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: Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.
problem_activity: Activity
The purpose of this activity is for students to see that to find what percentage one number is of another, divide them and then multiply by 100. First they find what percentage of 20 various numbers are, then they organize everything into a table. Using the table, students then describe the relationship they see between dividing the numbers by 20 and finding the percentage (MP8).
Launch
Depending on students' strengths in grade 5 work, it may be beneficial to offer some strategies for writing a fraction in an equivalent decimal form before students start working. Display the fraction
\(\frac{15}{25}\)
and ask students how they might think about writing it in decimal form. Possible strategies are writing the equivalent fraction
\(\frac{60}{100}\)
and knowing that "60 hundredths" is written 0.60 or 0.6. Another possibility is to write the equivalent fraction
\(\frac35\)
, which is equal to
\(\frac{6}{10}\)
and knowing that "6 tenths" can be written as 0.6.
Give students quiet think time to complete the activity and then time to share their explanations with a partner.
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure students access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Speaking: MLR8 Discussion Supports.
To amplify mathematical uses of language to communicate about equivalent fractions and percentages (e.g., 15 minutes is 75% of 20 minutes, 75% of 20 minutes is equivalent to
\(\frac{15}{20}\)
), invite students to use these phrases when stating their ideas, revoicing and rephrasing as necessary.
Design Principle(s): Support sense-making, Optimize output (for explanation)
Student Facing
A school held a jump-roping contest. Diego jumped rope for 20 minutes.
Jada jumped rope for 15 minutes. What percentage of Diego’s time is that?
Lin jumped rope for 24 minutes. What percentage of Diego’s time is that?
Noah jumped rope for 9 minutes. What percentage of Diego’s time is that?
Record your answers in this table. Write the quotients in the last column as decimals.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}, "_value": "time (minutes)"}, {"_attributes": {"scope": "col"}, "_value": "percentage"}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "39226"}, "_value": "\\(\\text{time} \\div 20\\)"}]}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "Diego"}], "td": [{"_value": "20"}, {"_value": "100"}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "failed-conversion"}, "_value": "\\(\\frac{20}{20} = 1\\text{ \u00a0 }\\)"}]}]}, {"th": [{"_attributes": {"scope": "col"}, "_value": "Jada"}], "td": [{"_value": "15"}, {}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "failed-conversion"}, "_value": "\\(\\frac{15}{20} = \\text{ \u00a0 \u00a0 \u00a0 \u00a0 \u00a0}\\)"}]}]}, {"th": [{"_attributes": {"scope": "col"}, "_value": "Lin"}], "td": [{"_value": "24"}, {}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "failed-conversion"}, "_value": "\\(\\frac{24}{20} = \\text{ \u00a0 \u00a0 \u00a0 \u00a0 \u00a0}\\)"}]}]}, {"th": [{"_attributes": {"scope": "col"}, "_value": "Noah"}], "td": [{"_value": "9"}, {}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "failed-conversion"}, "_value": "\\(\\frac{9}{20} = \\text{ \u00a0 \u00a0 \u00a0 \u00a0 \u00a0}\\)"}]}]}]}]}]}
What do you notice about the numbers in the last two columns of the table?
Student Response
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Activity Synthesis
Have students share their strategies for the first three problems. Then ask what they noticed about the last two columns. There are many ways to formulate the relationship. Make sure everyone sees that to find what percentage a number is of 20, divide it by 20 and then multiply by 100. A table showing the percentage for 1 minute might also help:
{"table": [{"_attributes": {"border": "1", "cellpadding": "1", "cellspacing": "1", "style": "width:500px"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}, "_value": "time (minutes)"}, {"_attributes": {"scope": "col"}, "_value": "percentage"}]}]}], "tbody": [{"tr": [{"th": [{"_value": "Diego"}], "td": [{"_value": "20"}, {"_value": "100"}]}, {"th": [{"_value": "unit rate"}], "td": [{"_value": "1"}, {"_value": "5"}]}, {"th": [{"_value": "Jada"}], "td": [{"_value": "15"}, {"_value": "75"}]}, {"th": [{"_value": "Lin"}], "td": [{"_value": "24"}, {"_value": "120"}]}, {"th": [{"_value": "Noah"}], "td": [{"_value": "9"}, {"_value": "45"}]}]}]}]}
Note that to find the percentage for 1 minute, we divide 100 by 20, and to find the percentage for any number of minutes, we multiply the result by that number of minutes. That is the same as dividing the number of minutes by 20 and multiplying by 100.


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