Open Middle, Exponent Properties and Desmos Scientific

How do you help students learn exponent properties? I have shifted over time from presenting properties with practice to exploration using open middle problems. At the end of the year our classes take an assessment called the Geometry Readiness Test. In all four of my algebra sections the top area of strength for each of the classes was exponents. I was a little surprised. If anything I was concerned that students had spent little time practicing exponent properties on worksheets or homework. Instead we pushed the our depth of knowledge by exploring a series of open middle problems followed by a short skill practice. I am writing to share the set of problems we used.

I use this set of problems throughout an exponent properties unit. They were the “fire up” to open the class. I use desmos scientific. It is a great tool for students can keep track of previous results as they make new attempts.

Open Middle Problem Notes
Place the digits 1 to 9 in the boxes below. Each digit may only be used once. Find the largest result. Find the smallest result.

xp1

 

 

 

 

 

 

 

 

Use before introducing scientific notation.

These answers should generate the need for scientific notation. Students will see results in scientific notation. Expect students to be puzzled by the output. Pause the class early on to wonder about the format of the results they are seeing.

Which is larger?

1.88×1029 or  2.51×1027t

Which is smaller?

1.97×10-13 or 6.05×10-13

Wonder why it is better to have (749)(658) as opposed to (759)(648). Shouldn’t we make the number as large as possible for the 9? Hopefully a student wonders but if they don’t you should. What is happening here?

If you have time have students place  0, 2.51×1027 and 1.88×1029 on a number line

Use each of the digits 1 – 9 only once to create the largest and smallest product.

xp2

 

 

 

 

 

 

 

 

Useful before multiplying numbers in scientific notation and the product of powers property.

Compare

(7.53×109)(6.421×108)

(7.53×108)(6.421×109).
Why is the result the same? Shouldn’t the one with the 7.53 get the power of 9?

An opportunity to discuss the commutative and associative properties of multiplication.
There is also something to wonder about the remaining 7 digits and the arrangement that makes the largest or smallest product. What math is happening underneath here?

Use each of the digits 1 – 9 only once to create the largest and smallest power

xp3

 

 

Useful prior to negative powers and power of power properties
A chance to consider both negative powers and a power to a power.
Is there a way to rewrite this problem that would be helpful? Sometimes I see a student who does division instead of the negative powers.
What is the effect of the negative?
When going for the smallest, Why is it helpful to have a 9 as the outside power?
Use each of the digits 1 – 9 only once to create the largest and smallest product

xp4

 

An opportunity to consider fractional exponents before introducing the property.
What is the best strategy for large numbers?
How does the denominator effect the result?
Why can’t we find really small results?
Use each of the digits 1 – 9 only once to create the largest and smallest sum

xp5

 

 

 

 

 

 

 

 

 

Useful before encountering addition of numbers in scientific notation.

Listen for students noticing the relationship between the expression and the result. Ask what patterns they noticed.
8.7×109+5.4×106+2.1×103=8.7054021×109

What is going on with addition?

Going for smallest can lead to an interesting discussion.

8.9×101+6.7×102+4.5×103 has the same results when we flip the 5 and 6.

8.9×101+6.7×102+4.5×103= 8.9×101+5.7×102+4.6×103
Why? Are there other pairs I can flip and still get the same results?

Fill in the boxes using digits 1 through 9 to make the biggest 3 digit number. A digit can only be used once.

xp7

There are several other exponent problems at the open middle site. here are a few.

From Open Middle – Robert Kaplinsky

 

 

Fill in the boxes using digits 1 through 9 to make the biggest number. The digits in the power and the digits in the result must all be different.

xp8

Follow up to previous. Does it help to remove the restriction of three digits?

 

 

 

Find 3 positive integers that add up to 10. Place each number into one of the blanks to find the largest possible result.

xp6

From Open Middle – Zack Miller

Which three numbers? Which has the most impact?
If negative integers were allowed, would it be helpful?

 xp9 Not an open middle but expect a good debate and disequilibrium among students.
 xp10 Not an open middle. This AMC8 problem should generate some good discussion among groups.

Last year we used the first 5 open middle problems to open classes. We also used the bottom two to generate discussion. You might choose to do the largest one day and the smallest the next or have students choose whether they are going for large or small. In groups two students might go big and two small and then they collaborate together.

I realize that there is a downside to having 9 blank spots. When using open middle with equations we didn’t set up that many blanks. There are some very good problems on the open middle site that can meet the same goal with fewer blanks. I included a few that we have used previously. In the case of exponent properties, using all 9 digits helped generate really large numbers or numbers really close to zero forcing results in scientific notation.

Beyond open middle problems, I need to share a unit opening group 3-Act related to stacking paper to the moon and a unit closing activity followed by 3-act inspired by the book The King’s Chessboard.

What do you think? Where have you used open middle problems with students?

 

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