Year 7 Maths Acceleration Study Notes

Year 7 Maths Acceleration

1. The Distributive Law

Expanding Brackets

• Lesson 3: Algebraic Techniques Last lesson's recap: In algebra, the variable 'x' can often be omitted in certain products without causing confusion.

  • Example 1: $2 imes 3y = 6y$

  • Example 2: $3 imes y = 3y$

  • Here, $2 imes 3 = 6$ and $2 imes 3y = 6y$ mean both follow multiplicative principles.

  • Explanation of omission: The 'x' sign can be omitted when it appears between a product of pronumeral algebraic terms in brackets.

  • Example: $4 imes (x + 2)$ can be simplified to $4(x + 2)$ without confusion.

  • Expanded form: $4(x + 2) = 4 imes (x + 2) = (x + 2) + (x + 2) + (x + 2) + (x + 2)$

  • Simplification of repeated terms:

  • $= x + x + x + x + 2 + 2 + 2 + 2 = 4x + 8$

Concept Check 1.1

• Use the distributive law to expand the following:

  • a) $5(b + c)$

  • Expanded: $5b + 5c$

  • b) $2(6 + 4mn)$

  • Expanded: $12 + 8mn$

  • c) $-3(4x + 5y)$

  • Expanded: $-12x - 15y$

  • d) $-(3c - 2h)$

  • Expanded: First change sign:

    • $-1(3c - 2h) = -3c + 2h$

  • e) $m(9 - 5m + 8n)$

  • Expanded: $9m - 5m^2 + 8mn$

  • f) $-a(3a - 4 + 5b)$

  • Expanded: $-3a^2 + 4a - 5ab$

  • g) $7f(5 - 2e - f)$

  • Expanded: $35f - 14fe - 7f^2$

  • h) $-4xz(x - y + 2z)$

  • Expanded: $-4xz^2 + 4xyz - 8x^2z$

  • i) $-5xy^2(7xy - 3x^2y - 2xy^2)$

  • Expanded: $-35xy^3 + 15x^3y + 10xy^4$

2. Simplifying Multiple Brackets

Lesson 3: Algebraic Techniques

• When expanding expressions with multiple brackets, the approach is to work from the innermost bracket outwards.

• Example: Expand and simplify $2x + y - (5x + 3y - (2x - 4y)) - 7x$:

  • Step 1: Identify inner brackets.

  • Step 2: Expand inner brackets:

  • $2x + y - (5x + 3y - 2x + 4y) - 7x$

  • Step 3: Combine like terms:

  • Simplified: $2x + y - 5x - 3y + 2x - 4y - 7x$

  • Final result: $-8x - 6y$

• Additional Example: Expand and simplify the expression involving multiple brackets:

  • Simplifying $9igra{2b - c + (3b - c)} - 4(3b - 7c)$:

  • Leading to further simplification:

    • Combined: $…$ (Provide completed workings similar to above)

3. Equivalent Expressions

Lesson 3: Algebraic Techniques

• Equivalent expressions have different forms but represent the same value or quantity.

  • Examples:

  • $(x + y)$ is equivalent to $1(x + y)$

    • Each term inside the brackets is multiplied by 1.

  • $-(x + y)$ equates to $-1(x + y)$

    • Each term inside is multiplied by -1.

  • Thus:

  • $(x + y) = 1(x + y) = x + y$

  • $-(x + y) = -1(x + y) = -x - y$

  • Significance: Understanding how to express algebraic expressions in equivalent forms is helpful in evaluating more complex expressions.

• Discussion prompts for students:

  • Explain the difference between $5 + (x + 3)$ and $5(x + 3)$:

  • $5 + (x + 3)
    eq 5(x + 3)$

Concept Check 2.1

• Compare expressions:

  • Is $x + yz$ equivalent to $x + yxz$?

  • Answer: No, they are not equivalent because:

    • $x + yz
      eq x + yxz$

• Additional checks as per lesson prompts:

  • Explanation task to determine relation between given expressions (Provide details similar to above example)

4. Translating Word Problems into Algebraic Expressions

Key Words

• Understanding key phrases for mathematical operations:

  • Addition (+):

  • Keywords: Sum, and, more than, total, plus, increased, greater than.

  • Subtraction (-):

  • Keywords: Minus, less, less than, difference, take away, reduce, decrease.

  • Multiplication (×):

  • Keywords: Times, product, multiply, double (×2), triple (×3).

  • Division (÷):

  • Keywords: Share, quotient, divide, halve (+2), third (-3).

• Examples of word phrases translated into algebraic expressions:

  • Example: "Three times the sum of x and 5" translates to $3(x + 5)$

  • Identify and underline key words corresponding to operations in provided phrases.

Completing Concept Checks

• Translate given scenarios into algebraic expressions based on assistance from parameters provided to students.