is incredibly challenging to determine if really large numbers are prime.
• Primes up to 7 were enough to identify primes up to 100. How much higher could they go? Have the class try to identify a few more primes. They need to make sure that the numbers are not divisible by some other prime like 11, 13, or 17.

Focus: Divisibility Rules for 2, 4, 5, 8, 10

• Before: Divisibility by 2, 5, and 10
• Write a large number, but leave a blank for the ones digit, for example:
59 84_. Note that you could go with even more digits.
• Explain that for a number to be divisible, it means that there is a number that can be divided without leaving a remainder. For example, 12 is divisible by 4 because 12 ÷ 4 = 3.
• Ask them what the missing digit could be so that the number is divisible by 10.
• All whole numbers which are multiplied by 10 have a 0 as the last digit, so 59 840 is divisible by 10.
• In a similar fashion, ask them to think about what they know about products to determine what the final digit could be to make the whole number divisible by:
• 5 (the final digit could be a 0 or a 5)
• 2 (the final digit must be an even number, i.e., 0, 2, 4, 6, or 8)
• During Part One: Divisibility by 4
• Ask the class to confirm that 32 is divisible by 4.
• Ask them to investigate in groups 132 is divisible by 4. Then investigate 332.
• After Part One:
• Have groups share their reasoning for how they know 132 is divisible by 4. For example:
• “We divided 132 by 4 and got 33.”
• “To divide by 4 you can take half of a half, so we did 132 to 66 to 33.”
• If none of the groups suggest it, ask the class to think about 132 as 100 + 32.
• Ask if 100 is divisible by 4. They should answer yes, that 4 groups of 25 make 100. So if 100 is divisible by 4, and 32 is divisible by 4, then 132 must be divisible by 4. Visually this can be seen as:
• Putting the parts together, there would be 4 groups where each has 25 + 8 = 33.

• Ask how this reasoning could be applied to show that 332 is divisible by 4.
• If 100 is divisible by 4, any number of hundreds is divisible by 4 as well. So because 300 and 32 are both divisible by 4, 332 must be divisible by 4.
• Write a bunch of digits and make the last two digits 32. Ask if this number is divisible by 4. For example:
• 25 982 361 532 is divisible by 4 because 25 982 361 500 is divisible by 4 and so is 32.
• During Part Two: Divisibility by 8
• Similar to the last activity, knowing that 32 is divisible by 8, ask the class to work in groups to investigate whether 132 is divisible by 8.
• If not, explain why not.
• Ask if there is a similar strategy that could work for dividing by 8.
• After Part Two:
• Have the class share what they figured out. For example:
• “100 cannot be divided by 8, so even though 32 is divisible by 8, 132 is not.”
• “We figured out that 1000 is divisible by 8, so if we have a 3-digit number that is divisible by 8, we can add any number of 1000s and still have a number that is divisible by 8,”
• As a class summarize the rules for divisibility by 2, 4, 5, 8, and 10.
• Have the class test several other numbers for divisibility by 2, 4, 5, 8, and 10.
• Challenge them to find a numbers that are:
• divisible by NONE of 2, 4, 5, 8, and 10.
• divisible by ALL of 2, 4, 5, 8, and 10.

