Content Deepening7th 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
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
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 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 Questioning/Wait Time Pair-Share Grouping/Engaging Problems Questioning/Wait Time Showing Thinking Grouping/Engaging Problems Grouping/Engaging Problems Encourage Reasoning Grouping/Engaging Problems Showing Thinking Questioning/Wait Time Grouping/Engaging Problems Pair-Share Questioning/Wait Time Grouping/Engaging Problems Showing Thinking Questions/Prompts for Groups Grouping/Engaging Problems Showing Thinking Grouping/Engaging Problems Grouping/Engaging Problems Encourage Reasoning Showing Thinking Allowing Struggle Questions/Prompts for Groups Encourage Reasoning Allowing Struggle Grouping/Engaging Problems Allowing Struggle Encourage Reasoning
Critical Focus Areas Ratios and Proportional Reasoning 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.
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
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?
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.
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?
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.
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?
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