Download SFUSD Unit A.6 Quadratic Equations

1
SFUSD Mathematics Core Curriculum Development Project
2014–2015
Creating meaningful transformation in mathematics education
Developing learners who are independent, assertive constructors of their own understanding
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015
2
Algebra 1
A.6 Quadratic Equations
Number
of Days
Lesson
Reproducibles
Number of
Copies
Materials
1
Entry Task
Rectiles (2 pages)
1 per student
Algebra tiles (optional)
7
Lesson Series 1
CPM CCA Lesson 8.1.1 (2 pages)
CPM CAA Lesson 8.1.2 (2 pages)
CPM CAA Lesson 8.1.3 (2 pages)
CPM CAA Lesson 8.1.4 (2 pages)
CPM CAA Lesson 8.1.5 (2 pages)
CPM CAA Lesson 8.2.2 (4 pages)
CPM CAA Lesson 8.2.3
1 per pair
1 per pair
1 per pair
1 per pair
1 per pair
1 per pair
1 per pair
Algebra tiles
1
Apprentice Task Quadratic Functions
Domino Cards (2 pages)
1 per student
1 per pair
Scissors
4
Lesson Series 2
CPM Algebra Connections, Lesson 10.2.4
CPM Algebra Connections, Lesson 10.3.1 (3 pages)
CPM Algebra Connections, Lesson 10.3.2 (3 pages)
CPM CCA Lesson 10.2.5 (6 pages)
1 per pair
1 per pair
1 per pair
1 per pair
Algebra tiles
2
Expert Task
Victory Celebration (2 pages)
1 per pair
Graph paper
4
Lesson Series 3
CPM CCA Lesson 9.1.2 (2 pages)
CPM CCA Lesson 9.1.2 HW (2 pages)
CPM CCA Lesson 9.1.3 (2 pages)
CPM CCA Lesson 9.1.3 HW (2 pages)
CPM CCA Lesson 9.1.4 (2 pages)
CPM CCA Lesson 9.1.4 HW (2 pages)
Graphs (2 pages)
1 per pair
1 per student
1 per pair
1 per student
1 per pair
1 per student
1 per student
Graphing calculators or software
1
Milestone Task
CLA 2 Constructed Response Questions
CLA 2 Performance Assessment – Falling Shoes
Provided by AAO Graph paper
Provided by AAO
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015
3
Unit Overview
Big Idea
Quadratic equations have equivalent forms (standard, factored, vertex) that reveal different properties of the relationship represented by the given equation.
These properties, along with the quadratic formula, can be applied to solve problems and interpret them in context.
Unit Objectives
● Students will be able to convert between equivalent forms of the quadratic equation by factoring, completing the square, and using the quadratic
formula.
● Students will be able to choose and produce the form of the quadratic equation that is best suited to solve a given problem or model a given situation
or context.
Unit Description
In this unit, students will explore equivalent forms of quadratic equations. Students will convert between equivalent forms of quadratic equations by factoring,
completing the square, and using the quadratic formula. In the first lesson series, students will use diamonds, generic rectangles and find the root(s) of a
quadratic equation by factoring. In the second lesson series, students will: 1. Find the root(s) of a quadratic equation by factoring, completing the square, and
using the quadratic formula, 2. Find the roots, vertex, maximum or minimum, axis symmetry, and produce a rough graph, and 3. Model a situation with a
quadratic equation and interpret in context.
CCSS-M Content Standards
The Complex Number System
Perform arithmetic operations with complex numbers
2
N.CN.1 Know there is a complex number i such that i = –1, and every complex number has the form a + bi with a and b real.
Use complex numbers in polynomial identities and equations
N.CN.7 Solve quadratic equations with real coefficients that have complex solutions.
Seeing Structure in Expressions
Interpret the structure of expressions
4
4
2 2
2 2
A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x – y as (x ) – (y ) , thus recognizing it as a difference of
2
2
2
2
squares that can be factored as (x – y )(x + y ).
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015
4
Write expressions in equivalent forms to solve problems
A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
A.SSE.3a Factor a quadratic expression to reveal the zeros of the function it defines.
A.SSE.3b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
Arithmetic with Polynomials and Rational Expressions
Understand the relationship between zeros and factors of polynomials
A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the
polynomial.
Creating Equations
Create equations that describe numbers or relationships
A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
Reasoning with Equations and Inequalities
Understand solving equations as a process of reasoning and explain the reasoning
A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption
that the original equation has a solution. Construct a viable argument to justify a solution method.
