Chapter 5 Summary: Using Numbers In Sensible Ways

Mental Computation

  • Helps develop flexibility with numbers and operations.
  • Gives a sense of control over numbers.

Distributive Property

  • a(b + c) = ab + ac
  • Applied to mental computation and division.

Associative Property

  • (a + b) + c = a + (b + c); (a · b) · c = a · (b · c)
  • Applied to mental computation.
  • Cannot be applied to division.

Pictures and Diagrams

  • Strip diagrams can be incorporated.
  • Example: finding percentages.

Over-Estimation

  • Used when needing to ensure having enough (money, materials).
  • ETAs often overestimate due to unforeseen events.
  • Trade workers include contingency for unexpected extra work.

Under-Estimation

  • Used for income to avoid financial issues.
  • Used in bidding scenarios (e.g., The Price is Right).

Estimation

  • Involves mental computation and rounding.
  • Number sense should sometimes override strict rounding.
  • Good estimators use various strategies and understand numbers deeply.
  • Understand the magnitude to gain experience in flexible rounding.

Rounding

  • Simplifies numbers by expressing them in terms of nearest unit.
  • General rule: look at the digit to the right of the desired place value.
  • Numbers less than 5 round down; 5 or higher round up.
  • Emphasis on teaching place value is crucial.

Rounding Large Numbers

  • Goal: "Slightly better than an educated guess".
  • Ten Thousands → Nearest Thousand
  • Thousands → Nearest Hundred
  • Hundreds → Nearest Ten
  • Tens → Nearest One

Estimation with Products

  • Rounding both up: overestimating.
  • Rounding both down: underestimating.
  • Rounding one up and one down: more accurate estimate.

Estimation with Differences And Quotients

  • Rounding both up OR both down is better.
  • Use number sense.
  • Example: Estimate 24,354 ÷ 13

Why Estimate

  • Essential in daily life; improves with practice.
  • Flexibility and number sense are crucial.
  • Helps verify calculator answers.

Benchmarking

  • Develops a personal feel for quantity size.
  • Compares to personally meaningful references.
  • Examples: money, population, time, distance, height, weight.

Examples of Benchmarking Problems

  • Relating large numbers to familiar contexts.
    • How many times can Usain Bolt run 100m in 10 minutes?
    • How many laps around a track is a mile?
    • How many classrooms can fill a stadium?
    • How many Dolly Partons to reach a basketball hoop?

Benchmarking Summary

  • Involves intelligent conclusions, not just guessing.
  • Requires number sense (relative and absolute size).
  • Useful for developing a personal feel for large/small numbers.