few_shots: Example:
Problem/activity:
Narrative
The purpose of this activity is for students to create composite shapes and compose new shapes from composite shapes. For example, students may find that a rectangular prism can be made from four triangular prisms or from eight smaller cubes. Students then discover that they can combine two of the composed rectangular prisms to make a new rectangular prism.
Required Materials
Materials to Gather
Geoblocks
Launch
Groups of 4
Give each group geoblocks.
Display a large cube.
“This cube can be built from other shapes. See how many ways you can build a cube using other shapes.”
3 minutes: partner work time
Share responses.
Activity
“Choose a different shape. Try to find smaller shapes you can use to build that shape.”
5 minutes: partner work time
Student Response
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Activity Synthesis
Display a cube and a composite cube—for example, a cube composed of four small cubes and one rectangular prism.
Demonstrate putting the shapes together.
“What new shape did I make with the cube and the cube I made?”
“Put your shapes together to make a rectangular prism.”
Display a rectangular prism and two triangular prisms.
“Find these shapes. Put them together. What new shape can you make?” (a cube, a rectangular block)


Mathematical concept/skill options: 
A. Reason with shapes and their attributes.
B. Understand similarity in terms of similarity transformations.
C. Use coordinates to prove simple geometric theorems algebraically.
D. Translate between the geometric description and the equation for a conic section.
E. Understand congruence and similarity using physical models, transparencies, or geometry software.
F. Analyze, compare, create, and compose shapes.
G. Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
H. Understand congruence in terms of rigid motions.
I. Classify two-dimensional figures into categories based on their properties.
J. Draw, construct, and describe geometrical figures and describe the relationships between them.
K. Explain volume formulas and use them to solve problems.
L. Prove theorems involving similarity.
M. Prove geometric theorems.
N. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
O. Define trigonometric ratios and solve problems involving right triangles.
P. Apply trigonometry to general triangles.
Q. Visualize relationships between two-dimensional and three-dimensional objects.
R. Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
S. Make geometric constructions.
T. Apply geometric concepts in modeling situations.
U. Solve real-world and mathematical problems involving area, surface area, and volume.
V. Graph points on the coordinate plane to solve real-world and mathematical problems.
W. Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
X. Understand and apply theorems about circles.
Y. Understand and apply the Pythagorean Theorem.
Z. Experiment with transformations in the plane.
AA. Find arc lengths and areas of sectors of circles.


Your response, in json format:
{
  "explanation": "This activity asks students to reason how shapes can combine to make other shapes.",
  "answer": "A"
}

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_lesson_009431

1
options: Use complex numbers in polynomial identities and equations.

Extend the properties of exponents to rational exponents.

Use properties of rational and irrational numbers.

Represent and model with vector quantities.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Perform operations on matrices and use matrices in applications.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Perform arithmetic operations with complex numbers.

Reason quantitatively and use units to solve problems.

Perform operations on vectors.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Compute fluently with multi-digit numbers and find common factors and multiples.

Represent complex numbers and their operations on the complex plane.

Apply and extend previous understandings of numbers to the system of rational numbers.
problem_activity: Activity
The purpose of this activity is to understand that the division facts for rational numbers are simply a consequence of the multiplication done previously. Students work several numerical examples relating multiplication to division, and then articulate a rule for the sign of a quotient based on the signs of the dividend and divisor (MP8).
Monitor for students who identify and describe the rule clearly.
Launch
Remind students that we can rearrange division equations to be multiplication equations, and vice versa. It may be useful to demonstrate with positive numbers if students struggle to recall this. For example, ask how we could rewrite
\(10 \div 2 = 5\)
as a multiplication equation. Students can also reference the multiplication table from a previous lesson, if needed.
Arrange students in groups of 2. Give students 4 minute of quiet work time followed by 2 minutes of partner discussion, then follow with whole-class discussion.
Engagement: Develop Effort and Persistence.
Connect a new concept to one with which students have experienced success. For example, reference examples from the previous lesson on signed multiplication to provide an entry point into this activity.
Supports accessibility for: Social-emotional skills; Conceptual processing
Reading: MLR6 Three Reads.
Use this routine to support students’ reading comprehension of the final question. In the first read, students read the text with the goal of understanding what the situation is about (e.g., Han and Clare are walking toward each other). In the second read, ask students to name the important quantities (e.g., Han’s velocity, Clare’s velocity, each person's distance from 0 in feet, elapsed time in seconds). After the third read, ask students to brainstorm possible strategies to determine where each person will be 10 seconds before they meet up, and when each person will be
\(-10\)
feet from the meeting place. This will help students connect the language in the word problem with the reasoning needed to solve the problem.
Design Principle(s): Support sense-making
Student Facing
Find the missing values in the equations
\(\text-3 \boldcdot 4 = \text{?}\)
\(\text-3 \boldcdot \text{?} = 12\)
\(3 \boldcdot \text{?} = 12\)
\(\text{?} \boldcdot \text-4 = 12\)
\(\text{?} \boldcdot 4 = \text-12\)
Rewrite the unknown factor problems as division problems.
Complete the sentences. Be prepared to explain your reasoning.
The sign of a positive number divided by a positive number is always:
The sign of a positive number divided by a negative number is always:
The sign of a negative number divided by a positive number is always:
The sign of a negative number divided by a negative number is always:
Han and Clare walk towards each other at a constant rate, meet up, and then continue past each other in opposite directions. We will call the position where they meet up 0 feet and the time when they meet up 0 seconds.
Han's velocity is 4 feet per second.
Clare's velocity is -5 feet per second.
Where is each person 10 seconds before they meet up?
When is each person at the position -10 feet from the meeting place?
Student Response
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Student Facing
Are you ready for more?
It is possible to make a new number system using
only
the numbers 0, 1, 2, and 3. We will write the symbols for multiplying in this system like this:
\(1 \otimes 2 = 2\)
. The table shows some of the products.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "43618"}, "_value": "\\(\\otimes\\)"}]}, {"_attributes": {"scope": "col"}, "_value": "0"}, {"_attributes": {"scope": "col"}, "_value": "1"}, {"_attributes": {"scope": "col"}, "_value": "2"}, {"_attributes": {"scope": "col"}, "_value": "3"}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "_value": "0"}], "td": [{"_value": "0"}, {"_value": "0"}, {"_value": "0"}, {"_value": "0"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "1"}], "td": [{}, {"_value": "1"}, {"_value": "2"}, {"_value": "3"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "2"}], "td": [{}, {}, {"_value": "0"}, {"_value": "2"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "3"}], "td": [{}, {}, {}, {}]}]}]}]}
In this system,
\(1 \otimes 3 = 3\)
and
\(2 \otimes 3 = 2\)
. How can you see that in the table?
What do you think
\(2 \otimes 1\)
is?
What about
\(3\otimes 3\)
?
What do you think the solution to
\(3\otimes n = 2\)
is?
What about
\(2\otimes n = 3\)
?
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
Have students refer to their previous work with signed multiplication to help them with the division statements.
Activity Synthesis
Select students who can articulate and explain what they have learned about the sign of a quotient based on the signs of the dividend and divisor (note that students do not need to use these words, they can just give examples). Highlight that the division facts are simply a consequence of the multiplication done previously. Moving forward, we are going to take these facts to be true for all rational numbers.


correct_option_index: 4