Talking Algebra - Digital Chalkboard

Report
Math /Algebra Talks:
Mental Math Strategies
Session Goal
To consider discussion-based activities that:
 Develop targeted CCSS Standards for
Mathematical Practice (SMPs).
 Expand number sense to develop
algebraic thinking.
 Build in formative assessment.
Session Outline
 Watch and discuss an elementary number talk.
 Try some number talks and algebra talks
together.
 Understand the purpose and structure of
number talks.
Key Components of Number Talks
 Classroom Environment and Community
 Classroom Discussions
 The Teacher’s Role
 The Role of Mental Math
 Purposeful Computation Problems
Overarching Goals of Number Talks
 Number Sense
 Place Value
 Fluency
 Properties
 Connecting Mathematical Ideas
Number Talks
 Whole-class activities centered around
mental math tasks.
 Students explain and justify multiple
solution strategies.
 Teacher acts as a facilitator.
 Time required: 5-10 minutes.
Let’s try this problem:
26 + 27
Remember this is a mental math
problem…
No paper or pencil…
An Elementary Number Talk
How would you mentally calculate.
32 x 15?
Try to find the product in two or more ways.
Video Clip
Classroom Discussions:
Using Math Talk to Help Students Learn
Students find 32 x 15
How does this
teacher…
 Use wait time?
 Assess understanding?
 Record student thinking?
How do these
students…
 Meet the Standards for
Mathematical Practice?
 Show algebraic thinking?
 Extend student thinking?
What aspects of this activity would you use
in your classroom?
Which would you change?
Compare and explain your reasoning.
7
8
12
13
6 7
10 10
1
3
1
8
2
5
1
4
5 18
6 16
2
5
2
8
6 8
7 9
3
5
1
2
7
8
12
13
What Makes it a Math Talk as opposed
to a Lesson?
• Understanding how numbers work, rather than learning various
skills.
• Empowers students to examine problems in their own way.
• Short term practice toward long term goals.
• Increased difficulty levels - encourages students to find more
efficient ways to solve problems.
• Never expect students to see the problem the “teacher’s” way.
• Not predictable.
• Don’t replace current curriculum or lesson; only 10-15 minutes of
each day.
Algebra Talks
At the secondary level, build on number sense
to make connections to algebra.
Choose a topic and build a scaffolded “string”
of mental math tasks.
Examples: percents, function concepts, solving
equations, and factoring patterns.
Percent String
Find and compare each pair of numbers. Be
ready to explain how you arrived at your
answers.
 60% of 40 and 40% of 60
 25% of 80 and 80% of 25
 5% of 110 and 110% of 5
 n% of 100 and 100% of n
Describe the pattern. Will this hold every time?
Why?
Guess My Rule
Input
Output
Guess My Rule
Input
Output
1
0
2
3
-3
8
5
24
x
x2 – 1
Equation String
Use only mental math to find a value for
the variable that makes the equation true.
Be prepared to explain your solution.
Equation String
 x+1=5
 x + ½ = 4½
 2x + ½ = 4½
 2(x + ½ ) = 9
 5.5 = 3x + 2.5
Product String
 Estimate each product first.
 Do not calculate until told to do so!
Product String
Estimate First
 19 x 21
 99 x 101
 199 x 201
 39 x 41
 299 x 301
 (n – 1)(n + 1)
Standards for Mathematical Practice
1.
Make sense of problems,
and persevere in solving
them.
2.
Reason abstractly and
quantitatively.
3.
Construct viable
arguments, and critique
the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools
strategically.
6.
Attend to precision.
7.
Look for and make use of
structure.
8.
Look for and express
regularity in repeated
reasoning.
Standards for Mathematical Practice
1.
Make sense of problems,
and persevere in solving
them.
2.
Reason abstractly, and
quantitatively.
3.
Construct viable
arguments and critique
the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools
strategically.
6.
Attend to precision.
7.
Look for and make use of
structure.
8.
Look for and express
regularity in repeated
reasoning.
How to Get Started
 Choose a topic, skill or problem that will be taught
in the next two weeks. This will be your target.
 Choose a starter question. It should involve a
prerequisite skill or topic. Make it accessible!
 How will you scaffold questions to build
complexity towards the target?
 What tools will be available to students?
 What will you listen for in student responses?
Some Examples
Talk Starters, and Targets
 Which is larger, 4/7 or
3/8?
 Estimate the number of
hairs on your head.
 Find 13% of 30 mentally.
 (x – 7)(x + 2) = 0
x2 + x
 Simplify 2
x + 2x +1
 Use estimation. Suggest the
equation of a parabola
passing through the points
below:
 I paid $54 for an item that
was discounted 40%. What
was my savings?
 Write a numerical
expression equal to 46.
Find as many as you can.
 Write an equation
equivalent to x = 6. Find as
many as you can.
Benefits of Math Talks
 Clarify thinking (MP 1,2).
 Consider and test other strategies to see if they
make sense (MP 1).
 Investigate and apply mathematical
relationships (MP 2,3,7,8).
 Build a repertoire of efficient strategies
(MP 1,3,5,8).
 Make decisions about choosing efficient
strategies for specific problems (MP 5,7,8).
Teachers don’t always understand
the student’s thinking. It’s OK to
say, “I’d like to study this further
and get back with you.”
“When kids listen to each other, they understand it
better than when they hear it directly from me. It
makes more sense to them…I really see a lot of
learning going on by children listening to the other
children, I really do. I mean I see some of the slower
kids really picking up on concepts…really learning a
lot from listening to other kids.”
Susan Gehn, first and third-grade teacher
Children’s Mathematics, CGI (98)
Thank you!
Teachers are the key to changing the
way students learn mathematics
-Dana and Yendol-Silva
Questions?
Contact us:
Madeleine Jetter
Department of Mathematics
Cal State University, San Bernardino
[email protected]
Vicky Kukurda
Instructional Services
Riverside County Office of Education
[email protected]
Recommended Resources
 Chapin, O’Connor and Anderson, Classroom
Discussions: Using Math Talk to Help Students
Learn. Math Solutions
 Anderson and Schuster, Good Questions for Math
Teaching: Why Ask Them and What to Ask,
Grades 5-8. Math Solutions
 Classroom Video Visits at
www.insidemathematics.org
 Parrish, Sherry, Number Talks: Helping Children
Build Mental Math and Computation Strategies,
Grade K-5. Math Solutions.

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