# Big IdeasBetter Understanding

Symphony Math® teaches the “big ideas” that illuminate mathematical understanding. Some of these ideas include:

- Quantity: Number
- Parts-to-Whole: Addition and Subtraction
- Hierarchical Grouping: Place Value
- Hierarchical Grouping: Multi-Digit Addition and Subtraction
- Repeated Equal Groupings: Multiplication and Division
- Repeated Equal Groupings: Fractions

#### Quantity: Understanding Number

We help students develop number conceptualization — the idea that the sequence of numbers is determined by each number’s magnitude.

Below, students show that they can work with sequences of large to small. They will move on to using other manipulatives, and then numbers to demonstrate their mastery of the number sequence. Next, students will move on to the use of vocabulary that expresses the ideas of ‘greater’, ‘less’, and ‘equal’.

#### Parts-to-Whole: Understanding Addition and Subtraction

We help students visualize Parts-to-Whole to comprehend numbers as an inter-related system. These ideas help kids grasp addition and subtraction, story problems, multiplication, division, and fractions.

We begin with simple addition and use number bars to show that two small bars can be combined to create a longer bar. Students then learn the ideas behind missing parts.

#### Hierarchical Grouping: Understanding Place Value

“Which number is larger, 79 or 81?”. This question could be difficult to students who do not grasp the idea of Hierarchical Groupings. Such students might think 79 is larger because 7 + 9 = 16, whereas the 8 + 1 = 9.

Symphony Math® helps students learn how to integrate the ones, tens, and hundreds place values; thus, they can understand 79 is indeed less than 81.

#### Hierarchical Groupings Coordinated with Parts-to-Whole: Understanding Multidigit Addition & Subtraction

Instead of teaching rote memorization of complex procedures, we help students learn advanced addition and subtraction concepts by understanding Parts-to-Whole and Hierarchical Groupings.

#### Repeated Equal Groupings: Understanding Multiplication and Division

When students memorize multiplication tables without understanding what they mean, they can be flustered by more challenging questions that require stronger problem solving.

By using onscreen manipulatives to build visual models of multiplication and division, we help students see the meaning of the operation and its relationship to addition and subtraction. This leads to a firmer grasp of multiplication and division, which overcomes the limits of rote memorization.

#### Repeated Equal Groupings Coordinated with Parts-to-Whole: Understanding Fractions

Fractions and related concepts such as ratios, decimals, and percents can be very confusing. We help students navigate the many counter-intuitive aspects of these principles by teaching the coordination of Equal Groupings with Parts-to-Whole.

We use visual models to convey the fundamental concepts and to help students connect fractions to their understanding of multiplication.