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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for free access to Student Response.
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:
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: Perform arithmetic operations with complex numbers.

Extend the properties of exponents to rational exponents.

Represent complex numbers and their operations on the complex plane.

Perform operations on matrices and use matrices in applications.

Use complex numbers in polynomial identities and equations.

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

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

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

Use properties of rational and irrational numbers.

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

Represent and model with vector quantities.

Reason quantitatively and use units to solve problems.

Perform operations on vectors.

Apply and extend previous understandings of numbers to the system of rational numbers.
problem_activity: Activity
In this activity, students interpret equations that represent situations (MP2). The purpose is for students to see that equations of the form
\(x + p = q\)
can be solved by adding the opposite of
\(p\)
to the equation, regardless of whether
\(p\)
is positive or negative. Students also see that equations of the form
\(px = q\)
can be solved by multiplying the equation by the reciprocal of
\(p\)
. Through this work, students see that the structure of equations can be used to reason about a path to a solution (MP7) even when negative values are included or when a variable can represent a negative number.
Launch
Give students 5–6 minutes of quiet work time followed by whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in 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
Reading: MLR6 Three Reads.
Use this routine to help students interpret the representations of the situation in this activity. Use the first read to help students comprehend the different stages of situation (e.g., hiking a mountain, temperature falls, cost of the trip) without using numbers. Use the second read to identify the important quantities by asking students what can be counted or measured (e.g., increased elevation 290 feet, now at 450 feet; temperature fell 4 degrees, now is at 32 degrees; 3 times as many students this year, now 42 students; cost is
\(\frac23\)
of last year’s, cost
$
32 this year). For the third read, ask students to brainstorm possible strategies to answer the given questions.
Design Principle(s): Support sense-making
Student Facing
The Hiking Club is on a trip to hike up a mountain.
The members increased their elevation 290 feet during their hike this morning. Now they are at an elevation of 450 feet.
Explain how to find their elevation before the hike.
Han says the equation
\(e + 290 = 450\)
describes the situation. What does the variable
\(e\)
represent?
Han says that he can rewrite his equation as
\(e=450 + \text-290\)
to solve for
\(e\)
. Compare Han's strategy to your strategy for finding the beginning elevation.
The temperature fell 4 degrees in the last hour. Now it is 21 degrees. Write and solve an equation to find the temperature it was 1 hour ago.
There are 3 times as many students participating in the hiking trip this year than last year. There are 42 students on the trip this year.
Explain how to find the number of students that came on the hiking trip last year.
Mai says the equation
\(3s=42\)
describes the situation. What does the variable
\(s\)
represent?
Mai says that she can rewrite her equation as
\(s=\frac13 \boldcdot 42\)
to solve for
\(s\)
. Compare Mai's strategy to your strategy for finding the number of students on last year’s trip.
The cost of the hiking trip this year is
\(\frac23\)
of the cost of last year's trip. This year's trip cost
$
32. Write and solve an equation to find the cost of last year's trip.
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
A number line is shown below. The numbers 0 and 1 are marked on the line, as are two other rational numbers
\(a\)
and
\(b\)
.
[IMG]
Decide which of the following numbers are positive and which are negative.
\(a-1\)
\(a-2\)
\(\text-b\)
\(a+b\)
\(a-b\)
\(ab+1\)
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students may be misled by words to add or multiply (or subtract or divide) by the wrong numbers. For example, the word "increased" in question 1 may lead students to simply add the numbers they see, while the words "three times as many" in question 4 may lead students to multiply the numbers in the problem. Encourage students to make sense of the situations by acting them out or using visual diagrams, which will help them understand the actions and relationships in the stories.
Activity Synthesis
Tell students: “We learned four things about the hiking trip in this activity: the students were climbing, the temperature was falling, there were more students this year than last, and the cost of the trip was less this year than last.” Then ask them:
“Think about how you knew what operation described the rise in elevation, fall in temperature, rise in number of students, and fall in the cost. How did you know whether the situation used adding or multiplying?”  (This conversation can highlight the problem with relying on “key words” . For example, when students see “times as many” , they might want to multiply the numbers they see in the problem. Encourage students to make sense of the situations by acting them out or drawing diagrams.)
“How did you decide how to solve for the unknown quantity?”
“What are some ways to know that a situation involves negative values?”


correct_option_index: 9