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Laurel County SSADay 4
Jim Moore
Jennifer McDaniel
Agenda• 8:00-8:30 Sign In, Warm-Up Problems
• 8:30-8:45 Outline Goals and Expectations
• 8:45-10:45 Break Out Sessions(Building Algebraic Thinking)
• 11:00-12:30 Lunch
• 12:30-1:30 Strategies to Engage Students
• 1:30-1:40 Break
• 1:40-2:30 Using Technology to Address the Common Core Standards
• 2:30-3:00 Wrap-Up, Reflections, and Evaluations
MEMORY BOX
Summer Review
Memory Box
• How might you use this strategy in your classroom?
• Can you name five “go to” strategies that you use already in your classroom to review material?
SSA Goals
- Assist teachers with diagnosing needs specific to learning goals in The Number System, Expressions and Equations and Functions.
- Have teachers design instructional experiences that incorporate technology, manipulatives, vocabulary strategies, various representations, communications and connections to engage and motivate students of all learning styles.
- Connect the Mathematical Practice Standards to the mathematical content
SSA Goals
- Teachers understand that strategies will also be useful as interventions for struggling students.
- Participants leave each training session with the instructional materials used in training.
SSA Goals
Some Trainer Stepping Stones to the Stated Goals:
- Creating familiarity with KCAS, Career and College Readiness Standards and successful EPAS performance.
To this end, trainers will work in partnership
with district leadership and MSLN leaders
to identify content focus and relatedassessments as indicators of
improvementover the three year project.
SSA Goals
- give teachers experiences/strategies needed to elicit higher level thinking skills that can be incorporated into their instruction.
- give teachers first hand experience with technology tools to use to support instruction (e.g. GeoGebra, Graphing Calculators, TI Nspire Teacher Edition Software, etc.).
- give teachers knowledge of and experience with tools to create a classroom climate to ensure that all students are engaged in the learning of mathematics
Day One• Stumbling Blocks -Prequisite Skills/retention/fear of
failure-past experiences; Lack of support from home, motivation, waiting to dropout; Content: generalizations, variables, moving beyond concrete stage, misuse of technology
• Technology-Inspire CX - Introduction
• Characteristics of Highly Effective Teaching and Learning (CHETL) Learning Environment States of Mind
• Integer Games and Number Line Activities
• Persistence in Problem Solving Connections to Mathematical Practice Standards Multiple Representations (NAGS)
Day 2
• Learning Styles & Student Engagement Task Rotations Question Museums Parallel and Open Tasks Conceptual Skills Practice Problems
Examples:
• Open Task: You multiply two integers.
The result is 50 less than one of them.
What might the two integers be?
Day 2
• Parallel Task:
Choice 1: Describe what 10 colored cubes you would put in a bag so that the probability of selecting a red one is high, but not certain.
Choice 2: Describe what 10 colored cubes you would put in a bag so that the probability of selecting a red one is 2/5.
Day 2
• Conceptual Skills Problem:
The average of four numbers is negative.
Explain your response to each:
(a) Can all four numbers be negative?
(b) Can all four numbers be positive?
(c) Can only two of the four numbers be positive?
(d) Can only three of the numbers be negative?
(e) Can only one of the four numbers be negative?
Day 2
• Writing Quality Questions• Open Questions, Parallel Tasks and
Conceptual Skills Practice Problems
Day 3
• Modeling with Mathematics
• Data Analysis Comparing TI-84 & CX
• GeoGebra (Free Technology) – Introduction as as dynamic geometry software, all types of graphing and data analysis
• Grade level grouping for a first effort in writing some open questions, parallel tasks and conceptual skills practice problems.
Feedback
Technology – requests for TI 84 and TI Nspire, GeoGebra, SmartBoard, Manipulatives, iPads
Hands on activities
Writing Quality Questions
Manipulatives
Follow Up
• Three Full Day On Site Sessions Sept. 28th
Oct. 25th
Nov. 29th
• Administration Support TBD
• Digital Presence
• Year 2 & Year 3
Laurel County SSADay 4
Middle School Breakout Session
Jim Moore
Jennifer McDaniel
Today’s Learning Targets
• I can use the border problem to build algebraic thinking in my classroom.
• I can determine high and low cognitive tasks.
• I can identify appropriate mathematical practices addressed during a high level cognitive task.
• I can develop a stop doing/start doing list to create high cognitive level tasks in my classroom.
Connecting Mathematical Ideas• Border Problem
The Border Problem
Without counting, use the information given in the figure above (exterior is 10 x 10 square; interior is an 8 x 8 square; the border is made up of 1x1 squares) to determine the number of squares needed for the border. If possible, find more than one way to describe the number of border squares.
• What about a 6 in by 6 in grid?
• What about a 15 in by 15 in grid?
• What about a 253 in by 253 in grid?
• What about an n inch by n inch grid?
• Create a verbal representation
• Use the verbal representation to introduce the notion of variable
• If n represents the number of unit squares on one side, give an algebraic expression for the number of unit squares in the border.
• Develop understanding of function, variables (independent and dependent) and graphing.
Border Problem Video
• Part One (Use printed transcripts to follow the dialogue)
• As you watch the video, concentrate specifically on the activity, the teacher, the students, and the learning environment.
