The Grade 7 Math curriculum as prescribed by the common core state standards is designed to extend the previously-learned math concepts with a focus on graphical interpretation and real-life application. Let’s explore how the contents of Grade 7 Math, which are divided among the following five domains, serve this purpose:
- The Number System
- Ratio and Proportional Relationships
- Expressions and Equations
- Statistics and Probability
The Number System
The standards under this domain build upon the intuitive idea of integers and fractions developed in Grade 6 to introduce directed numbers (i.e., integers and rational numbers taken together) and their operations to the student. Here, the students learn to recognize fractions, decimals, integers, and percentages as different forms of rational numbers. Also, keeping with the overall theme of Grade 7 Math, the focus is on graphically interpreting operations on directed numbers, and applying them in everyday contexts to demonstrate their relevance in real-life situations.
Arithmetic of rational numbers is usually taught as a set of abstract rules, leaving the students confused and resulting in the formation of conceptual gaps. At Cuemath, we enable the students to visualize operations on rational numbers graphically, and treat them as a natural extension of whole number arithmetic. For example, we use the number line to teach addition of integers using the number line as a visual aid as shown below:
As shown above, to add 5 to -2, we start at -2 on the number line and move 5 units to its right. Thus, our final position, i.e. the number 3, is nothing but the sum of -2 and 5.
Ratio and Proportion Relationships
The standards under this domain prescribe interpreting and analyzing a wide variety of real-life situations including interest, discount, taxes, etc. using the understanding of ratios and proportionality from Grade 6. Additionally, the students interpret proportional relationships geometrically by recognizing that changing the side-lengths of a geometric figure proportionally changes its size, but preserves its shape. This makes for an effective base upon which the concept of similarity is built in higher grades.
One of the key features of this domain is switching among different representations (tables, equations, graphs, etc.) of proportional relationships. This helps the students connect an equation (i.e., an algebraic construct) to a line (i.e., a geometric figure), and provides the foundation for the study of linear relationships in higher grades. This connection helps the students recognize that the unit rate of a proportional relationship determines the steepness of its graph. This provides the intuition behind the important concept of ‘slope’ and helps the students compare proportional relationships graphically as shown below:
The above graph shows how the price of cake at three bakeries varies with its weight. So, by comparing the steepness of these three lines, the students can easily identify the bakery that provides the best bargain.
Expressions and Equations
Given that the students learn rational numbers and their arithmetic in grade 7, they work with more sophisticated expressions, equations and inequalities in this domain. Also, the knowledge of arithmetic of rational numbers helps them solve equations and inequalities that require more algebraic manipulation than those covered in grade 6. At Cuemath, the student achieves these objectives by modelling a wide range of real-life scenarios as shown below:
The standards under this domain cover various geometrical concepts ranging from properties of angles at a point to constructing various geometrical figures. These, along with exposure to scale drawings, help the students study and prove more sophisticated geometrical results related to congruence and similarity in grade 8. Also, from computing the perimeter/area and surface area/volume of simple 2D and 3D shapes in Grade 6, the students graduate to computing these for composite shapes in Grade 7.
This grade also marks the introduction of the circumference and area of a circle to the students. Here, given that a mathematically rigorous proof of the formula for the area of a circle is out of scope at this grade-level, we demonstrate a visual proof to help them develop the required intuition:
As shown above, we divide the circle into 8 equal parts (or sectors) and form a rectangular arrangement with
a length of $$pi*r$$ units (i.e., half the circumference of the circle), and
a breadth of $$r$$ units (i.e., the radius of the circle).
This enables approximating the circle’s area as that of the rectangular arrangement, i.e. the product of its length and breadth, or $$pi*r$$. Then, we show that, by dividing the circle into more and more sectors, the rectangular rearrangement resembles an actual rectangle more and more closely, resulting in a more accurate approximation to the circle’s area.
Statistics and Probability
Under this domain, the students further their study of statistics by learning how random sampling can help reveal important attributes of a population. They also learn to use measures of center and variability to compare two populations.
One of the key features of this domain is the introduction and formalization of the big idea of probability. Here, the students are expected to understand the concepts of experimental and theoretical probability, and the difference between them. To help them do this, we use applets to emulate experiments like the tossing of a coin, die, etc. as follows:
Using this, they can repeat the experiment of flipping a fair coin as many times as they want, and observe the frequencies of heads and tails showing up. This helps them recognize that, as the number of trials increases, the experimental probabilities get closer and closer to the corresponding theoretical probabilities, i.e. 0.5 each.
The 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 the students love math and see its relevance in their day to day lives.
-By Rigved Jhunjhunwala