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 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 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_pos_0

7

fl_problem_001894_pos_0

HS_0
standard_description: Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
problem_activity: Activity
This activity gives students the opportunity to see that an estimate of the probability for an event should be close to what is expected from the exact probability in the long-run; however, the outcome for a chance event is not guaranteed and estimates of the probability for an event using short-term results will not usually match the actual probability exactly (MP6).
Launch
Tell students that the probability of a coin landing heads up after a flip is
\(\frac{1}{2}\)
.
Give students 5 minutes of quiet work time followed by a whole-class discussion.
Student Facing
For each situation, do you think the result is surprising or not? Is it possible? Be prepared to explain your reasoning.
You flip the coin once, and it lands heads up.
You flip the coin twice, and it lands heads up both times.
You flip the coin 100 times, and it lands heads up all 100 times.
If you flip the coin 100 times, how many times would you expect the coin to land heads up? Explain your reasoning.
If you flip the coin 100 times, what are some other results that would not be surprising?
You’ve flipped the coin 3 times, and it has come up heads once. The cumulative fraction of heads is currently
\(\frac{1}{3}\)
. If you flip the coin one more time, will it land heads up to make the cumulative fraction
\(\frac{2}{4}\)
?
Student Response
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Activity Synthesis
The purpose of the discussion is for students to recognize that the actual results from repeating an experiment should be close to the expected probability, but may not match exactly.
For the first problem, ask students to indicate whether or not they think each result seems surprising. For the second and third questions, select several students to provide answers and display for all to see, then create a range of values that might not be surprising based on student responses. Ask the class if they agree with this range or to provide a reason the range is too large. It is not important for the class to get exact values, but a general agreement should arise that
some
range of values makes sense so that there does not need to be exactly 50 heads from the 100 flips.
An interesting problem in statistics is trying to define when things get "surprising." Flipping a fair coin 100 times and getting 55 heads should not be surprising, but getting either 5 or 95 heads probably is. (Although there is not a definite answer for this, a deeper study of statistics using additional concepts in high school or college can provide more information to help choose a good range of values.)
Explain that a probability represents the expected likelihood of an event occurring for a single trial of an experiment. Regardless of what has come before, each coin flip should still be equally likely to lands heads up as tails up.
As another example: A basketball player who tends to make 75% of his free throw shots will probably make about
\(\frac{3}{4}\)
of the free throws he attempts, but there is no guarantee he will make any individual shot even if he has missed a few in a row.
Conversing, Reading: MLR2 Collect and Display.
As students share whether each result is surprising or not, write down the words and phrases students use to explain their reasoning. Listen for students who state that the actual results from repeating an experiment should be close to the expected probability. As students review the language collected in the visual display, encourage students to revise and improve how ideas are communicated. For example, a phrase such as: “If you flip a coin 100 times, it is impossible for the coin to land heads up all 100 times” can be improved with the phrase “If you flip a coin 100 times, it is very unlikely for the coin to land heads up all 100 times, but it is possible.” This routine will provide feedback to students in a way that supports sense-making while simultaneously increasing meta-awareness of language.
Design Principle(s): Support sense-making; Maximize meta-awareness


aligns: yes