The Teacher’s Strategy
The teacher used the experience of the 10 by 10 border problem to built algebraic understanding. She asked the students to think about a smaller square, 6 by 6, and asked the students to determine a set of equations of the 6 by 6 that matched the ways the students thought about the 10 by 10 square. They had to write new equations in the same manner that Sharmane, Colin and the others had in the first problem. Next the teacher asked the students to color a picture of the border problem, to match each equation and also write the process to find each total in a paragraph. Now she felt the students were ready to use algebraic notation to generalize each equivalent equation.
Video Discussion
• Why without talking?
• Why without writing?
• Why without counting one by one?
• Why not give them each a grid to facilitate their thinking?
• Why did the teacher act as the recorder for the arithmetic expressions?
Boaler, J. & Humphreys, C. (2005). Building on student ideas: The border problem, part I. Connecting mathematical ideas: Middle school video cases to support teaching and learning (pp.13-39). New Hampshire: Heineman Publications.
The Border Problem
Sharmane: 4•10 - 4 = 36
Colin: 10+9+9+8 = 36
Joseph: 10+10+8+8 = 36
Melissa: 10•10 - 8•8 = 36
Tania: 4•9 = 36
Zachery: 4•8 + 4 = 36
Border Problem Video
• Part two (Use printed transcript to follow the dialogue)
• As you watch the video, concentrate specifically on the activity, the teacher, the students, and the learning environment.
Student Equations Generalizing For Any
Size Square• 10+10+8+8=36
• Let x be the number of unit square along the side of the square.
• x + x + m + m = total
• x + x + (x-2) + (x-2) = total
Introducing Algebraic Notation
Moving from the specific to the general case.
Developing an understanding of variable and its uses.
Tying abstract ideas to concrete situations.
Fostering meaning to notation.
Developing the concept of equivalent expressions.
Encouraging efficiency and brevity in notation
The Border Problem allowed for most (if not all) students to develop an algebraic expression, which would calculate the square units in the border of a square frame. What I found is that many of the students did not naturally use a variable in their expression. In the future, I would require students to work with several different size square borders; then have them present their expressions while I compiled a list of correct ones. We would then look for similarities and as a Part II, I would have the expectation that generalizations be made, and that a variable represent the same “part” of different sized frames.
Teacher Reflections
Comparing Two Mathematical Tasks
Martha’s Carpeting Task
The Fencing Task
Comparing Two Mathematical Tasks
How are Martha’s Carpeting Task
and the Fencing Task the same and how are they different?
Martha’s Carpeting Task
Martha was re-carpeting her bedroom, which was 15 feet long and 10 feet wide. How many square feet of carpeting will she need to purchase?
The Fencing Task
• Ms. Brown’s class will raise rabbits for their spring science fair. They have 24 feet of fencing with which to build a rectangular rabbit pen to keep the rabbits. If Ms. Brown’s students want their rabbits to have
as much room as possible, how long would each of the sides of the pen be?
How long would each of the sides of the pen be if they had only 16 feet of fencing?
How would you go about determining the pen with the most room for any amount of fencing? Organize your work so that someone else who reads it will understand it.
Comparing Two Mathematical Tasks
• Think privately about how you would go about solving each task (solve them if you have time)
• Talk with you neighbor about how you did or could solve the task
Martha’s CarpetingThe Fencing Task
Solution Strategies: Martha’s Carpeting Task
Martha’s Carpeting TaskUsing the Area Formula
A = l x w
A = 15 x 10
A = 150 square feet
Martha’s Carpeting TaskDrawing a Picture
10
15
Solution Strategies: The Fencing Task
The Fencing TaskDiagrams on Grid Paper
The Fencing TaskUsing a Table
Length Width Perimeter Area
1 11 24 11
2 10 24 20
3 9 24 27
4 8 24 32
5 7 24 35
6 6 24 36
7 5 24 35
The Fencing TaskGraph of Length and Area
0
5
10
15
20
25
30
35
40
0 1 2 3 4 5 6 7 8 9 10 11 12 13
Length
Area
Comparing Two Mathematical Tasks
How are Martha’s Carpeting Task and the Fencing Task the same and how are they different?
Similarities and Differences
Similarities
• Both are “area” problems
• Both require prior knowledge of area
Differences
• The amount of thinking and reasoning required
• The number of ways the problem can be solved
• Way in which the area formula is used
• The need to generalize
• The range of ways to enter the problem
Similarities and Differences
Similarities
• Both are “area” problems
• Both require prior knowledge of area
Differences
• The amount of thinking and reasoning required
• The number of ways the problem can be solved
• Way in which the area formula is used
• The need to generalize
• The range of ways to enter the problem
A Critical Starting Point for Instruction
Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.
Stein, Smith, Henningsen, & Silver, 2000
The level and kind of thinking in which students engage determines what they will learn.
Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997
• There is no decision that teachers make that has a greater impact on students’ opportunities to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages students in studying mathematics.
Lappan & Briars, 1995
Task Analysis
• Use Cognitive level Handout and printed tasks
To Do List
To STOP Doing To START Doing
Laurel County SSADay 4
High School Breakout Session
Jim Moore
Jennifer McDaniel
The House that Jack Built
• Building Algebraic Thinking
Lunch
11:00-12:30
Ideas for engagement
• Turnover Cards
• Card Sorts
• Math Strings
Technology
• Human Wave/Pass the ball
Regression activity for graphing calculator
How does the CX compare to the TI-84?
Solving Equations using the graphing calculator.• Standard (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.
New Ideas: The hidden potential in QR Codes
Shared Resources
• Content Network Updates
• NROC website
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