A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
Solve equations and inequalities in one variable
A.REI.4 Solve quadratic equations in one variable.
2
A.REI.4a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p) = q that has the same
solutions. Derive the quadratic formula from this form.
2
A.REI.4b Solve quadratic equations by inspection (e.g., for x = 49), taking square roots, completing the square, the quadratic formula and factoring, as
appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and
b.
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015
5
Solve systems of equations
A.REI.7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the
2
2
points of intersection between the line y = –3x and the circle x + y = 3.
Interpreting Functions
Analyze functions using different representations
F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
F.IF.8a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015
6
Progression of Mathematical Ideas
Prior Supporting Mathematics
Students previously identified and explained the
parts of graphs of quadratic equations.
Students learned that there are different
representations of quadratic functions and that
different forms of the quadratic equation reveal
different information.
Students learned basic mathematics needed to
manipulate equations (exponents, negative
numbers, square roots and radicals, solving
equations, etc.).
Current Essential Mathematics
Future Mathematics
In algebra, students find the root(s) of a quadratic
equation by factoring, completing the square, and
the quadratic formula as well as discerning when to
use each method.
In Algebra 2, students will conduct more in-depth
explorations of imaginary numbers and connection
to roots of polynomials as well as being able to
write equivalent rational expressions.
Students find the roots, vertex, maximum or
minimum, and axis symmetry, and they produce a
rough graph.
Students model a situation with a quadratic
equation and interpret in context.
Students simplify radicals.
Students learned to multiply polynomials.
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015
7
Unit Design
All SFUSD Mathematics Core Curriculum Units are developed with a combination of rich tasks and lessons series. The tasks are
formative assessments of student learning. The tasks are designed to address four central questions:
1 Day
7 Days
1 Day
4 Days
2 Days
Lesson Series 3
Milestone
Task
Lesson Series 2
Expert
Task
Lesson Series 1
What do you already know?
What sense are you making of what you are learning?
How can you apply what you have learned so far to a new situation?
Did you learn what was expected of you from this unit?
Apprentice
Task
Entry Task
Entry Task:
Apprentice Task:
Expert Task:
Milestone Task:
4 Days
1 Day
Total Days: 20
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015
8
Entry Task
Rectiles
Apprentice Task
Forming Quadratics
Expert Task
Victory Celebration
Milestone Task
Falling Shoes
CCSS-M
Standards
A.SSE.3a, F.IF.8a
A.SSE.2, A.SSE.3
A.SSE.3b, A.CED.1, A.REI.2,
A.REI.4
A.CED.1, A.REI.2, A.REI.4
Brief
Description
of Task
Students will use algebra tiles to
find area and side lengths as a
lead into factoring.
Given the standard form of an
equation, students will produce
the factored form and make a
graph and produce the vertex
form and make a graph.
Given the starting height and initial
speed of a firework, students will
find information about the situation,
including a sketch, maximum
height, time it takes the fireworks to
reach the maximum height, etc.
Students use basic information
about two thrown shoes to extract
information about the situation
(connecting graphs to equations,
finding points of intersection, etc.).
Source
Rectiles, Silicon Valley
Mathematics Initiative - Course 1,
Spring 2012 Performance Test
Forming Quadratics FAL,
MARS/Shell 2012
IMP Year 2, p. 280, pp. 316–319
Teacher Created
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015
9
Lesson Series 1
Lesson Series 2
Lesson Series 3
CCSS-M
Standards
A.SSE.2, A.SSE.3a
A.APR.3
F.IF.8a
A.SSE.3b
A.REI.4a
F.IF.8a
A.CEDa
A.REI.1, A.REI.2, A.REI.4b, A.REI.7
N.CN.1, N.CN.7
Brief
Description
of Lessons
Themes: Factoring
A. Students will use the diamond method,
algebra tiles, and generic rectangles to factor
2
quadratic expressions in the form ax + bx + c.
B. Students will factor completely expressions
that have a common factor in each term.
C. Students will learn relationships with factoring
and the graph of quadratic equations in the form
2
y = ax + bx + c, for y = 0.
● For a = 1 diamond problems work CPMCCA 8-1 to 8-4, 8-6, 8-7,8-9, 8-10, 8-24.
● For a <> 1 diamond method with the
generic rectangle 8-13 to 8-16.
● Any method may be used to factor the
following: 8-24 to 8-27, 8-31.
● Factoring Completely: 8-37,8-38, 8-45 to
8-47
D. Zero Product Property
● 8.2.2 CPM 8-64 to 8-67
Themes: Completing the Square, Deriving the
Quadratic Formula
Themes: Solving Quadratic Equations With the
Quadratic Formula
Students will look at equivalent quadratic
equations and learn how to complete the
square, both with and without algebra tiles
and equation mats, then derive the quadratic
formula by completing the square.
