I. From the beginning of Hamilton's Essentials of Arithmetic: Higher Grades (7th & 8th grades), p. 25:

How to Solve Problems

I. Before you try to solve a problem you must find out exactly what it means. That is, you must consider:

a. What facts are stated or implied in the problem.

b. What kind of answer the question asked for.

c. By what steps the required answer can be found from the given facts.

II. The most important habit to acquire is accuracy. A wrong answer is worthless. Always test your work. Also make a mental estimate of the answer.

III. The second essential is rapidity. To secure rapidity, always choose the shortest method of work where several methods are possible. It is sometimes well to indicate the necessary operations before performing any of them. Then the work may often be shortened by cancellation.

1. Find the cost of 3 3/4 lb. of lamg at $.39 a pound.

Facts stated: Amount of lamb bought; price per pound.

Question: What is the cost of the total amount bought?

Mental estimate: 3 3/4 lb @ $.39 cost about 3 3/4 × $.40, or $1.50.

Written work: 3 3/4 = 15/4; 15/4 × $.39 = $1.47.

Test: 3 × $.39 = $1.17; 3/4 of $.39 = $.30; $1.17 + $.30 = $1.47.

2. Tell by mental estimates which is greater: 99 × $5 or 100 × $4; 7/8 or 5 3/4 or 6;

1 7/8 × 6 or 7.

3. How much cheaper is it to buy 3 dozen plums at 15 cents a dozen than at 3 for 5 cents?

4. How much is saved by buying 3 1/2 lb. of sugar for 35 cents instead of at 10 1/2 cents a pound?

II. From the beginning of Connected Mathematics 7th grade booklet, Accentuate the Negative, p. 17:

Mathematical Reflections

In this investigation, you worked with positive and negative numbers. You analyzed sequences of events in the MathMania game, looked at temperature, and extended the number line to represent numbers less than 0. You also learned how to decide whether one number is less than or greater than another number. These questions will help you summarize what you learned:

1. Describe what positive numbers, negative numbers, and 0 mean in terms of

a. keeping score in MathMania.

b. temperature readings.

2. Describe how you can compare the following types of numbers to decide which is greater. Use examples to illustrate your thinking.

a. two positive numbers

b. two negative numbers

c. a positive number and a negative number

3. Describe how to locate numbers on a number line. Use examples to illustrate your thinking. Be sure to include positive and negative numbers as well as fractions and decimals in your examples.

Think about your answers to these questions, discuss your ideas with other students and your teacher, and then write a summary of your findings in your journal.

III. Extra Credit:

Consider accuracy and rapidity in problem solving vs. mathematical reflections. What are your mathematical priorities? Can we have it both ways? Might one problem set do a better job than the other one does of encouraging accuracy, rapidity, and mathematical reflection?

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