Algebra in 6th grade math introduces students to basic algebraic concepts that will be essential for understanding more complex math in later years. Here’s a breakdown of the key areas covered:
1. Variables:
- Letters are used to represent unknown numbers (variables).
- Examples: x, y, a, b
- Variables allow us to write general rules and relationships between numbers.
2. Expressions:
- Expressions are combinations of numbers, variables, and operations (addition, subtraction, multiplication, division).
- Examples: 2x + 3, 5y – 1, a + b × 2
3. Evaluating Expressions:
- Substituting numerical values for variables to find the result of an expression.
- Example: If x = 4, then the value of 2x + 3 becomes 2(4) + 3 = 11
4. Simplifying Expressions:
- Combining like terms (terms with the same variable raised to the same power).
- Using the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right)
5. Equations:
- Statements that show equality between two expressions.
- The equal sign (=) indicates a balance between the left and right sides of the equation.
- Examples: x + 5 = 10, 2y – 1 = 7
6. Solving Equations:
- Finding the value of the variable that makes the equation true.
- Often involves isolating the variable using basic operations.
- Example: Solve x + 3 = 7. Subtract 3 from both sides to get x = 4.
7. Word Problems:
- Applying algebraic concepts to solve real-world problems.
- Translating word problems into mathematical equations and solving for the unknown.
Exercise 11.1
1. Find the rule which gives the number of matchsticks required to make the following matchsticks patterns. Use a variable to write the rule.
(a) A pattern of letter T as T
(b) A pattern of letter Z as Z
(c) A pattern of letter U as U
(d) A pattern of letter V as V
(e) A pattern of letter E as E
(f) A pattern of letter S as S
(g) A pattern of letter A as A
Ans :
(a) Pattern T:
- Two sticks are used to form a single T.
- Rule: Number of matchsticks = 2 (for the T)
(b) Pattern Z:
- Three sticks are used to form a single Z.
- Rule: Number of matchsticks = 3 (for the Z)
(c) Pattern U:
- Four sticks are used to form a single U.
- Rule: Number of matchsticks = 4 (for the U)
(d) Pattern V:
- Two sticks are used to form a single V (similar to T).
- Rule: Number of matchsticks = 2 (for the V)
(e) Pattern E:
- Five sticks are used to form a single E.
- Rule: Number of matchsticks = 5 (for the E)
(f) Pattern S:
- Five sticks are used to form a single S (similar to E).
- Rule: Number of matchsticks = 5 (for the S)
(g) Pattern A:
- Six sticks are used to form a single A.
- Rule: Number of matchsticks = 6 (for the A)
2. We already know the rule for the pattern of letters L, C and F. Some of the letters from Ql. (given above) give us the same rule as that given by L. Which are these? Why does this happen?
Ans :
Rules for the following letters:
- For L, the rule is 2n
- For C, the rule is 2n
- For V, the rule is 2n
- For F, the rule is 3n
- For T, the rule is 3n
- For U, the rule is 3n
We observe that the rule is the same for L, C, and V, as they each require only 2 matchsticks. The letters F, T, and U share the same rule, which is 3n, as they each require 3 matchsticks.
3. Cadets are marching in a parade. There are 5 cadets in a row. What is the rule which gives the number of cadets, given the number of rows? (use n for the number of rows.)
Ans :
Number of cadets = Number of cadets in a row × Number of rows
We can express this rule mathematically using the variable n for the number of rows:
Number of cadets = 5 × n
5 represents the number of cadets in a single row (given in the problem).
n represents the variable for the number of rows in the parade.
4. If there are 50 mangoes in a box, how will you write the total number of mangoes in terms of the number of boxes? (Use b for the number of boxes.)
Ans : The total number of mangoes in terms of the number of boxes can be written as:
50b
Here’s the explanation:
- 50: This represents the number of mangoes in a single box (given in the problem).
- b: This variable represents the number of boxes.
5. The teacher distributes 5 pencils per student. Can you tell how many pencils are needed, given the number of students? (Use s for the number of students.)