Solving Quadratic Equations with the Quadratic
Formula
CPM Core Connections Algebra, Chapter 8
CPM Core Connections Algebra, Chapter 10
CPM Algebra Connections, Chapter 10
CPM Core Connections Algebra, Chapter 9
MARS (Graphs)
Sources
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015
10
Entry Task
Rectiles
Students will find the area of shapes with algebra tiles
Mathematics Objectives and Standards
Framing Student Experience
Math Objectives:
● Students will be able to find area of polynomials in trinomial form.
● Students will be able to prepare for factoring trinomials.
Launch:
Make sure students understand the situation.
CCSS-M Standards Addressed: A.SSE.3, F.IF.8a
During:
Students perform the task, finding the area of the figure.
Potential Misconceptions:
● Area is length times width.
Closure/Extension:
Students create new shapes and find their areas.
Note: The task uses tiles with lengths of x and y. You might consider relabeling
the diagrams so the tiles have lengths of x and 1 to match the dimensions of
algebra tiles.
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015
11
Rectiles
How will students do this?
Focus Standards for Mathematical Practice:
5. Use appropriate tools strategically.
7. Look for and make use of structure.
Structures for Student Learning:
Academic Language Support:
Vocabulary: configuration, rectangular, width, length, area
Sentence frames:
Differentiation Strategies:
Provide students with algebra tiles to make their own configurations and figure out the area.
Participation Structures (group, partners, individual, other):
Students should do this task with partners.
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015
12
Lesson Series #1
2
Lesson Series Overview: Students factor quadratic expressions, ax + x + c using manipulatives and visually enhanced algebraic methods. Emphasis will be
on distinguishing methods based on a = 1 and a <> 1. Algebra tiles will be used to help with factoring and transition to transferring the visual information to a
generic rectangle. Students connect that factoring leads to solving for the x-intercepts of a quadratic function.
CCSS-M Standards Addressed: A.SSE.2, A.SSE.3a, A.APR.3, F.IF.8a
Time: 7 days
Lesson Overview – Day 1
Description of Lesson:
Introduction to factoring quadratics
Resources
Algebra Tiles
CPM-CCA Lesson 8.1.1: Core problems 8-1 to 8-5
(If you have CPM Algebra Connections textbooks, Lesson 8.1.1 is a
similar lesson.)
Lesson Overview – Day 2
Description of Lesson:
Factoring with generic rectangles
Lesson Overview – Day 3
Description of Lesson:
Factoring with special cases
CPM-CCA Lesson 8.1.2: Core problems 8-13 to 8-16
(If you have CPM Algebra Connections textbooks, Lesson 8.1.2 is a
similar lesson.)
Resources
CPM-CCA Lesson 8.1.3: Core problems 8-24 to 8-26
(If you have CPM Algebra Connections textbooks, Lesson 8.1.3 is a
similar lesson.)
Lesson Overview – Day 4
Description of Lesson:
Factoring Completely
Resources
Resources
CPM-CCA Lesson 8.1.4: Core problems 8-35 to 8-38
(If you have CPM Algebra Connections textbooks, Lesson 8.1.4 is a
similar lesson.)
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015
13
Lesson Overview – Day 5
Resources
Description of Lesson:
Factoring Shortcuts
CPM-CCA Lesson 8.1.5: Core problems 8-45 to 8-47
Lesson Overview – Day 6
Resources
Description of Lesson:
Opening: Students will learn relationship between factoring and the graph of quadratic
2
equations in the form y = ax + bx + c, for y = 0.
CPM-CCA Lesson 8.2.2: problems 8-64 to 8-68
Methods and Meanings p. 393
(If you have CPM Algebra Connections textbooks, Lesson 8.2.3 is a
similar lesson.)
Notes:
Make sure students remember difference between x- and y- intercepts and vertex.
Lesson Overview – Day 7
Resources
Description of Lesson:
2
Opening: Students will be given quadratics equations in the form y = ax + bx + c.
Students will factor and solve for x using the zero product property. They will also
graph/sketch parabolas for those equations.
CPM-CCA Lesson 8.2.2 and 8.2.3: problems 8-71, 8-73 can omit
finding vertex, 8-77
Notes:
You can also have students find the vertex if they have had experience with this already.
It is good fore them to know that its x-value lies halfway between the x-intercepts.
