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 to multiply a fraction or whole number by a fraction.

Answer: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

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_neg_0

7

fl_problem_001894_pos_0

HS_0
standard_description: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
problem_activity: Activity
Students have just learned the meaning of the
\(y\)
-intercept for a line and have been interpreting slope in context. In this activity, they investigate the
\(y\)
-intercept and slope together and investigate what happens when their values are switched.
In contexts like this one, the
\(y\)
-intercept and the slope come with natural units and understanding this can help graph accurately. Specifically, the
\(y\)
-intercept is the number of pages Lin read before she starts gathering data for the graph. The slope, on the other hand, is a rate: it’s the number of pages Lin reads
per day
.
Watch for students who understand the source of Lin’s error (confusing the
\(y\)
-intercept with the slope) and invite them to share this observation during the discussion.
Launch
Work time followed by whole-class discussion.
Representation: Internalize Comprehension.
Use MLR6 Three Reads to supporting reading comprehension of the word problem. Use the first read to orient students to the situation. Ask students to describe what the situation is about without using numbers (Lin’s reading assignment). Use the second read to identify quantities and relationships. Ask students what can be counted or measured without focusing on the values. Listen for and amplify the important quantities that vary in relation to each other in this situation: first 30 pages and 40 pages each day. After the third read, ask students to brainstorm possible strategies to answer the question, “What does the vertical intercept mean in this situation?”
Supports accessibility for: Language; Conceptual processing
Speaking: MLR8 Discussion Supports.
Use this to amplify mathematical uses of language to communicate about vertical intercepts, slope, and constant rate. Invite students to use these words when describing their ideas. Ask students to chorally repeat phrases that include these words in context.
Design Principle(s): Support sense-making, Optimize output (for explanation)
Student Facing
Lin has a summer reading assignment. After reading the first 30 pages of the book, she plans to read 40 pages each day until she finishes. Lin makes the graph shown here to track how many total pages she'll read over the next few days.
After day 1, Lin reaches page 70, which matches the point
\((1,70)\)
she made on her graph. After day 4, Lin reaches page 190, which does not match the point
\((4,160)\)
she made on her graph. Lin is not sure what went wrong since she knows she followed her reading plan.
[IMG]
Sketch a line showing Lin's original plan on the axes.
What does the
vertical intercept
mean in this situation? How do the vertical intercepts of the two lines compare?
What does the slope mean in this situation? How do the slopes of the two lines compare?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Jada's grandparents started a savings account for her in 2010. The table shows the amount in the account each year.
If this relationship is graphed with the year on the horizontal axis and the amount in dollars on the vertical axis, what is the vertical intercept? What does it mean in this context?
| year | amount in dollars |
|------|-------------------|
| 2010 |        600        |
| 2012 |        750        |
| 2014 |        900        |
| 2016 |       1050        |
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Ask students:
“How did your graph compare to Lin’s?” (It is steeper but starts off at 30 instead of 40 pages.)
“Which point does your graph have in common with Lin’s?” (The point
\((1,70)\)
, 70 pages read after one day.)
Invite selected students to share what’s the likely source of Lin’s error (confusing the
\(y\)
-intercept with the slope). In this context, the
\(y\)
-intercept is the number of pages Lin read before she starts counting the days (30), and the slope is the number of pages Lin reads per day (40).
Emphasize how the
\(y\)
-intercept and slope influence the graph of a line.
The
\(y\)
-intercept indicates where the line touches or crosses the
\(y\)
-axis.
The slope indicates how steep the line is.


aligns: yes