1 stage 1: the number sequence - itecnz.co.nzquantity is the big idea that describes amounts, or...

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Big Idea: Quantity

Quantity is the big idea that describes amounts, or sizes. It is a fundamental idea that refers to quantitative properties; the size of

things (magnitude), and the number of things (multitude).

Why is Quantity Important?

Quantity means that numbers represent amounts. If students do not possess an understanding of Quantity, their knowledge of

foundational mathematics will be undermined. Understanding Quantity helps students develop number conceptualization. In or-

der for children to understand quantity, they need foundational experiences with counting, identifying numbers, sequencing, and

comparing. Counting, and using numerals to quantify collections, form the developmental progression of experiences in Stage 1.

Children who understand number concepts know that numbers are used to describe quantities and relationships’ between

quantities. For example, the sequence of numbers is determined by each number’s magnitude, a concept that not all children

understand. Without this underpinning of understanding, a child may perform rote responses, which will not stand the test of

further, rigorous application. The developmental progression of experiences in Stage 1 help students actively grow a strong

number knowledge base.

Concept Standard Example Description

1.1: Sequencing K.CC.2 1, 2, 3, 4, ?

Children complete short sequences of visual displays of quantities beginning with 1. Subsequently, the sequence shows gaps which the students need to fill in. The sequencing tasks ask students to show that they have quantity and number names in order of magnitude and to associate quantities with numerals.

1.2: Identifying

NumberK.CC.3

Find the

number ‘3’

Students see the visual tool with a numeral beneath it. Students match the displayed quantity with its corresponding numeral. Then they see a visual tool with a quantity, and choose the numeral that matches the displayed amount.

1.3: Count

ForwardK.CC.2 4, 5, 6, 7, ?

The student sees the initial position in a sequence missing, then the last. He progresses to visual sequential displays which contain 2 missing amounts, which he fills in further showing his understanding of order.

1.4: Count

BackwardK.CC.2 9, 8, 7, 6, ? Count backward in the same progression of tasks as 1.3, again using numbers

stage 1: the number sequence

1

Stage 1 Learning Progression

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NZ Standard Year 1: - continue sequential patterns and number patterns based on ones

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Typewritten Text

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2

Big Idea: QuantityQuantity is the big idea that describes amounts, or sizes. It is a fundamental idea that refers to quantitative properties; the size

of things (magnitude), and the number of things (multitude).

Why is Quantity Important?Quantity means that numbers represent amounts. If students do not possess an understanding of Quantity, their knowledge of

foundational mathematics will be undermined. Understanding Quantity helps students develop number conceptualization. In

order for children to understand quantity, they need foundational experiences with counting, identifying numbers, sequencing,

and comparing. Counting, comparing the magnitude of collections, and using numerals to quantify collections, form the

developmental progression of experiences in Stage 2.

Children who understand number concepts know that numbers are used to describe quantities and relationships’ between

quantities. For example, the sequence of numbers is determined by each number’s magnitude, a concept that not all children

understand. Without this underpinning of understanding, a child may perform rote responses, which will not stand the test of

further, rigorous application. The developmental progression of experiences in Stage 2 helps students actively grow a strong

number knowledge base.

stage 2: more / less / same


2.1: Find ‘1 More’ K.CC.4 Find ‘1 more’ than 3

Find the next number without counting . Find ‘One More’ first with a visual representation of an amount, and then with a number name, without counting. Amounts are visually displayed, and then, a single numeral is given. The goal is for a student to build on the con-cept of order of magnitude, and to understand each successive number in sequence is ‘1 more than’ the number before; that ‘the next number after equals one more.’ Four means ‘the number that is 1 more than 3.’ Having ‘number-after’ knowledge asks the student to enter the sequence at any point and show next number without having to start at one.

2.2: Find ‘1 Less’ K.CC.4 Find ‘1 less’ than 8

This level continues the idea of sequence magnitude by first showing a visual of an amount, and then a number name-not in any sequence. It tasks the student with finding ‘the number that comes before’ or, ‘1 less.’ Having the ‘number before’ knowledge shows that the student can enter the sequence at any point and name ‘the number before.’ 2.1 and 2.2 introduce number relationships as the next learning progression after number size.


2.3: Find More K.CC.6 Find more than 3

Students apply their small-number recognition, their understanding of order, counting, quantity size, to a comparison of collections within the 0-9 range. The tasks allows students to use counting and ordering knowledge in a practical manner, and develop a concrete understanding of numerical relationships. A number farther along a number line represents a larger quantity. A collection of dots labeled ‘5’ is more than the collection labeled ‘2.’ A taller bar is more than a shorter bar; longer becomes equated with more in the bar context

2.4: Find Less K.CC.7 Find less than 3

This level continues the tasks and skills begun in 1.7, as needed in the concept of finding less, fewer, and shorter. As in 1.7, the different visual environments give breadth to using different comparative words.

2.5: Find Same K.CC.7 Find a number the same as 4

Continues a child’s intuitive understanding of relative amounts of more and less, and extends it to that of ‘same.’ The words used to describe the relationships move from “the same as” to, ‘equals’, with the use of the equals symbol to accompany number names.

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3Big Idea: Parts to Whole

Parts-to-Whole is the big idea that underlies addition and subtraction. The central concept is that there is a whole that can be

partitioned into a certain number of parts. If we combine the parts, they equal the whole. If the whole is 8, the parts could be 6

and 2. Combine the two parts (6 + 2), they equal the whole (8). We can change the order of the parts (2 + 6) and they still equal

the whole. We can also find several different ways of making a whole (8) out of two parts, such as 7 + 1 or 3 + 5, or three parts,

such as 4 + 3 + 1.

A part can be taken away from the whole leaving another part left over. The whole is 8, we take away 5, 3 is left over. A student

that has developed in-depth understanding of the Parts-to-Whole big idea can see addition and subtraction as different ways of

forming number relationships, often called “fact families,” or, related facts.

Why is Parts to Whole Important?

Understanding how numbers are related to each other signals that children are ready to experience that each number is more

than a distinct character; larger numbers, or wholes, are made up of smaller numbers, or parts. When the student sees the

iconic 5 dots on a number card, combined with an additional 2 dots, she can count or add on and know there are 7 dots in total.

The 5-length bar with a 2-length bar added on takes on the length that is the same as the 7-bar. Two jumps on the number line

past the 5 mark, is the same as 2 numbers past 5, which in turn shows that 7 is two more than 5. Children begin by changing

a small collection of dots, or bars, or number line jumps, to a larger amount by virtue of more dots, longer bars, or end points

farther along the number line.

stage 3: addition & subtraction to 5


3.1:

Addition: Missing Result to 5

K.OA.1 1 + 2 = ?

3.1 introduces Parts-to-Whole first using visual tools, and then by bridging to symbols. Equations appear for the first time in the program, as well as an extension of the equal sign, first seen in Stage 2.5. Number equations show the meaning of the + symbol, and how the = symbol is used in mathematics. The placement of the = symbol is both at the beginning and the “end” of so that students become familiar and flexible with both appearances while combining parts to make a new whole.The missing result comes after combining parts, or appears before parts are combined. Two sides of an equation are equal when the = sign is used.

3.2:

Addition: Missing Change to 5

K.OA.2 1 + ? = 3

Still keeping the numbers small, students are given the whole, and need to find what parts combine with another to make that whole. In 3.2, they see how numbers can be broken apart in a variety of ways. They encounter the idea that order doesn’t matter. And they that the quantity remains the same even when parts are broken up differently. Students encounter the same progression of environments offered in 3.1 with the results being reinforcement of part-part-whole and more fluency with the pairings to 5.


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3.3: Subtraction: Missing Result to 5 K.OA.2 3 - 2 = ?

After working on missing parts in 3.2, subtraction is introduced in 3.3. Consider 5 - 2. The iconic 5-dotted number card is presented, and then 2 dots are taken away, how many would be left over from the original quantity of 5? A 5-length bar is presented; determine which bar would be left over if a 2-length bar were taken away from that 5-length bar.

3.4: Subtraction: Missing Change to 5

K.OA.3 3 - ? = 1

The student who understands Parts-to-Whole approaches this problem in a different way. With 2 + 3, the student can mentally picture 2 combined with 3 as being equal to 5. When considering the result of 5 - 2, she knows that since 2 + 3 = 5, 5 - 2 equals 3. She can identify missing parts by using related combinations.

3.5: Adding 0 and 1 1.OA.5 7 + 0 = ?

In Stage 1.5, students experienced ‘1 more’ and in 3.5, they are asked to return to this concept, and extend it to ‘adding 1’, or, ‘+ 1.’ Students are also increasing their number values to 9, as well as considering the role of zero. As in 1.5, adding 0 and 1 aim to be automatic-found without counting-and further reinforcements of the size of things and interconnectedness of numbers in our number system.

3.6: Subtracting 0 and 1 1.OA.5 7 - 1 = ? As in 3.5, without counting.

3.7: Addition - Commutative Property

K.OA.2 5 + 2 = 2 + ?Taking the next step from SubLevels 3.4 and 3.5, students now work with combinations 0-9. Experiences with partitioning in 3.2 lead them to experience that addition is commuta-tive.

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10

10001001s

10s100s

1000s

1s10s

100s1000s4 stage 4:

ten as a unit

Big Idea: Hierarchical Groupings

Hierarchical Groupings is the idea that amounts can be grouped into a system of sets and subsets. We can count 11 objects and

group them into 1 ten and 1 one, or we can call them 11 ones. One hundred seventy-nine represents 1 hundred, 7 tens and 9

ones. It is the concept that a group of things can also be one thing – one unit – at the same time.

In order to progress mathematically students need to be able to perceive the place value structure of a number – they need to

be able to understand what larger numbers mean. They need to understand that 273 is 2 groups of one hundred and 7 groups

of tens and 3 ones. Without understanding this structure, 273 is just the number that comes after 272 and before 274. While

this may be correct, it does not represent the sufficient understanding children need in order to use what they know for more

complex number manipulations.

There are other examples of hierarchical groupings, further illustrating why this is an important Big Idea. Time, for example, is

organized in terms of seconds, minutes, hours, days and so on. Sixty seconds equals one minute, 60 minutes equals one hour,

and 24 hours equals one day. Time is a particularly complex system for children to master because it involves the base ten

number system as well as the unique hierarchical grouping of the temporal system. Furthermore, the grouping amounts for each

step of the hierarchy may be different; 60 minutes equals 1 hour, but 60 hours does not equal 1 day.

Why are Hierarchical Groupings Important?

A student who does not have a strong understanding of Hierarchical Groupings may have difficulty answering the question,

“Which number is larger, 79 or 81?” He might think 79 is larger because the 7 and the 9 combined make 16, while the 8 and

the 1 combined only make 9. Or the 9 is larger than the 1, concluding that 79 is more than 81. Such conceptions indicate a lack

of understanding that the 7 in 79 equals 7 tens and the 8 in 81 is the same as 8 tens. The understanding of how these num-

bers are composed in the base-ten place-value system is foundational to success with multi-digit addition and subtraction where

students must compose larger numbers in addition and decompose them for subtraction.

In Stage 4, the student sees what parts combine to make 10, composes and decomposes 10 in multiple ways, and experi-

ences 11-19 as a bundle of 10 and some more. As in Stage 3 with combinations 0-5, Symphony Math provides a multitude of

examples for the student to master combinations to 10. Ten is a benchmark number, and the tasks in Stage 4 are scaffolded

to help a student use it as such when composing and decomposing numbers. Counting can become a hindrance when working

with large numbers; Stage 4 aims to help students rely on strategies with 10 when solving problems instead.

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4.1: Introducing 10 K.OA.4 8 + 2 = ? Students find the number that makes 10 when added to any given number 0-9.

4.2: Combinations of 10 K.OA.4 1 + ? = 3

Providing additional opportunities to learn how numbers combine to make 10, SubStage 4.2 combines missing addends, missing sums, and the commutative property with the goal being that students become fluent with these pairs; more automatic in recognizing the parts of 10.

4.3: Ten Plus K.NBT.1 10 + 4 = ?

Once a child recognizes that ten ones make one unit of 10, he is ready to add on to 10 and enter the teens. Without counting, he forms a bundle of ten, and sees the leftovers as number that is that many more than 10. He is able to compose numbers from 11-19 into ones and some further ones.

4.4: Subtracting with 10 1.OA.4 3 - ? = 1

Connect addition and subtraction relative to the number 10. Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added on to 8.

