few_shots: Example:
Problem/activity:
Problem 1
The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions.
|                                                          | **           No solution          ** | **           Exactly 1 solution          ** | **           Infinite number of solutions          ** |
|----------------------------------------------------------|--------------------------------------|---------------------------------------------|-------------------------------------------------------|
|                       $${9=4x+7}$$                       |                                      |                                             |                                                       |
|                       $${y=3y+5}$$                       |                                      |                                             |                                                       |
|                    $${y={x\over2}}$$                     |                                      |                                             |                                                       |
|                     $${3x+6y-9=0}$$                      |                                      |                                             |                                                       |
|    $${x=y}$$                        $${1.25x=1.25y}$$    |                                      |                                             |                                                       |
| $${9.9=6x+8y}$$                        $${9x=2.5y-8.8}$$ |                                      |                                             |                                                       |
| $${11x-2y=1.5}$$                        $${11x-2y=2.5}$$ |                                      |                                             |                                                       |
Problem 2
Kim has a small container and a large container as shown.
[IMG]
It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container.
What is the size of each of the two containers?


Mathematical concept/skill options: 
A. Write expressions in equivalent forms to solve problems.
B. Understand properties of multiplication and the relationship between multiplication and division.
C. Interpret the structure of expressions.
D. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
E. Use polynomial identities to solve problems.
F. Understand the relationship between zeros and factors of polynomials.
G. Write and interpret numerical expressions.
H. Work with equal groups of objects to gain foundations for multiplication.
I. Perform arithmetic operations on polynomials.
J. Create equations that describe numbers or relationships
K. Generate and analyze patterns.
L. Analyze and solve linear equations and pairs of simultaneous linear equations.
M. Understand solving equations as a process of reasoning and explain the reasoning.
N. Apply and extend previous understandings of arithmetic to algebraic expressions.
O. Rewrite rational expressions.
P. Represent and solve problems involving multiplication and division.
Q. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
R. Reason about and solve one-variable equations and inequalities.
S. Understand and apply properties of operations and the relationship between addition and subtraction.
T. Understand the connections between proportional relationships, lines, and linear equations.
U. Multiply and divide within 100.
V. Gain familiarity with factors and multiples.
W. Use the four operations with whole numbers to solve problems.
X. Work with addition and subtraction equations.
Y. Represent and solve problems involving addition and subtraction.
Z. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
AA. Solve equations and inequalities in one variable.
AB. Solve systems of equations.
AC. Represent and analyze quantitative relationships between dependent and independent variables.
AD. Work with radicals and integer exponents.
AE. Represent and solve equations and inequalities graphically.
AF. Analyze patterns and relationships.
AG. Use properties of operations to generate equivalent expressions.
AH. Add and subtract within 20.


Your response:
Thought: This set of problems involves solving linear equations and pairs of linear equations.
Answer: L

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_001392

8
options: Extend the properties of exponents to rational exponents.

Represent complex numbers and their operations on the complex plane.

Perform operations on matrices and use matrices in applications.

Compute fluently with multi-digit numbers and find common factors and multiples.

Use complex numbers in polynomial identities and equations.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Use properties of rational and irrational numbers.

Perform operations on vectors.

Reason quantitatively and use units to solve problems.

Perform arithmetic operations with complex numbers.

Represent and model with vector quantities.

Apply and extend previous understandings of numbers to the system of rational numbers.
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
Teachers with a valid work email address can
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: 11