Focus: Divisibility Rules for 3, 6, 9

• Before: Divisibility by 3
• Have students work in pairs.
• Provide several multiples of 3, e.g., 24, 87, 156, and 261
• Add the digits of a number that is not divisible by 3, e.g., 142
• What do they notice about the sum of the digits of numbers divisible by 3 that is not true for the sum of digits of numbers that are not divisible by 3.
• Discuss what they noticed. What should emerge is that if the sum of the digits of a number is divisible by 3, then the number is divisible by 3.
• During: Divisibility by 9
• Rather than noticing a pattern as was done in the Before, this activity is for making sense about why the sum of digits leads to divisibility (or not) by 9.
• Ask students to represent a 3-digit number (e.g., 378) that is divisible by 9 on grid paper (similar to a base-ten block representation).
• Represent the sum of the digits by highlighting one square from each piece.
• Ask the class to think about how this representation helps us see why 378 is divisible by 9.
• After:
• Have students share their reasoning.
• Each 100 has 99 remaining blocks, and 99 is divisible by 9.
• Each 10 has 9 remaining blocks, and 9 is divisible by 9.
• The highlighted blocks sum to 18, and 18 is divisible by 9.
• Because all of the parts of 378 (99s, 9s, and sum of digits) are divisible by 9, the whole number 378 is divisible by 9.
• Generally, when we remove the sum of the digits from any number, what’s left is divisible by 9. So if the sum of the digits is also divisible by 9, the whole number is divisible by 9.
• The divisibility rule for 6 could be presented. Alternatively, you could do an activity on a hundred chart.
• Shade all of the multiples of 2, and circle all of the multiples of 3
• All of the multiples of 6 are both shaded and circled, so all numbers divisible by 6 are divisible by 2 AND by 3.
• As a class summarize the rules for divisibility by 3, 6, and 9.
• Have the class test several other numbers for divisibility by 3, 6, and 9.
• Write a large number with the last digit missing. Ask students to figure out what the last digit could be so that the number is divisible by 2, 3, 4, 5, 6, 8, 9, or 10.

Focus: Prime Factorization

• Before: From factors to prime factors
• Remind the class that every time they divide a number by a number, they reveal two factors of the number. For example, 90 is divisible by 3. , so 3 and 30 are factors of 90.
• Have the students work in pairs. Ask them to apply divisibility rules to determine other factor pairs of 90. They can then further divide the factors they found to find more factors.
• Have the class share which factor pairs they found:
• 1 x 90, 2 x 45, 3 x 30, 5 x 18, 6 x 15, 9 x 10
• Ask which factors of 90 are prime numbers [2, 3, 5]. Tell them that these are called prime factors.
• Show the class how they can use a factor tree to break a number down into its prime factors. Ask them along the way to suggest the factors for each branch.
• Once you have completed one factor tree, show that the factor tree could have been done using different factor pairs.
• In both trees, circle the prime factors.
• Write the product of factors in order. This is called the prime factorization of 90.
• Though not required, you may wish to show how the prime factorization can be expressed using exponents if it makes sense to your students.
• During: Prime factorization
• Have the class work in pairs to determine the prime factorization of 252. If any pairs are struggling with this number, you may provide a smaller number, or scaffold them through the process.
• Once some pairs finish, they could try other numbers.
• After:
• Have the class share one or more factor trees of 252.
• Confirm with the class how to express the prime factorization of 252.
• or
• Ask the class how they could check if the prime factorization is correct.
• “We could just multiply the numbers.”
• Have the class work in pairs or small groups to do the prime factorization of some more numbers. Some may decide to use an alternative strategy than a factor tree.
• For further practice, students could pair up. One student writes a prime factorization and then multiplies the numbers. They then ask their partner to see if they can figure out the prime factorization.

Focus: Common Factors and GCF

• Before: Review of factors and prime factorization.
• Do a Which One Doesn’t Belong Many different reasons may emerge, but ask students to think about reasons which involve factors or prime factorization.
• Sample responses include:
• 2x2x5: Only one which uses a prime factor more than once
• 3×7: Only one which does not have 2 as a prime factor
• 2x3x5: Only one that has three different prime factors
• 2×53: Only one that has a two-digit prime factor
• Expand the two products on the left to get 20 and 30. Ask students to determine the factors of each number, and suggest that using the prime factors may help.
• 20: 1, 2, 4, 5, 10, 20
• 30: 1, 2, 3, 5, 6, 10, 15, 30
• Show students how to organize the factors into a Venn Diagram:
• Explain that the numbers in the overlap of the circles are common factors. The largest of these (10) is the greatest common factor of 20 and 30. We can use GCF as an abbreviation for greatest common factor.
• During:
• Have the class work in pairs or groups to determine the GCF of each set of numbers. Encourage them to use a venn diagram if it helps. Some groups may develop other strategies as they work.
• The list below is roughly in order of complexity. It is okay if any group does not get through them all. Encourage them to get through as many as they can.
• Determine the GCF of:
• 6 and 9
• 10 and 14
• 4 and 12
• 12 and 18
• 16 and 24
• 24 and 36
• 72 and 108
• 12, 30, and 90
• After:
• Have the class share their answers and strategies. The answers are as follows:
• 6 and 9: GCF = 3
• 10 and 14: GCF = 2
• 4 and 12: GCF = 4
• 12 and 18: GCF = 6
• 16 and 24: GCF = 8
• 24 and 36: GCF = 12
• 72 and 108: GCF = 36
• 12, 30, and 90: GCF = 6
• Many groups may have found all of the factors of each number and then determined the GCF from the lists or using a Venn Diagram. Other strategies may also emerge. If they do not, you may present them as alternatives and invite students to make sense of and reflect on them. For example:
• We may be able to find the GCF by inspecting the numbers.
• e.g., I know that 6 is a factor of both 12 and 18 and there isn’t a higher factor of both.
• We can use prime factorization. For example:
24 = 2 x 2 x 2 x 3
36 = 2 x 2 x 3 x 3

