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: 
Represent addition and subtraction with objects, fingers, mental images, drawings (Drawings need not show details, but should show the mathematics in the problem. (This applies wherever drawings are mentioned in the Standards.)), sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

Answer: 
Yes. This task involves representing addition with objects such as number cards and dice.

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 and division and of fractions to multiply and divide rational numbers.

Answer: 
Yes. This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).

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_pos_0

K

fl_problem_001097_pos_0

7

fl_problem_001894_pos_0

HS_0
standard_description: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
problem_activity: Task
“Exponential decay” describes a situation in which a quantity decreases following a certain pattern. We’re going to investigate a situation that decays exponentially. Read through steps 1-6 below and answer questions a-c before carrying out the investigation.
1. Get your hands on 30 6-sided dice and put them in a container.
2. Find a safe place where you can roll 30 dice at once, like a tray (or the container itself, if it is large enough).
3. Roll the remaining dice (or shake the container thoroughly).
4. Remove any dice that display a "1" and set them aside.
5. Write down how many dice remain.
6. Go back to step 3 and repeat. Stop when you have rolled the dice ten times.
Pre-investigation questions:
When you roll 30 dice, how many 1’s do you predict you are likely to roll? Why?
After you remove the 1’s on the first roll, what fraction of the original number of dice do you predict will remain?
How many times do you predict you might have to roll the dice and remove the 1’s before you get to a roll where you don’t roll any 1’s?
Conduct the Investigation:
Construct a table to record the number of 1’s and the number of dice remaining after each roll, as shown.
{"table": [{"tbody": [{"tr": [{"td": [{"p": [{"_value": "Roll"}]}, {"p": [{"_value": "0"}]}, {"p": [{"_value": "1"}]}, {"p": [{"_value": "2"}]}, {"p": [{"_value": "3"}]}, {"p": [{"_value": "4"}]}, {"p": [{"_value": "5"}]}, {"p": [{"_value": "6"}]}, {"p": [{"_value": "7"}]}, {"p": [{"_value": "8"}]}, {"p": [{"_value": "9"}]}, {"p": [{"_value": "10"}]}]}, {"td": [{"p": [{"_value": "Number of 1\u2019s"}]}, {"_value": "X"}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}]}, {"td": [{"p": [{"_value": "Number of Dice Remaining"}]}, {"p": [{"_value": "30"}]}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}]}]}]}]}
Now for some mathematical modeling:
Use technology to create a scatter plot with number of dice remaining on the vertical axis and roll number on the horizontal axis. Discuss what you see.
Based on what you know about exponential functions and the behavior of dice, explain in words why an exponential model would be appropriate for this situation.
Based on what you know about exponential functions and 6-sided dice, write a function of the form \(d(x)=ab^x\) to model the relationship between the roll number, x, and the number of dice remaining, d(x). Your job here is to decide what numbers to use in place of a and b. This should not be based on the data you collected -- it is a mathematical model for how dice behave.
Graph your function on the same set of axes as your scatter plot. Compare and contrast the graphed model d(x) to the outcome of the experiment shown in the scatter plot.
Suppose someone repeated this activity, except they started with 100 8-sided dice instead of 30 6-sided dice. Write a new function that would model this situation.
Here is a tool that could be used in this task:


aligns: no