This week's issue of my local paper contains 8 pages.
- Page 1 has all the headline news and no advertisements.
- Page 2 has 1 advertisement that takes up 1/2 of the page.
- Page 3 has ads for 2 local businesses, one takes up 1/2 of the page and the other is the size of 1/6 of the page.
- Page 4 has a grocery ad that takes up 1/4 of the page.
- 5/12 of page 5 advertises local products.
- 5/6 of page 6 is real estate ads.
- Page 7 is all sports news.
- Page 8 has only 1/6 of the page for news, and the rest is restaurant and movie ads.
What fractional part of the whole paper is taken up with news and not advertising? If each page measures 11" x 16", what is the area of the entire paper available for printing the week's news? Bonus: How does this ratio of news to ads compare with that in the typical issue of the Colchester Chronicle?
This task assesses the students' ability to perform fraction computations, calculate area measurements, use reasoning and problem-solving skills, and connect problem-solving opportunities to real-life situations. Students should use models or diagrams to determine the total amount of ad space in this eight-page newspaper, thus determining the amount of space available for printing the news. Students should use their life experience and realize that there is white space on all newspaper pages and account for this in their solutions. The task also includes a challenge question that encourages students to take a look at our local paper and determine the ratio of ads to news found in this paper.
45 minutes
I gave this task as a homework assignment. The basic task could be accomplished in one 45-minute class period with the bonus done that evening. You could have copies of a local paper on hand and have students do the task in two-three class periods with the extension done as well.
There is a natural real-life connection, as students are encouraged to analyze the local newspaper for the ad to news ratio. Students might investigate this task while studying newspapers, printing or advertising. Students might even read some of the paper while they are at it!
I suggest using the disk copy of this task and changing the wording in the "Bonus" to reflect the name of your local paper. I suggest using a paper, which has the same dimensions as the task rather than a full sized national paper. In our area, each town has its own local paper. Perhaps yours does too.
This task was given at the end of a unit of study in which students had been finding common denominators and adding fractions using fraction bars. Several of the students used them to solve this problem (see the Practitioner exemplar). To adapt this task for students with lesser skills, you could have a four-page paper instead of eight, you could provide all fractions with common denominators, and/or you could eliminate the portion of the task dealing with area.
You could encourage all students to consider margins in their solutions to make the task slightly more complicated. You could also have more talented students investigate the cost of various sized ads and determine the revenue generated by the advertisements. You might want to invite a local newspaper editor or owner in to have a class discussion, or send a group of interested students out to interview the marketing manager of your local paper.
- Fraction bars
- Copies of local papers
- Calculators
There are three 1/2 pages of ads and four 1/2 pages for news. Total area of the eight pages is 1,408 square inches, so 792 square inches are available for the printing of news. If you consider a one-inch margin around the pages, then the available dimensions for printing would be 9" x 14" for an area of 126 square inches x 8 pages = 1008 square inches total space for all printing. This would only allow 567 square inches for the news.
Novice problem solvers as a group were not able to add the fractional parts of pages accurately, nor did they communicate their solutions adequately. The exemplar for this level shows a student who organized the data in a table with the fractions all converted to twelfths. The reader has no idea how this was done, nor is the reader able to determine where the 7/16 or the 9/16 answers come from. The solution number two is the correct area measure for the amount of space for printing news, but there is absolutely no evidence to support this answer. The student does not understand the concept of common denominators as evidenced in the statement, "My rule worked with this problem and with others because making all the denominators twelfths would work in any addition problems with fractions."
The Apprentice made mathematical errors, solved only parts of the problem, or were unsuccessful at communicating their solutions clearly. The work featured in this exemplar shows an incomplete solution. The approach used was workable. Using tables to organize the fractional parts of the pages and finding common denominators shows good reasoning and clear communication. The approach for finding the area of a page and the entire area of the paper worked and is easy to follow. The student never found the fractional part of the 1,408 square inches that was available for news rather than filled with ads. The student either stopped too soon or did not understand the question fully. This student did use good representations and did have a good command of mathematical language, despite the shortcomings of the solution. The reader has the sense that this student may not have read the problem carefully enough.
This student does a great job of finding the fractional parts of the paper, which are filled with ads and news. The approach for finding the total area available for news is efficient and successful. The student used good reasoning in verifying the solution in paragraph number two of the text. The student has great communication skills and uses accurate and appropriate mathematical representation. This piece of work is very representational of the majority of the Practitioners who did this task. Those whom I considered to be Experts went the extra distance and did the bonus problem, as well as the basic task.
Expert students got the correct answer, had a good representation and clearly described their processes. Many did the bonus for a comparison and saw that the local paper had more ads than the paper in the task. To my surprise, none of them considered the margins allowed in virtually all newspapers. This exemplar demonstrates clear understanding of the task and the bonus and takes it one step further by converting the fractions to percents. The table is accurate and clear in its comparison of the two papers. It should, however, have had an additional column for the fractions as given in the task. When I mentioned this to the student, the response was that the conversion had taken place in the student's head, as the task was read and I believe that to be true. The reader would, however, benefit from this information. The color-coded graphs were good for comparing the two newspapers at a glance. A next step for this student would be to look at USA Today® or any other regional paper and see if the hypothesis put forth in the last paragraph is correct.
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