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 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_pos_0

7

fl_problem_001894_pos_0

HS_0
standard_description: Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
problem_activity: Activity
The purpose of this activity is to understand that discrete values in a table can be used as evidence that a relationship is proportional and can be used to know for sure that a relationship is not proportional, but can’t be used to know for sure whether a relationship is definitely proportional. This activity builds on previous ones involving constant speed, but analyzes pace (minutes per lap) rather than speed (laps per minute). Explaining why the information given in the table is enough to conclude that Han didn't run at a constant pace but is not enough to know for sure whether Clare ran at a constant pace requires students to make a viable argument (MP3).
Launch
Keep students in the same groups of 2. Give 5 minutes of quiet work time, followed by 5 minutes of students discussing responses with a partner, followed by a whole-class discussion.
Conversing, Reading: MLR5 Co-Craft Questions.
Use this routine to help students interpret the language of proportional relationships, and to increase awareness of language used to talk about proportional relationships. Display the first sentence of this problem (“Han and Clare were running laps around the track. The coach recorded their times at the end of laps 2, 4, 6, and 8.”) and the tables of Han and Clare’s run, and ask students to write down possible mathematical questions that could be asked about the situation. If needed, provide students with the question starter “Why is. . .?” Invite students to share their questions with the class before revealing the task questions. Listen for and amplify any questions involving proportional relationships or mathematical language, such as “constant pace,” “not constant,” “proportional,” or “constant of proportionality.” This will help students produce the language of mathematical questions and start noticing whether or not there is a proportional relationship in this task prior to starting the task.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
Han and Clare were running laps around the track. The coach recorded their times at the end of laps 2, 4, 6, and 8.
Han's run:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "distance (laps)"}, {"_attributes": {"scope": "col"}, "_value": "time (minutes)"}, {"_attributes": {"scope": "col"}, "_value": "minutes per lap"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "2"}, {"_value": "4"}, {}]}, {"td": [{"_value": "4"}, {"_value": "9"}, {}]}, {"td": [{"_value": "6"}, {"_value": "15"}, {}]}, {"td": [{"_value": "8"}, {"_value": "23"}, {}]}]}]}]}
Clare's run:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "distance (laps)"}, {"_attributes": {"scope": "col"}, "_value": "time (minutes)"}, {"_attributes": {"scope": "col"}, "_value": "minutes per lap"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "2"}, {"_value": "5"}, {}]}, {"td": [{"_value": "4"}, {"_value": "10"}, {}]}, {"td": [{"_value": "6"}, {"_value": "15"}, {}]}, {"td": [{"_value": "8"}, {"_value": "20"}, {}]}]}]}]}
Is Han running at a constant pace? Is Clare? How do you know?
Write an equation for the relationship between distance and time for anyone who is running at a constant pace.
Student Response
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Anticipated Misconceptions
Students are likely to answer that Clare is running at a constant pace because the minutes per lap shown in the table are the same for each lap. Because we only have four data points in a table, spaced at 5-minute intervals, Clare could still be speeding up and slowing down between the recorded times. However, given the data, it is reasonable to assume Clare is running at a constant pace for the purpose of estimating times or distances.
Activity Synthesis
Invite students to explain why they think each person is or is not running at a constant pace. Point out to students that although the data points in the table for Clare are pairs in a proportional relationship, these four pairs of values do not guarantee that Clare ran at a constant pace. She might have, but we don't know if she was running at a constant pace between the times that the coach recorded.
Ask the following questions:
"Can you represent either relationship with an equation?" (The answer for Han is “no” and the answer for Clare is “yes, if she really ran at a constant pace between the points in time when the times were recorded.” Write the equation for Clare together:
\(t = 2.5 d\)
.)
“Are the pairs of values in the table for Clare's run still values from a proportional relationship if we calculate laps per minute instead of minutes per lap? How does that change the equation?” (Yes,
\(d = 0.4 t\)
.)


aligns: no