few_shots: Example 1:
Problem/activity:
You’re mixing ingredients for cookies. The recipe says to combine 6 tablespoons, or ¾ stick, of butter with 1 cup of sugar. You accidentally mix in a whole stick of butter (8 tablespoons) with the cup of sugar. How can you fix this?


Mathematical concept/skill options: 
A. Analyze proportional relationships and use them to solve real-world and mathematical problems.
B. Understand ratio concepts and use ratio reasoning to solve problems.


Thought: This problem focuses on the ratio between two ingredients. Students do not have to analyze proportional relationships in-depth, which is reserved for a higher grade level, but rather use ratios to find missing values.
Answer: B

Example 2:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Mathematical concept/skill options: 
A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
B. Reason quantitatively and use units to solve problems.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
D. Represent complex numbers and their operations on the complex plane.
E. Perform arithmetic operations with complex numbers.
F. Compute fluently with multi-digit numbers and find common factors and multiples.
G. Represent and model with vector quantities.
H. Perform operations on vectors.
I. Use properties of rational and irrational numbers.
J. Use complex numbers in polynomial identities and equations.
K. Extend the properties of exponents to rational exponents.
L. Know that there are numbers that are not rational, and approximate them by rational numbers.
M. Perform operations on matrices and use matrices in applications.
N. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.


Thought: This problem involves understanding fractions, multiplication, and division.
Answer: A

Example 3:
Problem/activity:
Problem 1
The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions.
|                                                          | **           No solution          ** | **           Exactly 1 solution          ** | **           Infinite number of solutions          ** |
|----------------------------------------------------------|--------------------------------------|---------------------------------------------|-------------------------------------------------------|
|                       $${9=4x+7}$$                       |                                      |                                             |                                                       |
|                       $${y=3y+5}$$                       |                                      |                                             |                                                       |
|                    $${y={x\over2}}$$                     |                                      |                                             |                                                       |
|                     $${3x+6y-9=0}$$                      |                                      |                                             |                                                       |
|    $${x=y}$$                        $${1.25x=1.25y}$$    |                                      |                                             |                                                       |
| $${9.9=6x+8y}$$                        $${9x=2.5y-8.8}$$ |                                      |                                             |                                                       |
| $${11x-2y=1.5}$$                        $${11x-2y=2.5}$$ |                                      |                                             |                                                       |
Problem 2
Kim has a small container and a large container as shown.
[IMG]
It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container.
What is the size of each of the two containers?


Mathematical concept/skill options: 
A. Write expressions in equivalent forms to solve problems.
B. Understand properties of multiplication and the relationship between multiplication and division.
C. Interpret the structure of expressions.
D. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
E. Use polynomial identities to solve problems.
F. Understand the relationship between zeros and factors of polynomials.
G. Write and interpret numerical expressions.
H. Work with equal groups of objects to gain foundations for multiplication.
I. Perform arithmetic operations on polynomials.
J. Create equations that describe numbers or relationships
K. Generate and analyze patterns.
L. Analyze and solve linear equations and pairs of simultaneous linear equations.
M. Understand solving equations as a process of reasoning and explain the reasoning.
N. Apply and extend previous understandings of arithmetic to algebraic expressions.
O. Rewrite rational expressions.
P. Represent and solve problems involving multiplication and division.
Q. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
R. Reason about and solve one-variable equations and inequalities.
S. Understand and apply properties of operations and the relationship between addition and subtraction.
T. Understand the connections between proportional relationships, lines, and linear equations.
U. Multiply and divide within 100.
V. Gain familiarity with factors and multiples.
W. Use the four operations with whole numbers to solve problems.
X. Work with addition and subtraction equations.
Y. Represent and solve problems involving addition and subtraction.
Z. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
AA. Solve equations and inequalities in one variable.
AB. Solve systems of equations.
AC. Represent and analyze quantitative relationships between dependent and independent variables.
AD. Work with radicals and integer exponents.
AE. Represent and solve equations and inequalities graphically.
AF. Analyze patterns and relationships.
AG. Use properties of operations to generate equivalent expressions.
AH. Add and subtract within 20.


Thought: This set of problems involves solving linear equations and pairs of linear equations.
Answer: L

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_000972

6

fl_problem_001097

7

fl_problem_001392

8
options: Write expressions in equivalent forms to solve problems.

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

Perform arithmetic operations on polynomials.

Analyze patterns and relationships.

Represent and solve equations and inequalities graphically.

Use the four operations with whole numbers to solve problems.

Interpret the structure of expressions.

Create equations that describe numbers or relationships

Apply and extend previous understandings of arithmetic to algebraic expressions.