Both prime factorizations have 2 x 2 x 3, so the GCF is 12

• Explain to the class that they have actually used common factors before but may not have realized it. In Grade 5 when they were finding equivalent fractions, at times they would have simplified fractions into simplest terms. A fraction can be reduced to simplest terms by dividing the numerator and denominator by the GCF of both. For example:
• Problems involving GCF will likely be done after learning about LCM, but you may wish to provide a GCF problem on this day as well. For example:
• If I have 9 roses and 12 carnations, what is the largest number of bouquets I can make so that each bouquet has the same flowers?
[The GCF is 3, so I can make 3 bouquets. Each bouquet will have 3 roses and 4 carnations.]

In the next week, students will explore multiples of a number using a strategy such as skip counting. They will then explore the multiples of two or more numbers to identify common multiples and the least common multiple (LCM). Finally they will work on solving contextualized problems involving factors and multiples. It is good to do these both factor and multiple problems together so that students have to decide which concept applies.

##### Suggestions for Assessment

By the end of Grade 6, students should be able to apply a strategy to determine all of the factors of a number, including applying divisibility rules. They should be able to identify common factors and the GCF of two (or more) numbers. They should be able to determine several multiples of a number, and identify common multiples of two (or more) numbers, including determining the LCM. Finally they should be able to decide whether to use factors or multiples to model and solve contextual problems.

Surrey Schools Numeracy Support: Video Series for Parents: Factors and Multiples

Mathigon:

nRich problems for factors and multiples

Number Visuals from YouCubed

Prime Climb from Math for Love

#### Key Number Concept 3: Ratios and Percents

##### Overview

Building on students’ understanding of multiplicative reasoning, students in grade 6 are introduced to how values relate and compare and represent their thinking using ratios. Students start to build their understanding of ratio relationships using concrete representations and then move  to making sense of abstract representation of ratio relationships. They learn that a ratio is an association between two quantities.

Students learn about the language used to describe the association shown in a ratio and use diagrams to illustrate the association. Proportional relationships are found in many different real world patterns.Proportional relationships that are multiplicative rather than additive are a type of

relationships that most students have not previously encountered before. Proportional reasoning is crucial to understanding higher mathematics,including linear equations and beyond. In this section, students begin to explore proportional reasoning by engaging with equivalent ratios in real-life contexts (recipes or colour mixtures) to visualize the sameness of two ratios.

As an extension of equivalent ratios, students begin to understand the concept of percentage as a ratio with the whole part as 100. By representing percentages using a hundredths grid, students begin to understand percent is a special ratio that compares a number to 100.

In later units of study, students will use percentages to make comparisons between quantities.

##### Number Sense Foundations:

The following concepts and competencies are foundational in supporting understanding of ratios in grade 6:

• Numbers and operations (multiplication and division)
• Factors and multiples
• Fraction equivalence and comparison
##### Progression:
Ratios
• Describe how values and quantities relate and compare using ratios
• Provide a concrete or pictorial representation for a given ratio
• Write a ratio from a given concrete or pictorial representation
• Express a given ratio in multiple forms, such as 3:5, or 3 to 5
• Identify and describe ratios from real-life contexts and record them symbolically ( relate to equivalent ratios eg. doubling a recipe doesn’t change the taste)
• Explain the part- to-whole and part-to-part ratios of a set (e.g., for a group of 3 girls and 5 boys, explain the ratios 3:5, 3:8, and 5:8
• Solve a given problem involving ratio
• Explore equivalent ratios and create them in several ways
• Use double number line to show how equivalent ratios can be identified
Percents
• explain that “percent” means “out of 100.”
• explain that percent is a ratio out of 100
• use concrete materials and pictorial representations to illustrate a given percent
• record the percent displayed in a given concrete or pictorial representation
• express a given percent as a fraction and a decimal
• identify and describe percents from real-life contexts, and record them symbolically
• solve a given problem involving percents
• Finding missing part (whole or percentage) using hundred grids or double number lines
##### Sample Week at a Glance

