Lecture Notes: Perimeter and Multiplication - Vocabulary Flashcards

Perimeter and Parallelograms

• Rim vs perimeter: the rim is the boundary; perimeter is the total around an object.
• Parallelogram/Rectangle properties:
  • Opposite sides are equal.
  • Perimeter of a parallelogram (or rectangle) with adjacent sides $a$ and $b$ is P = 2(a+b).
  • In a rectangle, opposite sides are equal (lengths match across).
• Quick example: if a rectangle has side lengths $a=17$ and $b=10$, then the perimeter is P = 2(17+10) = 54. cm.
• Area note: area will be covered in the next section; for a rectangle, area would be ext{Area} = a\cdot b (to be learned later).

Symbols and Basics of Multiplication

• Common symbols for multiplication:
  • Cross symbol: \times (the time symbol).
  • Juxtaposition (writing numbers next to each other) and sometimes a dot \cdot.
• Interpreting multiplication: a\times b means repeated addition: a\times b = a+a+\cdots +a\quad (b ext{ times}) or equivalently b\times a = b+b+\cdots +b\quad (a\text{ times}).
• Key properties:
  • Commutative: a\times b = b\times a
  • Associative: (a\times b)\times c = a\times (b\times c)
  • Multiplicative identity: a\times 1 = a\quad\text{and}\quad 1\times a = a
  • Zero property: a\times 0 = 0\quad\text{and}\quad 0\times a = 0

Models of Multiplication

• Number line model:
  • Example: 3\times 4 = 4+4+4 = 12.
  • Interprets as 3 groups of 4.
• Grouping model:
  • Example: 3 groups of 4 -> 12; or 4 groups of 3 -> 12.
• These models illustrate why multiplication is commutative (order of grouping or order of addition does not change total).

Multiplication with Large Numbers: Expanded Notation (Partial Products)

• When numbers are large, use expanded forms to avoid regrouping mistakes.
• Example: 27\times 253 = (20+7)\times(200+50+3)
  • Expand numbers: 27=20+7,\quad 253=200+50+3
  • Partial products:
  • 7\times 3 = 21
  • 7\times 50 = 350
  • 7\times 200 = 1400
  • 20\times 3 = 60
  • 20\times 50 = 1000
  • 20\times 200 = 4000
  • Sum: 21+350+1400+60+1000+4000 = 6831
  • Result: 27\times 253 = 6831

Lattice Method

• Draw a rectangle with one factor on the top and the other on the side (split digits if needed).
• Multiply digits in corresponding cells and place products in the cell (tens and ones separated as needed).
• Sum diagonals (carry as necessary) to get the final product.
• Example: 27\times 255 = 6885
  • Top digits: 2 and 7; Side digits: 5 and 5.
  • Compute cell products (e.g., 2\times5=10\$, 7\times5=35\$ for each row, etc.).
  • Add diagonals to obtain the final result: 27\times 255 = 6885.
• Pros/cons: lattice reduces regrouping mistakes but uses more paper; expanded/partial products use less paper but requires careful summation.

Quick Takeaways

• Perimeter of a parallelogram/rectangle uses all four sides; opposite sides are equal.
• Multiplication can be represented by several symbols; core idea is repeated addition.
• Properties to memorize: commutative, associative, multiplicative identity, and zero property.
• For large numbers, use expanded notation (partial products) or lattice method to organize the multiplication.