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: 
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_neg_0

K

fl_problem_001097_pos_0

7

fl_problem_001894_pos_0

HS_0
standard_description: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
problem_activity: Activity
Students have seen that the lengths of corresponding segments in a figure and its scaled copy vary by the same scale factor. Here, they learn that in such a pair of figures,
any
corresponding distances—not limited to lengths of sides or segments—are related by the same scale factor. The side lengths of the polygons in this task cannot be easily determined, so students must look to other distances to compare.
Students must take care when they identify corresponding vertices and distances. As students work, urge them to attend to the order in which points or segments are listed.
If students are not sure what to make out of the values in the table (for the second question), encourage them to consider the corresponding distances of two figures at a time. For example, ask: What do you notice about the corresponding vertical distances in
\(IJKL\)
and
\(EFGH\)
? What about the corresponding horizontal distances in those two figures?
Teacher Notes for IM 6–8 Math Accelerated
Adjust this activity to 15 minutes. Conclude the activity synthesis by displaying this image:
[IMG]
Ask, “Here are two quadrilaterals. Kiran says that Polygon
\(EFGH\)
is a scaled copy of
\(ABCD\)
, but Lin disagrees. Do you agree with either of them?” (Lin is correct, because the corresponding distances are not multiplied by the same number. Kiran may have noticed that the corresponding angles are equal and thought this meant the polygons are scaled copies. I noticed that the scale factors for the corresponding sides are not the same.
\(AB\)
and
\(EF\)
are related by a scale factor of
\(\frac23\)
, but
\(DC\)
and
\(HG\)
are related by a scale factor of
\(\frac12\)
.)
After a brief quiet think time, select students to share their responses. The goal of this discussion is to make clear that angle measurements and distances are both important when deciding whether two polygons are scaled copies.
Launch
Arrange students in groups of 2. Ask if they can tell the lengths of segments
\(GF\)
or
\(DC\)
from the grid (without using rulers). Explain that they will explore another way to compare length measurements in scaled copies.
Give students 2–3 minutes of quiet work time for the first two questions, and 1 minute to discuss their responses with a partner before continuing on to the last question.
Student Facing
Each of these polygons is a scaled copy of the others. You already checked their corresponding angles.
[IMG]
The side lengths of the polygons are hard to tell from the grid, but there are other
corresponding distances
that are easier to compare. Identify the distances in the other two polygons that correspond to
\(DB\)
and
\(AC\)
, and record them in the table.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "_value": "quadrilateral"}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["distance that", "corresponds to"], "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "138"}, "_value": "\\(DB\\)"}]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["distance that", "corresponds to"], "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "33"}, "_value": "\\(AC\\)"}]}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "20"}, "_value": "\\(ABCD\\)"}]}], "td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "40908"}, "_value": "\\(DB = 4\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "40949"}, "_value": "\\(AC = 6\\)"}]}]}, {"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "142"}, "_value": "\\(EFGH\\)"}]}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "167"}, "_value": "\\(IJKL\\)"}]}], "td": [{}, {}]}]}]}]}
Look at the values in the table. What do you notice?
Pause here so your teacher can review your work.
The larger figure is a scaled copy of the smaller figure.
[IMG]
If
\(AE = 4\)
, how long is the corresponding distance in the second figure? Explain or show your reasoning.
If
\(IK = 5\)
, how long is the corresponding distance in the first figure? Explain or show your reasoning.
Student Response
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for free access to Student Response.
Anticipated Misconceptions
Students may list the corresponding vertices for distances in the wrong order. For example, instead of writing
\(IK\)
as the distance corresponding to
\(DB\)
, they may write
\(KI\)
. Remind students of the corresponding points by asking, “Which vertex in
\(IJKL\)
corresponds to
\(D\)
? Which corresponds to
\(B\)
?” and have them match the order of the vertices accordingly.
Activity Synthesis
Display the completed tables for all to see. To highlight how all distances in a scaled copy (not just the side lengths of the figure) are related by the same scale factor, discuss:
How does the vertical distance in
\(ABCD\)
compare to that in
\(EFGH\)
? How do the horizontal distances in the two polygons compare? Do the pairs of vertical and horizontal distances share the same scale factor?
How do the vertical distances in
\(EFGH\)
and
\(IJKL\)
compare? What about the horizontal distances? Is there a common scale factor? What is that scale factor?
What scale factor relates the corresponding lengths and distances in the two drawings of the letter W?
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between figures. For example, mark corresponding line segments using the same color.
Supports accessibility for: Visual-spatial processing
Speaking: MLR7 Compare and Connect.
Use this routine to call attention to the different ways students may identify scale factors. Display the following statements: “The scale factor from
\(EFGH\)
to
\(IJKL\)
is 3,” and “The scale factor from
\(EFGH\)
to
\(IJKL\)
is
\(\frac13\)
.” Give students 2 minutes of quiet think time to read and consider whether either or both of the statements are correct. Invite students to share their initial thinking with a partner before selecting 2–3 students to share with the class. In this discussion, listen for and amplify any comments that refer to the order of the original figure and its scaled copy, as well as those who identify corresponding vertices and distances. Draw students’ attention to the different ways to describe the relationships between scaled copies and the original figure.
Design Principle(s): Maximize meta-awareness


aligns: no