Ans :
Number of pencils = Pencils per student × Number of students
We can express this mathematically using the variable s for the number of students:
Number of pencils = 5s
Explanation:
- 5 represents the number of pencils the teacher distributes to each student (given in the problem).
- s represents the variable for the total number of students.
6. A bird flies 1 kilometre in one minute. Can you express the distance covered by the bird in terms of is flying time in minutes? (Use t for flying time in minutes.)
Ans :
Distance covered in 1 minute = 1 km. The flying time = t minutes.
Distance covered:
- For t = 1 is 1 x 1 km
- For t = 2 is 1 x 2 km
- For t = 3 is 1 x 3 km
∴ The rule is 1 x t km, where t represents the flying time in minutes.
7. Radha is drawing a dot Rangoli (a beautiful pattern of lines joining dots with chalk powder. She has a dots in a row. How many dots will her rangoli have for r rows? How many dots are there if there are 8 rows? If there are 10 rows?
Ans :
Number of rows = r Number of dots in a row drawn by Radha = 8
Therefore, the number of dots required:
- For r = 1 is 8 x 1
- For r = 2 is 8 x 2
- For r = 3 is 8 x 3
∴ The rule is 8r, where r represents the number of rows.
- For r = 8, the number of dots is 8 x 8 = 64
- For r = 10, the number of dots is 8 x 10 = 80
8. Leela is Radha’s younger sister. Leela is 4 years younger than Radha. Can you write Leela’s age in terms of Radha’s age? Take Radha’s age to be x years.
Ans : Since Leela is 4 years younger than Radha, and Radha’s age is represented by the variable x, we can express Leela’s age as:
Leela’s age = Radha’s age – 4 years
Substituting Radha’s age with the variable x:
Leela’s age = x years – 4 years
Therefore, Leela’s age can be written as (x – 4) years.
9. Mother has made laddus. She gives some laddus to guests and family members, still 5 laddus remain. If the number of laddus mother gave away is l, how many laddus did she make?
Ans :
Given that the number of laddus given away is l,
and the number of laddus left is 5,
∴ The total number of laddus made by mother is l + 5.
10. Oranges are to be transferred from larger boxes into smaller boxes. When a large box is emptied, the oranges from it fill two smaller boxes and still 10 oranges remain outside. If the number of oranges in a small box are taken to be x, What is the number of oranges in the larger box?
Ans :
x: Number of oranges in a small box (given).
10: Number of oranges remaining outside after emptying a large box (given).
Total oranges = Oranges in small boxes + Remaining oranges
Total oranges = 2x oranges + 10 oranges
Therefore, the large box contains 2x + 10 oranges.
11. (a) Look at the following matchstick pattern of square. The squares are not separate. Two neighbouring squares have a common matchstick. Observe the patterns and find the rule that gives the number of matchsticks in terms of the number of squares.
(Hint: If you remove the vertical stick at the end, you will get a pattern of Cs)
(b) Following figure gives a matchstick pattern of triangles. As in Exercise 11(a) above, find the general rule that gives the number of matchsticks in terms of the number of triangles.
Ans :
(a) Let n be the number of squares.
∴ The number of matchsticks required:
- For n = 1: 3n + 1 = 3 x 1 + 1 = 4
- For n = 2: 3n + 1 = 3 x 2 + 1 = 7
- For n = 3: 3n + 1 = 3 x 3 + 1 = 10
- For n = 4: 3n + 1 = 3 x 4 + 1 = 13
∴ The rule is 3n + 1, where n represents the number of squares.
(b) Let n be the number of triangles.
∴ The number of matchsticks required:
- For n = 1: 2n + 1 = 2 x 1 + 1 = 3
- For n = 2: 2n + 1 = 2 x 2 + 1 = 5
- For n = 3: 2n + 1 = 2 x 3 + 1 = 7
- For n = 4: 2n + 1 = 2 x 4 + 1 = 9
∴ The rule is 2n + 1, where n represents the number of triangles.