Prior to this week of lessons, students practiced operations with fractions and spent time exploring equivalent fractions

Fill in each day using a 3-Part Lesson model and a variety of lesson structures.

Embed computational fluency in different colour or icon

Introduction to Ratios

Key Math Terms:

• Ratio
• Set
• Term
• Part-to-part ratio
• Part-to-whole ratio

Pair Investigation: In groups of two, come up with all the ways you can think of that the items in the image can be compared. Share your thinking with your learning partner and draw each comparison.

Share your thoughts with another pair of students. What was the same? What was new?

Teacher led class discussion: what are the different ways that the items in the picture were compared?

Explain that the term ratio can be used to describe a comparison of two quantities of the same type (item, unit, animal, etc.).

All the items in the picture can be referred to as a set.

Show that some groups decided to separate the items in the set and compare them to each other; Part-To-Part Ratio

Some groups decided to look at one specific item in the set and compare it to all the items in the set: Part-To-Whole

We can represent ratios in different ways: as a fraction (part-to-whole), in words, or by using symbols (numbers and a colon)

Invite students to revisit their recordings from the introductory investigation and to record the ratios they represented in drawings  in one or two other ways.

Closing Circle: where in real-life might we see the use of ratios?
Why might they be useful?

Playing with Ratios

Math Talk

Source: Amy Hoelscher

Invite students to share their process for determining a solution and to ensure they include the key terms in their explanation. Students are invited to compare ratios in this example, without having a formal introduction to equivalent ratios. Take note of the various strategies students share and the connections they make to equivalent fractions. Revisit this example when exploring equivalent fractions more explicitly.

Small Group Investigation: What is being compared in each ratio?

1. 3:4
2. 4/7
3. 3 to 7
4. 4:3

Teacher led class discussion: debrief introductory activity assessing for understanding of key concepts and key terms.

Using manipulatives to

Introduction to Equivalent Ratios

Key Math Term:

• Equivalent Ratios

Pair Investigation: Students use colour hues to model, construct, and communicate equivalent ratios.

Materials: water and food colouring (red and blue). Pre-mix two solutions (one drop of food colouring per cup) for each pair of students.

Procedure: provide students with pre-mixed solutions and ask them to record observations of each and to make predictions about what might happen if the solutions are combined. Students combine the solutions in a 1 (red): 2 (blue) ratio and record observations. They can then play with various recipes to determine how the ratios of the color mixtures impact the hue. The activities here reinforce the idea that scaling a recipe up (or down) requires scaling the amount of each ingredient by the same factor

Students should notice that the colour doesn’t change when ratios are increased proportionately (equivalent ratios).

Extension: students can be invited to create certain hues and identify the ratio of the solutions that was necessary to create them.

Closure: What makes ratios equivalents? All ratios that are equivalent can be generated by multiplying both terms of the ratio  by the same number.

Exit slip:Explain why the pair of ratios is an equivalent ratio, or draw a diagram that shows why.

4:5 and 8:10

Exploring Equivalent Ratios

Number Talk Routine: Find the quotients mentally.

150÷2

150÷4

150÷8

Locate and label the quotients on the numberline.

Note: This number talk helps students think about what happens to a quotient when the divisor of a fraction, or a term of a ratio is doubled.

Small Group Investigation:

Provide students with the following problem to determine a solution for.

George is planting a border around his garden. He plants 5 yellow daisies for every 3 red tulips. How many tulips would he plant for each number of daisies.

1. 10 daisies
2. 15 daisies
3. 20 daisies

Invite students to record their thinking and solution and to share their process for determining a solution with the class.

Invite students to share how the concept of equivalent ratios applies to this problem.

