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: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
problem_activity: Activity
In this activity, students are asked to interpret in context the slope and vertical intercept of a linear model given a scatter plot and the equation for a linear model that fits the data well. The linear model is also used to interpolate and extrapolate information about the data in context.
Monitor for students who:
estimate values based on the graph
use the equation for the linear model to find the values
Launch
Arrange students in groups of 2.
Student Facing
The scatter plot shows the sale price of several food items,
\(y\)
, and the cost of the ingredients used to produce those items,
\(x\)
, as well as a line that models the data. The line is also represented by the equation
\(y = 3.48x + 0.76\)
.
[IMG]
What is the predicted sale price of an item that has ingredients that cost \$1.50? Explain or show your reasoning.
What is the predicted ingredient cost of an item that has a sale price of \$7? Explain or show your reasoning.
What is the slope of the linear model? What does that mean in this situation?
What is the
\(y\)
-intercept of the linear model? What does this mean in this situation? Does this make sense?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Although the grid in the picture does not look like squares, each line in both the horizontal and vertical directions are 0.5 apart. Tell students who are confused to look at the values listed on the axes and identify the values attached to a few of the grid lines.
Activity Synthesis
The purpose of this discussion is for students to describe where to find the slope and vertical intercept from a scatter plot and interpret the values in terms of the context.
Select previously identified students to share in the order listed in the narrative. Ask students:
“Are the values obtained from looking at the graph close to the values calculated using the equation?” (Yes, the values are close.)
“Which method will you use to predict values based on the model?” (Maybe a mixture of both methods. Using the equation produces a more precise value from the model, but it is also important to look at the graph to get a sense of how well the model fits the data where I am predicting.)
“Why do you think the intercept for the model is not
\((0,0)\)
?” (Companies may still try to charge money for items that cost nothing to produce. They need to pay for the salaries of their workers, research and development, as well as other things that require the company to make money more than just the cost of making the item. It may also be the case that it does not make sense to interpret the
\(y\)
-intercept, since we have no evidence to believe that the same linear relationship will hold there.)
Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct.
Before students share their interpretations of the slope, present an ambiguous explanation. For example: “The slope means that as sales increase, the cost of ingredients increases.” Ask students to identify inaccuracies, critique the reasoning, and write a correct explanation. As students discuss with a partner, listen for students who identify and clarify the ambiguous language in the statement. Invite students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use to describe what happens to the sale price when the cost of ingredients increases by
$
1. This helps students evaluate, and improve on, the written mathematical arguments of others, as they clarify the meaning of slope in context.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Action and Expression: Internalize Executive Functions.
Provide students with a two-column graphic organizer. Label one column “graph” and another column “equation.” Ask students to transfer their answers into the corresponding column on their organizer. When the whole group shares, they can continue to use the organizer capturing the answer if anyone used the alternative method for each question.
Supports accessibility for: Language; Organization


aligns: no