Download Looking for Structure and Repeated Reasoning in Algebra

Looking for Structure and Repeated Reasoning in Algebra
Josh Chesler ‐ California State University, Long Beach
Tyler Boogar ‐ Garden Grove Unified School District
CMC‐S • November 6, 2015
The title of this talk…
Looking for Structure and Repeated Reasoning in Algebra
SMP 7: Look for and make use of structure
SMP 8: Look for and express regularity in repeated reasoning
Is this “Algebra” as:
a CCSSM conceptual category?
a high school class?
What do we mean by “Algebra”?
• 6 Conceptual Categories: Number and Quantity, Algebra, Functions, Modeling, Geometry, Statistics and Probability
• The Modeling category has no standards
• The CDE Algebra I and II frameworks include standards from Number and Quantity, Algebra, Functions, Statistics and Probability (and starred Modeling standards). • Yes, Algebra and Functions are separate conceptual categories. • Why?
• For the purposes of this talk, we will focus on the Algebra conceptual category.
• Why? Time constraints. • Though, looking for structure and looking for repeated reasoning are mathematical practices. They can and should be done within any conceptual category.
What is meant by “look for…”?
SMP 7: Look for and make use of structure
SMP 8: Look for and express regularity in repeated reasoning
• What do the SMP mean by “look for”?
• How do we help our students “look for…”?
• Once they’ve found it, then what?
These practices may seem harder to assess and harder to teach than skills‐based mathematics.
So far, we’ve payed very careful attention to the language of the CCSSM and the mathematical language we’ve chosen.
Mathematical Practices
• Do students ever ask you: How did you know to do that?
• How did you know?
Examples:
• Clearing denominators
6
1 by rewriting as a quadratic in 3
• Factoring 9
• 9
6
1
3
2 3
1
3
1 3
Has the structure:
∎
2∎ 1
1
Goals
SMP 7: Look for and make use of structure
SMP 8: Look for and express regularity in repeated reasoning
• Gain some more insight into SMP 7 & 8 by exploring a few High School Algebra standards. • Explore the questions from the previous slide: What do the SMP mean by “look for”? How do we help our students “look for…”? Once we’ve found it, then what?
Disclaimer: SMP 7 & 8 are broad practices which help define what it means to do mathematics. We will not explore the entirety of their breadth or depth. Practice Standard 7
• Look for and make use of structure
• Mathematically proficient students look closely to discern a pattern or structure.
• They also can step back for an overview and shift perspective.
• They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects
Quoted from the introductory pages of the CCSSM.
Four Algebra Domains
Algebra Domains
• Seeing Structure in Expressions
• Arithmetic with Polynomials & Rational Expressions
• Creating Equations
• Reasoning with Equations & Inequalities
From the Illustrative Mathematics Project
• illustrativemathematics.org
2
4and 4 .
• Consider the following algebraic expressions: a. Use the figures below to illustrate why the expressions are equivalent
b. Find some algebraic deductions of the same result. A‐SSE: Seeing Structure in Expressions Cluster: Interpret the structure of expressions. Standard: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Practice Standard 7
• Look for and make use of structure
• Mathematically proficient students look closely to discern a pattern or structure.
• They also can step back for an overview and shift perspective.
• They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects
Quoted from the introductory pages of the CCSSM.
More examples from A‐SSE. Seeing Structure in Expressions 2 Clusters within this domain
• Interpret the structure of expressions.
• Write expressions in equivalent forms to solve problems.
Problems On Algebra
from Algebra: Form and Function
You plan to drive 300 miles at 55 miles per hour, stopping for a two‐hour rest. You want to know , the number of hours the journey is going to take. Which of the following equations would you use? (p. 28)
a 55
190
b 55 2
300
c 55
2
300
d 55
2
300
Seeing Structure – Example 1
You plan to drive 300 miles at 55 miles per hour, stopping for a two‐hour rest. You want to know , the number of hours the journey is going to take. Which of the following equations would you use?
What can you see in the structure of the expression on the left side of the equation: 55
2
300?
A‐SSE.1 Interpret expressions that represent a quantity in terms of its context.
a) Interpret parts of an expression, such as terms, factors, and coefficients.
b) Interpret complicated expressions by viewing one or more of their parts as as the product of and a a single entity. For example, interpret 1
factor not depending on .
