Content Deepening 7th Grade Math February 6, 2014 Jeanne Simpson AMSTI Math Specialist

Welcome Name School Classes you teach What are you hoping to learn today?

He who dares to teach must never cease to learn. John Cotton Dana The new COS offers many opportunities for us to learn – new content, new teaching strategies, higher expectations for students, filling in gaps during the transition.

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 Surface Area & Volume Statistics Fractions Probability

acos2010.wikispaces.com Electronic version of handouts Links to web resources

Five Fundamental Areas Required for Successful Implementation of CCSS Collaboration Instruction Content Intervention Assessment

How do we teach? Instruction Content Intervention Assessment Collaboration Instruction Deep conceptual understanding Collaborative lesson design Standards for Mathematical Practice Content Fewer standards with greater depth Understanding, focus, and coherence Common and high-demand tasks Intervention Common required response to intervention framework response Differentiated, targeted, and intensive response to student needs Student equity, access, and support Assessment PLC teaching-assessing-learning cycle In-class formative assessment processes Common assessment instruments as formative learning opportunities

Standards for Mathematical Practice Mathematically proficient students will: SMP1 - Make sense of problems and persevere in solving them SMP2 - Reason abstractly and quantitatively SMP3 - Construct viable arguments and critique the reasoning of others SMP4 - Model with mathematics SMP5 - Use appropriate tools strategically SMP6 - Attend to precision SMP7 - Look for and make use of structure SMP8 - Look for and express regularity in repeated reasoning

SMP Proficiency Matrix   Students: (I) Initial (IN) Intermediate Advanced 1a Make sense of problems Explain their thought processes in solving a problem one way. Explain their thought processes in solving a problem and representing it in several ways. Discuss, explain, and demonstrate solving a problem with multiple representations and in multiple ways. 1b Persevere in solving them  Stay with a challenging problem for more than one attempt. Try several approaches in finding a solution, and only seek hints if stuck. Struggle with various attempts over time, and learn from previous solution attempts. 2 Reason abstractly and quantitatively Reason with models or pictorial representations to solve problems. Are able to translate situations into symbols for solving problems. Convert situations into symbols to appropriately solve problems as well as convert symbols into meaningful situations. 3a Construct viable arguments  Explain their thinking for the solution they found. Explain their own thinking and thinking of others with accurate vocabulary. Justify and explain, with accurate language and vocabulary, why their solution is correct. 3b Critique the reasoning of others.  Understand and discuss other ideas and approaches. Explain other students’ solutions and identify strengths and weaknesses of the solution. Compare and contrast various solution strategies and explain the reasoning of others. 4 Model with Mathematics Use models to represent and solve a problem, and translate the solution to mathematical symbols. Use models and symbols to represent and solve a problem, and accurately explain the solution representation. Use a variety of models, symbolic representations, and technology tools to demonstrate a solution to a problem. 5 Use appropriate tools strategically  Use the appropriate tool to find a solution.  Select from a variety of tools the ones that can be used to solve a problem, and explain their reasoning for the selection. Combine various tools, including technology, explore and solve a problem as well as justify their tool selection and problem solution. 6 Attend to precision  Communicate their reasoning and solution to others.  Incorporate appropriate vocabulary and symbols when communicating with others.  Use appropriate symbols, vocabulary, and labeling to effectively communicate and exchange ideas. 7 Look for and make use of structure Look for structure within mathematics to help them solve problems efficiently (such as 2 x 7 x 5 has the same value as 2 x 5 x 7, so instead of multiplying 14 x 5, which is (2 x 7) x 5, the student can mentally calculate 10 x 7. Compose and decompose number situations and relationships through observed patterns in order to simplify solutions. See complex and complicated mathematical expressions as component parts. 8 Look for and express regularity in repeated reasoning Look for obvious patterns, and use if/ then reasoning strategies for obvious patterns. Find and explain subtle patterns. Discover deep, underlying relationships, i.e. uncover a model or equation that unifies the various aspects of a problem such as discovering an underlying function.

