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
Teachers with a valid work email address can
click here to register or sign in
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: Understand the relationship between zeros and factors of polynomials.

Use the four operations with whole numbers to solve problems.

Gain familiarity with factors and multiples.

Represent and analyze quantitative relationships between dependent and independent variables.

Understand and apply properties of operations and the relationship between addition and subtraction.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Write and interpret numerical expressions.

Analyze patterns and relationships.

Use properties of operations to generate equivalent expressions.

Represent and solve problems involving addition and subtraction.

Multiply and divide within 100.

Create equations that describe numbers or relationships

Work with addition and subtraction equations.

Write expressions in equivalent forms to solve problems.

Represent and solve equations and inequalities graphically.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Understand properties of multiplication and the relationship between multiplication and division.

Generate and analyze patterns.

Perform arithmetic operations on polynomials.

Add and subtract within 20.

Reason about and solve one-variable equations and inequalities.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Understand the connections between proportional relationships, lines, and linear equations.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Understand solving equations as a process of reasoning and explain the reasoning.

Represent and solve problems involving multiplication and division.

Work with equal groups of objects to gain foundations for multiplication.

Use polynomial identities to solve problems.

Solve systems of equations.

Solve equations and inequalities in one variable.

Work with radicals and integer exponents.

Interpret the structure of expressions.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Rewrite rational expressions.
problem_activity: Activity
This task is analogous to a previous activity with positive exponents. The number line strongly encourages students to think about how to change expressions so they all take the form
\(b \boldcdot 10^k\)
, where
\(b\)
is between 1 and 10, as in the case of scientific notation. The number line also is a useful representation to show that, for example,
\(29 \boldcdot 10^\text{-7}\)
is about half as much as
\(6 \boldcdot 10^\text{-6}\)
. The last two questions take such a comparison a step further, asking students to estimate relative sizes using numbers expressed with powers of 10.
As students work, look for different strategies they use to compare expressions that are not written as a product of a number and
\(10^{\text-6}\)
. Also look for students who can explain how they estimated in the last two problems. Select them to share their strategies later.
Launch
Arrange students in groups of 2. Give students 5 minutes of quiet work time, followed by partner discussion and whole-class discussion. During partner discussion, ask students to share their responses for the first two questions and reach an agreement about where the numbers should be placed on the number line.
Action and Expression: Develop Expression and Communication.
Maintain a display of important terms and vocabulary. During the launch take time to review the visual display of rules for exponents.
Supports accessibility for: Memory; Language
Speaking: MLR8 Discussion Supports.
Display sentence frames to support students as they describe how they compared and estimated the difference in magnitude between the pairs of values
\(29⋅10^{\text-7}\)
and
\(6⋅10^{\text-6}\)
or
\(7⋅10^{\text-8}\)
and
\(3⋅10^{\text-9}\)
. For example, “First I ____ , then I ____ .” or “_____ is (bigger/smaller) because ______ .”
Design Principle(s): Optimize output (for explanation)
Student Facing
[IMG]
Label the tick marks on the number line.
Plot the following numbers on the number line:
A.
\(6 \boldcdot 10^{\text -6}\)
B.
\(6 \boldcdot 10^{\text -7}\)
C.
\(29 \boldcdot 10^{\text -7}\)
D.
\((0.7) \boldcdot 10^{\text -5}\)
Which is larger,
\(29 \boldcdot 10^{\text -7}\)
or
\(6 \boldcdot 10^{\text -6}\)
? Estimate how many times larger.
Which is larger,
\(7 \boldcdot 10^{\text -8}\)
or
\(3 \boldcdot 10^{\text -9}\)
? Estimate how many times larger.
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
Students may have trouble comparing negative powers of 10. Remind these students that, for example,
\(10^\text{-5}\)
is 5 factors that are
\(\frac{1}{10}\)
and
\(10^\text{-6}\)
is 6 factors that are
\(\frac{1}{10}\)
, so
\(10^\text{-5}\)
is 10 times larger than
\(10^\text{-6}\)
.
Students may also have trouble estimating how many times one larger expression is than another. Offer these students an example to illustrate how representing numbers as a single digit times a power of 10 is useful for making rough estimations. We have that
\(9 \boldcdot 10^\text{-12}\)
is roughly 50 times as much as
\(2 \boldcdot 10^\text{-13}\)
because
\(10^\text{-12}\)
is 10 times as much as
\(10^\text{-13}\)
and 9 is roughly 5 times as much as 2. In other words,
\(\displaystyle \frac{9 \boldcdot 10^\text{-12}}{2 \boldcdot 10^\text{-13}} \approx \frac{10 \boldcdot 10^\text{-12}}{2 \boldcdot 10^\text{-13}} = 5 \boldcdot 10^{\text-12 - (\text-13)} = 5 \boldcdot 10^1 = 50\)
Activity Synthesis
Select previously identified students to explain how they used the number line and powers of 10 to compare the numbers in the last two problems.
One important concept is that it’s always possible to change an expression that is a multiple of a power of 10 so that the leading factor is between 1 and 10. For example, we can think of
\(29 \boldcdot 10^\text{-7}\)
as
\((2.9) \boldcdot 10 \boldcdot 10^\text{-7}\)
or
\((2.9) \boldcdot 10^\text{-6}\)
. Another important concept is that powers of 10 can be used to make rough estimates. Make sure these ideas are uncovered during discussion.


correct_option_index: 30