While students share their processes and solutions, illustrate the use of a table and patterns to determine the ratios.

Ask students to design a problem that requires the use of equivalent ratios to determine a solution. Provide constraints to the problem (eg. real-life context, numbers to 100, etc.) Students invite another pair in the class to determine a solution to their problem.

Applying Ratios

Class investigation: What is being compared in this ratio?

Display a set of items and corresponding ratios that might apply to the context.

Example:

(Note: image from Surrey School District: Parent Video Series)

Future learning opportunities with ratios may introduce double number lines and how they may be used to identify equivalent ratios. Students may also be invited to explore ratios in recipes and making adjustments in recipes using ratios.

##### Suggestions for Assessment

By the end of grade six, students will be able to interpret different representations of ratios, situations involving equivalent ratios and tables of equivalent ratios. They will explain reasoning about equivalence of ratios and they will be able to compare representations of ratios. In addition, students will  describe and represent ratio associations, represent doubling/tripling quantities in a ratio, represent equivalent ratios, and justify whether ratios are or aren’t equivalent and why information is needed to solve a ratio problem.

Comparing Quantities and Ratios Math Talk

https://howweteach.com/comparing-quantities-ratios-math-talk/

Surrey Schools Numeracy Support: Video Series for Parents

Rod Ratios Interactive Activity

https://nrich.maths.org/4782

#### Key Number Concept 4: Multiplication and Division with Decimals

##### Overview

In grade 6, learners draw on their understanding of patterns in a place value chart to read, write, compare and round decimals, and to estimate and determine sums, differences, products and quotients.

Students learn how to use their knowledge of decimal place value and operations with whole numbers to determine decimal products and quotients.  As with addition and subtraction of decimals, students see that for multiplication and division, they can use what they know about whole number multiplication and division if they pay close attention to place value. So, they apply their understanding of whole-number operations to add, subtract, multiply, and divide decimal numbers that build on the relationship between multiplication and division, multiplication and addition, and division and subtraction. In grade 6, learners focus on multiplying and dividing decimals by whole numbers.

##### Number Sense Foundations:

The following concepts and competencies are foundational in supporting understanding of multiplying and dividing decimals.

• Place value patterns: in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
• Place value understanding is used to read, write, round, compare, order, add, subtract, multiply and divide decimals.
• Place value reasoning is used to decide whether sums, differences, products and quotients are reasonable
• Multiplication and Division to three digits, including division with remainders
• understanding the relationships between multiplication and division, multiplication and addition, and division and subtraction
• using flexible computation strategies (e.g., decomposing, distributive principle, commutative principle, repeated addition, repeated subtraction)
##### Progression:
• Review relationship between place value, fractions and decimals to read, write (standard and expanded form), and compare decimals.
• Represent decimals using concrete materials (Place Value Mat, Grid Paper, Base Ten Blocks, Cuisenaire Rods, number line), and represent decimals pictorially.
• Express a given pictorial or concrete representation as a decimal (eg. 150 shaded squares on a thousandth grid can be expressed as 0.150)
• Compare and order a given set of decimals (to thousandths) by placing them on a number line that contains benchmarks; including tenths, hundredths, and thousandths, or by using place value or by using equivalent decimals. Students can use other strategies to compare and order but should be able to explain their thinking.
• Review strategies for adding and subtracting decimals to thousandths. Students should try to demonstrate as many ways as they can concretely, pictorially, and symbolically. They should use flexible computation strategies involving taking apart and combining numbers.
• Students might use: Base 10 Blocks, Place-Value Mat, add whole numbers and estimate to place decimal point, using the standard algorithm, number line, etc.
• Use estimation strategies for predicting the product or quotient of decimals (decimal benchmarks, front-end estimation, compatible numbers)
• Apply concepts of multiplication (groups of, array, area model, distributive principle, commutative principle, repeated addition, etc.) and division (sharing, grouping, decomposing, repeated subtraction, etc.) to develop processes for multiplying and dividing decimals. Students may use knowledge of the standard algorithm as one method to multiply/divide decimals.
##### Sample Week at a Glance

Conceptual understanding of decimals requires students to connect decimals to whole numbers and fractions. Prior to this week, students engaged in tasks that had them reviewing how to read decimals as fractions and how to represent them using visuals (eg. shading parts of a 100th-grid) to understand parts of a whole; reinforcing that decimals and fractions are different ways of representing the same value. These learning experiences particularly focused on students representing decimals concretely (using materials such as Base 10 Blocks), pictorially, and symbolically and making connections between these representations. Students reviewed and practiced various strategies for estimating the sum and difference of decimals as a means of using number sense to determine the reasonableness of an answer.  Key vocabulary terms have been displaced on an anchor chart that will be circled back to this week: decimal point, place value, product, quotient, divisor, dividend, factor, etc.

