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 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: 
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_neg_0

7

fl_problem_001894_neg_0

HS_0
standard_description: Classify objects into given categories; count the numbers of objects in each category and sort the categories by count. (Limit category counts to be less than or equal to 10.)
problem_activity: Activity
Note: if it is possible in your local environment, we recommend using the digital version of this activity.
This activity is intended to help students identify correspondences between parts of a table, graph, and an equation of a line through the origin in the first quadrant. Students are guided to notice:
Any pair of positive values
\((a, b)\)
determine a proportional relationship.
Given a point
\((a,b)\)
other than the origin on the graph of a line through the origin, the constant of proportionality is always
\(\frac{b}{a}\)
.
In an equation
\(y=\frac{b}{a} x\)
that represents the relationship, the constant of proportionality appears as the coefficient of
\(x\)
.
The constant of proportionality is the
\(y\)
-coordinate when
\(x\)
is 1, that is,
\((1,\frac{b}{a})\)
is a point on the graph.
In the print version, students plot one point and draw a ray that starts at the origin and passes through their point. They also create a table and an equation to represent the relationship. They respond to a series of questions about their representations. Then, they compare their representations of their relationship with the representations of other relationships created by members of their group.
In the digital version (recommended), students interact with a dynamic sketch while recording observations and responding to prompts. In the sketch, students are able to manipulate the graph of a proportional relationship, while changes to an associated table and equation automatically update.
Digital
Launch
For the print version: Arrange students in groups of 3. Assign each student in each group a letter: A, B, or C. Provide access to rulers.
For the digital version: Arrange students in groups of 2–3 and have them complete the task.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention
Speaking, Writing: MLR2 Collect and Display.
While pairs or groups are working, circulate and listen to student talk about the connections they see between the tables, characteristics of the graphs, and the equations. Write down common or important phrases you hear students say about the connections onto a visual display. Throughout the remainder of the lesson, continue to update collected student language and remind students to borrow language from the display as needed. This will help students read and use mathematical language during paired and whole-class discussions.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Student Facing
Explore the graph. Start by dragging the gray bar on the left across the screen until you can see both the table and the graph. Notice the values in the table and the coordinates of the labeled point. Grab the point and move it around.
What stays the same and what changes in the table? in the equation? on the graph?
Choose one row in the table and write it here. To what does this row correspond on the graph?
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "50"}, "_value": "\\(x\\)"}]}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "29"}, "_value": "\\(y\\)"}]}]}]}], "tbody": [{"tr": [{"td": [{}, {}]}]}]}]}
Grab and drag the point until you see the equation
\(y=\frac 32 x\)
.
Do not move the point. Choose three rows from the table, other than the origin. Record
\(x\)
and
\(y\)
, and compute
\(\frac yx\)
.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "50"}, "_value": "\\(x\\)"}]}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "29"}, "_value": "\\(y\\)"}]}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "9657"}, "_value": "\\(\\frac{y}{x}\\)"}]}]}]}], "tbody": [{"tr": [{"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}]}]}]}
What do you notice? What does this have to do with the equation of the line?
Do not move the point. Check the box to view the coordinates
\((1, ?)\)
. What are the coordinates of this point? What does this correspond to in the table? What does this correspond to in the equation?
Drag the point to a different location. Record the equation of the line, the coordinates of three points, and the value of
\(\frac yx\)
.
Equation of the line: _______________________________
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "50"}, "_value": "\\(x\\)"}]}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "29"}, "_value": "\\(y\\)"}]}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "9657"}, "_value": "\\(\\frac{y}{x}\\)"}]}]}]}], "tbody": [{"tr": [{"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}, {"td": [{}, {}, {}]}]}]}]}
Based on your observations, summarize any connections you see between the table, characteristics of the graph, and the equation.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
For the print version: Arrange students in groups of 3. Assign each student in each group a letter: A, B, or C. Provide access to rulers.
For the digital version: Arrange students in groups of 2–3 and have them complete the task.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention
Speaking, Writing: MLR2 Collect and Display.
While pairs or groups are working, circulate and listen to student talk about the connections they see between the tables, characteristics of the graphs, and the equations. Write down common or important phrases you hear students say about the connections onto a visual display. Throughout the remainder of the lesson, continue to update collected student language and remind students to borrow language from the display as needed. This will help students read and use mathematical language during paired and whole-class discussions.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Student Facing
Your teacher will assign you
one
of these three points:
\(A=(10,4)\)
,
\(B=(4,5)\)
,
\(C=(8,5)\)
.
[IMG]
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math", "math-repaired"]}, "_value": "\\(x\\)"}]}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math", "math-repaired"]}, "_value": "\\(y\\)"}]}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math", "math-repaired"]}, "_value": "\\(\\frac{y}{x}\\)"}]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "0"}, {}, {"_value": "NA"}]}, {"td": [{"_value": "1"}, {}, {}]}, {"td": [{"_value": "2"}, {}, {}]}, {"td": [{"_value": "3"}, {}, {}]}, {"td": [{"_value": "4"}, {}, {}]}, {"td": [{"_value": "5"}, {}, {}]}, {"td": [{"_value": "6"}, {}, {}]}, {"td": [{"_value": "7"}, {}, {}]}, {"td": [{"_value": "8"}, {}, {}]}, {"td": [{"_value": "9"}, {}, {}]}, {"td": [{"_value": "10"}, {}, {}]}]}]}]}
On the graph, plot and label
only
your assigned point.
Use a ruler to line up your point with the origin,
\((0,0)\)
. Draw a line that starts at the origin, goes through your point, and continues to the edge of the graph.
Complete the table with the coordinates of points on your graph. Use a fraction to represent any value that is not a whole number.
Write an equation that represents the relationship between
\(x\)
and
\(y\)
defined by your point.
Compare your graph and table with the rest of your group. What is the same and what is different about:
your tables?
your equations?
your graphs?
What is the
\(y\)
-coordinate of your graph when the
\(x\)
-coordinate is 1? Plot and label this point on your graph. Where do you see this value in the table? Where do you see this value in your equation?
Describe any connections you see between the table, characteristics of the graph, and the equation.
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?
The graph of an equation of the form
\(y = kx\)
, where
\(k\)
is a positive number, is a line through
\((0,0)\)
and the point
\((1,k)\)
.
Name at least one line through
\((0,0)\)
that cannot be represented by an equation like this.
If you could draw the graphs of
all
of the equations of this form in the same coordinate plane, what would it look like?
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
When all students have completed the activity, ask them to share their responses with a partner or small group. After a minute, go around the room asking each group to share one thing from the last question, and display these for all to see. When each group has shared, ask if there were any important observations that were missed.
Ensure that all the important connections are highlighted:
A graph of a line through the origin and passing through the first quadrant represents a proportional relationship.
The value of
\(\frac ba\)
computed from any point
\((a,b)\)
on that line (other than the origin) is the constant of proportionality.
An equation of the relationship is given by
\(y = kx\)
where
\(k\)
is
\(\frac ba\)
for any point
\((a,b)\)
on the graph other than the origin.


aligns: no