Rewrite rational expressions.

Work with addition and subtraction equations.

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

Reason about and solve one-variable equations and inequalities.

Generate and analyze patterns.

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

Represent and solve problems involving multiplication and division.

Work with radicals and integer exponents.

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

Represent and analyze quantitative relationships between dependent and independent variables.

Add and subtract within 20.

Solve equations and inequalities in one variable.

Understand the relationship between zeros and factors of polynomials.

Gain familiarity with factors and multiples.

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

Solve systems of equations.

Represent and solve problems involving 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.

Multiply and divide within 100.

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

Use properties of operations to generate equivalent expressions.

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

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

Use polynomial identities to solve problems.
problem_activity: Activity
This activity revisits a polynomial students first encountered in an earlier lesson in this unit, relating the polynomials to the integers. The purpose of this activity is to make clear that technology allows us to identify solutions to equations that can be tedious to do by hand. The last question reinforces this point by asking students to write their own polynomial and find the
\(x\)
value that corresponds to a certain output, since they will need to estimate the solution from a graph.
The question about possible factors is meant to preview the work of the next several lessons in which students will identify factors of polynomial expressions written in standard form and eventually show that for a polynomial
\(p(x)\)
, if
\(p(a)=0\)
, then
\((x-a)\)
is a factor of
\(p(x)\)
.
Launch
Display functions
\(p\)
and
\(q\)
for all to see. Ask, “Without using a graphing calculator, what is the most points of intersection the two functions could have? The least?” Encourage students to make sketches to justify that 3 is the most and 1 is the least.
Provide access to devices that can run Desmos or other graphing technology.
Speaking, Reading: MLR5 Co-Craft Questions.
Use this routine to spark students’ curiosity about a polynomial written in standard form. Display only the first sentence of this problem (“Consider the functions . . . .”) and ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the activity’s questions. Listen for and amplify any questions involving visualizing the graph, points of intersection, or possible factors. This will help students create the language of mathematical questions before feeling pressure to produce solutions.
Design Principle(s): Maximize meta-awareness; Support sense-making
Engagement: Develop Effort and Persistence.
Connect a new concept to one with which students have experienced success. For example, to help students see how questions 1 and 2 are related, ask them to find the solutions for an equation such as
\(x^2-4=12\)
. Then, invite them to briefly reflect on the steps used. Next, ask them what they would do differently if they were told to find the
\(x\)
-values where two functions
\(f(x)= x^2-4\)
and
\(g(x)=12\)
intersect. Ask students to notice what elements were similar in each process.
Supports accessibility for: Social-emotional skills; Conceptual processing
Student Facing
Consider the functions
\(p(x) = 5x^3+6x^2+4x\)
and
\(q(x) = 5640\)
.
Use graphing technology to find a value of
\(x\)
that makes
\(p(x)=q(x)\)
true.
For the
\(x\)
-value at the point of intersection, what can you say about the value of
\(5x^3+6x^2+4x-5640\)
?
What does your answer suggest is a possible factor of
\(5x^3+6x^2+4x-5640\)
?
Write your own polynomial
\(m(x)\)
of degree 3 or higher.
Use graphing technology to estimate the values of
\(x\)
that make
\(m(x)= q(x)\)
true.
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Anticipated Misconceptions
If students have trouble using the value of
\(5x^3+6x^2+4x-5640\)
at the
\(x\)
-value where the functions intersect to find a possible factor of
\(5x^3+6x^2+4x-5640\)
, encourage them to think about the relationship between factors of an expression and
\(x\)
-values for which the value of the expression is 0.
Activity Synthesis
The goal of this discussion is for students to see how to use technology to answer an algebraic question involving a higher degree polynomial. Invite students to share the polynomial
\(m(x)\)
that they wrote. If possible, display several versions of
\(m(x)\)
for all to see, and use the graph to estimate the
\(x\)
-values where
\(m(x)=q(x)\)
. If this is not possible, ask students to share the solutions they estimated. Students should understand that not all polynomials can be factored by hand, and that technology is a valuable tool for helping us interpret such polynomials.
Then ask students to suggest possible factors of
\(5x^3+6x^2+4x-5640\)
. After a few suggestions, ask students how they could prove something is a factor. It is important that students understand that for now, the only way to prove something like
\((x-10)\)
is a factor of
\(5x^3+6x^2+4x-5640\)
is to find the quadratic function that satisfies
\(5x^3+6x^2+4x-5640=(x-10)(ax^2+bx+c)\)
, where
\(a\)
,
\(b\)
, and
\(c\)
are real numbers. Tell students that the next lessons are about how to do this.


correct_option_index: 4