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(USCC) Standards of 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 standards of Grade 5 and ways to implement the same.

**GRADE 5**

Grade 5 is very crucial in the learning journey of a student as this is the year when students consolidate their mathematical understanding of place value, geometry, fractions, and measurement built through concrete and pictorial representations in the early primary years and get prepared for more abstract topics such as negative numbers, algebra, coordinate geometry and mensuration in grade 6 and beyond. Keeping this transition from primary school to middle school in mind, the Common Core lays down the standards for grade 5 mathematics with increased focus on generalization of place value to decimal numbers and operations on fractions and decimals. The coverage of geometry and mensuration also expands as one moves to higher grades. Common Core introduces Coordinate Geometry and the idea of volume as an attribute of 3-dimensional objects for the first time in grade 5. Reading and interpretation of measurement data using line plots is also one of the key areas of focus.

In order to attain these grade level outcomes, the Cuemath Curriculum exposes students to various abstract concepts through visual representations and simulations.

The curriculum of Grade 5, cuts across 5 major domains:

- Operations and Algebraic Thinking
- Numbers and Operations in Base 10
- Numbers and Operations- Fractions
- Measurement and Data
- Geometry

**Operations and Algebraic Thinking**

Forming expressions using variables, performing operations on them and then simplifying them are important learning outcomes in algebra in grade 6. To be able to do so efficiently, in grade 5, the Common Core lays emphasis on simplifying numerical expressions using multiple operations with and without parentheses.

Till grade 4, students have not simplified expressions that have multiple operations. In grade 5, they are exposed to a set of rules to simplify expressions that have all the four operations together. The use of parentheses along with all four operations further complicates the matter.

Students are often led to memorise the following rules to simplify numerical expressions:

- An expression grouped within the parentheses is to be simplified first.
- Multiplication and Division must be given precedence over Addition and Subtraction.

However, it is quite natural for them to be curious to know why these rules are required in the first place.. They often have to be satisfied with a standard response that these are conventions in mathematics that we ought to follow.

While these rules are extremely critical, it is equally important to establish the need for these rules or conventions. The fact that the simplification of the same expression cannot lead to different answers must be understood first before memorising these rules.

So, in Cuemath, a step by step, guided approach is adopted to help students realize the need to follow these rules or conventions for simplifying operations. In addition to this, students are provided experience of observing how different expressions with the same number and same operators placed in different order, simplify to different numbers.

Here is an example from the Cuemath Curriculum, where students are expected to match the expressions to the statements (in words) they represent.

One may notice that the same numbers** 9, 18 and 63** and the same operations **subtraction and division** are used in all the three expressions. In two of the three expressions, we can see the use of parentheses.

However, each of the three expressions represent three completely different situations. When expressions are connected to mathematical or real world situations, the rules or conventions appear more relatable and meaningful to students.

In the next step, students are led to simplification of these expressions and that is when they see how each expression simplifies to different values.

Fluency in simplification of numerical expression lays a strong foundation for algebra in the higher grades.

**Numbers and Operations in Base 10**

At the end of grade 4, students are expected to read and write numbers up to 6 digits with ease and understand the place value of large numbers. So, in grade 5, Common Core recommends generalization of the idea of place value to multi digit numbers. Students are expected to recognise that *in a multi digit number, a digit in a place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. *

In Cuemath, the students observe how the place of a digit shifts to left or right as the number is multiplied by 10 or 1/10 respectively in a place value chart, using examples from the real world as illustrated in the example below.

**“A music company sold 400 copies of a music album in the first month of its release. However, by the end of the year the sale increased to 4000.
Let us observe how many times the sale of the number of copies increased from the first month of the release to the end of the year”**

In the above illustration, the student can see that as the digits in the number move one place to the left, the number increases 10 times.

Similarly, students also observe that as the digits shift one place to their right, the number reduces to one tenth of its value as shown in the illustration below.

With this understanding of place value of whole numbers, students can easily transition to place

value of decimal numbers later in this grade and perform operations on the same.