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5 stage 5: comparing numbers


5.1: Equals K.CC.7 8 = ?

Expanding on the introduction of the equal sign in 2.5, in which the student first sees and hears language like ‘the same as’ ‘as many as’ and then, ‘equals,’ in Stage 5 the student is asked to draw on material learned in Stages 1-3 involving part-whole, combinations to 10, the commutative property, and the actions of addition and subtraction, now with combina-tions to 20. Throughout Symphony Math, the equal sign appears in different positions to ensure that an equation with an unknown initial quantity is as familiar as the more com-mon equation with the result being the unknown. The answer to an equation does not have to be what follows, or comes after, an equal sign. The equal sign is a symbol that expresses the relationship, ‘is the same as.’ An understanding of the concept of equality is essential to successful algebraic thinking.

5.2: Greater Than K.CC.7 6 > ?

First with 1-digit numbers, then 2-digit numbers, students are scaffolded to the use of the shortcut symbol, >, for ‘greater than.’ As they move into 2-digit numbers, they use their knowledge of tens and ones for the comparisons. Being able to express which quantity is more than another also builds on the idea that larger numbers are made up of smaller numbers, a key idea in understanding number relationships.

5.3: Less Than K.CC.7 3 < ?

Finding ‘less’ when using pictorial tools is more apparent for the student than when com-paring numerals. As the student progresses to the sole use of numbers for the comparison ‘less than,’ she has shown her comfortability for the move out of the representations to the symbolic, by using the idea of the tens and ones hierarchy.

Big Idea: Quantity

Quantity is the big idea that describes amounts, or sizes. It is a fundamental idea that refers to quantitative properties; the size

of things (magnitude), and the number of things (multitude).

Why are Number Comparisons Important?

Quantity means that numbers represent amounts. If students do not possess an understanding of Quantity, their knowledge of

foundational mathematics will be undermined. Understanding Quantity helps students develop number conceptualization. In

order for children to understand quantity, they need foundational experiences with counting, identifying numbers, sequencing,

and comparing. Counting, comparing the magnitude of collections, and using numerals to quantify collections, formed the

developmental progression of experiences in Stages 1 and 2. Children who understand number concepts know that numbers

are used to describe quantities and relationships between quantities. An understanding of magnitude is vital to understanding

numbers.

In Stage 5, students take their growing understanding based on their experiences with numbers and number relationships, and

use mathematical symbols to compare magnitudes. They are asked to compare actual quantities in a variety of settings, and

use the <, >, and = symbols to do so.


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6Big Idea: Parts to Whole

Parts-to-Whole is the big idea that underlies addition and subtraction. The central concept is that there is a whole that can be

partitioned into a certain number of parts. If we combine the parts, they equal the whole. If the whole is 18, the parts could be 6

and 12. Combine the two parts (6 + 12), they equal the whole (18). We can change the order of the parts (12 + 6) and they still

equal the whole. We can also find several different ways of making a whole (18) out of two parts, such as 7 + 11 or 13 + 5, or

three parts, such as 4 + 13 + 1. Parts can be rearranged and maintain the whole.

A part can be taken away from the whole leaving another part left over. The whole is 17, we take away 9, 8 is left over. A student

that has developed in-depth understanding of the Parts-to-Whole big idea can see addition and subtraction as different ways

of forming number relationships, often called “fact families,” or, related facts. Stage 5 includes combinations to 20, which

necessitates adding on from ten. Ten, the focus of Stage 3, is an integral part of the Part-to-Whole hierarchy, and leads the way

to Place Value concepts.


Understanding how numbers are related to each other signals that children are ready to experience that each number is more

than a distinct character; larger numbers, or wholes, are made up of smaller numbers, or parts. When the student sees the

iconic 8 dots on a number card, combined with an additional 4 dots, she can count or add on and know there are 12 dots in

total. The 8-length bar with a 4-length bar added on takes on the length that is the same as the 10-bar and 2-bar. Four jumps on

the number line past the 8 mark is the same as 8 numbers past 4, which in turn shows that 12 is four more than 8.

In Stage 6, children use the unit of ten as they work with numbers in the teens. The representations of 10, in the different Sym-

phony Math environments, offer a dynamic visual that aids students’ understanding and automaticity with how parts combine

to make a whole. The more familiar students become with the unit of ten, the less they need to count or calculate when adding

and subtracting with ten. The connection between counting on and counting back, with addition and subtraction, is more appar-

ent, and more efficient.

stage 6: addition & subtraction to 20


6.1: Addition: Miss-ing Result to 20 1.OA.1 5 + 6 = ?

Students continue to build on combinations of 10 and the teens as they work with numbers to 20. Doubles play a role in this work, as does their foundation with smaller, known parts now called upon. As in Stage 3.1, students witness the equal sign in a variety of positions. They will also be asked to provide multiple solutions to problems, in order to extend their familiarity with the interrelatedness of numbers; part-part-whole.

6.2: Addition: Miss-ing Change to 20 1.OA.8 5 + ? = 11

Stage 6.2 continues a student’s development of his foundational numeracy through examples that call on his knowledge of missing parts. As he works through more and more tasks, he becomes able to identify a missing part when he knows the total and one part. Stage 6.2 further explores the principle that small numbers are parts of larger numbers, and that when numbers are broken apart and recombined in a variety of ways, the wholes remain the same.


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6.3: Subtraction: Missing Result to 20

1.OA.1 11 - 5 = ?

As children work on their number combinations, some will come more easily than others. Symphony Math is not striving for a memorization of number combinations, but rather, the inter-connectedness of Parts-to Wholes. In Stage 6.3, the program builds on two types of number combinations. One, knowing that 12 + 2 = 14 can lead to 14 - 2 = 12. And the other, that taking away one part leaves the other part: 14 - 2 = 12 and therefore, 14 - 12 = 2.

6.4: Subtraction: Missing Change to 20

1.OA.8 11 - ? = 6

The student who understands Parts-to-Whole approaches this problem in a different way. With 5 + 6, the student can mentally picture 5 combined with 6 as being equal to 11. When considering the result of 11 - 5, she knows that since 5 + 6 = 11, 11 - 5 equals 6. She can identify missing parts by using related combinations.

6.5: Fact Families 1.OA.3

7 + 4 = ?

4 + 7 = ?

11 - 7 = ?

11 - 4 = ?

Stage 5.5 reinforces how numbers relate to one another. Through a rich visual environ-ment, and then with numbers, children have ample opportunities to engage in the world of related facts so that such combinations become both clear and facile. When they see that 14 + 5 = 19 in a visual model, it becomes more apparent that any order in which 14 and 5 are combined will still yield 19. And when one of these particular parts of 19 are taken away, it will consistently yield the other known part of the pair. As with previous material, the equal sign will move around, so that students experience the variety of order in which equations may be written.

6.6: 3-Part Addition and Subtraction 1.OA.6 3 + 4 + 2 = ?

Students solve problems that call for addition of three whole numbers, and use strate-gies such as: counting on, making ten, decomposing a number leading to a ten; using the relationship between addition and subtraction, and creating equivalent but easier or known sums; (adding 8 + 9 by creating the known equivalent 8 + 8 + 1 = 16 + 1= 17). The properties of operations are applied as strategies to add and subtract.

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10

10001001s

10s100s

1000s

1s10s

100s1000s7 stage 7:

tensBig Idea: Hierarchical Groupings

Hierarchical Groupings is the idea that amounts can be grouped into a system of sets and subsets. We count 11 objects and

group them into 1 ten and 1 one, or we can call them 11 ones. Seventy five represents 7 tens and 5 ones.

In order to progress mathematically students need to be able to perceive the place value structures of a number. They need to

understand what larger numbers mean. Up to Stage 7, students are immersed in numbers to 20; counting, comparing, combin-

ing, composing, and decomposing with multiple solution strategies. Tens are shown as a unit of 10 ones, and then combined

with some additional ones to make numbers in the teens. Quantitative questions, involving combining and taking away are

related to addition and subtraction, and ten is presented as a strategic amount to use when solving problems.

In Stage 7, Tens are the counters. Using a similar progression as previously, Stage 7 highlights Ten as an important anchor in the

number system. Place value concepts begin to take shape visually and meaningfully.


Children need experiences with groups of ten as a foundation for the base ten place value system. In Stage 4, they are intro-

duced to ten as a unit. In Stage 7, ten is the basis for the 2-digit numbers they will encounter, the numbers made up of varying

groups of ten. A group of things is also a single thing: twenty such things are also two groups of ten. Stage 7 allows children to

stay in this introductory environment of tens for as long as they need. When they have shown their fluency in this concept, they

progress to place value.

Understanding of the tens place value is developed by challenging students to solve problems where they have to compose and

decompose numbers such as 10, 20, and 30.



7.1: Identifying 10s 1.NBT.2 Create ‘80’Students were introduced to ten in Stage 4. In Stage 7.1 they show their recognition of how varying bundles of ten combine to make multiples of ten up to ninety. They explore the idea that numbers such as 10, 20, 30, etc., are groups of tens with zero ones.

7.2: Making 10s 1.NBT.2 20, 30, 40, ?

Building on number sense, students work on the order of the decade numbers when counting by tens. The sequence is presented first beginning with 10, and then from any number no higher than 90 when students demonstrate they do not have to begin with 10. Decade sequences have one value missing, and then two values missing. In addition, the placement of the missing positions shift from the last place missing to the first place missing to any place missing as they complete sequences.

7.3: Counting Forward and Backward by 10s

1.NBT.2 70, 60, 50, ?Using the same order of task presentations as 6.2, students are asked to count the decades backwards by 10s, with numbers 10-90.

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7.4: Find ‘10 more’ 1.NBT.5 Make ‘10 more’ than 50

Relative magnitude is explored as students determine, and then know without counting, numbers ‘ten more’ than any decade number 0-80.

7.5: Find ‘10 less’ 1.NBT.5 Make ‘10 less’ than 40

Relative magnitude is explored as students determine, and then know without counting, numbers ‘ten less’ than any decade number 10-90.

7.6: Related 1s and 10s combinations 2.OA.2

2 + 5 = ?

20 + 50 = ?

While not all students have mastered their combinations to 10, they work on larger combinations even so. In 7.6, their familiarity and variety of experiences with these smaller combinations come to bear as they consider what happens when adding related pairs that are 10 times larger. Students are expected to use mental strategies when adding and subtracting, one of which is creating easier or known sums to solve a problem with larger numbers. 7.6 allows the students to see how related smaller parts that create a whole come into play when they are ten times larger and therefore create a whole that is ten times larger. Such proportional parts and wholes will come in to play as students work with larger numbers and more advanced mathematics.

7.7: Combinations of 100 1.NBT.4 100 = 30 + ?

Since students are expected to combine hundreds, tens, and ones as a computation strategy, knowing combinations of ten and one hundred automatically serve them to this end. Providing additional opportunities to learn how numbers combine to make 100, Stage 7.7 combines missing addends, missing sums, and the commutative property with the goal being that students become fluent with these pairs; more automatic in recognizing the parts of 100. Gaining automaticity with parts of 100 reinforces an efficient calculation strategy.

7.8: Adding Teens and Ones 1.OA.6 13 + 6 = ?

Composing numbers using ten as a benchmark number shows more flexibility than a dependence on counting.

One sort of composition is that of relying on known combinations of parts, 0-10, even when confronted with teens and some more. A student who knows 4 + 3 = 7, shown 14 + 3, can reference his knowledge of the parts and wholes to see the total is 10 + the known combination of 4 + 3, or 7. 14 + 3 = 10 + 7.

7.9: Comparing 10s 1.NBT.2 30 ? 50

Comparing two digit decade numbers shows an understanding of quantities of ten. In order to successfully say which of two numbers is larger, smaller, or same, children examine the amounts of tens and use the greater than (>) and less than (<) and equal sign appropri-ately.

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stage 8: addition & subtraction with 10s

Stage 8: Page 1 of 2

Big Idea: Hierarchical Groupings with Parts-to-Whole

As its name implies, Hierarchical Groupings with Parts-to-Whole is a complex idea -- one that involves the coordination of two

earlier big ideas. It also illustrates the hierarchal nature of mathematics and how a poor foundation is likely to interfere with

the learning of later concepts. For this big idea students must coordinate their knowledge of Hierarchal Groupings and Parts-to-

Whole. Hierarchical Groupings is the idea that amounts can be grouped into a system of sets and subsets. These sets can be

combined to form a whole, an amount that is equal to the sum of its parts.

Why is this So Important?

