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: 
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Answer: 
No. This problem does not directly teach how side ratios in right triangles lead to definitions of trigonometric ratios. Instead, it involves solving a trigonometric function.

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_neg_0

HS_0
standard_description: Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
problem_activity: Activity
This activity revisits a context seen previously. Students solved problems like this as early as grade 3 without formulating them in terms of ratios or rates (“If 1 cup of rice serves 3 people, how many people can you serve with 12 cups of rice?”). In this activity, they ultimately find an equation for the proportional relationship.
As students find missing values in the table, they should see that they can always multiply the number of food items by the constant of proportionality. When students see this pattern (MP7) and represent the number of people served by
\(x\)
cups of rice (or
\(s\)
spring rolls) as
\(3x\)
(or
\(\frac{1}{2}s\)
), they are expressing regularity in repeated reasoning (MP8).
Only one row in each table is complete. Based on their experience in the previous lesson, students are more likely to multiply the entries in the left-hand column by 3 (or
\(\frac{1}{2}\)
) than to use scale factors, at least for the first and third rows. If they do, they are more likely to see how to complete the last row in each table.
Some students might use unit rates: If 1 cup of rice can serve 3 people, then
\(x\)
cups of rice can serve
\(3x\)
people. So, students have different ways to generate
\(3x\)
as the expression that represents the number of people served by
\(x\)
cups of rice: completing the table for numerical values and continuing the pattern to the last row; or finding the unit rate and using it in the case of
\(x\)
cups.
Monitor students for different approaches as they are working.
Launch
Tell students that this activity revisits a context they worked on in an earlier lesson.
Student Facing
A recipe says that 2 cups of dry rice will serve 6 people. Complete the table as you answer the questions. Be prepared to explain your reasoning.
How many people will 1 cup of rice serve?
How many people will 3 cups of rice serve? 12 cups? 43 cups?
How many people will
\(x\)
cups of rice serve?
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "cups of dry rice"}, {"_attributes": {"scope": "col"}, "_value": "number of people"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {}]}, {"td": [{"_value": "2"}, {"_value": "6"}]}, {"td": [{"_value": "3"}, {}]}, {"td": [{"_value": "12"}, {}]}, {"td": [{"_value": "43"}, {}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "50"}, "_value": "\\(x\\)"}]}, {}]}]}]}]}
A recipe says that 6 spring rolls will serve 3 people. Complete the table as you answer the questions. Be prepared to explain your reasoning.
How many people will 1 spring roll serve?
How many people will 10 spring rolls serve? 16 spring rolls? 25 spring rolls?
How many people will
\(n\)
spring rolls serve?
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "number of spring rolls"}, {"_attributes": {"scope": "col"}, "_value": "number of people"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {}]}, {"td": [{"_value": "6"}, {"_value": "3"}]}, {"td": [{"_value": "10"}, {}]}, {"td": [{"_value": "16"}, {}]}, {"td": [{"_value": "25"}, {}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "749"}, "_value": "\\(n\\)"}]}, {}]}]}]}]}
How was completing this table different from the previous table? How was it the same?
Student Response
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Anticipated Misconceptions
If students have trouble encapsulating the relationship with an expression, encourage them to draw diagrams or to verbalize the relationship in words.
Activity Synthesis
Select students who have used the following approaches in the given order:
Recognize that to move from the first column to the second, you multiply by 3.
Say that 3 is the constant of proportionality.
Recognize that the equation
\(2 \boldcdot  {?} = 6\)
can be used to find the constant of proportionality algebraically.
Say that the 3 can be interpreted as the number of people per 1 cup of rice.
If no student sees that these insights are connected to prior work, explicitly connect them with the lessons of the previous two days. At the end of the discussion, suggest to students that we let
\(y\)
represent the number of people who can be served by
\(x\)
cups of rice. Ask students to write an equation that gives the relationship of
\(x\)
and
\(y\)
. (This builds on work from grade 6, but it may be rusty.) Be sure to write the equation
\(y = 3x\)
where all students can see it and help students interpret its meaning in the context: “To find
\(y\)
, the number of people served, we can multiply the number of cups of rice,
\(x\)
, by 3.”
Representation: Internalize Comprehension.
Use color and annotations to illustrate student thinking. As students describe their approaches, use color and annotations to scribe their thinking on a display.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Conversing: MLR7 Compare and Connect.
As students explain their different approaches to completing the tables, ask “What is similar, what is different?” about their methods. Draw students’ attention to the different ways the constant of proportionality was represented across the different strategies. For instance, when discussing first question, ask “Where do you see the 3 in each approach?” These exchanges strengthen students’ mathematical language use and reasoning based on constant of proportionality.
Design Principle(s): Cultivate conversation; Maximize meta-awareness


aligns: no