SMP Instructional Implementation Sequence Think-Pair-Share (1, 3) Showing thinking in classrooms (3, 6) Questioning and wait time (1, 3) Grouping and engaging problems (1, 2, 3, 4, 5, 8) Using questions and prompts with groups (4, 7) Allowing students to struggle (1, 4, 5, 6, 7, 8) Encouraging reasoning (2, 6, 7, 8)

SMP Proficiency Matrix   Students: (I) Initial (IN) Intermediate Advanced 1a Make sense of problems Explain their thought processes in solving a problem one way. Explain their thought processes in solving a problem and representing it in several ways. Discuss, explain, and demonstrate solving a problem with multiple representations and in multiple ways. 1b Persevere in solving them  Stay with a challenging problem for more than one attempt. Try several approaches in finding a solution, and only seek hints if stuck. Struggle with various attempts over time, and learn from previous solution attempts. 2 Reason abstractly and quantitatively Reason with models or pictorial representations to solve problems. Are able to translate situations into symbols for solving problems. Convert situations into symbols to appropriately solve problems as well as convert symbols into meaningful situations. 3a Construct viable arguments  Explain their thinking for the solution they found. Explain their own thinking and thinking of others with accurate vocabulary. Justify and explain, with accurate language and vocabulary, why their solution is correct. 3b Critique the reasoning of others.  Understand and discuss other ideas and approaches. Explain other students’ solutions and identify strengths and weaknesses of the solution. Compare and contrast various solution strategies and explain the reasoning of others. 4 Model with Mathematics Use models to represent and solve a problem, and translate the solution to mathematical symbols. Use models and symbols to represent and solve a problem, and accurately explain the solution representation. Use a variety of models, symbolic representations, and technology tools to demonstrate a solution to a problem. 5 Use appropriate tools strategically  Use the appropriate tool to find a solution.  Select from a variety of tools the ones that can be used to solve a problem, and explain their reasoning for the selection. Combine various tools, including technology, explore and solve a problem as well as justify their tool selection and problem solution. 6 Attend to precision  Communicate their reasoning and solution to others.  Incorporate appropriate vocabulary and symbols when communicating with others.  Use appropriate symbols, vocabulary, and labeling to effectively communicate and exchange ideas. 7 Look for and make use of structure Look for structure within mathematics to help them solve problems efficiently (such as 2 x 7 x 5 has the same value as 2 x 5 x 7, so instead of multiplying 14 x 5, which is (2 x 7) x 5, the student can mentally calculate 10 x 7. Compose and decompose number situations and relationships through observed patterns in order to simplify solutions. See complex and complicated mathematical expressions as component parts. 8 Look for and express regularity in repeated reasoning Look for obvious patterns, and use if/ then reasoning strategies for obvious patterns. Find and explain subtle patterns. Discover deep, underlying relationships, i.e. uncover a model or equation that unifies the various aspects of a problem such as discovering an underlying function. Pair-Share Questioning/Wait Time Grouping/Engaging Problems Questioning/Wait Time Grouping/Engaging Problems Showing Thinking Grouping/Engaging Problems Encourage Reasoning Grouping/Engaging Problems Showing Thinking Questioning/Wait Time Grouping/Engaging Problems Pair-Share Questioning/Wait Time Grouping/Engaging Problems Grouping/Engaging Problems Questions/Prompts for Groups Showing Thinking Showing Thinking Grouping/Engaging Problems Grouping/Engaging Problems Showing Thinking Allowing Struggle Encourage Reasoning Questions/Prompts for Groups Allowing Struggle Encourage Reasoning Grouping/Engaging Problems Allowing Struggle Encourage Reasoning

Instruction Content Intervention Assessment What are we teaching? Collaboration Instruction Deep conceptual understanding Collaborative lesson design Standards for Mathematical Practice Content Fewer standards with greater depth Understanding, focus, and coherence Common and high-demand tasks Intervention Common required response to intervention framework response Differentiated, targeted, and intensive response to student needs Student equity, access, and support Assessment PLC teaching-assessing-learning cycle In-class formative assessment processes Common assessment instruments as formative learning opportunities

Critical Focus Areas Ratios and Proportional Reasoning Standards 1-3 Applying to problems Graphing and slope Standards 1-3 Number Systems, Expressions and Equations Standards 4-10 Geometry Scale drawings, constructions, area, surface area, and volume Standards 11-16 Statistics Drawing inferences about populations based on samples Standards 17-20 Probability – Standards 21-24

Geometry Draw, construct, and describe geometrical figures and describe the relationships between them. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Surface Area and Volume 7.G.6 - Solve real-world and mathematical problems involving area, volume and surface area of two- and three- dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

7.G.6 - Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

Accelerated 8.G.9 – Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

“Know the formula” means: According to Bill McCallum, 7th graders need to be able to find the surface area of pyramids, but not the volume. “Know the formula” means: Having an understanding of why the formula works Being able to use the formula to solve a problem without being told which formula to use Surface area formulas are not the expectation. Students should recognize that finding the area of each facing and adding the areas will give the surface area. For volume, focus on the area of the base times the height. Pyramids are considered a type of cone for the 8th grade standard.

