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:
Thought: 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: Reason about and solve one-variable equations and inequalities.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Represent and analyze quantitative relationships between dependent and independent variables.

Create equations that describe numbers or relationships

Generate and analyze patterns.

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

Work with radicals and integer exponents.

Write expressions in equivalent forms to solve problems.

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

Solve systems of equations.

Add and subtract within 20.

Use properties of operations to generate equivalent expressions.

Perform arithmetic operations on polynomials.

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

Interpret the structure of expressions.

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

Rewrite rational expressions.

Use the four operations with whole numbers to solve problems.

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

Represent and solve problems involving multiplication and division.

Analyze patterns and relationships.

Represent and solve problems involving addition and subtraction.

Solve equations and inequalities in one variable.

Multiply and divide within 100.

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

Write and interpret numerical expressions.

Gain familiarity with factors and multiples.

Represent and solve equations and inequalities graphically.

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

Use polynomial identities to solve problems.

Work with addition and subtraction equations.

Understand the relationship between zeros and factors of polynomials.

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

Work with equal groups of objects to gain foundations for multiplication.
problem_activity: Activity
By now, students recognize that when a quadratic equation is in the form of
\(\text{expression} = 0\)
and the expression is in factored form, the equation can be solved using the zero product property. In this activity, they encounter equations in which one side of the equal sign is not 0. To make one side equal 0 requires rearrangement. For example, to solve
\(x(x+6)=8\)
, the equation needs to be rearranged to
\(x(x+6)-8=0\)
. Yet because the expression on the left is no longer in factored form, the zero product property won’t help after all and another strategy is needed.
Students recall that to solve a quadratic equation in the form of
\(\text{expression} = 0\)
is essentially to find the zeros of a quadratic function defined by that expression, and that the zeros of a function correspond to the horizontal intercepts of its graph. In the case of
\(x(x+6)-8=0\)
, the function whose zeros we want to find is defined by
\(x(x+6)-8\)
. Graphing
\(y=x(x+6)-8\)
and examining the
\(x\)
-intercepts of the graph allow us to see the number of solutions and what they are.
Launch
Arrange students in groups of 2 and provide access to graphing technology. Give students a moment to think quietly about the first question and then ask them to briefly discuss their response with their partner before continuing with the rest of the activity.
Action and Expression: Provide Access for Physical Action.
Support effective and efficient use of tools and assistive technologies. To use graphing technology, some students may benefit from a demonstration or access to step-by-step instructions.
Supports accessibility for: Organization; Memory; Attention
Student Facing
Han is solving three equations by graphing.
\((x-5)(x-3)=0\)
\(\displaystyle (x-5)(x-3)=\text-1\)
\(\displaystyle (x-5)(x-3)=\text-4\)
To solve the first equation,
\((x-5)(x-3)=0\)
, he graphed
\(y=(x-5)(x-3)\)
and then looked for the
\(x\)
-intercepts of the graph.
Explain why the
\(x\)
-intercepts can be used to solve
\((x-5)(x-3)=0\)
.
What are the solutions?
To solve the second equation, Han rewrote it as
\((x-5)(x-3)+1=0\)
. He then graphed
\(y=(x-5)(x-3)+1\)
.
Use graphing technology to graph
\(y=(x-5)(x-3)+1\)
. Then, use the graph to solve the equation. Be prepared to explain how you use the graph for solving.
Solve the third equation using Han’s strategy.
Think about the strategy you used and the solutions you found.
Why might it be helpful to rearrange each equation to equal 0 on one side and then graph the expression on the non-zero side?
How many solutions does each of the three equations have?
Student Response
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click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
The equations
\((x-3)(x-5)=\text-1\)
,
\((x-3)(x-5)=0\)
, and
\((x-3)(x-5)=3\)
all have whole-number solutions.
Use graphing technology to graph each of the following pairs of equations on the same coordinate plane. Analyze the graphs and explain how each pair helps to solve the related equation.
\(y=(x-3)(x-5)\)
and
\(y=\text-1\)
\(y=(x-3)(x-5)\)
and
\(y=0\)
\(y=(x-3)(x-5)\)
and
\(y=3\)
Use the graphs to help you find a few other equations of the form
\((x-3)(x-5)=z\)
that have whole-number solutions.
Find a pattern in the values of
\(z\)
that give whole-number solutions.
Without solving, determine if
\((x-5)(x-3)=120\)
and
\((x-5)(x-3)=399\)
have whole-number solutions. Explain your reasoning.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
If students enter the equation
\((x-5)(x-3)=0\)
into their graphing technology, they may see an error message, or they may see vertical lines. The lines will intersect the
\(x\)
-axis at the solutions, but they are clearly not graphs of a quadratic function. Emphasize that we want to graph the function defined by
\(y=(x-5)(x-3)\)
and use its
\(x\)
-intercepts to find the solution to the related equation.
All
the points on the two vertical lines do represent solutions to the equation (because the points along each vertical line satisfy the equation regardless of the value chosen for
\(y\)
), but understanding this is beyond the expectations for students in this course.
Activity Synthesis
Invite students to share their responses, graphs, and explanations on how they used the graphs to solve the equations. Discuss questions such as:
“
Are the original equation
\((x-5)(x-3)=\text-1\)
and the rewritten one
\((x-5)(x-3)+1=0\)
equivalent?” (Yes, each pair of equations are equivalent. In that example, 1 is added to both sides of the original equation.)
"Why might it be helpful to rearrange the equation so that one side is an expression and the other side is 0?" (It allows us to find the zeros of the function defined by that expression. The zeros correspond to the
\(x\)
-intercepts of the graph.)
“What equation would you graph to solve this equation:
\((x-4)(x-6)=15\)
?” (
\(y=(x-4)(x-6)-15\)
) “What about
\((x+3)^2-1=5\)
?" (
\(y=(x+3)^2-6\)
)
Make sure students understand that some quadratic functions have two zeros, some have one zero, and some have no zeros, so their respective graphs will have two, one, or no horizontal intercepts, respectively.
Likewise, some quadratic equations have two solutions, some have one solution, and some have no real solutions. (Because students won’t know about numbers that aren’t real until a future course, for now it is sufficient to say “no solutions.”)


correct_option_index: 27