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.


Standard options: 
A. 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.
B. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
C. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
D. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
E. Fluently add and subtract within 5.


Thought: This task involves representing addition with objects such as number cards and dice.
Answer: A

Example 2:
Problem/activity:
Task
 Arrange the fractions in order from least to greatest. Explain your answer with a picture.
 $\frac{1}{5}, \frac{1}{7}, \frac{1}{3} $
 $\frac{2}{5}, \frac{2}{7}, \frac{2}{3} $
 $\frac{5}{6}, \frac{3}{6}, \frac{1}{6} $
 $\frac{5}{12}, \frac{8}{12}, \frac{4}{12} $

Standard options: 
A. Understand a fraction as a number on the number line; represent fractions on a number line diagram.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
B. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
C. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.


Thought: By arranging the fractions in this problem, students learn to compare fractions by reasoning about their size.
Answer: B

Example 3:
Problem/activity:
Problem 1
Solve using the standard algorithm. If you get stuck, use an area model and/or the partial products algorithm to help. Then assess the reasonableness of your answer.
a.
$$5,803\times46$$
b.
$$63\times2,116$$
Problem 2
Dunkin' introduced a croissant doughnut in 2014. In the first three months of its debut, they sold 7,870 dozen of them per day. How many croissant doughnuts were sold in the first three months?


Standard options: 
A. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
B. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
C. Fluently multiply multi-digit whole numbers using the standard algorithm.


Thought: This set of problems involves multi-digit whole numbers because the numbers in each problem are four and two digit integers, and the mathematical calculations involved in both problems focus on multiplication. The first problem explicitly asks the student to solve its subproblems using the standard algorithm.
Answer: C