Stages 1-7 in Symphony Math emphasize combinations to 20, and then the underlying structure of numbers to 100. Before

students move on to numbers beyond 100, they must develop an understanding of quantity, the composition of numbers and

their inter-relatedness, and the organization of tens and ones. Such underpinnings are vital to a student’s ability to flexibly and

fluently apply organization and meaning to future work and operations with larger numbers and fraction and decimal numbers.

In Stage 8, Symphony Math extends the idea of ten as a unit to multiple groups of ten. Ten is a benchmark number, and com-

binations with multiples of ten are scaffolded in such way to elucidate how ‘tens and some more’ form the numbers 10-100.

Stage 8 extends the behavior of ten as a benchmark number in order to compose and decompose numbers 10-100. Making

tens is a key strategy children are encouraged to use; and decade numbers behave much as one ten does-make as many

bundles of ten as possible, and then add on the leftovers.



8.1: Place Value Addition: Missing Result

1.OA.8 20 + 3 = ?

Symphony Math models provide the visual manipulatives in which groups of ten are distinguishable from groups of ones. When a student shows she is ready not to have to count all the ones, combining tens and some ones becomes automatic; looking at a visual representation of 27, she is able to count 2 tens, and know the 2 references 2 bundles of ten or twenty. The leftover ones appear physically different than the representation of tens. Thus the total becomes increasingly automatic.

8.2: Place Value Addition: Missing Change

1.NBT.4 20 + ? = 23

Given the whole, the student uses what she’s learned regarding the relationship between addition and subtraction, making groups of ten, and hierarchical groupings in order to find the missing part. She can complete the part that makes the equation between the three whole numbers true, thus showing that she sees the relationship of the three numbers to each other.

8.3: Place Value Subtraction: Missing Result

1.OA.8 23 - 3 = ?

As her experiences with combinations of parts and wholes continues, as well as her grasp on the structure of ten and groups of ten, the student can work on 8.3 with good under-standing. She is asked to think of subtracting ones from a given group of tens combined with that same number of ones ( 47 - 7). Symphony Math’s visual tools model how groups of tens and some ones leave those same groups of tens, once the ones are taken away.(47 - 7 = 40)

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8.4: Place Value Subtraction: Missing Change

2.OA.1 23 - ? = 20

Given the whole, the student uses what she’s learned regarding the relationship between addition and subtraction, making groups of ten, and hierarchical groupings in order to find the missing part. She can complete the part that makes the equation between the three whole numbers true, thus showing that she sees the relationship of the three numbers to each other.

8.5: Parts-to-Whole with 1s and 10s 2.NBT.5 56 = ? + ? + 6

Stage 8.5 challenges the student to decompose a 2-digit number into varying parts of tens and ones, in a multitude of ways. Based on place value, properties of operations, and the relationship between addition and subtraction, she has the chance to put her knowledge of number composition to the test. Given a number 73, she may need to find 2 missing addends + 3 that combine to make the whole; or 2 additional missing addends + 30 to compose that same whole.

8.6: Parts-to-Whole with 1s and 10s (+10)

1.NBT.5 69 + 10 = ?

Given any 2-digit number, students are asked to show they can add 1 ten more. Stage 7.6 highlights how ten is not an ordinary number; they are not adding 1 more, but an additional 1 group of ten. The significant properties of ten make a difference to the structure of the original number. The visual tools in Symphony Math make the results of this action apparent, thus omitting the need to count to find the answer. Adding ten becomes a mental strategy.

8.7: Parts-to-Whole with 1s and 10s (-10)

1.NBT.5 56 - 10 = ?

Given any 2-digit number, students are asked to show they can take away 1 ten. As in Stage 8.6, ten is highlighted as a special number, one that behaves differently than subtracting 1. With the initial aid of visual manipulatives, students can subtract 10 and see the results without counting. ‘Minus 10’ is a developmentally sophisticated strategy students use to subtract. Through the examples in Symphony Math, subtracting ten becomes an action students can do mentally.

8.8: Adding With Multiples of 10-No Regrouping

1.NBT.4 61 = ? + 21

Students work on their automaticity with composing with multiples of tens. They use number relationships learned in earlier stages so that such combinations become fluent. Stage 8.8 furthers previous work with the concept of adding tens and tens and ones and ones. Eventually adding multiples 10 becomes a key mental strategy.

8.9: Comparing 2-Digit Numbers 1.NBT.2 67 > ?

Students compare two two-digit numbers based on meanings of the tens and ones digits, and show the results using symbols >, =, <. They use their understanding of the hierarchi-cal order of groups, and the order of digits to compare two sets of numbers based on each’s amounts of tens and ones.

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10

10001001s

10s100s

1000s

1s10s

100s1000s9 stage 9:

hundreds

Big Idea: Hierarchical Groupings


group them into 1 ten and 1 one, or we can call them 11 ones. Seventy five represents 7 tens and 5 ones.

Stages 1-8 in Symphony Math emphasize combinations to 20, and then the underlying structure of numbers to 100. Before

students move on to numbers 100 and beyond, they are developing an understanding of the composition of numbers, their

inter-relatedness, and the hierarchical organization of tens and ones. Such underpinnings are vital to a student’s ability to

flexibly and fluently apply organization and meaning to future work and operations with larger numbers.

After successfully developing an understanding of tens, students progress to the hundreds. Stage 8 follows the similar

succession of tasks as previous levels do, when 1s and 10s are introduced. With a parallel pedagogical structure, Symphony

Math aims for students to think of one hundred in much the same way they think of ten; with one hundred being a natural

extension of ten.


The idea that a group is also a single unit is a complex one. Progressing from learning to count one number for one object, then,

that ten objects are called one thing, and now, that ten things, or even one hundred things, are also a singular unit, is a complex

one. Because Symphony Math ensures that students actively participate in making meaning for quantities up to 100, their first

encounter with 100 is built on a foundation for which it makes sense and is solid.

Stage 9 is the introduction to the 3-digit number. Children see how a number like 500 may be 5 hundreds, 50 tens, and 500

ones all at the same time. The Symphony Math visual environments allow them to experience how 3-digit numbers are organized

and thus lay further foundational support for future manipulations with these quantities.



79.1: Identifying 100s 2.NBT.1 Create ‘500’

In Stage 7 student show their recognition of how varying bundles of ten combine to make multiples of ten up to ninety. In 9.1, students do the same with bundles of 100 up to 900. They experience that three digits create a number in the hundreds, after they see that 10 bundles of ten, or 100 ones create one hundred. They explore the idea that numbers such as 100, 200, 300, etc., are groups of hundreds with zero tens and zero ones.

9.2: Making 100s 2.NBT.2 200, 300, ?

Building on number sense, students work on the order of numbers in the hundreds when counting by one hundred. The sequence is presented first beginning with 100, and then from any number no higher than 900 when students demonstrate they do not have to begin with 100. Sequences of the hundreds have one value missing, and then two values missing. The idea is to demonstrate their knowledge that counting by 100 means that values increase by the quantity of 100.

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9.3: Counting Forward and Backward by 100s

2.NBT.2 70, 60, 50, ?Using the same order of task presentations as 9.2, students are asked to count the hundreds backwards by 100s, with numbers 100-900.

9.4: Find ‘100 more’ 2.NBT.1

Make ‘100 more’ than 500

Relative magnitude is explored as students determine, and then know without counting, numbers ‘one hundred more’ than any given multiple of hundreds 100-900.

9.5: Find ‘100 less’ 2.NBT.2 Make ‘100 less’ than 400

Relative magnitude is explored as students determine, and then know without counting, numbers ‘one hundred less’ than any given multiple of hundreds 100-900.

9.6: Related 1s, 10s, and 100s combinations

2.MD.6

2 + 5 = ?

20 + 50 = ?

200 + 500 = ?

While not all students have mastered their combinations to 10, they work on larger combinations even so. In 7.6, their familiarity and variety of experiences with these smaller combinations come to bear as they consider what happens when adding related pairs that are 10 times larger. Students are expected to use mental strategies when adding and subtracting, one of which is creating easier or known sums to solve a problem with larger numbers. 7.6 allows the students to see how related smaller parts that create a whole come into play when they are ten times larger and therefore create a whole that is ten times larger. Such proportional parts and wholes will come in to play as students work with larger numbers and more advanced mathematics.

9.7: Adding 2-Digit and 1-Digit Numbers

2.MD.6 730 + 50 = ?

The combinations to 20 are called on as students see the behavior of a 2-digit number added to a 1-digit number. One he knows 5 + 3 = 8, he can use this part-whole pairing to quickly calculate 15 + 3, 65 + 3 or 95 + 3. Stage 9.7 shows him that the tens remain a constant grouping, and the ones combine in the same way each time. After many experi-ences, beginning with the visual representation of this action, the part-part-whole may be automatic.

9.8: Comparing 100s 2.NBT.1 300 ? 500

Comparing two multiples of 100 shows an understanding of quantities of one hundred. In order to successfully say which of two numbers is larger, smaller, or same, children exam-ine the amounts of hundreds and use the greater than (>) and less than (<) and equal sign appropriately. 9.8 requires children to identify the number of hundreds in 3-digit multiples of 100. They explore the idea that numbers such as 100, 200, 300, etc., are groups of hundreds with zero tens and zero ones.

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stage 10: addition & subtraction with 100s


As its name implies, Hierarchical Groupings with Parts-to-Whole is a complex idea -- one that involves the coordination of two

earlier big ideas. It also illustrates the hierarchal nature of mathematics and how a poor foundation is likely to interfere with

the learning of later concepts. For this big idea students must coordinate their knowledge of Hierarchal Groupings and Parts-to-

Whole. Hierarchical Groupings is the idea that amounts can be grouped into a system of sets and subsets. These sets can be


Why is this So Important?

The Part-to-Whole combinations in Stage 10 highlight the structure of numbers, which students will be able to use when

performing more complex operations with larger numbers. In Stage 10, Symphony Math extends the idea of one hundred as a

unit to multiple groups of one hundred making many such units. One hundred is a benchmark number, and combinations with

multiples of one hundred are scaffolded in such way to elucidate how ‘one hundred and some tens more’ form the numbers

100-1000. Stage 9 extends the behavior of one hundred as a benchmark number in order to compose and decompose numbers

100-1000. Making hundreds is a key strategy children are encouraged to use; and multiples of 100 behave much as one

hundred does-make as many bundles of one hundred as possible, and then add on the leftovers. Students add and subtract

with multiples of 100, and experience how quantities change, without needing to recombine using place value concepts of

exchange.



10.1: Place Value Addition: Missing Result

2.NBT.7 200 + 30 = ?

Symphony Math models provide the visual manipulatives in which groups of one hundred are distinguishable from groups of tens. When a student shows she is ready not to have to count all, combining hundreds and some tens becomes automatic; looking at a visual representation of 270, she is able to count 2 hundreds, and know the ‘2’ references 2 bundles of one hundred or two hundred. The leftover tens appear physically different than the representation of hundreds. Thus the total becomes increasingly automatic.

10.2: Place Value Addition: Missing Change

2.MD.6 200 + ? = 230

Given the whole, the student uses what she’s learned regarding the relationship between addition and subtraction, hierarchical grouping, and making groups of hundreds and tens, in order to find the missing part. She is called on to demonstrate that she can identify how tens and tens combine, separately from hundreds and hundreds. She can complete the part that makes the equation between the three whole numbers true, thus showing that she sees the relationship of the three numbers to each other.

10.3: Place Value Subtraction: Missing Result

2.NBT.7 230 - 30 = ?

As her experiences with combinations of parts and wholes continues, as well as her grasp on the structure of one hundred and groups of ten, the student can work on 9.3 with good understanding. She is asked to think of subtracting tens from a given group of hundreds combined with that same number of tens ( 470 - 70). Symphony Math’s visual tools model how groups of hundreds and some tens leave those same groups of hundreds, once the tens are taken away ( 470 - 70 = 400).

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10.4: Place Value Subtraction: Missing Change

2.MD.6 230 - ? = 200

Given the whole, the student uses what he’s learned regarding the relationship between addition and subtraction, making groups of ten, and hierarchical groupings in order to find the missing part. He can complete the part that makes the equation between the three whole numbers true, showing that he sees the relationship of the three numbers to each other.

10.5: Parts-to-Whole with 1s, 10s, and 100s

2.NBT.5 563 = 200 + ? + ?

