Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

 Exploring Representation
 Try It Yourself: Typical Week | Equations Related to Data | Problem Reflection | Your Journal

Equations are a symbolic form of representation. They are compact means of communicating about relationships, and they may help students develop additional understanding of percents or other numbers that relate one quantity to another.

We frequently make statements like "I spend one-fourth as much time socializing as I do working" or "I spend twice as much time on family duties than I do on myself."

 1. Look again at Ms. Lee's circle graph. What relationships do you notice between the number of hours spent on Family (F) and the number of hours spent on Sleep (S)? Make some statements about this graph, including a percent in at least one, and write some appropriate equations. Show Answer
 Sample Answer: Times as many: Sleep (S) = 56 hrs and Family (F) = 14 hrs; 56/14 = 4. So, 4 times as many hours are spent on Sleep (S) as on Family (F). This can be written as S = 4 • F. Similarly, 1/4 as many hours are spent on Family (F) as on Sleep (S). So, F = (1/4) • S. Because 4 = 4.00 = 400%, and 1/4 = 25%, we can say, "The amount of sleep is 400% as much as the amount of family time" and also, "The amount of family time is 25% as much as the hours spent sleeping." Or, in other words, S = 400% • F, and also, F = 25% • S. This type of comparison, which involves multiplication, and/or ratios, is called relative comparison: Values are compared relative to the whole, or to one another. Another type of comparison is absolute comparison: Only absolute values are compared. This type of comparison uses the words "fewer" or "more," as in the examples below: Fewer: Forty-two fewer hours are spent on Family than on Sleep, or, S - 42 = F. More: Sixteen more hours are spent on Sleep than on Work, W + 16 = S.
 2. Gather your information about how you spend a typical week, and make several statements to compare the various categories. Write simple equations to represent these statements. Use both relative (that is, "times as many") and absolute (that is, "more hours" and "fewer hours") comparisons. Notice how you make use of the chart in order to formulate your statements. Show Answer
 Sample Answer: Answers will vary. Here are two examples: (1) Using a relative comparison, I spend about four times as much time on Family (F) as I do on household Chores (C); 28 hours = 4 • 7 hours, so F = 4C. (2) Using an absolute comparison, I spend two hours more per week on chores (C) than on exercise (E); 8 = 2 + 6, or C = 2 + E.
 3. Take one of your relative comparison statements from above, and turn your data into a ratio (for example, your ratio will be W/S or W:S, if you did the work/sleep comparison). Divide this and change it to a percent. Then use the percent to write an equation, and read the statement aloud to yourself. Check to make sure that your equation works, and that it accurately represents your data. Can you explain in words what the percentage represents? Try several more if you need additional practice. Show Answer
 Sample Answer: One example is to compare the amount of hours spent on leisure and family combined to the hours spent working: (F + L)/W = 24/52 = 46%. The equation can also be written as (F + L) = 46% • W, and read as a statement that says, "The number of hours spent on family and leisure activities combined is 46 percent of the number of hours spent working." Making sensible equations is an important part of addressing the Representation Standard in the upper elementary grades. Note that when fractions such as 4/24 or 8/24 are changed to percents, we often round the result to an approximate amount, for example, 4/24 = 16.666 . . .% and 24/72 = 1/3 = 33.333 . . .%. We use a special symbol when we say "is approximately equal to": 4/24 ≈ 17%, and 8/24 = 1/3 ≈ 33%.

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