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: Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
problem_activity: Activity
This activity continues studying dilations on a circular grid, this time focusing on what happens to points lying on a polygon. Students first dilate the vertices of a polygon as in the previous activity. Then they examine what happens to points on the sides of the polygon. They discover that when these points are dilated, they all lie on a side of another polygon. Just as the image of a grid circle is another circle, so the dilation of a polygon is another polygon. Moreover the dilated polygon is a scaled copy of the original polygon. These important properties of dilations are not apparent in the definition.
Monitor for students who notice that the sides of the scaled polygon
\(A'B'C'D'\)
are parallel to the sides of
\(ABCD\)
and that
\(A'B'C'D'\)
is a scaled copy of
\(ABCD\)
with scale factor 2. Also monitor for students who notice the same structure for
\(EFGH\)
except this time the scale factor is
\(\frac{1}{2}\)
. Invite these students to share during the discussion.
Digital
Launch
Provide access to geometry toolkits. Tell students that they are going to dilate some points. Before they begin, demonstrate the mechanics of dilating a point using a center of dilation and a scale factor. Tell students, “In the previous activity, each point was dilated to its image using a scale factor of 3. The dilated point was three times as far from the center as the original point. When we dilate point
\(D\)
using
\(P\)
as the center of dilation and a scale factor of 2, that means we’re going to take the distance from
\(P\)
to
\(D\)
and place a new point on the ray
\(PD\)
twice as far away from
\(P\)
.” Display for all to see:
[IMG]
If using the digital activity, demonstrate the mechanics of dilating using the applet. You can also use the measurement tool to confirm.
Representation: Internalize Comprehension.
Begin with a physical demonstration of the process of dilating a point using a center of dilation and a scale factor to support connections between new situations and prior understandings. Consider using these prompts: “How does this build on the previous activity in which the main task was to find distances and scale factor?” or “How does the point D’ correspond to the points D and P?”
Supports accessibility for: Conceptual processing; Visual-spatial processing
Conversing, Reading: MLR2 Collect and Display.
Circulate and listen to students as they make observations about the polygon with a scale factor of 2 and the polygon with a scale factor of
\(\frac12\)
. Write down the words and phrases students use to compare features of the new polygons to the original polygon. As students review the language collected in the visual display, encourage students to clarify the meaning of a word or phrase. For example, a phrase such as “the new polygon is the same as the original polygon but bigger” can be clarified with the phrase “the new polygon is a scaled copy with scale factor 2 of the original polygon.” A phrase such as “the polygons have the same angles” can be clarified with the phrase “each angle in the original polygon is the same as the corresponding angle in the new polygon.” This routine will provide feedback to students in a way that supports sense-making while simultaneously increasing meta-awareness of language.
Design Principle(s): Support sense-making; Maximize meta-awareness
Student Facing
Here is a polygon
\(ABCD\)
Dilate each vertex of polygon
\(ABCD\)
using
\(P\)
as the center of dilation and a scale factor of 2.
Draw segments between the dilated points to create a new polygon.
What are some things you notice about the new polygon?
Choose a few more points on the sides of the original polygon and transform them using the same dilation. What do you notice?
Dilate each vertex of polygon
\(ABCD\)
using
\(P\)
as the center of dilation and a scale factor of
\(\frac{1}{2}\)
.
What do you notice about this new polygon?
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
Provide access to geometry toolkits. Tell students that they are going to dilate some points. Before they begin, demonstrate the mechanics of dilating a point using a center of dilation and a scale factor. Tell students, “In the previous activity, each point was dilated to its image using a scale factor of 3. The dilated point was three times as far from the center as the original point. When we dilate point
\(D\)
using
\(P\)
as the center of dilation and a scale factor of 2, that means we’re going to take the distance from
\(P\)
to
\(D\)
and place a new point on the ray
\(PD\)
twice as far away from
\(P\)
.” Display for all to see:
[IMG]
If using the digital activity, demonstrate the mechanics of dilating using the applet. You can also use the measurement tool to confirm.