Seeing Structure – Example 2
The revenue to a bookstore from selling 1000 20
T‐shirts at dollars each is
1000 20
Suppose that each T‐shirt costs the bookstore $3 to make.
a) Write an expression for the cost of making the T‐Shirts.
b) Write an expression for the profit, which is revenue minus the cost.
c) For what values of is the profit positive? (p. 264)
Example 2 ‐ CCSSM connections
• A‐SSE.2: Use the structure of an expression to identify ways to rewrite it. For example, –
, thus recognizing it as a difference of squares that can be see – as .
factored as –
• A‐SSE.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
3 1000 20
• Profit = 1000 20
• A‐SSE.2: See the two terms as having a common factor and use the distributive law. Profit = 3 1000 20
20
3 50
• A‐SSE.3: See profit as the product of two factors. Profit is negative when exactly one of the factors is negative. The first factor changes sign at 3, the second at 50.
•
• 3
•
50: negative profit
50: positive profit
3: negative profit
Example 2, extended
• We determined that, if the price of a T‐shirt is set at $ , the bookstore’s 3 1000 20 . Now, suppose the bookstore wants to profit will be maximize profit. They want to choose the price that will yield the greatest profit. The best form of this expression for figuring what price to set is which of the following equivalent expressions:
a
3 1000 20
b 20
3 50
c 20
1060
3000
d 20
26.5 11045
e 20
3
50
f
1000 20
3 1000 20
CCSSM Connections
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) Factor a quadratic expression to reveal the zeros of the function it defines.
b) Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
Repeated Reasoning
Repeated Reasoning
We’ll see:
• Repeated reasoning across courses and grades (global)
• The CCSSM emphasize coherence.
• Repeated reasoning within a problem context (local)
Global Example 1: Solving equations
CCSSM Algebra Domain: Reasoning with Equations and Inequalities A ‐
REI
Understand solving equations as a process of reasoning and explain the reasoning
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.
2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
It’s always the same question…
3
6
4
3
0
What are all the values of which make this true?
4
3
0
What are all the values of which make this true?
What are all the values of which make this true?
3
3
0
4
3
1
3
2
3
1
What are all the pairs ,
that make this true?
What are all the pairs ,
that make this true?
What are all the pairs ,
that make this true?
What are all the pairs ,
that make this true?
What are all the triples , ,
that make this true?
What are all the pairs , that make this true? Or, is it all triples , , which make the equation true? By convention, it’s typically the former. 0
10
2
What are all the pairs ,
which make both equations true?
What are all the values of which make this true?
3
6
3
2
If an equation in one variable is true for all values of the variable make it true, then the equation is an identity or property.
… and it’s always a set.
3
6
4
3
0
The set of all which make this true.
4
3
0
The set of all which make this true.
The set of all which make this true. 3
3
The set of all , y which make this true.
3
0
4
The set of all , y which make this true.
3
2
3
The set of all , y which make this true.
1
10
2
6
3
The set of all , y, z which make this true.
The set of all , y which make this true. Or, is The set of all , y, k which make this true. By convention, it’s typically the former. 0
3
Remember, a relation is a set of ordered pairs. Some relations are functions. The set of all , y which make this true.
1
3
6
The set of all , y which make both equations true.
2
The set of all which make this true.
SBAC 11 (Seeing the big picture helps)
Given:
3
1
0
Select values of and to complete the statements about the solutions to the given equation.
Choices: 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5
a. The equation has no solutions when
____ and ___
b. The equation has an infinite number of solutions when ___and
___.
c. When
1 and 2, ___.
Grade 11 SBAC
What does a solution look like for an equation of two variables? An equation is shown. How many solutions are there?
3
Use the Add Point tool to plot three solutions to this equation on the coordinate grid.
Repeated reasoning…
With an equation of one variable 6th grade 6.EE.B.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? With more complicated equations and inequalities of one variable in high school.
With equations/inequalities of two variables in high school.
With systems of equations/inequalities in high school.
SMP 7: Look for and make use of structure
SMP 8: Look for and express regularity in repeated reasoning
Another quick example of global repeated reasoning
Algebra Domains
• Seeing Structure in Expressions
• Arithmetic with Polynomials & Rational Expressions
• Creating Equations
• Reasoning with Equations & Inequalities
Students first explore arithmetic with whole numbers, then rational numbers, then integers, then decimals… and now polynomial and rational expressions! The reasoning is repeated and we can look for and make use of structure.
Arithmetic with Polynomial & Rational Expressions
One of the four domains listed in the Algebra Standards of the CCSS is “Arithmetic with Polynomials & Rational Expressions”.
The same algebraic structure.
2 of the seven standards in this domain are:
• Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
• (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
Repeated Reasoning within a problem context
• A square lattice can be made from toothpicks as in the examples above. What is the relationship between the number of toothpicks and the size of the square?