Stage 10.5 challenges the student to decompose a 3-digit number into varying parts of hundreds, tens and ones, in a multitude of ways. Based on place value, properties of operations, and the relationship between addition and subtraction, he has the chance to put his knowledge of number composition to the test. Given a number 743, he may need to find 2 missing addends + 3 that combine to make the whole; or 2 additional missing ad-dends + 30 to compose that same whole. Thus the student is challenged to demonstrate that numbers are more than place holders; they are instead able to be reorganized in more than one way while still maintaining their same value.

10.6: Parts-to-Whole with 1s, 10s, 100s (+100)

2.NBT.8 690 + 100 = ?

Given any 3-digit number, students are asked to show they can add 1 hundred more. Stage 10.6 highlights how one hundred is not an ordinary number; they are not adding 1 more, but an additional 1 group of 1 hundred. The significant properties of 100 make a differ-ence to the structure of the original number. The visual tools in Symphony Math make the results of this action apparent, thus omitting the need to count to find the answer. Adding 100 becomes a mental strategy.

10.6: Parts-to-Whole with 1s, 10s, 100s (-100)

2.NBT.8 560 - 100 = ?

Given any 2-digit number, students are asked to show they can take away 100. As in Stage 10.6, 100 is highlighted as a special number, one that behaves differently than subtract-ing 1. With the initial aid of visual manipulatives, students can subtract 100 and see the results without counting. ‘Minus 100’ is a developmentally sophisticated strategy students use to subtract. Through the examples in Symphony Math, subtracting 100 becomes an action students can do mentally.

10.8: Comparing 3-Digit Numbers 2.NBT.4 ? < 463

Students compare two 3-digit numbers based on meanings of hundreds, tens and ones digits, and show the results using symbols >, =, <. They use their understanding of the hierarchical order of groups, and the order of digits to compare two sets of numbers based on each’s amounts of hundreds, tens and ones.

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stage 11: foundations for multiplication

Big Idea: Repeated Equal Groupings

The Repeated Equal Groupings big idea builds upon the Parts-to-Whole idea. With Repeated Equal Groupings, the whole is

not only broken into parts but broken into a specific number of parts and each part is of equal size. For example, there are 30

students in the class (the whole). The teacher divides the class into groups of 6 (the equal groups). These equal groups are

repeated 5 times to equal the whole.

Repeated equal groupings is the big idea that underlies multiplication and division. Multiplication consists of taking a part (the

multiplicand) and repeating it a certain number of times (the multiplier) to equal the whole (the product). Division consists of a

whole (the dividend) that is partitioned into a certain number of equal groups (the divisor) that is equal to the size of the parts

(the quotient).

Why are Repeated Equal Groupings Important?

It is possible for students to memorize multiplication facts without understanding what they mean. Students can also use skip

counting to correctly solve multiplication problems without appreciating the significance of how they arrived at the correct

answer. In addition to being able to immediately recall multiplication and division facts, students also need to be able to

understand what these operations mean.

Students who understand equal groupings have a better chance of memorizing multiplication and division number relationships

because they have a conceptual basis to support their learning. For example, they are more likely to see the connections

between, 7 x 4, 4 x 7, 28 ÷ 7, and 28 ÷ 4 . Equal groupings is also fundamental to fractions. Students who are only relying on

memorized facts and skip counting strategies may have difficulty understanding the meaning of fractions and how they build

upon multiplication and division.

Stages 11 and 13 develop an understanding of grouping and partitioning by building on the Parts-to-Whole concepts established

in Stages 3, 6, 8, and 10. They reinforce the concept of repeated groupings - that multiplication represents repeated addition

and division represents repeated subtraction. Previous Symphony Math Stages help students develop their conceptual

understanding of what these operations mean, and then help students learn the number relationships through systematic

practice and evaluation.

Stages 11 and 13 address number relationships with products and dividends up to thirty. This approach uses the repeated

addition model of multiplication and the repeated subtraction model of division. In turn, the models help students understand

the connection between addition and multiplication and between subtraction and division.

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11.1: Skip Count-ing 2.NBT.2 2, 4, 6, ?, ?

Students count on from any number and say the next few numbers that come afterwards, counting by 2s, 5s, 10s, 20s, 50s, and 100s. Skip counting is introduced to help students work towards multiplication concepts. It is not yet true multiplication because they don’t keep track of the number of groups counted, yet, they can experience that when counting by 2s, 5s, 10s, 20s and so on, they are counting by groups with that amount in each group.

11.2: Adding 2s, 3s, and 4s 2.OA.4 2 + 2 + 2 = ? Using the array formation, students find the sum of equal addends up to 5 addends.

11.3: Equal Groups 2.OA.4 ‘Create 4 groups of 2’

Focusing on story problems, students show that they can model quantities presented in a given story. They use Symphony Math tools to show that they understand groups versus the numbers per group. No calculations are necessary since the goal is to show an under-standing and interpretation of problem quantities. In time, learning how to model story problem quantities with a visual model will help students solve these problems.


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12Big Idea: Hierarchical Groupings with Parts-to-Whole

Hierarchical Groupings with Parts to Whole is a complex idea-one that involves the coordination of two earlier big ideas. It

also illustrates the hierarchal nature of mathematics and how a poor foundation is likely to interfere with the learning of later

concepts. For this big idea students must coordinate their knowledge of Hierarchal Groupings and Parts-to-Whole. Let’s look at

the following problem: 268 + 453 = ?

Two hundred sixty eight is one part. The other part is 453. Combined they form the whole. Conceptually the student needs to

understand she is looking for a missing whole. Procedurally, she must understand that when she combines the 8 and the 3 this

yields a 10 and a 1. The ten will be combined with the other tens (6 and 5). Or, she will make the most amount of hundreds

possible by combining the 200 and the 400, and seeing if 60 and 50 together can yield an additional group of 100. Having es-

tablished that another 100 is possible, with 10 left over, she then looks to see if the 8 and the 3 make a ten, which they do. She

combines the two tens, the most that is possible, and sees what is leftover. And so on. She can see that her method of parts

composition, 700 + 20 + 1 will sum to a whole of 721. In these ways, the student is working with both the Hierarchical Group-

ings idea and the Part-to-Whole idea. Not only is solving multi-digit addition and subtraction problems procedurally complex, but

it is also conceptually complex relative to the preceding concepts in the curriculum. It is one of the first periods in mathemati-

cal learning where many students begin to apply complex procedures they do not fully understand.


The student who has not developed an in-depth understanding of these big ideas is likely to rely only on the step-by-step

application of procedures without the supporting understanding of what these procedures mean and why they work. When he

produces a nonsensical answer, the mistake is not apparent because he has stopped looking at the numbers as representing

meaningful quantities. He is simply manipulating symbols as best he can according to that procedure.

With the groundwork established especially in Stages 7 - 10, students understand the structure of tens and hundreds, and

thus can apply their knowledge in a way that makes sense when solving problems.

stage 12: regrouping with 2- and 3-digits


12.1: Regrouping with 1- and 2-digits 2.OA.1 19 + 5 = 20 + ?

Students complete equations with unknowns in different positions, and must solve by composing tens first. For example, 13 + 9 = 20 + ?. Combining tens first is recommended as a preferred place value strategy, as is decomposing into tens.

12.2: Regrouping to 100 2.NBT.5 60 + ? = 47 + 19

Students complete equations by regrouping, using a place value strategy of combining all the tens first. With the unknown moving from position to position, the progression of challenges follow 12.1.

12.3: Regrouping with 2-digits: Subtraction

2.NBT.5 33 - 5 = 20 + ? Students complete subtraction equations, presented as in 12.2, in which they first need to attend to the resulting tens.


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12.4: Regrouping with 2-Digit Numbers: Missing Change

2.NBT.5 39 + ? = 60 + 8

The unknown in 12.4 is the missing change, for example:

75 + ? = 90 + 7

Students are challenged to understand how the tens and some ones are composed in the equation, given an initial amount with which to work. They may decompose into tens, or think addition, for examples of solution strategies.

12.5: Subtracting 2-Digit Numbers: Missing Change

2.NBT.5 50 + 7 = 71 - ?In the challenge of 12.5, students may combine one side of the equality in order to determine the relationship of addition and subtraction quantities, or, solve by decomposing tens first.

12.6: Regrouping to 1000: Addition 2.NBT.7

247 + 386 =

600 + ? + 3

Students extend the work from 12.1-12.5 to two 3-digit numbers. As always in the Sym-phony Math progression, students have ample experiences using concrete materials and pictorial representations to support their work.

12.6 also references composing and decomposing tens, making a 100, breaking apart a 10, or creating an easier problem.

12.7: Regrouping to 1000: Subtrac-tion

2.NBT.7712 - 30 =

600 + ? + 2

Students are challenged in a few ways in Stage 12.7. They are using their understanding of the relationship between addition and subtraction to construct both a quantity and an equality. They are demonstrating their knowledge of hundreds, tens and ones and how such parts combine to make a whole. And, they must analyze the result of number decomposition in order to successfully compute the answer in an unorthodox-looking way.

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stage 13: multiplication & division









(the quotient).








between, 7 x 4, 4 x 7, 28 ÷ 7, and 28 ÷ 4 . Stages 11 and 13 develop an understanding of grouping and partitioning by building

on the Parts-to-Whole concepts established in Stages 3, 6, 8, and 10. They reinforce the concept of repeated groupings - that

multiplication represents repeated addition and division represents repeated subtraction. Previous Symphony Math Stages help

students develop their conceptual understanding of what these operations mean, and then help students learn the number

relationships through systematic practice and evaluation.

Stage 13 builds on the Repeated Equal Groupings big idea from Stage 11, expanding from additive thinking to multiplicative

thinking. It is important that students understand the difference between the two. Together with the repeated equal groupings

of a quantity, multiplicative thinking includes that the relationship between the two numbers involved is constant. Multiplicative

thinking is the important groundwork on which an understanding of place value for whole numbers and decimal numbers,

multiplication and division, fractions, operations with decimals and fractions, percentages, and ratio and proportion is built.

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13.1: Multiplication: Unknown Product

3.OA.1 4 x 2 = ?

Unknown Product is the first problem structure in Stage 13. Students interpret 3 x 7 as the total number of objects in 3 groups of 7 objects each, for example. They learn to recognize multiplication as a way to determine the total number of objects when there are a specific number of groups with the same number of objects in each group.

13.2: Multiplication: Unknown Number of Groups

3.OA.3 ? x 2 = 8

The problem structures of 13.2 and 13.3 are progressively more challenging than 13.1. It is important for students to internalize the difference between quotients (number of groups) and factors (size of each group). While 3 x 4 and 4 x 3 have the same product, the number sentences have different meanings that are important for students to understand, especially when they begin to apply multiplication in real-world situations.

13.3: Multiplication: Unknown Group Size

3.OA.4 4 x ? = 8 Where 13.2 focuses on a missing number of equal groupings, 13.3 emphasizes the unknown size of groups, or the multiplicand.

13.4: Division: Missing Result 3.OA.4 8 ÷ 2 = ?

In their introduction to division, it is important that students see a connection between multiplication and division. Partitioning models focus on how many objects are in each group when the groups are equal. As students work on problems in 13.4, they begin to see how the ‘fact family’ relationships learned in work with multiplication are applicable to division situations in an inverse way.

13.5: Missing Dividend 3.OA.6 8 ÷ ? = 4

13.5 is an opportunity for students to use their knowledge of multiplication and its connection to division. Missing dividend problems are another way of asking them to multiply the two given factors to find a product.

13.6: Missing Divisor 3.OA.6 ? ÷ 2 = 4 Students determine the unknown by relating the 3 numbers in the same ‘fact family’ to

each other to make the visual representations and then equations true.

13.7: Multiplication and the Commutative Property

3.OA.5 4 x 6 = 6 x ?

Students were first introduced to the commutative property of addition in Stage 3. The same rule applies to multiplication. As students work on tasks that show 2 x 3 = 6 and 3 x 2 = 6, they begin to see the constancy in the quantities and subsequently, the relationship these 3 numbers have to each other.

13.8: Multiplication and the Distributive Property

3.OA.5 2x3 + 5x3 = ?Students break up numbers into smaller numbers. The distributive property can be a way for students to both find an easier way to compute, as well as show their understanding of how smaller parts behave when multiplying.


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stage 14:introduction to fractions

Big Idea: Repeated Equal Groupings with Parts-to-Whole

The big idea of Repeated Equal Groupings with Parts-to-Whole is the most complex of the big ideas addressed so far. Similar to

Repeated Equal Groupings this big idea involves repeating an equal sized group, or partitioning an amount into equal groups.

