options: Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.
problem_activity: Activity
The purpose of this activity is for students to build fluency expressing the product of two complex numbers in the form
\(a+bi\)
, where
\(a\)
and
\(b\)
are real numbers. In order to do this, students must use the fact that
\(i^2=\text-1\)
.
Look for students who answer the last question as
\(13+0i\)
as opposed to 13 to highlight during discussion.
Representation: Internalize Comprehension.
Activate or supply background knowledge. Display the table provided in the launch. As students share their explanations, use color-coding and annotations to highlight the connections between the expression, the values in the table, and the final product after combining like terms. Students can refer back to the chart throughout the activity.
Supports accessibility for: Visual-spatial processing
Launch
Tell students that they are now going to multiply complex numbers together. Display the expression
\((3 + 2i)(\text- 4 - 5i)\)
for all to see and give students 1 minute of quiet think time to consider how they would find the product. Then, display this table for all to see:
|                            | 3 | \(\boldsymbol{2i}\) |
|----------------------------|---|---------------------|
|             -4             |   |                     |
| \(\boldsymbol{\text- 5i}\) |   |                     |
After a brief time to consider the diagram, select students to explain where each of the values came from. Ask students, “Now that we have
\(\text- 12 - 8i - 15i - 10i^2\)
, what do we do in order to write the number in the form
\(a+bi\)
?” (We know
\(i^2 = \text-1\)
, so the sum of these is
\(\text- 12 - 8i - 15i +10 = \text- 2 - 23i\)
.)
Student Facing
Write each product in the form
\(a + bi\)
, where
\(a\)
and
\(b\)
are real numbers.
\((\text- 3 + 9i)(5i)\)
\((8 + i)(\text- 5 + 3i)\)
\((3 + 2i)^2\)
\((3 + 2i)(3 - 2i)\)
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
On October 16, 1843, while walking across the Broom Bridge in Dublin, Ireland, Sir William Rowan Hamilton came up with an idea for numbers that would work sort of like complex numbers. Instead of just the number
\(i\)
(and its opposite
\(\text- i\)
) squaring to give -1, he imagined three numbers
\(i\)
,
\(j\)
, and
\(k\)
(each with an opposite) that squared to give -1.
The way these numbers multiplied with each other was very interesting.
\(i\)
times
\(j\)
would give
\(k\)
,
\(j\)
times
\(k\)
would give
\(i\)
, and
\(k\)
times
\(i\)
would give
\(j\)
. But the multiplication he imagined did not have a commutative property. When those numbers were multiplied in the opposite order, they’d give the opposite number. So
\(j\)
times
\(i\)
would give
\(\text- k\)
,
\(k\)
times
\(j\)
would give
\(\text- i\)
, and
\(i\)
times
\(k\)
would give
\(\text- j\)
. A
quaternion
is a number that can be written in the form
\(a+bi+cj+dk\)
where
\(a\)
,
\(b\)
,
\(c\)
, and
\(d\)
are real numbers.
Let
\(w=2+3i-j\)
and
\(z=2i+3k\)
. Write each given expression in the form
\(a+bi+cj+dk\)
.
\(w+z\)
\(wz\)
\(zw\)
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
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
\(i^2=\text-1\)
to write products in the form
\(a+bi\)
, where
\(a\)
and
\(b\)
are real numbers.
Select previously identified students to share their responses to the last question, for which
\(13+0i\)
is the technically correct response. Discuss the idea that numbers like 13 or
\(5i\)
don’t need to be written as
\(13+0i\)
and
\(0+5i\)
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
\(a+bi\)
where
\(b=0\)
.


correct_option_index: 8