## Monday, October 31, 2011

### Introducing Exponents

In the second semester of Algebra 1, students encounter exponents and the rules of exponent multiplication and division. The power of exponents and the huge effect of a continual increase in the rate of increase (which is what an exponent greater than one provides) need introduction as well. This lesson uses the NROC video presentation from Algebra 1—An Open course, Unit 7, Lesson 1, Topic 1: Rules of Exponents followed by a concrete example wherein the number 2 is raised to the powers of zero through 64. The example comes from a classic math story in which grains of rice are doubled on each successive square of a chessboard resulting in the last square receiving 2^64 grains of rice. Students then work with the numbers from this example to explore just how big a value 2^64 really is.

Learning Objective(s)
• Understand exponents
• Be able to simplify basic fractions involving variables raised to exponents
• Be able to multiply variables with exponents
• Be able to explain how raising to a power is different than multiplying a number by that power.

Assessment Type
This is an introductory assignment that presents the basics of exponent use with constants and variables. It provides a formative assessment of student understanding of exponents and is designed for the 55minute high school algebra class.

Assignment Details
1. 15min--Play the video presentation from Algebra 1—An Open course, Unit 7, Lesson 1, Topic 1: Rules of Exponents which explains basic exponent concepts of exponent notation, multiplication and division. Students should take notes and then summarize key points as a whole class at the end. After this, check for understanding with a few example problems and assign any textbook work for homework. Be sure that students are NOT just multiplying by 2 or using 2x and x^2 interchangeably.

2. 10min--Tell a version of the Rice Grain Story, where someone asks a sultan (or raja or king) for payment in the form of rice on a chessboard. One grain on the first square, 2 grains on the second, 4 grains on the third, 8 on the 4th and so on. Write this as 2^0, 2^1, 2^2, and so on. Ask if this payment is more valuable than a pound of gold on each square and have students discuss. (Someone might have to look up how much a pound of gold is worth.)

Now, bring out one or more chessboards and a large rice sack to illustrate. Have student volunteers count out grains of rice to fit on the board until they can’t anymore. How many squares did that take?

Get someone to figure out how much rice would be on the last square. (If you have a computer projector, you can use this to do a quick aside on using a spreadsheet for calculations. Otherwise handheld calculators work.) 2^64 is 18446744073709551616. Yikes!

Redo problem with 3 grains of rice (3^0, 3^1, 3^2…) to make sure students don’t confuse the doubling for correct use of exponents. This also shows just how fast exponents increase in size…and how useful they are as 2^64 is easier to write out then the huge number it becomes.

3. 20min--Have students try to get a sense of how big 2^64 is. They can go about it several different ways depending on interest and what you have available. A small group of students could use a computer spreadsheet to see just how much numerically summing from 2^2 to 2^64 is. Meanwhile, another group can approach this as a storage problem. How much space would 2^64 grains of rice take up in a cubic warehouse? Have the students actually calculate this by filling a small box (shoebox for example) with rice. Have a measuring tape and half cup, quarter cup, measures available. Other students could try to compare the number of grains calculated to how much rice is grown yearly in the world. Finally, how much would that all weigh? Final numbers could be put up on a single poster displaying the results. Anyone finished begins on homework. To get a sense of what they’ll come up with, try this page.

4. 10min--Assign your cleanup people. Anyone not cleaning up is helping create the class results on one large poster and/or writing an answer to this Exit Slip Prompt: A student of mine once said, “5^2 is 10, 5^3 is 15, and 5^4 is 20. It’s so easy!” Could you kindly explain to this student where they went wrong and help them find a trick or slogan that will let them remember how powers work. Students who finish can share aloud. Summarize findings and dismiss the class!

Instructor Notes

• When doing example problems at the beginning of class, be sure to do a squared problem and a cubic problem and mention the relationship between squaring and the area of squares and cubing and the volume of cubes.

