There has been a growing need for a focused and coherent mathematics curriculum in the United States to improve achievement in mathematics. To address this need, the United States Common Core Standards (USCC) for mathematics for each grade have been laid out. The standards are described in detail to help educators plan appropriate pedagogical approaches to ensure achievement of outcomes. These standards are drawn in from research in math education and models of best mathematical practices across the world. 

Here are the highlights of the grade specific standards.

GRADE 3
In grade 3, students' knowledge of numbers and operations is extended further. Developing an understanding of the core concepts such as multiplication and division within 100 as well as fractions is given utmost importance. In measurement, students are introduced to the concept of area of 2D shapes. Students are further encouraged to analyse 2D shapes to understand their similarities and differences. Aligned with the USCC recommendations, the Cuemath Curriculum makes use of visual models for introduction as well as application of these concepts as this not only solidifies the understanding of the concepts but also hones the skill of mathematical reasoning.

In order to facilitate mastery of abstract concepts by young learners, Cuemath adopts a Concrete-Pictoria-Abstract (CPA) approach. The curriculum is aided with concrete models and visuals as much as possible to make the transition from concrete to abstract concepts seamless.

The curriculum for Grade 3 focuses on 5 major domains:

1. Operations and Algebraic Thinking
2. Numbers and Operations in Base Ten
3. Numbers and Operations—Fractions
4. Measurement and Data
5. Geometry

Operations and Algebraic Thinking
By the end of Grade 3, students are made well-versed in all four operations and their properties and patterns. USCC recommends that different kinds of real life problems be presented for solving, enabling the students to select the operation to be performed based on reasoning as against the key words used, which can often be misleading. 
Here is an example from the Cuemath curriculum that challenges the use of ‘keywords’ in word problems.

The word 'times' is often associated with multiplication. So, a majority of the students may end up multiplying 36 by 9 if they use the keywords approach to solve this problem, which is incorrect. To nip this problem in the bud, USCC recommends the use of models to help the students visualise the problem. The problem above can be modelled using an array as shown above.

Students are able to visualize from the model that they know the number of objects in each row and that they need to find the number of rows the 36 stamps can be divided equally into to find the number of sips of juice that Mandy will take. 

36 ÷ 9 = 4. So, Mandy will take 4 sips of her juice.  
The use of different models makes it easy for students to understand what the real life problems represent, what each operation stands for and ensures apt application of these operations in real life. With this experience early, students will not struggle to frame an expression or an equation from a given situation when they transition to algebra in middle school.

Numbers and Operations in Base Ten
Estimation is an important skill that the students will use in their daily lives. One concept that is important for estimation is rounding off numbers, which the students learn in grade 3. USCC recommends the use of place value to round off the whole numbers to the nearest 10 or 100. A common approach to teaching rounding off is having the students look at the digit on the right of the place the number is being rounded off to.

However, it is important for students to understand why this is done instead of learning this as a rule. In Cuemath, the emphasis has always been on the ‘why’ behind the ‘what’. Unless the ‘why’ behind this rule is internalized, students will struggle to generalize this same rule to round off larger numbers. An example of a visual model below from the Cuemath curriculum shows how the rounding off technique works. 

The numbers are represented on the number line and the students are encouraged to use the same to reason which hundred they will be rounded off to based on the hundred they are closest to. Students learn to use the number that lies 'halfway' between the two hundreds and analyse whether the number being rounded off is less or more than halfway between the two hundreds to find the closest hundred. 

This knowledge is used to lead them to the rules for rounding off, which ensures better understanding and retention through reasoning. 

Numbers and Operations—Fractions
In Grade 3, students are introduced to fractions for the first time. To be able to perform operations on fractions and understand decimals in the later grades, it is crucial that the foundation of this concept is strong. 

One concept of fractions where the students often face difficulty is comparing fractions that have the same numerator or denominator. From their understanding of whole numbers, it comes naturally to them that 2 is less than 3 and both of them are less than 4. However, this knowledge becomes a roadblock when the students compare the unit fractions with these numbers as the denominators. They are unable to understand how 1/4 is less than 1/3 and both of them are less than 1/2 even though 4 is greater than 3 and 2.

So, USCC recommends the use of fraction models and number lines which help the students visualize them as equal parts of a whole. 

Use of common, real life objects in representing fractions helps the students compare their sizes and develop a sense of reasoning. In Cuemath, students are made to compare fractions visually using familiar real life objects. With this approach, students are able to successfully visualize how the parts become smaller as the denominator increases, enabling them to compare fractions that have the same numerator or denominator. 

Measurement and Data
In Grade 3, students are introduced to the concept of area and perimeter. Traditionally, students are taught to calculate the area of a rectangle using the formula length × breadth which the students find hard to retain and recollect at later stages. So, USCC recommends that the concept of area is connected to the concept of multiplication. 

In Cuemath, students are eased into the concept of area through activities where they are made to cover the shapes using unit squares which are then counted to find the area of the shape. 

This enables the students to visualize rectangles as arrays made of unit squares, enabling them to connect the concept of area to multiplication.

As a result, students are able to successfully connect area to multiplication and justify using multiplication to determine the area of a rectangle as length × breadth.
 
Geometry
By the end of this grade, students are able to extend their knowledge of 2D shapes and compare and classify them on the basis of sides and angles. Instead of rote memorisation of their properties and classifications, it is important that the students are encouraged to observe these shapes and to arrive at the conclusions. 
Here is an example from the Cuemath curriculum below.

This enables the students to see that a rectangle and a square share a lot of common attributes. A square has the same properties as a rectangle and both of them together belong to the category quadrilaterals. The observation skills that they develop through this activity help them to conclude that a shape can belong to more than one category.  At the same time, they are also able to acknowledge that quadrilaterals are not just limited to the shapes that they have learned about and that there are quadrilaterals that may not fit into any of the categories.
 
Cuemath curriculum provides learning experiences of important concepts through concrete and pictorial representations and connects mathematics with the lives of students. Cuemath conducts Live Online Math classes. This makes them love math and see its relevance in their day to day lives. Thus, with a strong understanding students are expected to transition to grade 4 with greater motivation and confidence.

-By Anshul Sharma  
 Anshul is the Project Lead for the primary Math curriculum for the US. With a Master's Degree in Commerce and a diploma in teaching up her sleeve, she is passionate about education and feels that it is important for the students to understand the 'why' behind the 'how' of math concepts. She has been in this field for 11 years with expertise in all phases of the Indian and International curriculum. In her free time, she enjoys spending time with her family.