It also involves a trio of parts to the whole. Not only do students have to keep track of the whole and its equal groups as with

multiplication and division, but they also have to be mindful of the number of parts relative to the whole.

For example, with multiplication and division we need to work with the total, the size of the parts and the number of parts. There

are 30 students in the class (the whole equals 30), the teacher divides the class into 5 groups (the number of parts equals 5),

there are six students in each group (the size of the parts equals 6). This leads us to 30 divided by 5 equals 6.

Let’s take the same Equal Groupings sample and add the Parts-to-Whole component. What if the teacher said that 1/5 of the

class had siblings in lower grades: how can we determine how many students have siblings in lower grades? The fraction of

one-fifth is a parts-to-whole representation. It means one part out of five parts. To determine how many students is equal to 1/5

of 30 we can first use Equal Groupings. Five equal groups of 30 means that there are 6 students in each group. Therefore, 1/5

of 30 equals 6.

It is for this reason that an in-depth understanding of Repeated Equal Groupings with Parts-to-Whole is fundamental to under-

standing fractions. Students need to understand that the big idea of Parts- to -Whole is fundamental to understanding fractions.

Students need to understand the big idea of Parts-to-whole developed with addition and subtraction as well as the big ideas

of Repeated Equal Groupings developed with multiplication and division. These two big ideas coordinated together give us

Repeated Equal Groupings with Parts-to-Whole, the foundational idea for fractions. This perspective also helps us understand

why fractions can be so difficult for students. Not only do they need to have mastery of the proceeding big ideas, but they need

to coordinate them together.


When students are confronted with the complexity of fractions and do not have sufficient mastery of the necessary big ideas,

they may resort to memorized strategies that lead to correct answers. They learn rules like “find the common denominator, then

add.” Or, “flip it and multiply.” They might remind themselves that “the bigger the denominator the smaller the fraction.” While

these memorized strategies may lead to correct answers, on their own they do not lead to deep understanding of fractions.

Symphony Math launches students into fraction concepts by insuring they construct knowledge based on ideas of magnitude

and numeracy, rather than on procedural steps.

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14.1: Making a Whole 3.NF.1 Make ‘Thirds’.

Symphony Math introduces students to fractions with the idea that any whole can be divided into any number of equal parts, and that all these parts together make up that 1 whole. In visual form, the big idea of Equal Groupings with Parts-to-Whole are displayed with a new tool, the fraction bar.

14.2: Unit Fractions of 1 3.NF.1 Create 1/3.

Students are introduced to unit fractions using generally familiar fractions of 1/2, 1/3 and 1/4. Using the fraction bar, which sits on top of a number line calibrated from 0 to 1, students partition into these familiar fractional parts. The whole resembles a rectangle, and students color in one part after partitioning into the designated equal parts. They then see one such part written in fraction form with a 1 in the numerator.

14.3: Non-Unit Fractions of 1 3.NF.2 1/4 + 1/4 = ?

Using the Symphony Math visual environment, students watch how they partition for example, 1/4 and 1/4 and end up with 2/4. They use unit fractions to create non-unit fractions, using the part-part-whole model explicitly. In this way, students see how repeated equal groupings with parts-to-whole unveil before their eyes. They witness copies of unit fractions combining to form non-unit fractions.

14.4: Whole Numbers as Fractions

3.NF.3 4 = 4/?

This concept relates to fractions as division problems, where the fraction 3/1 is 3 wholes divided into one group. The visual environment of Symphony Math helps students under-stand the concept of a/1: how any number can be divided into 1 group: and thus still be that same number. It sets the stage for work where a set of any number of objects can be divided into any number of groups.

14.5: Comparing Fractions 3.NF.3 2/5 ? 3/5

As stated above, while students have learned that 8 > 4, now 1/8 < 1/4. Putting fractions with either the same numerator or the same denominator within the same fraction bar /number line model helps students visualize the size of the parts. Thus they can make comparisons. Such comparisons build on the crucial idea of magnitude and numeracy with fraction numbers.

14.6: Equivalent Fractions 3.NF.3 1/2 = ?/4

Students’ work with equivalent fractions throughout 14.6 is supported with visual fraction models. In this way, the important concept of equivalent fractions is established with a strong foundation. As students go on to work with generating equivalent fractions and decimal equivalencies they have strong visual experiences.


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stage 15: multiply & divide to 100









(the quotient).








between, 7 x 4, 4 x 7, 28 ÷ 7, and 28 ÷ 4 . Stages 11 and 13 develop an understanding of grouping and partitioning by building

on the Parts-to-Whole concepts established in Stages 3, 6, 8, and 10. They reinforce the concept of repeated groupings-that

multiplication represents repeated addition and division represents repeated subtraction. Previous Symphony Math Stages help

students develop their conceptual understanding of what these operations mean, and then help students learn the number

relationships through systematic practice and evaluation.

In addition to further practice with multiplication and division to 100, Symphony Math goes beyond the rote ‘memorize-tables’

approach. Number relationships are key in order that students observe and experience how numbers behave in relation with

each other. When students work through a problem like 5 x 12 = 60, they will see that 12 x 5 = 60, and that 5 x 10 + 5 x 2 = 60

as well as that 60 ÷ 5 = 12 and so on.

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15.1: Multiplication to 100 3.OA.3 7 x 9 = ?

Unknown Product is the first problem structure in Stage 15. Students interpret 3 x 7 as the total number of objects in 3 groups of 7 objects each, for example. They learn to recognize multiplication as a way to determine the total number of objects when there are a specific number of groups with the same number of objects in each group.

15.2: Multiplication: Unknown Factor / Group Size to 100

3.OA.3 ? x 9 = 63

The problem structures of 15.2 and 15.3 are progressively more challenging than that of finding an unknown product. Because students have experienced many instances of the equal sign up to this point, they may think, on seeing ? x 4 = 12, that some number of groups of 4 is the same as 15.

15.3: Multiplication: Unknown Number of Groups to 100

3.OA.4 7 x ? = 63 Where 15.2 focuses on a missing size of the equal groupings, 15.3 emphasizes an unknown number of groups, or the factor.

15.4: Division: Missing Result to 100

3.OA.4 63 ÷ 9 = ?

In their introduction to division, it is important that students see a connection between multiplication and division. Partitioning models focus on how many objects are in each group when the groups are equal. As students work on problems in 15.4, they begin to see how the ‘fact family’ relationships learned in work with multiplication are applicable to division situations in an inverse way.

15.5: Missing Dividend to 100 3.OA.6 63 ÷ ? = 7

15.5 is an opportunity for students to use their knowledge of multiplication and its connec-tion to division. Missing dividend problems are another way of asking them to multiply the two given factors to find a product.

15.6: Missing Divisor to 100 3.OA.6 ? ÷ 9 = 7 Students determine the unknown by relating the 3 numbers in the same ‘fact family’ to

each other to make the visual representations and then equations true.

15.7: Multiplication and the Commutative Property to 100

3.OA.5 12 x 5 = 5 x ?

Students were first introduced to the commutative property of addition in Stage 3. The same rule applies to multiplication. As students work on tasks that show 2 x 3 = 6 and 3 x 2 = 6, they begin to see the constancy in the quantities and subsequently, the relationship these 3 numbers have to each other.

15.8: Multiplication and the Distributive Property to 100

3.OA.5 6x9 + 3x9 = ?Students break up numbers into smaller numbers. The distributive property can be a way for students to both find an easier way to compute, as well as show their understanding of how smaller parts behave when multiplying.


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stage 16: multiply & divide with 1/10/100

Big Idea: Repeated Equal Groupings and Hierarchical Groupings








(the quotient).


group them into 1 ten and 1 one, or we can call them 11 ones. Seventy five represents 7 tens and 5 ones. When an under-

standing of hierarchical groupings is combined with repeated equal groupings, students can begin to understand the relation-

ship between 4 x 7, 4 x 70, and 4 x 700.






In Stage 16, students experience how numbers grow multiplicatively when using 1s, 10s, and then 100s. Rather than simply

learning a procedure of ‘add a zero’, students are able to visualize the relationship between multiplying similar place values. Six

groups of ‘2-dot’ cards looks similar to six groups of ‘20-dot’ cards, except the dots each have different values (1 and 10). Six

groups of ‘2-bars’ and six groups of ‘20-bars’ also have similarities, but the magnitude of the two sets of groups is very apparent.

And the number line that shows different groups of place values will also have a similar look, especially if the end points of the

number line are represented appropriately. Stage 16 consolidates ideas that have been used in every previous Stage, and by

the end, students will be prepared to approach multi-digit multiplication and division tasks with a high level of understanding.

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16.1: Multiply by 1, 10, 100: Missing Result

4.NBT.11 x 3 = ?1 x 30 = ?1 x 300 = ?

Students experience how numbers grow multiplicatively when using 1s, 10s, and then 100s as the multiplier. Rather than have students use the short cut, “add a zero” when multiplying by 10, Symphony Math shows students, through the use of visual aids, how numbers grow in an understandable manner. They are learning that a number to the left of another number is 10 times larger, and a number to the right is 10 times smaller.

Students will be able to use mental math strategies with reason, sense making, and appreciation.

16.2: Multiply by 1, 10, 100: Unknown Factor

4.NBT.17 x ? = 77 x ? = 707 x ? = 700

Students continue multiplying with 1s, 10s, and 100s, using what they have learned about the relationship between these powers of ten to supply missing factors.

16.3: Divide by 1, 10, 100: Missing Result

4.NBT.18 ÷ 1 = ?80 ÷ 10 = ?800 ÷ 100 = ?

Students progress with powers of 10 into division, where they have the opportunity to apply their knowledge of the relationship between multiplication and division. They can see division as an unknown factor problem. In the problem 90 ÷ 10 = ?, a student can ask herself, “what do I multiply by 10 to make 90,” for example.

16.4: Divide by 1, 10, 100: Unknown Factor

4.NBT.18 ÷ ? = 180 ÷ ? = 10800 ÷ ? = 100

Similarly to 16.3, 16.4 builds on students’ understanding of how multiplication and division are connected. 16.4 also continues developing the powers of ten in the place value system as students decompose 90 into 9 groups of 10, for example, to solve 90 ÷ ? = 10.

16.5: Multiplying 1s and 10s

3.NBT.3 7 x 60 = ?

Students extend their work with 1s, 10s, and 100s to multiplying multiples of 10. As in 16.1, the “adding a zero” approach is turned into something solid: in the problem 6 x 40, students will see 6 groups of 4 tens, or 24 tens. They will further come to understand that 24 tens is another name of 240, and so create a deeper understanding of the place value system.

The more students experience that numbers have different names, and that they can rename any 3-digit number into hundreds or tens or ones, for example, the more flexible they will be when working with large quantities and with decimals numbers.


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stage 17:add & subtract unit fractions






For example, with multiplication and division we work with the total, the size of the parts, and the number of parts. There are

30 students in the class (the whole equals 30), the teacher divides the class into 5 groups (the number of parts equals 5), and






of 30 equals 6.








Why are Repeated Equal Groupings and Parts-to-Whole Important?

Because of the complexity of fractions, and other related concepts such as ratios, decimals and percents, students need to

understand the coordination of Equal Groupings with Part-to-Whole. An understanding of this big idea will help students better

navigate the many counter-intuitive and confusing aspects of fractions.

Fractions are difficult for students to learn not only because of its conceptual complexity but also because many of its features

appear contradictory to what students have learned previously regarding whole numbers. For example, up to now in their

experience with whole numbers, students have learned that two distinct numbers indicate different amounts. The concept

of equivalence in fractions indicates that 2/2 equal one whole, and 4/4 equal 2/2, and so on. They have leaned that 2 does

not equal 1, 4 does not equal 2. But with fractions 2/2 does equal 1, and these equivalent fractions are the same point on a

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number line! In fact, this is the first time in their experience that numbers do not signal a count. Rather, fractional numbers are a

part relative to a whole.

A further confusing idea may be that experiences multiplying whole numbers up to now yielded bigger amounts. When dividing,

smaller amounts resulted. But when multiplying proper fractions, smaller amounts result, and when dividing with proper fractions,

the result is bigger. The before-learned properties of how numbers behave does not apply to fractional numbers. Equally

confusing is the fact that fractions with the same numerals can equal different amounts. A student might expect that 1/2 always

equals 1/2. But the student must consider 1/2 of what? One half of $100 does not equal 1/2 of $50.