Representation: Internalize Comprehension.
Begin with a physical demonstration of the process of dilating a point using a center of dilation and a scale factor to support connections between new situations and prior understandings. Consider using these prompts: “How does this build on the previous activity in which the main task was to find distances and scale factor?” or “How does the point D’ correspond to the points D and P?”
Supports accessibility for: Conceptual processing; Visual-spatial processing
Conversing, Reading: MLR2 Collect and Display.
Circulate and listen to students as they make observations about the polygon with a scale factor of 2 and the polygon with a scale factor of
\(\frac12\)
. Write down the words and phrases students use to compare features of the new polygons to the original polygon. As students review the language collected in the visual display, encourage students to clarify the meaning of a word or phrase. For example, a phrase such as “the new polygon is the same as the original polygon but bigger” can be clarified with the phrase “the new polygon is a scaled copy with scale factor 2 of the original polygon.” A phrase such as “the polygons have the same angles” can be clarified with the phrase “each angle in the original polygon is the same as the corresponding angle in the new polygon.” This routine will provide feedback to students in a way that supports sense-making while simultaneously increasing meta-awareness of language.
Design Principle(s): Support sense-making; Maximize meta-awareness
Student Facing
Here is a polygon
\(ABCD\)
.
Dilate each vertex of polygon
\(ABCD\)
using
\(P\)
as the center of dilation and a scale factor of 2. Label the image of
\(A\)
as
\(A’\)
, and label the images of the remaining three vertices as
\(B’\)
,
\(C’\)
, and
\(D’\)
.
Draw segments between the dilated points to create polygon
\(A’B’C’D’\)
.
What are some things you notice about the new polygon?
[IMG]
4. Choose a few more points on the sides of the original polygon and transform them using the same dilation. What do you notice?
5. Dilate each vertex of polygon
\(ABCD\)
using
\(P\)
as the center of dilation and a scale factor of
\(\frac{1}{2}\)
. Label the image of
\(A\)
as
\(E\)
, the image of
\(B\)
as
\(F\)
, the image of
\(C\)
as
\(G\)
and the image of
\(D\)
as
\(H\)
.
6. What do you notice about polygon
\(EFGH\)
?
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?
Suppose
\(P\)
is a point not on line segment
\(\overline{WX}\)
. Let
\(\overline{YZ}\)
be the dilation of line segment
\(\overline{WX}\)
using
\(P\)
as the center with scale factor 2. Experiment using a circular grid to make predictions about whether each of the following statements must be true, might be true, or must be false.
\(\overline{YZ}\)
is twice as long
\(\overline{WX}\)
.
\(\overline{YZ}\)
is five units longer than
\(\overline{WX}\)
.
The point
\(P\)
is on
\(\overline{YZ}\)
.
\(\overline{YZ}\)
and
\(\overline{WX}\)
intersect.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students may think only grid points can be dilated. In fact, any point can, but they may have to measure or estimate the distances from the center. Grid points are convenient because you can measure by counting.
Activity Synthesis
Display the original figure and its image under dilation with scale factor 2 and center
\(P\)
.
[IMG]
Ask selected students to share what they notice about the new polygon. Ensure that the following observations are made. Encourage students to verify each assertion using geometry tools like tracing paper, a ruler, or a protractor.
The new figure is a scaled copy of the original figure.
The sides of the new figure are twice the length of the sides of the original figure.
The corresponding segments are parallel.
The corresponding angles are congruent.
Ask students what happened to the additional points they dilated on polygon
\(ABCD\)
. Note that a good
strategic
choice for these points are points where
\(ABCD\)
meets one of the circles: in these cases, it is possible to double the distance from that point to the center without measuring. The additional points should have landed on a side of the dilated polygon (because of measurement error, this might not always occur exactly). The important takeaway from this observation is that dilating the polygon’s vertices, and then connecting them, gives the image of the entire polygon under the dilation.


aligns: no