To access students’ prior knowledge of adding and subtracting decimals, try an instructional routine such as, Would you Rather, or invite students to participate in an open-ended task that encourages extended responses and could expect them to reason and justify.

The following, Would you Rather, prompt allows students to demonstrate their competencies and knowledge of place value, decimals, properties of operations, equivalence (when using an identified time frame), etc.; while practicing flexible strategies.

While sharing ideas as a class, be sure to notice and name the relevant key ideas, while extending and elaborating using language and processes/strategies.

Because decimals can be found in many places in our daily lives, creating an open-ended mathematical question based on student interest could also generate prior knowledge while acquiring and building background knowledge.

Example of an open-ended question:

Identify a personal hobby/interest/passion. List items that you use when engaged in that hobby/interest/passion. You have been given \$350 to spend on items that will help you either improve your skills in your hobby/interest/passion or provide more opportunity for you to participate in it. Use as much money as possible to purchase specific goods/services. Provide proof of cost (pictures of items online with their associated cost). Let’s say that taxes are included in the cost of the goods/services. You must buy doubles/multiples of at least one item. Show your thinking in pictorial and symbolic form. Justify your purchase decisions.

Estimating the Product of a Decimal and a Whole Number

Estimating is an important competency for students when working with decimals to be able to judge and determine the reasonableness of their solutions.  John Van de Walle suggests that instruction on computation with decimals must start with estimating. If students can accurately estimate products and quotients, they are more likely to correctly place the decimal point when determining products and quotients in a variety of ways or refine solutions after recalculating.

State, “The digits in the product of the two numbers are shown below in the multiplication question. Without multiplying, decide where the decimal point should go.”

This activity can be implemented like a Number Talk. Ask the students to share what they think the answer is and record the range of responses. In sharing their strategies, students may then share their “proof” or justification of a solution by saying, “I think it is ____ because _____.”

During class discussion, extend and elaborate thinking by reinforcing that we can ignore the decimals at first. Find the product of the two factors without the decimals, and then estimate where the decimal point should go. You might show all the possible places the decimal could go and then narrow down for reasonableness.

82

x4

328

To estimate, find the product of the whole numbers and determine where the decimal point should go based on reasoning. Since the product of 8 and 4 is 32, the product of 8.2×4 should be close to 32, as well.

8×4=32

So,  8.2 x 4= 32.8

Note other strategies students used and compare to the strategy above. Some of the strategies that may be demonstrated: front-end estimating, using decimal benchmarks, or compatible numbers. Try to notice and name them during the discussion. Why do they work, or not work? What are other things we should consider when estimating products?

Extension: Is our estimate of 32 an overestimate, or underestimate? How do you know? Why is it important to know if an estimate is an over/underestimate?

Applying what we know:

1. How would you estimate 6.23 x 5? Is your estimate and overestimate, or an underestimate? How do you know?

Here, students are placing the decimal in the factors, rather than in the product. It is possible to have more than one correct response. Ask students what solutions they came up with and record their responses. 1.8×145 and 18×14.5, both satisfy the solution. Ensure various strategies have been discussed.

Closing Discussion:

Why might it be helpful for us to estimate the answer to a math question before using a strategy to find the exact answer? Where could you use estimation of products in daily life?

Determining the Product of a Decimal and a Whole Number

Marian Small’s Open Question with an extension: Choose a two-digit number that does not end in zero. Explain how you would use mental math to multiply it by 6. Then, show in as many visual ways as you can how to check your answer.