• Take a minute to think about this… we’re going to cut you off, though. (Sorry.)
lattice, • If you can count the toothpicks the same way in the 1st, 2nd, 3rd, … you are using repeated reasoning. Students need to look for this repeated reasoning. Then, you can express that regularity by writing an expression for lattice
the number of toothpicks in the SMP 7: Look for and make use of structure
SMP 8: Look for and express regularity in repeated reasoning
One solution
SMP 7: Look for and make use of structure
SMP 8: Look for and express regularity in repeated reasoning
• In the first lattice, there are 2 rows of 1 horizontal toothpick and 2 columns of 1 vertical toothpick. So there are 2 1 2 1
toothpicks.
• In the second lattice, there are 3 rows of 2 horizontal toothpicks and 3 columns of 2 vertical toothpicks. So there are 3 2 3 2
toothpicks.
• In the third lattice, there are 4 rows of 3 horizontal toothpicks and 4 columns of 2 vertical toothpicks. So there are 4 3 4 3
toothpicks.
1 rows of horizontal toothpicks • In the lattice, there are and 1 columns of vertical toothpicks. So there are 1
1
toothpicks.
There is value in NOT simplifying these expressions. They reflect our reasoning, there is meaning in their structure and their structure helps us express regularity.
Scaffolding Technique for Generalizing SMP 7: Look for and make use of structure
(Driscoll p. 94)
SMP 8: Look for and express regularity in repeated reasoning
• Show several examples of a generalizable phenomenon
• 4
• 5
• 8
4
5
8
16 and 3
25 and 4
64 and 7
5
6
9
15
24
63
• Discuss what is common to these
• Have students produce some additional examples of the “same kind”
• Give the students some large‐number cases, such as 256 256 65,536. What is the product of 255 257?
• Have them try “undoing”, e.g, 16 16 256, __ __ 255?
• Ask student to find a general rule that applies to all of the foregoing exercises.
• Ask students to show that rule is always true.
Driscoll, M. (1999). Fostering Algebraic Thinking: A Guide for Teachers, Grades 6‐10. Heinemann, 361 Hanover Street, Portsmouth, NH 03801‐3912.
Toothpicks – Working Backward
What counting strategy did the student use who wrote the following expression?
A.
1
1
B. 4
1
1
C. 4
1
1
Which of these was useful?
D. 2
2
SMP 7: Look for and make use of structure
SMP 8: Look for and express regularity in repeated reasoning
Are all of those expressions equivalent?
JC6
More Practice – look for structure
An orbiting spaceship releases a probe that travels directly away from Earth. The probe’s distance (in km) from Earth after seconds is given by = 600 + 5 . What does 600 represent in that equation? What does 5 represent in that equation? What does 5 represent in that equation? Do the same for these:
1.The cost, $ , of hiring a repairman for hours is given by 50 25 .
2.The cost, $ , of renting a limousine for h hours above the 4 hour minimum is given by 300 100 .
3.The monthly charge of a cell phone is 25 0.06 dollars, where n is the number of minutes used.
4.The value of an antique is 2500 80 dollars, where n is the number of years since the antique is purchased.
5.The distance, , in meters from the shore, of a surfer riding a wave is given by 120 5 , where t is the number of seconds since she caught the wave
Scenarios from: McCallum, W. G., Connally, E., & Hallett, D. H. (2014). Algebra: Form and function. AMC, 10, 12.
Slide 42
JC6
Should probably skip this... unless there's time.
Josh Chesler, 10/22/2015
Some Observations
SMP 7: Look for and make use of structure
SMP 8: Look for and express regularity in repeated reasoning
• SMPs 7 & 8 often go hand in hand.
• These are practices. We can invoke them in diverse contexts, but students need to be supported in looking for structure and repeated reasoning.
• After we find structure, we make use of it.
• After we find repeated reasoning, we attempt to express it mathematically.
Goals Revisited
• Gain some more insight into SMP 7 & 8 by exploring a few High School Algebra (& Functions) standards as a means to discuss.
• Explore these questions: What do the SMP mean by “look for”? How do we help our students “look for…”? Once they’ve found it, then what?
Last Thoughts
• There’s much more to structure, there’s much more to repeated reasoning.
• We focused just on some Algebra standards.
• These are practices used throughout mathematics.
• “Look for” may sound fuzzy… but that’s okay. • In the 2007 Handbook of Research on Mathematics Teaching and Learning, Hiebert and Grouws* surveyed research that links features of teaching with students’ understanding. They identified two key features:
Feature 1: Teachers and students attend explicitly to concepts
Feature 2: Students struggle with important mathematics
* In “The Effects of Classroom Mathematics Teaching on Student Learning”
Thank you.
We’ll post these slides on the CMC‐S conference website.
Josh Chesler ‐ [email protected]
Tyler Boogar ‐ [email protected]
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