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 following problem or activity 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: Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
problem_activity: Optional activity
This activity is optional. Use this activity to allow students to practice analyzing graphs of exponential functions involving
\(e\)
and setting an appropriate graphing window on their graphing technology in order to produce useful graphs.
Students have graphed exponential functions and analyzed the graphs to solve problems prior to this point. Though the functions here use base
\(e\)
, the analytical work is fundamentally the same as that when the base is another number. Students will also have opportunities to graph exponential functions with base
\(e\)
later in the unit.
Launch
Provide access to devices that can run Desmos or other graphing technology.
Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve their written responses to the first question. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “How did you get . . . ?”, “Why did you . . . ?”, and “How do you know . . . ?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify their reasoning for the population of bacteria 10 hours after the scientist put the initial bacteria in the dish.
Design Principle(s): Optimize output (for justification); Cultivate conversation
Student Facing
Use graphing technology to graph the function defined by
\(f(t) = 50 \boldcdot e^{(0.25t)}\)
. Adjust the graphing window as needed to answer these questions:
The function
\(f\)
models the population of bacteria in
\(t\)
hours after it was initially measured. About how many bacteria were in the dish 10 hours after the scientist put the initial 50 bacteria in the dish?
About how many hours did it take for the number of bacteria in the dish to double? Explain or show your reasoning.
Use graphing technology to graph the function defined by
\(p(t)=192 \boldcdot e^{(0.014t)}\)
. Adjust the graphing window as needed to answer these questions:
The equation models the population, in millions, in the U.S.
\(t\)
years after 1964. What does the model predict for the population of the U.S. in 1974?
In which year does the model predict the population will reach 300 million?
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?
Research what the population of the U.S. was in the year the model predicted 300 million people. How far off was the model? What factors do you think account for the actual population in that year being different from the prediction of the model? In what year did the U.S. actually reach 300 million people?
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
Focus the discussion on how students set the graphing window and how they used the graphs to answer the questions.
Depending on the graphing technology used, students may use the following strategies:
Visually estimate the value of the output given a certain input, or the input given an output.
Use a tracing tool to trace the graph and identify the coordinates when the input or output has a certain value.
To find an unknown input: graph a horizontal line at a given
\(y\)
-value (for example,
\(y=100\)
, a doubled bacteria population), find the intersection of that line and the graph of the exponential function, and identify the
\(x\)
-value at that intersection.
To find an unknown output: use a table or type something like
\(b(10)\)
after
\(b\)
was defined.
Some students may also notice that they could calculate the output value given an exponential equation and an input value, but had to rely on the graph to find an unknown input value because they were unsure how to solve for
\(x\)
when the equation has
\(e\)
as a base. Tell students that in a future lesson they will learn how to use logarithms to solve this type of equation algebraically similar to how they have used logarithms previously.


aligns: no