Connected Mathematics Building Boxes Designing Rectangular Boxes Prisms and Cylinders Cones, Spheres, and Pyramids Scaling Boxes

Connected Mathematics Understand volume as a measure of filling an object and surface area as a measure of wrapping an object. Design and use nets to visualize and calculate surface area of prisms and cylinders. Explore patterns among the volumes of cylinders, cones, and spheres. Develop strategies for finding the volumes of square pyramids, prisms, cylinders, cones, and spheres directly and by comparison with known volumes. Understand that three-dimensional figures my have the same volumes but quite different surface areas. Understand how changes in one or more dimensions of a rectangular prism or cylinder affects the prism’s volume and surface area. Extend students’ understanding of similarity and scale factor to three-dimensional figures. Use surface area and volume to solve a variety of real- world problems.

Surface Area and Volume Building a Box Patch Tool

Surface Area and Volume Changing Surface Areas Packing to Perfection

Changing Surface Area

Changing Surface Areas

Packing to Perfection Is there a relationship between surface area and volume? Can rectangular prisms with different dimensions have the same volume? Do rectangular prisms with the same volume have the same surface area?

Packing to Perfection

Packing to Perfection

Statistics Use random sampling to draw inferences about a population. Draw informal comparative inferences about two populations.

Understanding 7.SP.1 – Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

Pizza or Broccoli?

At a nearby school, teachers decided to get rid of pizza Fridays At a nearby school, teachers decided to get rid of pizza Fridays. After a survey of all teachers, counselors, and administrators, it was overwhelmingly decided that pizza would be replaced with broccoli with ranch dip. Is this fair???

After surveying 83 students in 3 classes, 70% responded that girls should be allowed to go to lunch two minutes early every day and boys will go at the regular time. Do you think this is an accurate statistic? Who do you think the sample population was?

tie Bias

Group Roles Each group will need to assign the following roles: Facilitator – keeps group on task and ensures equal participation Materials Manager – collects and returns materials Recorder – writes group answer on chart paper Reporter – presents group answer to the class

Group Work Discuss and complete the handout as a group. Begin with the multiple choice questions. Choose one biased survey to present to the class on chart paper. Include the following in your presentation: Original survey Why you think it is biased How you would correct it

Create a Survey Question Population Sample group

Exit Ticket In a poll of Mrs. Simpson’s math class, 67% of the students say that math is their favorite academic subject. The editor of the school paper is in the class, and he wants to write an article for the paper saying that math is the most popular subject at the school. Explain why this is not a valid conclusion, and suggest a way to gather better data to determine what subject is most popular.

Use random sampling to draw inferences about a population. 7.SP.2 – Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.

Counting Trees The diagram shows some trees in a tree farm. The circles show old trees and the triangles show young trees. Tom wants to know how many trees there are of each type, but says it would take too long counting them all, one by one.

The Tree Farm

Collaborative Work: Joint Solution Share your method with your partner(s) and your ideas for improving your individual solution. Together in your group, agree on the best method for completing the problem. Produce a poster, showing a joint solution to the problem. Make sure that everyone in the group can explain the reasons for your chosen method, and describe any assumptions you have made.

Analyzing Sample Responses to Discuss Does the approach make mathematical sense? What assumptions has the student made? How could the solution be improved? What questions could you ask the student, to help you understand their work?

Sample Responses to Discuss: Laura

Sample Responses to Discuss: Wayne

Sample Responses to Discuss: Amber

Draw informal comparative inferences about two populations. 7.SP.3 – Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

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 Handout 23-28

Create a distribution of nine data points on your number line that would yield a mean of 5. Work in groups of 2-4. Pass out copies so they can keep their handouts clean. Share results. What might the data represent? What limitations are there in only knowing the mean?

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. What are some possible contexts of data for the set? How did you determine the order? What is the best order?