Have students share their strategies and thinking. Pay particular attention to naming them and identifying the specific steps in each. Additionally, add or reinforce the language that is to be practiced. Students may demonstrate understanding of combining numbers, repeated addition, standard algorithm, and other ways to multiply whole numbers.  Students may use number lines, an array, base-ten blocks, 10th/100th grids, etc. to represent their thinking visually. This discussion is also an opportunity to reinforce and elaborate key concepts that can be built on when multiplying decimals by whole numbers.

Extend Thinking:

How can the strategies we used to multiply the whole numbers in the previous activity help us to multiply 3 x 0.4?

Record student thinking and focus on importance of practicing multiple strategies to determine solutions and to determine connections between the concrete, pictorial and symbolic representations. Possible strategies and representations:

Using Base-Ten Blocks

Using a Place Value Mat and Base-Ten Blocks

Estimating the Quotient of a Decimal and a Whole Number,

Determining the Quotient of a Decimal and a Whole Number

Show in as many ways as you can how to divide 126 by 3. You can use pictures, symbols, and materials.

Have students share their strategies and thinking. Pay particular attention to naming them and identifying the specific steps in each. Additionally, add or reinforce the language that is to be practiced, specifically for the parts of the question (dividend, divisor, quotient). Students may demonstrate understanding of decomposing, repeated subtraction, standard algorithm, and other ways to divide decimals by whole numbers.  Students may use number lines, base-ten blocks, 10th/100th grids, area models, etc. to represent their thinking visually. This discussion is also an opportunity to reinforce and elaborate key concepts that can be built on when dividing decimals by whole numbers. Students may demonstrate their understanding of the relationship between multiplication and division (multiplication is the inverse operation of division) to show their thinking.

Extend Thinking: How can the strategies we used to divide the whole numbers in the previous activity help us to divide 1.26 by 3?

Record student thinking and focus on importance of practicing multiple strategies to determine solutions and to determine connections between the concrete, pictorial and symbolic representations. Possible strategies and representations:

Using Base-Ten Blocks

How might we share 1.26 equally into three groups?

Start by representing 1.26 with base-ten blocks

Notice that it will be difficult to split the red block up into physical parts, so what can we do?

Represent the whole as 10 tenths!

Decimals in the Real World

To make meaningful connections to decimals in context, invite students to participate in an open-ended task, like a numeracy task. This one, created by Peter Liljedahl, provides many entry points into the problem and allows for various solutions, while requiring justification.

liljedahl, P. (2010) Hamster beta fish – peter liljedahl, Numeracy Tasks. liljedahl. Available at: https://peterliljedahl.com/wp-content/uploads/NT-The-Class-Pet.pdf (Accessed: March 3, 2023).

Students can use a variety of strategies to represent thinking. To ensure there is some pictorial representation, suggest that students use either number lines, base-ten blocks, 100th/10th grids, etc. to show some of their thinking. Students would benefit from a supporting graphic organizer, place value mats, etc.

Alternatively, create a plan and design project for a school context; such as a school garden, a beach/park clean-up, etc. that requires students to estimate, measure, use multiplication and division of decimals by a whole number.

This week is about applying various strategies to multiply and divide decimals.  After this week and throughout the year, students should have regular practice using what they have learned when working with money in Financial Literacy and when working with shapes and measurement.

Additionally, it is recommended that further exploration of the connection between fractions and decimals be explored where students may move flexibly and fluidly between them when using various strategies to perform operations with decimals.

##### Suggestions for Assessment

By the end of Grade 6, students will be able to draw on their ability to move flexibly between–and within–concrete, pictorial, and symbolic representations of decimals to multiply decimals by 1-digit multipliers and they will be able to divide decimals by 1-digit natural numbers using multiple strategies. They will be able to use their understanding of the relationship between multiplication and division, and multiplication and addition, and division and subtraction to perform operations with decimals. An emphasis should be placed

https://www.edutopia.org/article/strategies-teaching-students-estimate/

SD38 High-Yield Routines Video on Number Talks

Estimation

Estimation180 (Andrew Stadel) for a daily estimation challenge

https://estimation180.com/

Estimation Clipboard (Steve Wyborney) – Each image invites students to think about what number is represented, as a gradually released set of clues is provided.

https://stevewyborney.com/2018/04/the-estimation-clipboard/

Would You Rather Math

https://www.wouldyourathermath.com/

Open Questions