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015
14
Apprentice Task
Forming Quadratics
What will students do?
Mathematics Objectives and Standards
Framing Student Experience
Math Objectives:
● Students will be able to convert the standard form of a quadratic
equation into factored form.
● Students will be able to convert the standard form of a quadratic
equation into vertex form.
● Students will be able to explain how these forms are different.
Launch:
Have students work on “Quadratic Functions” handout.
During:
Give pairs of students each of the “Domino Cards” to cut out and match.
Students convert quadratic equations in standard form to factored form and
vertex form and match them to the graphs of the equations.
CCSS-M Standards Addressed: A.SSE.2, A.SSE.3
Potential Misconceptions:
● Students may connect numbers in equations with numbers on the
graph incorrectly.
2
● Students may factor incorrectly (by just changing x to x and bx to b,
for example)
● Students may connect negative coefficients to 3rd and 4th quadrant
on the graph.
Closure:
Debrief the lesson, assign competence, and review common misconceptions
(or call student to share answers).
Extension:
Have students group graphs/equations into similar groups and write
observations (open-ended: graphs with negative a coefficient, graphs that are
above/below x-axis, etc.).
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015
15
Forming Quadratics
How will students do this?
Focus Standards for Mathematical Practice:
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning;
Structures for Student Learning:
Academic Language Support:
Vocabulary: quadratic equation, coefficient, constant term, standard form for quadratics, simplifying, exponent, factored form
Sentence frames:
The factors of this equation are ________________.
If students get stuck:
What should we do next?
To clarify:
How did you do that?
First, we _________. Then, we ____________.
Differentiation Strategies:
 Remediation: Some students may not complete all problems, focusing instead on key problems. Review factoring integers with students. Review
diamonds with students.
 Challenge: Have students look for patterns among equations.
Participation Structures (group, partners, individual, other):
Partners with help from rest of group. Option for Participation Quiz.
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015
16
Lesson Series #2
Lesson Series Overview: Day 1: Types of solutions and completing the square. Day 2: Generalize completing the square. Day 3: Derive the quadratic formula.
CCSS-M Standards Addressed: A.SSE.3b, A.REI.4a, F.IF.8a
Time: 4 days
Lesson Overview – Day 1
Resources
Description of Lesson:
Opening: Show students graphs of quadratic equations with different types of solutions.
Discuss what it looks like to have two real roots, one real root, and zero real roots.
Classwork: Students use algebra tiles and equation mats to connect equivalent
quadratic equations and explore completing the square.
Opening: Show graphs of CPM Core Connections Algebra, Lesson
10.2.4, Problem 10-60, parts a, b, and d.
Classwork: CPM Algebra Connections, Lesson 10.3.1, Problems
10-99 to 10-104
Notes:
Students need to know diamonds, algebra tiles, equation mat. You may want to
practice/front load with warm-ups/Do Nows.
Lesson Overview – Day 2
Resources
Description of Lesson:
Students find ways to complete the square without using algebra tiles.
Notes:
You may want to give students homework to help prepare/front load for deriving the
quadratic equation on Day 3 (equations, rational terms, square roots, squares).
Classwork/Practice: CPM Algebra Connections, Lesson 10.3.2,
Problems 10-110 to 10-112 (Table from Standard Form to Perfect
Square Form; looks more intermediate)
Reference/guided practice: CPM Algebra Connections, Lesson
10.4.1, Methods and Meanings (Completing the Square), Problem
10-125
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015
17
Lesson Overview – Days 3 and 4
Resources
Description of Lesson:
Students use Completing the Square to derive the quadratic formula. After practicing
using the quadratic formula, they are introduced to imaginary numbers.
Day 3: Derivation of the Quadratic Formula
CPM Core Connections Algebra, Lesson 10.2.5, Problems 10-74 to
10-75, 10-81.
Notes:
Problem 10-74 can be redone as a graphic organizer for students to fill in.
Day 4: Imaginary Numbers
CPM Core Connections Algebra, Lesson 10.2.5, Problems 10-76 to
10-80.
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015
18
Expert Task
Victory Celebration
What will students do?
Mathematics Objectives and Standards
Framing Student Experience
Math Objectives:
● Students manipulate a quadratic model of a real-life situation to find
the maximum and roots of the parabola.
● Students interpret the equivalent forms on a quadratic function and its
solutions in terms of the situation it represents.
Adapted from IMP Year 2 pp. 280, 317–319
CCSS-M Standards Addressed: A.SSE.3b, A.CED.1, A.REI.2, A.REI.4
During:
Students work in groups to perform the tasks described on the second page.