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 3 1 3 2 1 3 4 6 Put the data in a table. Use the MAD to re-order the data sets. 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?

Draw informal comparative inferences about two populations. 7.SP.4 – Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.

Illustrative Mathematics Illustrative Mathematics provides guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students experience in a faithful implementation of the Common Core State Standards, and by publishing other tools that support implementation of the standards. http://www.illustrativemathematics.org/

Offensive Linemen ```````````````````````````````````````````````````````````````````````````````````````````````````````````````````````````````

X Marks the Spot

Shakespeare vs. Rowling Handout 29-32 (lesson plan) 33-36 (student sheets) I have clean copies of student sheets. Raise your hand if you have ever read a book by J.K. Rowling. Raise your hand if you have ever read a play by William Shakespeare. Let them work in groups on this task. Ask the after questions on page 30 of handout. Why do you think so many more kids have read Harry Potter books than Shakespeare’s plays?

The Number System Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Mathematics consists of pieces that make sense; they are not just independent manipulation/skills to be practiced and memorized – as perceived by many students. These individual pieces progress through different grades (in organized structures we called “flows”) and can/should be unified together into a coherent whole. Jason Zimba, Bill McCallum

Fractions Difficulty with learning fractions is pervasive and is an obstacle to further progress in mathematics and other domains dependent on mathematics, including algebra. It has also been linked to difficulties in adulthood, such as failure to understand medication regimens. National Mathematics Panel Report, 2008

Fractions “Students who are asked to practice the algorithm over and over…stop thinking. They sacrifice the relationships in order to treat the numbers simply as digits.” Imm, Fosnot, Uittenbogaard (2012)

Unit Fractions

Fraction Multiplication in Grade 5

Fraction Multiplication in Grade 5

Fraction Multiplication in Grade 5

Fraction Multiplication in Grade 5

5th Grade Division

5th Grade Division

5th Grade Division Problems How much chocolate will each person get if 3 people share ½ pound equally?

5th Grade Division Problems How many 1/3 cup servings are in 2/3 cups of raisins?

Fraction Division in Grade 6 6.NS.1 – Interpret and compute quotients of fractions, and solve word problems involving division of fractions, e.g., by using visual fraction models and equations to represent the problem. Examples: Create a story context… Use a visual fraction model to show the quotient… Explain division using its relationship with multiplication Sample problems

6th Grade Division

6th Grade Division

Probability

Investigate chance processes and develop, use, and evaluate probability models. 7.SP.5 – Understand that the probability of a chance event is a number between 0and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

Two bags of jellybeans I have two bags. Both contain red and yellow   There are more red jellybeans in bag A than in bag B. If I choose one jellybean from each bag I am more likely to choose a red one from bag A than from bag B.

True, False or Unsure? Take turns to select a card and decide whether it is a true or false statement. Convince your partner of your decision. It is important that you both understand the reasons for the decision. If you don’t agree with your partner, explain why. You are both responsible for each other’s learning. If you are both happy with the decision, glue the card onto the paper. Next to the card, write reasons to support your decision. Put to one side any cards you are unsure about.

Investigate chance processes and develop, use, and evaluate probability models. 7.SP.6 – Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times. 7.SP.7 – Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. 7.a – Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. 7.b – Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?

Investigate chance processes and develop, use, and evaluate probability models. 7.SP.8- Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. 8.a – Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. 8.b – Represent sample spaces for compound events using method such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event. 8.c – Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

What Do You Expect? Compound Probability

How is probability used in real life?

Theoretical probability Experimental probability Outcome Vocabulary Event Theoretical probability Experimental probability Outcome

What’s in the Bucket?

determine the fraction What’s in the Bucket? Without looking in or emptying the bucket, how could we determine the fraction of blocks that are red, yellow, or blue?

Problem 1.1 How many blocks drawn by your class were blue? How many were yellow? How many were red? Which color block do you think there are the greatest of in the bucket? Which color block do you think there are the least number of?

Problem 1.1 Based on our experimental data, predict the fraction of blocks in the bucket that are blue, that are yellow, and that are red.

Problem 1.1 How do the fraction of blocks that are blue, yellow, and red compare to the fractions of blue, yellow, and red drawn during the experiment?

Match / No-Match Rules Spin the spinner twice for each turn. If both spins land on the same color, you have made a MATCH. Player A scores 1 point. If the two spins land on different colors, you have made a NO-MATCH. Player B scores 2 points.

