Mathematics and the Physical World. by Morris Kline

By Morris Kline

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Section E: Order of Operations 33 E Order of Operations Arithmetic Trees While working on the home repair cost problems, Enrique wrote this arrow string to find the cost of having three workers for two hours of repairs. ؋ 37 2 ⎯⎯→ ⎯⎯ ؉ 25 ⎯⎯→ ⎯⎯ ؋3 ⎯⎯→ ⎯⎯ Karlene was working with Enrique, and she wrote this expression. 2 ؋ 37 ؉ 25 ؋ 3 Karlene found an answer of 149. Enrique is very surprised. 5. a. How did Karlene find 149 as her answer? b. Why is Enrique surprised? Karlene and Enrique decide that the number sentence 2 × 37 + 25 × 3 is not necessarily the same as the arrow string.

In your notebook, copy and complete the arithmetic trees. a. 12 b. 3 2 ؋ ____ ؊ ____ 24 4 c. 5 3 7 ، ؉ ؋ ____ ____ ____ ؋ ____ 8 2 ؊ ____ ؋ ____ 2. Make or design an arithmetic tree and find the answer. a. 5 ؋ 6 b. 5) ؋ 6 c. 15 ، (2 ؋ 2 ؉ 1) Suzanne took her cat to the veterinarian for dental surgery. ) Before the surgery, the veterinarian gave Suzanne an estimate for the cost: $55 for anesthesia, $30 total for teeth cleaning, $18 per tooth pulled, $75 per hour of surgery, and the cost of medicine.

Sample response: ؉2 ؉ 32 998 ⎯⎯→ 1,000 ⎯⎯⎯→ 1,032 This string is easier because when you add 2 to 998, you get lots of zeros that are easy to work with. It’s easy to add numbers to 1,000. 3. Check your answer with a classmate. Sample response: (long) ؉ 31 ؉ 19 232 ⎯⎯⎯→ 263 ⎯⎯⎯→ 282 ؉ 50 (short) 232 ⎯⎯⎯→ 282 Short strings are easier when the total of the numbers over the arrows is a multiple of 10 or a number between 1 and 10. 4. Check your answer with a classmate. Sample response: ؉ 98 (short) 232 ⎯⎯⎯→ 330 (long) ؉ 100 ؊2 232 ⎯ ⎯⎯→ 332 ⎯⎯⎯→ 330 Longer strings are easier when the total of the numbers over the arrows is not a multiple of 10 or a number between 1 and 10.

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