Content Deepening6th Grade Math January 24, 2014 Jeanne Simpson AMSTI Math Specialist
Welcome • Name • School • What are you hoping to learn today?
He who dares to teach must never cease to learn. John Cotton Dana
Goals for Today • Implementation of the Standards of Mathematical Practices in daily lessons • Understanding of what the CCRS expect students to learn blended with how they expect students to learn. • Student-engaged learning around high-cognitive-demand tasks used in every classroom.
Agenda • Statistics and Probability • Progression • Standards Analysis • Resources • High-Cognitive Demand Tasks • Expressions and Equations • Inequalities • Resources • Standards of Mathematical Practice • Fractions
acos2010.wikispaces.com • Electronic version of handouts • Links to web resources
The Structure is the standards The natural distribution of prior knowledge in classrooms should not prompt abandoning instruction in grade level content, but should prompt explicit attention to connecting grade level content to content from prior learning. To do this, instruction should reflect the progressions on which the CCSSM are built. For example, the development of fluency with division using the standard algorithm in grade 6 is the occasion to surface and deal with unfinished learning with respect to place value. Much unfinished learning from earlier grades can be managed best inside grade level work when the progressions are used to understand student thinking. • http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/#more-422
Knowledge gaps This is a basic condition of teaching and should not be ignored in the name of standards. Nearly every student has more to learn about the mathematics referenced by standards from earlier grades. Indeed, it is the nature of mathematics that much new learning is about extending knowledge from prior learning to new situations.For this reason, teachers need to understand the progressions in the standards so they can see where individual students and groups of students are coming from, and where they are heading. But progressions disappear when standards are torn out of context and taught as isolated events. • http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/#more-422
Learning Progression Jigsaw • Read your assigned section • K-5 Data, pages 1-5 • 6-8 Overview, pages 6-7 • Grade 6, pages 8-10 • Grade 7, pages 11-14 • Chart paper • Summarize what needs to be learned. • How can this document help you in your classroom? • Be prepared to share
Analysis Tool 6.SP.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.
What resources do you already have for teaching statistics? AMSTI Units Textbook Other Activities How do these match the standards?
SP Resources • Raisin Activity • MARS – Mean, Median, Mode • CMP • Data About Us • Common Core Investigations • Lessons for Learning • How MAD are You? • Shakespeare vs. Rowling
Raisin Activity • Count the number of raisins in your box. • Make a box plot for the number of raisins in each brand’s box. • Find the median, range, and interquartile range for each brand. • Make a dot plot of the data. Find the mean and the mean absolute deviation.
Computer Games: Ratings Imagine rating a popular computer game. You can give the game a score of between 1 and 6.
Computer Games: Ratings Rate the game Candy Crush with a score between 1 and 6.
Matching Cards • Each time you match a pair of cards, explain your thinking clearly and carefully. • Partners should either agree with the explanation or challenge it if it is unclear or incomplete. • Once agreed stick the cards onto the poster and write a justification next to the cards. • Some of the statistics tables have gaps in them and one of the bar charts is blank. You will need to complete these cards.
Sharing Posters • One person from each group visit a different group and look carefully at their matched cards. • Check the cards and point out any cards you think are incorrect. You must give a reason why you think the card is incorrectly matched or completed, but do not make changes to the card. • Return to your original group, review your own matches and make any necessary changes using arrows to show if card needs to move.
How MAD are You?(Mean Absolute Deviation) • Fist to Five…How much do you know about Mean Absolute Deviation? • 0 = No Knowledge • 5 = Master Knowledge
Create a distribution of nine data points on your number line that would yield a mean of 5.
Card Sort • Which data set seems to differ the least from the mean? • Which data set seems to differ the most from the mean? • Put all of the data sets in order from “Differs Least” from the mean to “Differs Most” from the mean.
The mean in each set equals 5. Find the distance (deviation) of each point from the mean. Use the absolute value of each distance. 3 1 3 3 6 2 4 1 3 Find the mean of the absolute deviations.
How could we arrange the nine points in our data to decrease the MAD? • How could we arrange the nine points in our data to increase the MAD? • How MAD are you?
An effective mathematical task is needed to challenge and engage students intellectually.
Comparing Two Mathematical Tasks Solve Two Tasks: Martha’s Carpeting Task The Fencing Task
How are Martha’s Carpeting Task and the Fencing Task the same and how are they different? Comparing Two Mathematical Tasks
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
Do the differences between the Fencing Task and Martha’s Carpeting Task matter? Why or Why not? Comparing Two Mathematical Tasks
Criteria for low cognitive demand tasks • Recall • Memorization • Low on Bloom’s Taxonomy
Criteria for high cognitive demand tasks • Requires generalizations • Requires creativity • Requires multiple representations • Requires explanations (must be “worth explaining”)
Patterns of Set up, Implementation, and Student Learning Task Set Up Task Implementation Student Learning A. High High High B. Low Low Low C. High Low Moderate Stein & Lane, 2012
Routinizing problematic aspects of the task • Shifting the emphasis from meaning, concepts, or understanding to the correctness or completeness of the answer • Providing insufficient time to wrestle with the demanding aspects of the task or so much time that students drift into off-task behavior • Engaging in high-level cognitive activities is prevented due to classroom management problems • Selecting a task that is inappropriate for a given group of students • Failing to hold students accountable for high-level products or processes (Stein, Grover & Henningsen, 2012) Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands
Scaffolding of student thinking and reasoning • Providing a means by which students can monitor their own progress • Modeling of high-level performance by teacher or capable students • Pressing for justifications, explanations, and/or meaning through questioning, comments, and/or feedback • Selecting tasks that build on students’ prior knowledge • Drawing frequent conceptual connections • Providing sufficient time to explore (Stein, Grover & Henningsen, 2012) Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands
“Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.” (Stein, Smith, Henningsen, & Silver, 2011) “The level and kind of thinking in which students engage determines what they will learn.” (Hiebert et al., 2011)
Activities • Inequalities • Look at standards. What is required? • Illustrative Mathematics tasks • MARS – Laws of Arithmetic lesson • Arithmetic with whole-number exponents • Order of operations • Finding area of compound rectangles by evaluating expressions • Math-Magic • Using variables
What are students asked to do with inequalities? • 6.EE.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? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. • 6.EE.6 – Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. • 6.EE.7 – Solve real-world and mathematical problems by writing and solving equations of the form x + p = q andpx = q for cases in which p, q and x are all nonnegative rational numbers. • 6.EE.8 – Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
Log Ride (6.EE.5) A theme park has a log ride that can hold 12 people. They also have a weight limit of 1500 lbs per log for safety reasons. If the average adult weights 100 lbs and the log itself weights 200, the ride can operate safely if the inequality 150A + 100C + 200 < 1500 is satisfied (A is the number of adults and C is the number of children in the log ride together). There are several groups of children of differing numbers waiting to ride. If 4 adults are already seated in the log, which groups of children can safely ride with them? Group 1: 4 children Group 2: 3 children Group 3: 9 children Group 4: 6 children Group 5: 5 children
Fishing Adventures (6.EE.8) Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can hold at most eight people. Additionally, each boat can only carry 900 pounds of weight for safety reasons. • Let p represent the total number of people. Write an inequality to describe the number of people that a boat can hold. Draw a number line diagram that shows all possible solutions. • Let w represent the total weight of a group of people wishing to rent a boat. Write an inequality that describes all total weights allowed in a boat. Draw a number line diagram that shows all possible solutions.