Students may also find the names of fractions to be confusing. Previously students have understood late numbers in the

counting sequence to represent larger quantities. Twelve is greater than 2 for example. However with fractions, 1/12 is not

greater than 1/2.

When some students are confronted with the complexity of fractions and they do not have sufficient mastery of the necessary big

ideas they may resort to memorized strategies that lead to correct answers. They learn rules like “find the common denominator,

then add.” Or, “flip it and multiply.” They might remind themselves that “the bigger the denominator the smaller the fraction.”

While these memorized strategies may lead to correct answers, on their own they do not lead to deep understanding of fractions.

Symphony Math continues with fraction concepts by insuring students construct knowledge based on ideas of magnitude and

numeracy, rather than on procedural steps.


In Stage 17, Symphony Math expands on the four concepts that comprise the Equal Groupings with Parts-to-Whole big idea:

* the whole is divided

* there are a specific number of parts

* the parts are of equal size

* the parts equal the whole

In Stage 17, Symphony Math introduces fractions with denominators up to 10. Using the same rich visual environment as Stage

17, students continue to experience a systematic foundation on which to build their understanding of Equal Groupings with Parts

to Whole.

In addition to making equivalencies and comparisons with different numerators and different denominators, students add

and subtract with fractions. In Stages 17.3-17.6 students are challenged to find missing parts, missing wholes, and missing

relationships. By solving problems from the perspective of missing parts, missing wholes, and missing relationships, students

come to a better understanding of the Equal Groupings with Parts-to-Whole big idea.

The Symphony Math visual learning environment explicitly connects adding and subtracting fractions with students’ prior

experiences composing and decomposing quantities.

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17.1: Equivalent Fractions 4.NF.1 1/2 = ?/8.

Students see that the pieces that represent a fraction in a fraction bar can be further subdivided; the amount doesn’t change, but the way we name the fraction does. Fractions with different names can also claim the identical point on a number line; it is the result of subdividing jumps into smaller (or larger) proportional jumps.

17.2: Comparing Fractions 4.NF.2 3/5 ? 4/7

Fraction magnitudes are key to an understanding of fractions. As students increase their deep understanding of fraction magnitudes, they will be more successful with fraction computation. In Stage 17.2, the method students use to determine the magnitude of fractions is first focused on visual fraction models: the area model and the number line model which have consistently been part of their fraction environment in Symphony Math.

17.3: Addition with Fractions: Missing Result

4.NF.3a 1/4 + 1/4 = ?

Students experience the action of combining unit fractions to make a non-unit fraction. Just as 1 + 1 + 1 = 3, 1/4 + 1/4 + 1/4 = 3/4. Addition of fractional parts actively unfold in the fraction bar + number line milieu, making addition of unit fractions explicit. Students see what happens when several copies of the unit fractions are added together.

17.4: Addition with Fractions: Missing Change

4.NF.3a 1/4 + ? = 2/4

In Stages 17.3-17.6 students are challenged to find missing parts, missing wholes, and missing relationships. By solving problems from the perspective of missing parts, missing wholes, and missing relationships, students come to a better understanding of the Equal Groupings with Parts-to-Whole big idea

17.5: Subtraction with Fractions: Missing Result

4.NF.3a 2/4 - 1/4 = ? As stated above, continue to work with Parts-to-Whole with fractions. In Stage 17.5, students take away from the whole to produce a smaller part.

17.6: Subtraction with Fractions: Missing Change

4.NF.3a 2/4 - ? = 2/4Students visualize the decomposition of non-unit fraction into unit fractions, helping them understand the amount of change they need to make up the difference in a missing change problem.

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stage 18:non-unit fractions






For example, with multiplication and division we work with the total, the size of the parts, and the number of parts. There are

30 students in the class (the whole equals 30), the teacher divides the class into 5 groups (the number of parts equals 5), and






of 30 equals 6.








Why are Repeated Equal Groupings and Parts-to-Whole Important?









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number line! In fact, this is the first time in their experience that numbers do not signal a count. Rather, fractional numbers are a

part relative to a whole.

A further confusing idea may be that experiences multiplying whole numbers up to now yielded bigger amounts. When dividing,

smaller amounts resulted. But when multiplying proper fractions, smaller amounts result, and when dividing with proper fractions,

the result is bigger. The before-learned properties of how numbers behave does not apply to fractional numbers. Equally

confusing is the fact that fractions with the same numerals can equal different amounts. A student might expect that 1/2 always

equals 1/2. But the student must consider 1/2 of what? One half of $100 does not equal 1/2 of $50.

Students may also find the names of fractions to be confusing. Previously students have understood late numbers in the

counting sequence to represent larger quantities. Twelve is greater than 2 for example. However with fractions, 1/12 is not

greater than 1/2.

When some students are confronted with the complexity of fractions and they do not have sufficient mastery of the necessary big

ideas they may resort to memorized strategies that lead to correct answers. They learn rules like “find the common denominator,

then add.” Or, “flip it and multiply.” They might remind themselves that “the bigger the denominator the smaller the fraction.”

While these memorized strategies may lead to correct answers, on their own they do not lead to deep understanding of fractions.



Stage 18 Learning ProgressionIn Stage 17 students experienced how copies of unit fractions add together to create non-unit fractions, (where numerators are

greater than 1). Stage 18 continues the work begun in Stage 18, with the computation of non-unit fractions with results less

than or equal to 1 whole.

In this way, students remain in a challenging but familiar environment before they work with fractions more than 1 whole

(improper fractions and mixed numbers). Thus they will be on familiar ground when Stage 20 introduces these concepts for the

first time.

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18.1: Addition with Non-Unit Fractions: Missing Result

4.NF.3b 2/5 + 3/5 =?

Students use their experiences with combining unit fractions from Stage 17.3 to combine non-unit fractions.

In Stage 18, students also call on the structure of whole number addition, and their grow-ing fluency with number combinations. Just as 3 + 5 = 8, 3/9 + 5/9 = 8/9.

18.2: Addition with Non-Unit Fractions: Missing Change

4.NF.3b 2/6 + ? = 5/6

Students use their experiences with combining unit fractions from Stage 17.3 and 17.4 to combine non-unit fractions. When the result of that combination is known, students supply the missing parts that combine to make that sum.

In Stage 18.2, students also call on the structure of whole number addition, and their growing fluency with number combinations. Just as 3 + ? = 8, 3/9 + ?/9 = 8/9, and the missing part is 5, or 5/9.

18.3: Subtraction with Non-Unit Fractions: Missing Result

4.NF.3b 3/3 - 2/3 = ?

Students use their experiences combining unit fractions from Stage 17.5 to decompose a fraction amount into its parts and determine what is left.

In Stage 18.3, students also call on the structure of whole number subtraction, and their growing fluency with number combinations. Just as 8 - 3 = 5, 8/9 - 3/9 = 5/9.

18.4: Subtraction with Non-Unit Fractions: Missing Change

4.NF.3b 7/8 - ? = 3/8

Students use their experiences combining unit fractions from Stage 17.5 and 17.6 to decompose a fraction amount into its parts and determine what is missing.

In Stage 18.4, students also call on the structure of whole number subtraction, and their growing fluency with number combinations. Just as 8 - ?= 5, 8/9 - ?/9 = 5/9.with the missing change being 3 or 3/9.

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stage 19:decimals

Big Idea: Hierarchical Groupings with Parts-to-Whole of Equal Groupings

Decimals are introduced for the first time in Symphony Math Stage 19. Students will also experience fractions with

denominators of 100. In Symphony Math, the topics of decimals and fractions are linked. They share visual models, and they are

shown to represent the same concept.

Because Symphony Math makes sure the student has learned the prerequisites for each step in the progression of new skills,

decimals may be seen as a new symbolic representation of previously learned concepts rather than a new concept that is

unrelated to other material.

Why are These Concepts Important?

The Part-to-Whole combinations in Stage 19 highlight the structure of decimal numbers. In Stage 19, students are introduced

to decimal numbers, which present a new system of representation for them. While fractions and decimals represent the same

idea, decimal numbers depend on hierarchical groupings.

In Stage 17 students learn:

* The whole is divided

* There are a specific number of parts

* The parts are of equal size

* The parts equal the whole

In Stage 19, those equal parts are in groups of 10s and multiples of 10s. In this way, Stage 19 extends the place value behavior

of numbers to non-whole numbers using the number line model. As students extend the base-ten system to non whole numbers,

they will see the same 0-1 number line divided into 10 equal parts and then 100 equal parts.

Extending the idea that 1 whole can also be ten tenths or one hundred hundredths, students see how decimal numbers are

constructed as they extend the base-ten number system. A natural extension of this big idea is that a whole can be divided into

10 parts or 50 parts or in Stage 19, 100 parts. And that 100 of those parts will equal the same whole.

Students also encounter how fraction numbers and decimal numbers have a connection and can be used to name the same

quantities. The number line model broken into both fractional parts and decimal parts helps students see equivalencies.

Familiar fractions and Symphony Math models help students make important connections between fractions and decimals.

In Stage 19, students are given opportunities to explore the ways in which fractions and decimals can both express any given

amount with different looking representations.

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19.1: Sequencing Decimals 4.NF.5 0.4, 0.5, ?, 0.7

Student sequence decimals on a number line. The sequencing tasks, first with 10ths then with 100ths, ask students to show they associate quantities with numerals and order of magnitude. Hundredths are thus encountered as an extension of previous work with tenths, with the number line now being divided into 100 parts.

19.2: Identifying Decimals 4.NF.5 Find ‘36

hundredths’.

Students match the displayed quantity with its corresponding decimal numeral. Next they supply the missing name for the designated decimal amount. Thus students demonstrate they can identify decimal quantities.

19.3: Equivalence with 10ths and 100ths

4.NF.6 3/10 = ?/100Students experience the equivalency of 6/10 and 60/100 and move on to correspond-ing decimal equivalencies of 0.6 and 0.60. In this way, students continue to experience that fractions and decimal numbers represent the same mathematical quantity.

19.4: Addition with 10ths and 100ths 4.NF.6 3/10 + 6/100 = ?

Once students experience and practice equivalencies with tenths and hundredths, they can use similar strategies to add fractions with these two different denominators. Build-ing on Substage 19.3, students can change fractions with denominators of 10 into their equivalent fraction with a denominator of 100. They then use what they learned about adding fractions in Stage 17 and Stage 18 to combine them.

19.5: Decimal Notation with 10ths and 100ths

4.NF.6 52/100 = ?

Students look at fraction numbers with denominators of 10 and 100 and express them as decimal numbers. They may see 50/100 and using the visual models, would create 0.50 as an equivalency. The calibrated number line helps students see decimals and fractions together, as both are labeled to show their identical locations.

19.6: Comparing Decimal Numbers 4.NF.7 0.76 ? 0.8

When considering the same whole, students compare decimal numbers to hundredths. Students can reason about their size by using visual models. When confronted with a comparison like 0.6 and 0.49, students apply lessons from Stage 19 to reason about their relative size, rather than be fooled into thinking more digits signifies a higher value, a common misconception.


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stage 20:fractions greater than 1 whole


Stage 20 is the first time students experience fractions greater than 1 whole . They see such quantities in what is commonly

called ‘improper fraction’ format as well as in mixed number format. While students have undoubtedly seen and used terms like

‘5 and 1/2’ or ‘3 and 3/4,’ they have not necessarily understood that eleven halves, or three wholes + another three quarters is

the same quantity as 5 1/2 and 3 3/4 respectively.

Students need to understand that the big idea of Parts- to -Whole is fundamental to understanding fractions. Students need

to understand the big idea of Parts-to-whole developed with addition and subtraction as well as the big ideas of Repeated

Equal Groupings developed with multiplication and division. These two big ideas coordinated together give us Repeated Equal

Groupings with Parts-to-Whole, the foundational idea for fractions. This perspective also helps us understand why fractions can

be so difficult for students. Not only do they need to have mastery of the proceeding big ideas, but they need to coordinate them

together.

Just as students develop foundational numeracy throughout their Symphony Math experiences, Stage 20 guides them through

familiar and increased contact with fractions. And because of their underpinnings with fractions, fraction composition and

decomposition is more of an extension of their past exposure rather than a stretch.

Why are Repeated Equal Groupings so Important?









number line! In fact, this is the first time in their experience that numbers do not signal a count. Rather, fractional numbers are

a part relative to a whole.

When some students are confronted with the complexity of fractions and they do not have sufficient mastery of the necessary

big ideas they may resort to memorized strategies that lead to correct answers. They learn rules like “find the common

denominator, then add.” Or, “flip it and multiply.” They might remind themselves that “the bigger the denominator the

smaller the fraction.” While these memorized strategies may lead to correct answers, on their own they do not lead to deep

understanding of fractions.