• To keep things interesting for successive class periods have them pretend to fill chessboards with 3^0, then 3^1, 3^2 of 4^0, 4^1, 4^2 and so on instead of powers of 2. Alternately, do not show the next class more than a glimpse at the previous class’s poster. Make it known that class posters will be voted on the next day (classes cannot vote for their own poster). The poster voted best will be kept on the wall.

• If there are not enough rice tasks to go around, a group could be assigned to summarize the morning’s exponent notes in a class poster as well.

• Show the effect of a negative exponent the next day (or the day after the posters have been voted on, as then you’ll have large scrap paper) as folding paper in half repeatedly. The How many times do students think the paper can be folded before it becomes too thick to fold?

• During summarizing the exit slip results, “Exponents are repeated multiplying of a number. Multiplication is repeated adding of a number,” is a slogan that helps many students.

Rubric
As this is a formative assignment, participation is what will be graded here. Any student who stayed on task and turned in a complete exit slip (or signed their name to the results poster or was a cleanup person) should receive full participation credit for the day. If kept in an organized notebook, notes from the morning video can be graded on a later day.

5pts--Arrived on time, stayed on task, and participated with class.
5pts--Turned in correctly completed exit slip. (If students had time to discuss their answers at the end, you could require students to revise their answers and grade correctness more strictly. Don’t forget that some students were assigned cleanup or poster creation during this time, and those students should receive credit for those activities instead of the slip.)
Total= 10pts

You can also grade an activity like this with a rough “Plus, check, minus, zero,” format where a plus is worth 100% credit, a check is 75%, minus is 50%, and zero, 0%.

nick said...

Multiplying fractions with exponents?
Hi everyone, I'm watching an online instructor on MyMathLab going over rules of exponents. He does one thing near the end that I just don't get.

For the following problem...

8y^6/y^6 multiplied times y^2/25x^-4 (that's 8y to the sixth over y to the 6th times y squared over 25 x to the negative fourth)

Using the quotient rule for exponents he subtracts the negative four from 6 (6-(-4)) to make 8y^10th on top BUT he subtracts the y exponent from the numerator from the y exponent in the denominator? In other words instead of subtracting the y exponent of six from two, he does the opposite. His final solution is 8x^10/25y^4 while I came up with 8x^10/25y^-4 ...which would then lead me to put y^4 in the numerator to make 8x^10y^4/25

In fact he very quickly refers to it as a shortcut of sort but doesn't explain it, he just does it and blazes on to the next problem. I really want to understand this before I move ahead. Thanks :)

BTW if anyone is using MyMathLab for school (Intermediate college algebra). It is the Chapter 5, section 1 lecture at approx. 15:45 or so.

sgrape said...

Nick, first, is the problem (8x^6)/(y^6) times (y^2)/(25x^-4)? If so, then to combine the x factors, you would subtract the exponents (6 - (-4)) giving x^10 in the numerator. Applying the quotient rule for exponents to the y factors would give you y^(2-6) or y^-4 power in the numerator. Since you have a negative exponent on y in the numerator, that can be rewritten as y^4 in the denominator giving you a final answer of (8x^10)/(25y^4).

The rule x^a/x^b = x^(a-b) means x^(a-b) is in the numerator, or like x^(a-b) over 1. So when (a-b) is negative, then you really have x to the opposite of that power in the denominator.

Hope that makes sense!

Laura Bost said...

Hi Nick,
Thanks for your question. Could you check it for some typos in the problem you posted?

I believe you've accidentally swapped in a "y" in the numerator first fraction where an "x" was intended.

(8x^6/y^6)(y^2/25x^-4)= (8x^10)/(25y^4)

Is the correct solution. Since you are leaving your answer in the denominator, it is the exponent from the denominator that leads in the subtraction of the exponents, 6-2, not 2-6.

Does that make sense? When simplifying fractions with exponents, you lead the exponent subtraction with the exponent that came from the same place as where you plan to write your result.

An overall review of exponent rules is available here:
http://nroc.remote-learner.net/mod/scorm/player.php?a=653&currentorg=&scoid=1609

And a video solving the specific problem you requested (assuming I have correctly written down your question) is here: