CCGPS Mathematics Unit-by-Unit Grade Level Webinar 8 th Grade Unit 5: Linear Functions October 30, 2012 Session will be begin at 8:00 am While you are waiting, please do the following: Configure your microphone and speakers by going to: Tools – Audio – Audio setup wizard Document downloads: When you are prompted to download a document, please choose or create the folder to which the document should be saved, so that you may retrieve it later.
Transcript
CCGPS MathematicsUnit-by-Unit Grade Level Webinar
8th GradeUnit 5: Linear Functions
October 30, 2012
Session will be begin at 8:00 amWhile you are waiting, please do the following:
Configure your microphone and speakers by going to:Tools – Audio – Audio setup wizard
Document downloads:When you are prompted to download a document, please choose or create the folder to which the document should be saved, so that you may retrieve it later.
• Intent and focus of Unit 5 webinar.• Framework tasks.• GPB sessions on Georgiastandards.org.• Standards for Mathematical Practice. • Resources. • http://ccgpsmathematics6-8.wikispaces.com/• CCGPS is taught and assessed from 2012-2013 and beyond.
Wiki/Email Questions• Our 8th grade teachers have a question: The standard seems vague to
them. It says to evaluate square roots and cubed roots. They were not sure how deep to go with this standards. With cubed roots do they need to go into simplifying of cubed roots.Do they need to go into adding and subtracting of square roots?
Estimate - Yes! Rewrite (Simplify) - No!
Is y = 2x + 1 a linear function?Is y = 2x² + 1 a linear function?
Activate your Brain
Mathematical CommunicationThe development of students’ mathematical communication shifts in precision and sophistication throughout the primary, junior and intermediate grades, yet the underlying characteristics remain applicable across all grades.
CBS Mathematics
Mathematical CommunicationMathematical communication is an essential process for learning mathematics because through communication, students reflect upon, clarify and expand their ideas and understanding of mathematical relationships and mathematical arguments.
Ontario Ministry of Education
Mathematical Communication•Developing effective mathematical communication•Categories of mathematical communication•Organizing students to think, talk, and write•Updating the three-part problem-solving lesson
Gallery WalkMath CongressBansho (Board Writing)
Mathematical Communication“Because mathematics is so often conveyed in symbols, oral and written, communication about mathematical ideas is not always recognized as an important part of mathematics education. Students do not necessarily talk about mathematics naturally; teachers need to help them learn how to do so.”
Cobb, Wood, & Yackel
Mathematical Communication“The role of the teacher during whole-class discussion is to develop and to build on the personal and collective sense-making of students rather than to simply sanction particular approaches as being correct or demonstrate procedures for solving predictable tasks.”
Stein, Engle, Smith, & Hughes
Mathematical CommunicationWhen teacher talk dominates whole-class discussion, students tend to rely on teachers tobe the expert, rather than learning that they can work out their own solutions and learn from other students.
CBS Mathematics
What’s the big idea?•Deepen understanding of ratios and proportional relationships.•Deepen understanding of functions.•Develop understanding of a linear function.•Derive the equation of a linear function.•Standards for Mathematical Practice
ImitativeBad Math?
Passive/receptive Bad Math?
Minimal student explanations, comparisons
Bad Math?
Passive Active
Transmission Challenging
Research - CommunicationResearch tells us that student interaction – through classroom discussion and other forms of interactive participation – is foundational to deep understanding and related student achievement. But implementing discussion in the mathematics classroom has been found to be challenging.
Dr. Catherine D. Bruce
Research - Communication•The value of student interaction•Challenges the teachers face in engaging students•The teacher’s role•Five strategies for encouraging high-quality student interaction
1. The use of rich math tasks2. Justification of solutions3. Students questioning one
another4. Use of wait time5. Use of guidelines for Math Talk
Coherence and Focus• K-7th
Operations with rational numbers and algebraic thinkingExpressions and solving equationsRatios and proportional reasoningFunctions (8th grade Unit 4)
• 9th-12th TransformationsWorking with various functionsSolving various systems
Examples & Explanations
The graphs below show the distance two cars have traveled along the freeway over a period of several seconds. Car A is traveling 30 meters per second. What is a reasonable estimate for the speed of car B?
Adapted from Illustrative Mathematics 8.EE.5 Comparing Speeds in Graphs and Equations
Examples & Explanations
The graphs below show the distance two cars have traveled along the freeway over a period of several seconds. Car A is traveling 30 meters per second. What is a reasonable estimate for the speed of car B?
Examples & Explanations
The graphs below show the distance two cars have traveled along the freeway over a period of several seconds. Car A is traveling 30 meters per second. What is a reasonable estimate for the speed of car B?
Examples & Explanations
The graphs below show the distance two cars have traveled along the freeway over a period of several seconds. Car A is traveling 30 meters per second. What is a reasonable estimate for the speed of car B?
Examples & Explanations
Kell has a job after school. The table below shows how much he got paid every day last week.
