Estimating for Accuracy

Session 1

Sense with Percents

1. Harry notices his bank charges him 18.7% per year interest on his credit card.  He owes $2897.31 and his credit card statement says he owes $45.15 interest this month.
   Discuss why 1/5 of $3000 = $600 is a good first estimate for the yearly interest.  So the yearly interest is actually less than $600.  So the monthly interest is less than $50.  So $45.15 looks reasonable.
   Discuss methods of calculating key percentages 10% (multiply by 1/10), 20% (multiply by 1/5), 25% (multiply by 1/4) etc.
2. Student Exercises. Estimate these answers

3. Combining estimates from 2 steps.
   Discuss calculating 14.8% of $440.71
4. Student Exercises. Estimate these answers

5. Student Exercises. Decide whether each of these answers is reasonable or definitely wrong by estimating.
   26% of $3908 = $1019.99                   52.7% of $496.13 = $231.46
   9.6% of $36,414 = $3,641.40  1.2% of $480,414 = $5,764.97
   34% of $101,408 = $34,478.72

Session 2

Estimating Products

1. Discuss why 51.8 x 50.76 ≈ 50 x 50 =2500 and why the exact answer is smaller than 2500 using 1 significant figure approximations (to exploit knowledge of the times table).
2. Student Exercises.  Estimate these products.  Is the answer more or less than the exact answer? 

3. Discuss why 49.8 x 60.7 ≈ 50 x 60 = 3000 and why the estimate is actually close to the exact answer. (50 is a little more than 49.8 and 60 is a little less than 60.7, and these effects compensate).
4. Student Exercises. Estimate these products

5. Estimating when the numbers are 'mid-range' eg rounding 251.8 to 300 or 200 during estimation is equally sensible.
   Discuss estimating 36.4 x 24.8 
   On the low side the estimate is 40 x 20 = 800 
   On the high side the estimate is 40 x 30 = 1200 
   On average (800 + 1200) = 1000 is reasonable.
6. Student Exercises. Use 'high' and 'low' estimates and average them to estimate the following:

7. Discuss rounding when both numbers are in the 'middle'.
   For example 24 x 36 ≈ 20 x 40 = 800
   One "up" and one "down" should give a good estimate.
8. Student Exercises. Estimate the products:

Session 3

Division estimates, and detecting addition/subtraction nonsense answers.

1. To estimate   rounding to   is not very successful.  A better criterion with division is round the division to 1 sig. fig. and then search for a 2 sig. fig. approximation that is an answer to a multiplication table.
   So discuss why  ≈    = 6.
   Discuss why the answer is actually bigger than 6.
2. Student Exercises. Estimate the answers 

3. A student wrote down         and works it out as 417 by following the rule 'add vertically in columns'.  Discuss an estimation strategy that show this answer nonsense.  For example answer is clearly less than 40 + 40 = 80 so 417 is nonsense.

4. Student exercises. Quickly argue why every one of these answers is nonsense by estimation.

A student who is not really very good at subtraction writes
Discuss the obvious error in his method.  He refuses to believe he is wrong.  Discuss how to try to convince him of the error of his ways by estimating 41.8 – 27.9 ≈ 42 – 28 = 14.

Student Exercises. Demonstrate why each of these subtractions must be wrong by estimation.