In this partner activity, students take turns matching equivalent expressions using the fact that . They build on a previous activity in which they multiplied imaginary numbers and saw how this changed the numbers’ representation on the complex plane. Fluency with strategically using the fact that will be an important skill in the next activity when students multiply numbers that have both real and imaginary parts. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and to critique the reasoning of others (MP3).

If students get stuck with the expressions or , consider saying:

• “What powers of do you already know a value for?”

• “How could writing out the repeated factors, for example, writing as , help you to match with an equivalent expression?”

Once all groups have completed the matching, here are some questions for discussion: 

• “Which matches were tricky? Explain why.” ( was tricky because there were several negative signs to keep track of and I had to double-check whether the final answer was negative or positive.)
• “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?” (I thought was -9 because I thought the 3 was squared too. I had to go back and just square the and then multiply it by 3.)

Ask groups to explain their reasoning for several matches, especially why . If not brought up by students, make sure to discuss that it’s possible to use the fact that to make equivalent expressions.
 

The purpose of this activity is for students to build fluency expressing the product of two complex numbers in the form , where and are real numbers. In order to do this, students must use the fact that .

Look for students who answer the last question as and students who answer 13, and highlight these during discussion.

The key takeaway is that the product of complex numbers is another complex number, and we can see this by using usual arithmetic along with the fact that to write products in the form , where and are real numbers.

Select previously identified students to share their responses to the last question. Discuss the idea that numbers like 13 or don’t need to be written as and in order to be recognizable as complex numbers. Writing complex numbers as a single term is okay; it’s something they did for a long time before they knew that all real numbers are complex numbers of the form where . It can be helpful to be flexible in writing numbers in either format. For example, when using the strategy to multiply complex numbers using tables, the question here using will match the other questions better if it is first written as .

If there is time, it can be helpful to compare multiplying complex numbers to multiplying two-digit numbers. Display the expression and invite students to share how they would do the multiplication without a calculator. Display their thinking for all to see. Most methods will involve some form of partial products such as , , , and , which result from distributing from the expression . Point out that these partial products can be understood as “8 hundreds, 12 tens, 10 tens, and 15 ones.” To find the sum of these to get the actual product of , we might need to rewrite some of these values in other ways, such as rewriting 12 tens as 1 hundred and 2 tens. Therefore the sum of the partial products is 10 hundreds, 3 tens, and 5 ones or 1,035.

The whole process is similar to finding a product like . First, we find the partial products: . Then, we rewrite the last partial product to find the sum. .