Mariko has a job that pays $7 per hour.Who would make more for working 10 hours? Explain.How can you see who makes more per hour just by looking at the graphs? Explain.
Adapted from Illustrative Mathematics 8.EE.5 Who Has The Best Job?
Examples & Explanations
Kell:
Mariko:
Adapted from Illustrative Mathematics 8.EE.5 Who Has The Best Job?
Examples & Explanations
By graphing the two functions, the one with the greater slope will be paid more per hour.
The slope for Kell is 8.4 and the slope for Mariko is 7. Therefore Kell gets paid more per hour.
Adapted from Illustrative Mathematics 8.EE.5 Who Has The Best Job?
Examples & Explanations
The figure shows the lines l and m described by the equations y = 4x – c and y = 2x + d respectively. They intersect at the point (p, q).
Adapted from Illustrative Mathematics 8.EE Equations of Lines
y l
m
(p,q)
x
Examples & Explanations
The figure shows the lines l and m described by the equations y = 4x – c and y = 2x + d respectively. They intersect at the point (p, q).
If you trace line l out to the point with x-coordinate p + 2, and I do the same with line m. How much greater would the y-coordinate of your ending point be than mine?
Adapted from Illustrative Mathematics 8.EE Equations of Lines
y l
m
(p,q)
x
Examples & Explanations• Line l has slope 4. So each increase of
one unit in the x-value produces an increase of 4 units in the y-value.
Adapted from Illustrative Mathematics 8.EE Equations of Lines
y l
m
(p,q)
x
Examples & Explanations• Line l has slope 4. So each increase of
one unit in the x-value produces an increase of 4 units in the y-value.
• Moving 2 units to the right will cause an increase of 8 units up.
Adapted from Illustrative Mathematics 8.EE Equations of Lines
y l
m
(p,q)
x
Examples & Explanations• Line l has slope 4. So each increase of
one unit in the x-value produces an increase of 4 units in the y-value.
• Moving 2 units to the right will cause an increase of 8 units up.
• The line m has slope 2. So each increase of 2 units in the x-value produces an increase of 4 units in the y-value.
Adapted from Illustrative Mathematics 8.EE Equations of Lines
y l
m
(p,q)
x
Examples & Explanations• Line l has slope 4. So each increase of
one unit in the x-value produces an increase of 4 units in the y-value.
• Moving 2 units to the right will cause an increase of 8 units up.
• The line m has slope 2. So each increase of 2 units in the x-value produces an increase of 4 units in the y-value.
• Therefore, your y-value would be 4 units (8 – 4) units larger than my y-value.
Adapted from Illustrative Mathematics 8.EE Equations of Lines
y l
m
(p,q)
x
Examples & Explanations
Graph the functions y = 2x + 1 and y = 2x2 + 1, and list as many differences between the two functions as you can.
Adapted from Illustrative Mathematics 8.F.3 Introduction to Linear Functions
Examples & Explanations
Graph the functions y = 2x + 1 and y = 2x2 + 1, and list as many differences between the two functions as you can.
Adapted from Illustrative Mathematics 8.F.3 Introduction to Linear Functions
Examples & Explanations• The first graph is a straight line, the
second graph is curved.• The first function has negative y-values,
the second function does not.• The first function has a constant slope, the
slope of the second function changes.• Each y-value in the first function has
exactly one x-value. In the second function, most y-values have 2 possible x-values.Adapted from Illustrative Mathematics 8.F.3 Introduction to Linear Functions
Is y = 2x + 1 a linear function?Is y = 2x² + 1 a linear function?
Activate your Brain
Is y = 2x + 1 a linear function?Is y = 2x² + 1 a linear function?
•Students do not necessarily talk about mathematics naturally; teachers need to help them learn how to do so.•The role of the teacher during whole-class discussion is to develop and the build on the personal and collective sense-making of students.•…learning that they can work out their own solutions and learn from other students.
Activate your Brain
Resource List
The following list is provided as a sample of available resources and is for informational purposes only. It is your responsibility to investigate them to determine their value and appropriateness for your district. GaDOE does not endorse or recommend the purchase of or use of any particular resource.
• Common Core Resources SEDL videos - http://bit.ly/RwWTdc or http://bit.ly/yyhvtc Illustrative Mathematics - http://www.illustrativemathematics.org/ Dana Center's CCSS Toolbox - http://www.ccsstoolbox.com/Common Core Standards - http://www.corestandards.org/ Tools for the Common Core Standards - http://commoncoretools.me/Phil Daro talks about the Common Core Mathematics Standards - http://bit.ly/URwOFT
~Thank you! Thank you! Thank you! This webinar was great, and the site has great resources that I can use tomorrow! I just shared it with everyone at my school! It is like going to a Common Core Conference and receiving all the materials for every session and having them in one place! I love it!
~I watch so many math videos for our common core lessons and I am speechless, how awesome all these small video clips are.
~Thanks for this. I attended the webinar last week and really like this site. I'm planning on having a PL session at school on Thursday.