Experimental Probability of Match number of turns that are matches total number of turns

Experimental Probability of No-Match number of turns that are no-matches total number of turns

What are the possible outcomes in this game? Color on 1st spin – Color on 2nd spin Are all outcomes equally likely?

Theoretical Probability of No-Match number of outcomes that are no- matches total number of turns

Theoretical Probability of Match number of outcomes that are matches total number of turns

Compare the experimental and theoretical probabilities for match and for no-match. Is Match/No-Match a fair game? If you think the game is fair, explain why. If you think it is not fair, explain how the rules could be changed to make it fair.

Making Purple RED BROWN GREEN BLUE YELLOW ORANGE

Expected Value Compound Probability Design and use a simulation to generate frequencies for compound events. [7-SP8c]

Investigation 3 Expected Value On April 14, 1993, during halftime of a basketball game between the Chicago Bulls and the Miami Heat, Don Calhoun won $1 million by making a basket from the free-throw line at the opposite end of the court. Don was chosen at random to attempt the basket as part of a promotional contest. A Sports Illustrated article explained: The odds against one randomly chosen person given one shot from the opposite foul line and making it are considered astronomical. Scottie Pippen admitted that after practice one day he and Michael Jordan tried to hit the shot but couldn’t. (Source: Bessone, Lisa, “Sports People: Don Calhoun.” Sports Illustrated, April 26, 1993, vol 48.) What Do You Expect Inv. 3

Will he hit it or miss it? In the section finals, William’s basketball team is 1 point behind with 6 seconds left. William has just been fouled, and he is in a one-and-one free- throw situation. This means that William will try one free throw. If he makes it, he tries a second free throw. If he misses the first time, he does not get a second try. William’s free-throw average is 60%.

One-and-One Free Throws What are the possible scores a player can make in a one-and-one free-throw situation? How can each score be made? How would you design an experiment to analyze this situation?

Will he hit it or miss it? In the section finals, William’s basketball team is 1 point behind with 6 seconds left. William has just been fouled, and he is in a one-and-one free- throw situation. This means that William will try one free throw. If he makes it, he tries a second free throw. If he misses the first time, he does not get a second try. William’s free-throw average is 60%.

One-and-One Free Throws What are the possible scores a player can make in a one-and-one free-throw situation? How can each score be made? How would you design an experiment to analyze this situation?

Which is a 60% shooter more likely to score? 0 points 1 point 2 points

Use the spinner to simulate William’s one- and-one situation 20 times Use the spinner to simulate William’s one- and-one situation 20 times. Record the result of each trial.

Make an area model for this situation using a 10 x 10 grid. What is the theoretical probability that William will score 0 points? That he will score 1 point? That he will score 2 points? Compare these with the experimental probabilities we found.

Suppose William’s free- throw average is 70% Suppose William’s free- throw average is 70%. How does this affect the outcome?

Suppose William’s free- throw average is 70% Suppose William’s free- throw average is 70%. How does this affect the outcome? How does this diagram help you predict how many times William will score 2 points in 100 one- and-one situations? In 200 one-and-one situations?

Investigation 3.2 Finding Expected Value In Problem 3.1, you looked at the probabilities of different outcomes of Nishi’s one-and-one free-throw situation. You might have been surprised about which outcome is most likely. In this Problem, you will look at the number of points Nishi can expect to make each time she is in a one-and-one free-throw situation. What Do You Expect Inv. 3 111

Standards of Mathematical Practice Standard 1: Make sense of problems and persevere in solving them. Standard 2: Reason abstractly and quantitatively. Standard 3: Construct viable arguments and critique the reasoning of others. Standard 4: Model with mathematics. Standard 5: Use appropriate tools strategically. Standard 6: Attend to precision. Standard 7: Look for and make use of structure. Standard 8: Look for and express regularity in repeated reasoning.

High Cognitive vs. Low Cognitive

An effective mathematical task is needed to challenge and engage students intellectually.