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The new material in Stage 20 presents students with fractions where the whole that is greater than one. Up to now, students’

experience in Symphony Math’s fraction work has been with parts of 1 whole. Whereas a whole greater than one is nothing new

for students, the way the whole looks and the way it is composed, when using fractions, will be explored. Students scaffold from

fraction addition in Stages 17 and 18 to combinations that create mixed numbers and numerators larger than denominators (

often called “improper fractions”). Their prior work decomposing fractions into sums of unit fractions, Stage 17, and working

with whole numbers as fractions from Stage 14, have built the foundations on which students’ work with fractions greater than 1

whole is founded.



20.1: Composing Fractions Greater Than 1 Whole

4.NF3c Make ‘3/2’

In Stage 14, students used unit fractions to compose both non-unit fractions and whole number amounts. In Stage 18, students continued the progression of fraction addition and further experienced the behavior of fractions when they are combined.

In Stage 20, students maintain this natural progression to combine fractions with like denominators with results that are greater than 1 whole. The visual model will aid their comprehension that it is possible to have a fraction with a numerator greater than its denominator and help students make sense of an understandable quantity in this newly written form.

20.2: Composing Mixed Numbers 4.NF.3c Make ‘2 1/3’

Mixed numbers are introduced for the first time in Stage 20.2. Because Symphony Math provides students with a firm underpinning in both whole number combinations and fraction combinations, students are able to apply and extend these experiences.

In Substages 14.2 and 14.3, students created both unit and non-unit fractions. In Stage 17 and Stage 18 they added and subtracted fractions. In Substage 20.2 students continue to have visual models to reference such prior experiences and extend them to create mixed numbers. Visual modeling continues to provide a powerful tool to help students turn improper fractions into mixed numbers.

It is important that students understand that 2 1/4 is the same amount as 2 + 1/4, as well as 9 fourths. The Symphony Math environments contribute to this fundamental understanding.

20.3: Decomposing Fractions Greater Than 1 Whole

4.NF.3c 5/3 = 2/3 + ?

With a problem such as 5/3 - 2/3 = ?, students can call on the structure of whole number subtraction, and their growing fluency with number combinations to understand the similarity to subtracting 5 - 3, much as they did with fraction subtraction in Stage 18.3. Students use their experiences combining unit fractions from Stage 17.5 to decompose a fraction amount into its parts and determine what is left. The fact that the numerator is greater than the denominator does not change the behavior of the fraction parts, as students will see. The emphasis continues to be on fraction decomposition into its parts.

20.4: Decomposing Mixed Numbers

4.NF.3b 2 1/3 = 3/3 + ?

Students show they understand that a mixed number behaves like a whole number in that it can be decomposed and recombined in different ways and still maintain the same whole. The quantity 2 1/3 can be achieved with 3/3 + 3/3 + 1/3 or 1 + 2/3 + 2/3 for examples. Students can reference similar work from Stage 10 where they were asked to recognize the relationships of parts that exist in any number they are working with and flexibly apply this understanding when adding and subtracting. In Stage 20.4, students are challenged to analyze the result of number decomposition in order to successfully compute the answer in a variety of ways that may seem uncommon.

Because students have extensive experiences with part-part-whole, the idea that any quantity can be expressed as the sum of any of its parts is normal. Fraction numbers behave like whole numbers in this way. In Stage 20.4, students demonstrate their knowledge of how fraction parts can be regrouped into an equivalent whole by showing that they recognize many different parts, the equivalency of the parts,and how hierarchical structures come into play.

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stage 21:standard algorithm: addition and subtraction


Hierarchical Groupings with Parts to Whole is a complex idea-one that involves the coordination of two earlier big ideas. It

also illustrates the hierarchal nature of mathematics and how a poor foundation is likely to interfere with the learning of later

concepts. Hierarchical Groupings is the idea that amounts can be grouped into a system of sets and subsets. These sets can be


Building on Stage 12, students in Stage 21 now possess the mathematical underpinnings to be able to apply their place value

understanding of numbers as they engage the standard algorithm process for multi-digit addition and subtraction. The vertical

representation, introduced in Stage 7, shows place value columns in which the potentially complex procedure for addition and

subtraction makes sense.

Why is Parts to Whole so Important?

The student who has not developed an in-depth understanding of these big ideas is likely to rely only on the step-by-step

application of procedures without the supporting understanding of what these procedures mean and why they work. When he

produces a nonsensical answer, the mistake is not apparent because he has stopped looking at the numbers as representing

meaningful quantities. He is simply manipulating symbols as best he can according to that procedure.

With the groundwork established especially in Stages 7 - 12, students understand the structure of tens, hundreds, and

thousands and can thus apply their knowledge in a way that makes sense as they use the standard algorithm for the first time,

supported by Symphony visual environments.

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21.1: 2-digit and 1-digit Addition 4.NBT.4 87 + 9 = ?

Students work with the vertical place value presentation and see how to solve the problem using the standard algorithm process. Dot-cards provide familiar visual models for regrouping in the standard way. Combining 2-digit numbers with 1-digit numbers is a recognized task for students and thus a safe entry point for the introduction of standard algorithm procedure.

21.2: 2-digit and 1-digit Subtraction 4.NBT.4 87 – 9 = ?

Students work with the vertical place value presentation and see how to solve a subtraction problem using the standard algorithm process. Dot-cards provide familiar visual modes for decomposing in the standard way. Subtracting a 1-digit number from a 2-digit number is a recognized task for students and thus a safe entry point for the introduction of the standard algorithm.


Students work with the vertical place value presentation and see how to solve a double-digit addition problem using the standard algorithm process. Regrouping ones into tens and tens into hundreds are actions students encounter in multiple formats in the Symphony Math program. Those experiences provide students with mathematical flexibility and understanding which forms the foundation that allows them to appreciate the efficiency of the standard algorithm.


Students work with the vertical place value presentation and see how to solve a double-digit subtraction problem using the standard algorithm process. Decomposing tens into ones is an action that students encounter in multiple formats in the Symphony Math program. Those experiences provide students with mathematical flexibility and under-standing which forms the foundation that allows them to appreciate the efficiency of the standard algorithm.


In Stage 12.6, students regrouped two 3-digit numbers by composing and decomposing tens, making a 100, breaking apart 10, or creating an easier problem. In Stage 21.5, students can see the efficiency of the standard algorithm as they follow and make sense of the procedures entailed.


In Stage 12.7, students demonstrate their knowledge of hundreds, tens and ones and how such parts decompose to make a new whole. In Stage 21.6, students apply this knowledge when they subtract two 3-digit numbers using the steps involved in the standard algorithm for subtraction.

21.7: 3-Part Addition with Mixed Digits

4.NBT.4 368 + 78 = ?

Students use the standard algorithm for addition to compute the sum of three addends of 1-, 2-, and 3-digits, necessitating that they use their place value knowledge to add and regroup accordingly. Consistent with Stage 21, numbers are presented in the vertical column format and accompanied by a visual environment.

21.8: 4-digit Subtraction 4.NBT.4 1,470 – 258 = ?

Multi-digit subtraction using the standard algorithm poses a challenge for students. Students prior experiences decomposing large quantities come into practice now as they see the vertical presentation of the problem: With the myriad tasks Symphony Math provides for students in earlier Stages, supported by visuals, the standard subtraction algorithm may be appreciated for its simplicity.

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stage 22: expanded mode multiply/divide





(the quotient).





Students in Stage 22 see a task- 7 x 36- presented vertically next to an area model. They label one side of the area model with

7, and the 36 on the other side can be partitioned into tens and ones, or, 30 and 6. In Symphony Math, students are familiar

with multiplying and dividing by multiples of ten ( Stage 16), and with place value partitions; in Stage 22, they partition the

number and use the distributive property to compute in steps 7 x 30 + 7 x 6. The resulting partial products appear, step by step,

under the vertical presentation of the problem as they go along. Using the visual area models, students are able to make sense

of and multiply and divide large quantities by adding partial products. Story problems are presented and students work them

using these same models.


In Stage 16, students experienced how numbers grow multiplicatively when using 1s, 10s, and 100s. Stage 22 consolidates

ideas that have been used in every previous Stage and provides an entry point for the multiplication and division of large

quantities using multiples of ten and expanded notation: harder problems can be broken down into their parts, and made

accessible. Thus is the groundwork laid for the standard algorithm students confront. Without an understanding of hierarchical

groupings, the computation processes in the standard algorithm may be procedural and lack sense-making. In Symphony

Math, use of the distributive property and expanded notation give students a foundation for the complex task of multiplying and

dividing larger quantities.

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22.1: 1-Digit x 2-digit 4.NBT.5 7 x 18 = ?

Students see a vertically presented multiplication problem next to an area model. They label the 1-digit factor along one side length of the box and decompose the 2-digit factor into tens and ones along the other side. An animated line accompanies this partition into tens and ones. Using the distributive property, students then compute partial products and watch the products automatically appear in the original vertically presented equation as they go along. These stacked partial products are added together for the final answer.

22.2: Division with Remainders 4.NBT.6 25 ÷ 8 = ? R ?

Students’ understanding of multiplication is able to be transferred to division with the use of the same visual area model. The problem 13 ÷ 4 = ? is a division problem, but the model with the grid is the same one they see when they multiply. In the division example, the dividend is inside the box, the divisor is along one side length, and the quotient, or missing factor, for the other side must be determined. In this way, the problem may be thought of as a missing factor problem and the connection between multiplication and division is explicit.

When there is a remainder, students can see the extra part of a box that is left over; the part that eventually can make a new whole.

22.3: 1-Digit x 3-Digits 4.NBT.5 8 x 106 = ?

Students see a vertically presented multiplication problem next to an area model. As in Stage 22.1, partial products appear stacked in the vertical problem as they go along, thus laying the groundwork for the standard algorithm students see in later grades.

22.4: Division Missing Factor and Divisor

4.NBT.6 102 ÷ 17 = ?Students see a division problem presented next to an area model. Advancing the idea that multiplication and division are inverse operations, in 22.4, the model for computing the quotient relies on the model for multiplication.

22.5: 2-Digits x 2-Digits 4.NBT.5 26 x 21 = ?

Using the distributive property and multiples of tens and hundreds, students have entry into a potentially difficult problem.The visual area model, and the stacked partial products appearing under the vertically presented multiplication, offer students sensible access to the standard algorithm when it appears later.


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stage 23:multiplying fractions and wholes


Students need to understand that the big idea of Parts- to -Whole is fundamental to understanding fractions. In addition,

students need to understand the big idea of Parts-to-whole developed with addition and subtraction as well as the big ideas




to coordinate them together. Students are called on to extend their understanding of whole number multiplication and how

fractions behave when they multiply these two quantities together.

Why are Repeated Equal Groupings so Important?




Fractions are difficult for students to learn not only because of the conceptual complexity but also because many of the features

appear contradictory to what students have learned previously regarding whole numbers. For example, with whole numbers,

students learn that when multiplied, their product is larger than either of the number or size of groups with which they begin.

Initially, fraction multiplication appears inconsistent with what students know.

With repeated equal groupings of fractional amounts, 3 x 2/5, the product will be smaller than the number of groups [3] with

which they begin. In the situation 3 x 2/5, we can ask how many group sizes of 2/5s are in 3 groups? It takes 6 (1/5s) to make

3 groups, or 1 1/2 [2/5 groups]. The answer, 1 1/2, in isolation, is a smaller amount than 3, because it is not obvious that the

quantity of 1 1/2 tells us how many group sizes of 2/5s we are talking about.

In Stage 23, students are grounded by virtue of the visual fraction models that make Symphony Math. In addition to the number

line, students see a model of fraction bars atop a number line, introduced in Stage 17. Using visual models, students see the

pictorial representation of those groups of 2/5 size, and then what happens when 3 of the groups are repeated. Thus Symphony

Math helps students grapple with the significance and meaning of ‘the whole’ which can be elusive when dealing with fraction

operations.

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23.1: Whole Numbers x Unit Fractions

5.NF.4 1/4 x 32 = ?

In Stage 17, students learn that 4/5 = 1/5 + 1/5 + 1/5 + 1/5. A natural extension of understanding non-unit fractions is that 4/5 also equals 4 x 1/5. The fraction bar/num-ber line models show students that fraction parts can be counted, just as numbers are counted. If 4 cookies can be thought of as 4 groups of 1 cookie, so to can 4/5 be shown as 4 groups of 1/5 or 4 x 1/5.