Provide graph paper.
Potential Misconceptions:
● Students may think that the parabola represents the 2-dimensional
trajectory of the rocket rather than the function of the rocket’s height
over time.
Launch:
Use the Three Read Protocol (described on pages 10-11 of the Math Teaching
Toolkit) to introduce the situation described on the first page.
Closure/Extension:
Have groups create posters and have a gallery walk for the class to write
questions and comments.
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015
19
Victory Celebration
How will students do this?
Focus Standards for Mathematical Practice:
1. Make sense of problems and persevere in solving them.
4. Model with mathematics.
Structures for Student Learning:
Academic Language Support:
Vocabulary: launch, base, maximum
Differentiation Strategies:
Allow students to draw diagrams and graphs to help visualize the situation.
Participation Structures (group, partners, individual, other):
Students should work in groups of four.
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015
20
Lesson Series #3
Lesson Series Overview: Students start with the quadratic formula and use it to solve quadratic equations. They practice simplifying answers that are both
rational and irrational. They solve a simple system with a quadratic equation and a linear equation.
CCSS-M Standards Addressed: A.CED.a, A.REI.1, A.REI.2, A.REI.4b, A.REI.7, N.CN.1, N.CN.7
Time: 4 days
Lesson Overview – Day 1
Resources
Description of Lesson:
Students will solve quadratic equations with rational answers using the quadratic formula.
CPM Core Connections Algebra, Lesson 9.1.2, Problems 9-12 to
9-15 (9-15 uses graphing calculator), Methods and Meaning (The
Quadratic Formula), Problems 9-17 to 9-19
Notes:
If students need practice using graphing calculators, have them look at CPM Core
Connections Algebra Lesson 9.1.1 (Problems 9-1, 9-2).
Lesson Overview – Day 2
Resources
Description of Lesson:
Students look at more difficult applications of the quadratic formula.
Notes:
Do you teach how to simplify radicals?
Show imaginary numbers and show what a complex solution looks like.
CPM Core Connections Algebra, Lesson 9.1.3, Problems 9-24 to
9-26, Methods and Meanings (Solving a Quadratic Equation),
Problems 9-27 to 9-30, 9-33 (factoring polynomials)
(If you have CPM Algebra Connections textbooks, Lesson 8.3.2 is
a similar lesson.)
Lesson Overview – Day 3
Resources
Description of Lesson:
Students look at a variety of problems and choose the method that best suits the situation.
CPM Core Connections Algebra, Lesson 9.1.4 (p.429), Problems
9-34 to 9-37
(If you have CPM Algebra Connections textbooks, Lesson 8.3.3 is
a similar lesson.)
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015
21
Lesson Overview – Day 4
Resources
Description of Lesson:
Students solve a system with a quadratic equation and linear equation.
Intersection of Quadratic and Linear Equation
(“Graphs,” MAP 2007)
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015
22
Milestone Task
Falling Shoes
What will students do?
Mathematics Objectives and Standards
Framing Student Experience
Math Objectives:
● Students will write quadratic equations based on a situation.
● Students will say how different parameters (height, starting velocity,
etc.) affect the graph of a quadratic equation.
Launch:
Remind students of the work they’ve done earlier in this unit, especially in the
Expert Task. You might also show a video of a thrown or falling object, or throw
a shoe up in the air.
CCSS-M Standards Addressed: A.CED.1, A.REI.2, A.REI.4
During:
In the constructed response portion, students graph a quadratic function, solve
the related equation using at least two techniques (factoring and using the zero
product property, completing the square, applying the quadratic formula), and
explain the connection between the solutions and the graph.
In the performance assessment, students write quadratic functions for two
thrown shoes and determine their maximum height and when they will hit the
ground.
Potential Misconceptions:
● Students may put coefficients in the wrong place.
Closure/Extension:
Students who need an extra challenge can solve the quadratic equation in the
constructed response portion using a third technique, or they can write their
own shoe problem.
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015
23
Falling Shoes
How will students do this?
Focus Standards for Mathematical Practice:
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
Structures for Student Learning:
Academic Language Support:
Vocabulary: parabola, vertex, line of symmetry, intercepts, velocity
Differentiation Strategies:
 Remediation: Students may struggle with finding the coefficients. Some students may be given the generic form of a quadratic equation and
prompted/helped to fill in the coefficients.
 Challenge: What happens if one of the shoes is thrown later? Students can write their own problems. What if shoes were thrown on the moon (where
the gravity is)?
Participation Structures (group, partners, individual, other):
Individual
SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 2014–2015