Read an excerpt from the article: Handout 37

Comparing Two Mathematical Tasks Solve Two Tasks: Martha’s Carpeting Task The Fencing Task Handout 38-39 Participants work each task on their own first, and then share with partner at their table. Note various solution paths used by participants. Prepare to have them share solutions in the following order   Martha’s carpeting Formula Diagram Fencing Task Trial and error Ordered set of diagrams Information organized in a table Algebraic equation in vertex form Graph of parabola showing vertex Calculus using derivative Chose participants to share out to the whole group. Record solutions on chart paper. In small groups, have them complete Similarities and Differences table. Share out responses in whole group. Then ask, “Do the differences between the Fencing Task and Martha’s Carpeting Task matter? Why or why not?” SAS Secondary Mathematics Teacher Leadership Academy, Year 1

Comparing Two Mathematical Tasks How are Martha’s Carpeting Task and the Fencing Task the same and how are they different? Participants work each task on their own first, and then share with partner at their table. Note various solution paths used by participants. Prepare to have them share solutions in the following order   Martha’s carpeting Formula Diagram Fencing Task Trial and error Ordered set of diagrams Information organized in a table Algebraic equation in vertex form Graph of parabola showing vertex Calculus using derivative Chose participants to share out to the whole group. Record solutions on chart paper. In small groups, have them complete Similarities and Differences table. Share out responses in whole group. Then ask, “Do the differences between the Fencing Task and Martha’s Carpeting Task matter? Why or why not?”

Similarities and Differences 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 Way in which area formula is used: Martha’s Carpeting can be solved by knowing and using the area formula but this formula alone is not sufficient to solve the Fencing Task The need to generalize: Martha’s carpeting does not lead to a generalization but the Fencing Task does The range of ways to enter the problem: Martha’s Carpeting Task cannot be started by a student who does not know the formula for area; the Fencing Task can be started in other ways, such as sketches on graph paper. 119

Comparing Two Mathematical Tasks Do the differences between the Fencing Task and Martha’s Carpeting Task matter? Why or Why not? Then ask, “Do the differences between the Fencing Task and Martha’s Carpeting Task matter? Why or why not?” If there is time, you may want participants to generate similarities and differences before sharing the upcoming slide.

Does Maintaining Cognitive Demand Matter? YES Have participants discuss the challenges they experience in implementing cognitively challenging tasks that promote thinking, reasoning, and problem solving. At the end of the discussion, suggest that this article presents a framework for lesson planning that may help with these challenges

Handout 40

Criteria for low cognitive demand tasks Recall Memorization Low on Bloom’s Taxonomy

Developing Mathematical Practices Review each of the “higher demand” tasks: What mathematical practices do they engage students in using?

Criteria for high cognitive demand tasks Requires generalizations Requires creativity Requires multiple representations Requires explanations (must be “worth explaining”)

Lower-Level Tasks Memorization What are the decimal equivalents for the fractions ½ and ¼? Procedures without connections Convert the fraction 3/8 to a decimal

Higher-Level Tasks Doing mathematics Shade 6 small squares in a 4 X 10 rectangle. Using the rectangle, explain how to determine: A. The decimal part of area that is that is shaded; B. The fractional part of area that is shaded

Higher-Level Tasks Procedures with Connections Using a 10 x 10 grid, identify the decimal and percent equivalents of 3/5.

What causes high- level cognitive demand tasks to decline? Have participants discuss the challenges they experience in implementing cognitively challenging tasks that promote thinking, reasoning, and problem solving. At the end of the discussion, suggest that this article presents a framework for lesson planning that may help with these challenges

Patterns of Set up, Implementation, and Student Learning Task Set Up Task Implementation Student Learning A. High High High B. Low Low Low Have participants read and discuss the slide. Ask, What do you notice? What are the implications for instruction? Evidence gathered across scores of middle school classrooms in four QUASAR middle schools has shown that students who performed the best on project-based measures of reasoning and problem solving were in classrooms in which tasks were more likely to be set up and implemented at high levels of cognitive demand. Results from QUASAR also show that students who had the lowest performance on project assessments were in classrooms where they had limited exposure to tasks that required thinking and reasoning (Stein & Lane, 1996). C. High Low Moderate Stein & Lane, 2012 130

Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands 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) This slide identifies the classroom factors that researchers observed when high-level tasks declined during a lesson.

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) By contrast, this slide identifies the classroom factors that researchers observed when the cognitive demands of high-level tasks were maintained during a lesson.

“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)

Jeanne Simpson UAHuntsville AMSTI jeanne.simpson@uah.edu Contact Information Jeanne Simpson UAHuntsville AMSTI jeanne.simpson@uah.edu acos2010@wikispaces.com

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