In this way, SM links students’ prior experiences multiplying whole numbers with multiply-ing fractions.

23.2: Whole Numbers x Non-Unit Fractions

5.NF.4 5 x 3/5 = ?

To multiply non-unit fractions by a whole number, students extend their understanding of multiples of 1/b, Substage 23.1, and see that group sizes of a/b are also repeatedly added by the number of groups being considered: 4 x (2/7) = 2/7 + 2/7 +2/7 + 2/7= 8/7 = 8 x (1/7) = 1 1/7.

Visual fraction models allow students to make sense of the product of a whole number times a non-unit fraction. Modeling number line jumps, or fraction bar delineations, of [2/7] being repeated 4 times, make a robust pictorial for students. Students will be able to make their own similar models when they multiply non-unit fractions by whole numbers.

23.3: Fractions x Whole Numbers: Missing Part

5.NF.4 2/3 x ? = 18

Students may see “How many groups of 1/4 make 2 1/4? 8/4? Stage 23.3 employs different fraction formats, (mixed numbers, fractions with numerators larger than their denominators and fractions with larger denominators than numerators), and shows students that they repeatedly add fractional groups of any size a certain number of times until they arrive at the desired product. By watching number line jumps and fraction bar increases of specified group-sizes, students can see when they land on their target number. The problem structures of Stage 23.3 is more challenging than 23.2, yet the connection between multiplication and division, made in Stage 13, come to bear.

stage 24: magnitude and place value

Big Idea: Repeated Equal Groupings and Hierarchical GroupingsThe Repeated Equal Groupings Big Idea builds upon the Parts-to-Whole idea. With Repeated Equal Groupings, the whole is not

only broken into parts but broken into a specific number of parts and each part is of equal size. Repeated equal groupings is the

Big Idea that underlies multiplication and division. When multiplying, a part, (the multiplicand) is repeated a certain number of

times and when dividing, the whole (the dividend) is partitioned into a certain number of equal groups that is equal to the size of

the parts.


group them into 1 ten and 1 one, or we can call them 11 ones. At this level of mathematics, students can appreciate that our

base-ten place value system is built on powers of 10; a critical building block of the system. The 1 in the tens place of the

number 11 is ten times the 1 in the ones place. The magnitude of a base-ten number part is always 10 times more or 10 times

less than the number part directly to its left or right. The 3 in the number 635 is different from the 3 in 390 because the 3 in

635 represents 3 tens while the 3 in 390 represents 3 hundreds, and is also 10 times the 3 in 635.

With an understanding of hierarchical groupings, students continue to appreciate how place value effects number magnitude.

They also come to recognize the relationship between whole numbers and fractional numbers, and consider how decimal frac-

tions relate to adjacent numbers on their right and left.

Why are Hierarchical Groupings Important?In Stage 16, students experienced how numbers grow multiplicatively when using 1s, 10s, and 100s. Rather than simply learn-

ing a procedure of ‘add a zero’, in Stage 24 students are able to visualize the relationship between multiplying similar place

values by powers of 10. Students will see a dot cards, and how multiplication by ten increases the value of each dot in a multi-

plicand by 10. Similary, they see how division affects each dot in an inverse way. With the goal of highlighting how the number of

zeroes increase and decrease in our base-ten place value system, the place value dot cards show students the mathematics of

the relationships. Through all, the multiplicative relationship of the place value system is clarified and emphasized.

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.Concept Standard Example Description

24.1: Multiply by 10, 100, and 1000 5.NBT.1 7.01 x 100 = ?

Students experience how numbers grow multiplicatively when using 1s, 10s, and then 100s as the multiplier. Rather than have students use the short cut, “add a zero” when multiplying by 10, Symphony Math uses visual aids to show how numbers grow in an understandable manner. Dot cards allow students to see what 10 times a num-ber looks like and then 100 times that same number and observe how the number of zeroes change. Multiplicative language, “Make a number 10 times as large..,” is used.

24.2: Multiply by 1/10 and 1/100 5.NBT.1 392 x 1/100= ?

Students grow their understanding of the inverse relationships between multiplication and division. Multiplicative language, “Make a number 1/10 as large as…,” is used. With visual modeling, students more readily understand the essential connection between multiplying by 1/10 and dividing by 10.

24.3: Divide by 10 and 100 5.NBT.1 35.6 ÷ 10= ?

Building on previous Stages, students are shown how they can move back and forth between decimal and fraction notations. They use their understanding of unit frac-tions to compare decimal places. The use of fractional language, ‘1/10 as large,’ can be applied to both the fraction and decimal tasks as they work on comparisons and see how the resulting product is less than the multiplicand.


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stage 25: decimals to thousandths

Big Idea: Hierarchical Groupings with Parts-to-Whole of Equal Groupings

In Symphony Math, the topics of decimals and fractions are linked, as reflected in Stage 24. They share visual models, and they

are shown to represent the same concept. Thus students can use fractions and decimals interchangeably. While fractions and

decimals represent the same idea, decimal numbers depend on hierarchical groupings.

Because Symphony Math makes sure the student has learned the prerequisites for each step in the progression of new skills,

decimal values to thousandths builds on previous work with fractions, expanded notation, and a deeper understanding of place

value.

Using the Symphony Mat visual models, students extend their understanding of decimals notation for factions and addition of

fractions with denominators of 10 and 100 to decimals to 1000ths. They identify, read, and compare decimal numbers as well

as write them in fractional form in expanded notation. Stage 25 guides students to an understanding of equivalence of decimals

(20.4 = 20.40, 5 = 5.0).

Why are These Concepts Important?

The Part-to-Whole combinations in Stage 25 highlight the structure of decimal numbers. Extending the idea that 1 whole can

also be ten tenths or one hundred hundredths, students see how decimal numbers are constructed as they extend the base-

ten number system. A natural extension of this big idea is that a whole can be divided into 10 parts or 50 parts or, in Stage

25, 1000 equal parts. And that 1000 of those parts will equal the same whole. Students build on foundational place value

understandings from Stage 24 as they consider expanded notation to thousandths, and thus, the relationship of adjacent

whole number and fractional number quantities. Expanded notation further displays the number of zeros and placement of the

decimal point.

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25.1: Decimal Notation 5.NBT.3 2×100 + 3×1 +

5x(1/10)

Students look at fraction numbers with denominators of 10, 100 and 1000 and express them as decimal numbers. They may see 52/1000, use the visual models, create 0.052 as an equivalency; and vice versa, decimals to fractions. The calibrated number line helps students see decimals and fractions together, as both are labeled to show their identical locations.

25.2: Add and Subtract Decimals 5.NBT.7 4.09 + 3.4 = ?

Students expand their use of the Standard Algorithm to work with decimals. The alignment and magnitude of the visual models helps students understand the purpose of ‘aligning the decimal point.’

25.3: Decimal Comparison 5.NBT.3 3.56 ? 3.5

When considering the same whole, students compare decimal numbers to thousandths. Students can reason about their size by using visual models. When confronted with a comparison like 0.56 and 0.509, students apply lessons from Stage 25 to reason about their relative size, rather than be fooled into thinking more digits signifies a higher value.


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stage 26: expanded mode X÷ with decimals





(the quotient).





Students in Stage 26 see a task- 7 x 8.5- presented vertically next to an area model. They label one side of the area model with

7, and the 8.5 on the other side can be partitioned into ones and tenths, or, 8 and .5. In Symphony Math, students are familiar

with multiplying and dividing by multiples of ten ( Stage 24), and with place value partitions; in Stage 26, they partition the

number and use the distributive property to compute in steps 7 x 30 + 7 x 6. The resulting partial products appear, step by step,

under the vertical presentation of the problem as they go along. Using the visual area models, students are able to make sense

of and multiply and divide large quantities by adding partial products. Story problems are presented and students work them

using these same models.


In Stage 24, students experienced how numbers expand and contract multiplicatively when multiplying and dividing tenths,

tens, hundreds, etc.. Stage 26 consolidates ideas that have been used in every previous Stage and provides an entry point for

the multiplication and division of large quantities using multiples of ten and expanded notation: harder problems can be broken

down into their parts, and made accessible. Thus is the groundwork laid for the standard algorithm students confront. Without

an understanding of hierarchical groupings, the computation processes in the standard algorithm may be procedural and lack

sense-making. In Symphony Math, use of the distributive property and expanded notation give students a foundation for the

complex task of multiplying and dividing larger quantities.

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26.1: 1-Digit x Decimals 5.NBT.5 7 x 9.3 = ?

Students see a vertically presented multiplication problem next to an area model. They label the 1-digit factor along one side length of the box and decompose the decimal into ones and tenths along the other side. An animated line accompanies this partition into tens and ones. Using the distributive property, students then compute partial products and watch the products automatically appear in the original vertically presented equation as they go along. These stacked partial products are added together for the final answer.

26.2: 3-Digit x Decimals 5.NBT.6 127 * 0.6 = ?

Students see a vertically presented multiplication problem next to an area model. As in Stage 26.1, partial products appear stacked in the vertical problem as they go along, thus laying the groundwork for the standard algorithm students see in later grades.

26.3: Division with Decimals 5.NBT.5 4.8 ÷ 3 = ?

Students see a division problem presented next to an area model. Advancing the idea that multiplication and division are inverse operations, in 26.3, the model for computing the quotient relies on the model for multiplication.

26.4: Decimal x Decimal 5.NBT.5 9.8 x 4.3 = ?

Using the distributive property and multiples of tens and hundreds, students have entry into a potentially difficult problem.The visual area model, and the stacked partial products appearing under the vertically presented multiplication, offer students sensible access to the standard algorithm when it appears later.


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Visual Models: Manipulatives and Worksheets

Worksheet Purpose Instructions

Manipulatives Use a visual model to represent the concept. Create bars, dot cards, or number lines for each item.

Bridge Connect symbols to their visual representations. Create objects, numbers, and symbols to complete each item.

Symbols Understand the concept at the abstract level. Create numbers and symbols to complete each item.

Apply Extend understanding to real-life problem solving.

1) Read the story presented at the top of the page.

2) Create a number model of the full solution.

3) Write the number sentence that matches the model.

Using the Extra Practice Worksheets

The Symphony Math Worksheets provide extended practice using the Multiples Ways of Knowing from the Symphony Math

program. Students should work through all worksheets in the order given:

Group Learning

The Symphony Math Extra Practice materials are designed to promote a conversation about the Big Ideas in math. One-on-one or

small group instruction with the materials is recommended for students who need more time to make connections between the

mathematical concepts in the Stage and the application of those concepts in their math curriculum.

Extension Worksheets

Symphony Extensions are an addtional set of worksheets for each Stage that include a mixture of Symphony like and novel tasks, to help teachers ensure students have a full understanding of each Stage's concepts and can applythem outside of the program.

Number Bars, Dot Cards, Number Lines and Fraction Bars

Visual models such as Number Bars, Dot Cards, Number Lines and Fraction Bars are used in both online activities and in offline in Worksheets. See subsequent pages for examples of the Stages these are used in.

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Symphony Bars: Stages 1-6

0 1 2 3 4 5 6 7 8 9 10

Dot Cards: Stages 1-6Dot Cards: Stages 1-6

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Dot Cards - Up to Hundreds at Stages 7-8

Dot Cards - Up to Thousands at Stages 9-13,15-16, 21, 24-25

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Symphony Bars: Ones & 10 - Stages 7-13, 15-16

1 2 3 4 5 6 7 8 9 10

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60 70 80 90 100

10 20 30 40 50

60 70 80 90 100

Symphony Bars: Tens - Stages 7-13, 15-16

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Symphony Bars: Hundreds - Stages 9-13, 15-16

900

100

300200100

400500600700800900100

0

300200100

400500600700800900100

0

Teacher Overview

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Fraction Bars - Stages 14, 17-18, 20

Teacher Overview

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Fraction Bars and Number Lines - Stage 19

Teacher Overview

Cop

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201

4 Sy

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Lear

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LLC

. Per

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Fraction Bars - Stage 23

Teacher Overview

Cop

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201

4 Sy

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Lear

ning

LLC

. Per

mis

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is g

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.

Symphony Dot Cards: Decimals - Stage 25

Janine

Typewritten Text

Janine

Typewritten Text

Janine

Typewritten Text

1 stage 1: the number sequence - itecnz.co.nzquantity is the big idea that describes amounts, or...

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