**Numbers and Operations- Fractions**

With the understanding of the meaning of fraction and addition and subtraction of the same with like denominators, students transition to grade 5. In this grade, the Common Core recommends understanding of addition and subtraction of fractions with unlike denominators.

However, this topic is often challenging for students.

Let us look at a question on subtraction of fractions with unlike denominators from the Cuemath Diagnostic Test with students’ responses.

The correct answer to this question is 1/16, as a fraction equivalent to ¾ with denominator 16 is 12/16 and 12/16 subtracted from 13/16 is 1/16.

Out of 547 students who attempted the test, 47% have chosen the correct answer. However, about 33% have chosen 10/12 as the correct answer.

What could be the reason?

Let’s try to analyze!

The probable reason for this could be that the students subtracted the numerators and the denominators separately to arrive at the answer. This happens mainly because the students tend to overgeneralize subtraction of whole numbers and treat numerators and denominators of fractions as separate numbers.

Hence, in Cuemath, the focus always is to ensure that the students first understand why a procedure works. So, instead of overemphasizing the procedure of making the denominators the same to add or subtract fractions with unlike denominators, a visual approach is adopted to establish why the denominators need to be made the same to subtract ¼ from ⅓ as illustrated below.

In the example above, students can relate that to subtract unlike fractions the unequal parts must be further divided till the size of all the parts of the whole are equal. Only then the second fraction can be subtracted from the first. Students then understand why the denominators must be made equal.

After thorough understanding of addition and subtraction of fractions is built, Common Core recommends multiplication and division of fractions as important topics to be covered in grade 5.

**Measurement and Data**

In addition to solving problems on interconversion of units of measurement, in grade 5, students are expected to understand volume as an important attribute of 3D-shapes. Just like the understanding of area is built using composition of unit squares, Common Core recommends introduction of volume by combining unit cubes with no gaps and overlaps as shown in the examples below.

Once this understanding is built, they eventually learn to derive the formula of finding volume of a rectangular prism through visualization, instead of memorizing it as **length x width x height**.

Here is an example from the Cuemath Curriculum that illustrates how the students derive the formula to find the volume of a rectangular prism using step by step visualization.

With the above process, it is quite easy for the student to understand that the volume of a rectangular prism can be calculated as the product of its length, width and height.

In addition to measuring volumes, the US Common core also recommends drawing and interpreting line plots using measurement data nearest to ½ , ¼ and ⅛ of a unit.

Tasks such as measuring the lengths of different objects and drawing the line plots to show how many objects are approximately less than or more than a given length help students visualize data. Once this understanding is built, students are expected to go one step ahead and compare how the representation of the data varies when the lengths are approximated to the nearest ½ unit, ¼ unit and ⅛ unit.

**Geometry**

In grades 3 and 4 students learn about properties of different types of quadrilaterals. However, in grade 5, the students are encouraged to enhance their geometric thinking by providing justification about how one category of quadrilateral can also belong to another category. For example, they learn to justify why a square is both a rectangle as well as a rhombus. They then learn to represent these relationships using Venn diagrams as shown below.

In addition to this, students are first time in grade 5 introduced to representing points as ordered pairs in the coordinate plane in the first quadrant. With this understanding they learn to find distance between two points and eventually find the area and perimeter of polygons drawn on the coordinate plane. In addition to this students also learn to analyze relationships between two numeric quantities by graphing them on the coordinate plane.

Although the complexity of the topics in grade 5 increases, with appropriate pedagogical approaches, the Cuemath Curriculum enables students to master the standards laid down by the US Common core with ease. With increased level of motivation and confidence, students become better prepared to delve deeper into more abstract topics from grade 6 onwards. **Cuemath conducts Live Online Math classes. **This makes them love math and see its relevance in their day to day lives.

*-By Joyita Banerjee*

*Joyita heads the curriculum for K-6 and assessments at Cuemath. She has a Master's degree in education and a Post-graduate diploma in Information Technology. With over 15 years of experience, she has expertise in designing and developing curriculum for students and training teachers in math education and educational assessments. Her passion lies in helping children from different social classes master math concepts. In her free time, she loves reading, listening to music and spending time with her family.*