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: im_task_001477

K

im_task_000157

3

fl_problem_000674

5
options: Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

Verify experimentally the properties of rotations, reflections, and translations:
- Lines are taken to lines, and line segments to line segments of the same length.
- Angles are taken to angles of the same measure.
- Parallel lines are taken to parallel lines.
problem_activity: Activity
The purpose of this activity is to begin to think of a dilation with a scale factor as a rule or operation on points in the plane. Students work on a circular grid with center of dilation at the center of the grid. They examine what happens to different points on a given circle when the dilation is applied and observe that these points all map to another circle whose radius is scaled by the scale factor of the dilation. For example, if the scale factor is 3 and the points lie on a circle whose radius is 2 grid units, then the dilated points will all lie on a circle whose radius is 6 units. Students need to explain their reasoning for finding the scale factor (MP3).
Students discover that the circular grid is a powerful tool for representing dilations and they will continue to use the circular grid as they study what happens when dilations are applied to shapes other than grid circles.
In the digital activity, students encounter some new tools and the circular grid.
Digital
Launch
Ask students if they have ever seen a pebble dropped in a still pond, and select students to describe what happens. (The pebble becomes the center of a sequence of circular ripples.) Display the image from the task statement, and ask students to think about how it is like a pebble dropped in a still pond. Demonstrate that distance on the circular grid is measured by counting units along one of the rays that start at the center,
\(P\)
. Use MLR 8 (Discussion Supports) to draw students’ attention to a few important words in the task:
“When we say ‘on the circle,’ we mean on the curve or on the edge. (We do not mean the circle’s interior.)”
“Remember that a ray starts at a point and goes forever in one direction. Their rays should start at
\(P\)
and be drawn to the edge of the grid.”
If using the digital activity, you may want to demonstrate dilating a point before having students begin the task.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of these terms. Include the following terms and maintain the display for reference throughout the unit: circular grid, “on the circle,” ray, scale factor, and distance. Include a circular grid on the display, give examples, and label the features.
Supports accessibility for: Conceptual processing; Language
Student Facing
The larger Circle d is a
dilation
of the smaller Circle c.
\(P\)
is the
center of dilation
.
Draw four points
on
the smaller circle using the Point on Object tool.
[IMG]
Draw the rays from
\(P\)
through each of those four points. Select the Ray tool, then point
\(P\)
, and then the second point.
[IMG]
Mark the intersection points of the rays and Circle d by selecting the Intersect tool and clicking on the point of intersection.
[IMG]
Complete the table. In the row labeled S, write the distance between
\(P\)
and the point on the smaller circle in grid units. In the row labeled L, write the distance between
\(P\)
and the corresponding point on the larger circle in grid units. Measure the distances between pairs of points by selecting the Distance tool, and then clicking on the two points.
[IMG]
|   | \(A\) | \(B\) | \(C\) | \(D\) |
|---|-------|-------|-------|-------|
| S |       |       |       |       |
| L |       |       |       |       |
The center of dilation is point
\(P\)
. What is the
scale factor
that takes the smaller circle to the larger circle? Explain your reasoning.
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
Ask students if they have ever seen a pebble dropped in a still pond, and select students to describe what happens. (The pebble becomes the center of a sequence of circular ripples.) Display the image from the task statement, and ask students to think about how it is like a pebble dropped in a still pond. Demonstrate that distance on the circular grid is measured by counting units along one of the rays that start at the center,
\(P\)
. Use MLR 8 (Discussion Supports) to draw students’ attention to a few important words in the task:
“When we say ‘on the circle,’ we mean on the curve or on the edge. (We do not mean the circle’s interior.)”
“Remember that a ray starts at a point and goes forever in one direction. Their rays should start at
\(P\)
and be drawn to the edge of the grid.”
If using the digital activity, you may want to demonstrate dilating a point before having students begin the task.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of these terms. Include the following terms and maintain the display for reference throughout the unit: circular grid, “on the circle,” ray, scale factor, and distance. Include a circular grid on the display, give examples, and label the features.
Supports accessibility for: Conceptual processing; Language
Student Facing
The larger Circle d is a
dilation
of the smaller Circle c.
\(P\)
is the
center of dilation
.
Draw four points
on
the smaller circle (not inside the circle!), and label them
\(E\)
,
\(F\)
,
\(G\)
, and
\(H\)
.
Draw the rays from
\(P\)
through each of those four points.
Label the points where the rays meet the larger circle
\(E’\)
,
\(F’\)
,
\(G’\)
, and
\(H’\)
.
[IMG]
4. Complete the table. In the row labeled c, write the distance between
\(P\)
and the point on the smaller circle in grid units. In the row labeled d, write the distance between
\(P\)
and the corresponding point on the larger circle in grid units.
|   | \(E\) | \(F\) | \(G\) | \(H\) |
|---|-------|-------|-------|-------|
| c |       |       |       |       |
| d |       |       |       |       |
5. The center of dilation is point
\(P\)
. What is the
scale factor
that takes the smaller circle to the larger circle? Explain your reasoning.
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
For question 5, students might think the scale factor is 4, because the distance between the smaller and larger circle for each point increases by 4. If this happens, ask students how many grid units circle
\(c\)
is from the center (2) and how many grid units circle
\(d\)
is from the center (6). Then remind them that
scale factor
means a number you multiply by.
Activity Synthesis
Ask students if they made a strategic choice of points, such as points that lie on the grid lines coming from the center point
\(P\)
. Why are these points good choices for dilating?
Ask students what they think would happen if a circle were dilated about its center with a scale factor of 2 or 4. (The result would be a circle with twice the radius and 4 times the radius, respectfully, all sharing the same center.)
Two important observations coming from the lesson are:
The scale factor for this dilation is 3 so distances from the center of the circles triple when the dilation is applied.
The large circle is the dilation of the small circle, that is each point on the circle with radius 6 units is the dilated image of a point on the circle of radius 2 units. (To find which one, draw the line from the point to the center and see where it intersects the circle of radius 2 units.)
Speaking, Listening: MLR8 Discussion Supports.
As students share the scale factor that takes the smaller circle to the larger circle, press for details in students’ reasoning by asking how they know the scale factor is 3. Listen for students’ explanations that reference the table with distances between the center of dilation and points on the circles. Amplify statements that use precise language such as, “The distance between point P and any point on the smaller circle is multiplied by a scale factor of 3 to get the distance between point P and the corresponding point on the larger circle.” This will a support rich and inclusive discussion about how the scale factor affects the distance between the center of dilation and points on the circle.
Design Principle(s): Support sense-making


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