few_shots: 
Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
[IMG]
Counters, 10 per pair of students
Decks of number cards 0-5 (with 4 of each number) or two 0-5 dice
Setup
This activity is done in pairs. The students sit side by side with 10-frame and deck of cards or dice. If they have cards, the cards are shuffled and placed face down in front of them.
Action
Player 1 flips a card or rolls the die. They place that many counters on the 10-frame. Player 2 flips a card or rolls the die. They place that many more counters on the 10-frame next to the counters already on the ten frame. The students count the total together.
A recording sheet can be added when the students have been introduced to equations and number sentences. For those students who can handle higher numbers, a deck of 0-10 cards and two 10-frames can be used.
This game can also be played alone.


Concept/skill: 
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

Answer: 
No. Though this task involves addition, it does not involve any word problems.

Example 2:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Concept/skill: 
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

Answer: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

Example 3:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Answer: 
Yes. An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: im_task_001477_neg_0

K

fl_problem_001097_neg_0

7

fl_problem_001894_pos_0

HS_0
standard_description: Solve quadratic equations with real coefficients that have complex solutions.
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:
{"table": [{"_attributes": {"align": "center", "border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}, {"_attributes": {"scope": "col"}, "_value": "3"}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math"], "data-png-file-id": "24901"}, "_value": "\\(\\boldsymbol{2i}\\)"}]}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "_value": "-4"}], "td": [{"_value": "-12"}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "24137"}, "_value": "\\(\\text- 8i\\)"}]}]}, {"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math"], "data-png-file-id": "24916"}, "_value": "\\(\\boldsymbol{\\text- 5i}\\)"}]}], "td": [{"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "24931"}, "_value": "\\(\\text- 15i\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "24946"}, "_value": "\\(\\text- 10i^2\\)"}]}]}]}]}]}
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
\(b=0\)
.


aligns: no