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Welcome to Day Four

Join us for a day of professional development focused on mathematical discourse, exploring topics such as ratio, proportion, and algebra. Learn effective practices for student reasoning and problem-solving. Share ideas and collaborate with other practitioners to improve your understanding and practice.

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Welcome to Day Four

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  1. Welcome to Day Four

  2. Agenda • 5 Practices for Mathematical Discourse • Ratio, Proportion, and Percent • Ratio, Proportion, and Algebra • Lessons and Reflection

  3. The Standards for Mathematical PracticeStudent Reasoning and Sense Making about Mathematics • Make sense of problems and persevere in solving them • Reason abstractly and quantitatively. • Construct viable arguments and critique the reasoning of others. • Model with mathematics. • Use appropriate tools strategically. • Attend to precision. • Look for and make use of structure. • Look for and express regularity in repeated reasoning.

  4. Bring your ideas… • As a group of professionals we have made a commitment to helping children attain success in life and a voice in the world. • Many times the best part of these kinds of professional development is simply the chance to share ideas, raise questions, and work with other practitioners to improve our own understandings and practice. • Please bring your stories of children’s learning with you.

  5. Our Socio-mathematical Norms • Listen intently when someone else is talking avoiding distractions • Persevere in problem solving; mathematical and pedagogical • Solve the problem in more than one way • Make your connections explicit - Presentation Ready • Contribute by being active and offering ideas and making sense • Limit cell phone and technology use to the breaks and lunch unless its part of the task. • Be mindful not to steal someone else’s “ice cream” • Respect others ideas and perspectives while offering nurturing challenges to ideas that do not make sense to you or create dissonance. • Limit non-mathematical and non-pedagogical discussions

  6. Presentation Norms • Presenters should find a way to show mathematical thinking, not just say it • Presenters should indicate the end of their explanation by stating something like “Are there any questions, discussion, or comments?” • Others should listen and make sense of presenters’ ideas. • Give feedback to presenters, extend their ideas, connect with other ideas, and ask questions to clarify understandings

  7. BREAK

  8. Comparing Mixtures Abby’s orange paint is made by mixing 1 cup of red paint for every 3 cups of yellow paint. Zack’s orange paint is made by mixing 3 cups of red paint for every 5 cups of yellow paint. How do the shades of their orange paints compare?

  9. Abby’s Orange Paint Additive vs. Multiplicative Thinking +1 +3 +1 +3 ×4 ×4

  10. ×3 Abby’s Orange Paint Within vs. Between Reasoning ×3 ×6 ×6

  11. Two Perspectives on Ratio • Ratio as a composed unit or batch • Ratio as fixed numbers of parts

  12. Representations of Ratios Physical Model Tape Diagram Ratio Table Double Number Line

  13. Mixing Paint Jessica gets her favorite shade of purple paint by mixing 2 cups of blue paint with 3 cups of red paint. How many cups of blue and red paint does Jessica need to make 20 cups of her favorite purple paint? Solve this using a ratio table, a tape diagram, and a double number line.

  14. Problem Solving with Ratios, Proportions, and Percent

  15. LUNCH

  16. Ratio, Proportion, and Algebra

  17. A Problem Carol spends 17 hours in a 2-week period practicing her culinary skills. How many hours does she practice in 5 weeks? Come up with as many ways of representing and solving this problem as you can think of and be prepared to share solutions.

  18. Proportions Why does “cross multiplication” work as a method for solving proportions?

  19. The Ball Bouncing Experiment

  20. The Problem/Task Which ball bounces “best” – a super ball, a golf ball, a tennis ball, or a ping pong ball?

  21. Ball Bouncing – Questions • What does the slope of the line of best fit represent in “real life” terms? • Which ball bounces the best from a freefall? How do you know? • What are the mean, median, mode, and range of the slopes for each ball?

  22. Ball Bouncing Activity • Students divided into teams of 3 or 4 • Each team given a tennis, ping pong, golf, and super ball and a meter stick • Students drop each ball from at least 5 different heights, recording the drop height (x) and the bounce height (y) for each • Data are organized into tables • Scatter plots are drawn for each ball

  23. Superball Scatterplot

  24. Ball Bouncing – Group Data

  25. Slope • In each case, the slope represents a rate of change – a ratio that can be used to determine bounce height for a ball dropped from any height. • Bounce heights can be determined by using proportions based on the slope (e.g., how high would we expect the tennis ball to bounce if dropped from 85 cm?).

  26. Wooden Cubes Task In this task, we make a comparison of two different types of wooden cubes by weighing them on a balance.

  27. Wooden Cubes Task • How do the slopes of the lines compare? • What does this tell us about the two types of wood? • Can we write an inequality that compares these slopes?

  28. Ohio Learning Standards Grade 6 6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px= q for cases in which p, q, and x are all nonnegative rational numbers. 6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

  29. Ohio Learning Standards Grade 7 7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example, if a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50.

  30. Ohio Learning Standards Grade 8 8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. 8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

  31. Message • The topics of ratio and proportion are not intended to be taught in a vacuum. • There are many opportunities within the study of “algebra” to address the issues of ratio and proportion and integrate these standards.

  32. Enlarging a Photo If you enlarge a 4 inch by 6 inch family photo to fit in an 8 inch by 10 inch frame, will everyone in the photo look the same?

  33. Similarity Tasks

  34. BREAK

  35. Math Content for our Classrooms • Each day we will spend time with grade level teams making lesson plans. • Each of us will make one plan that is part of a unit of plans the grade level team is working on. • Each plan must have the following: • Connected mathematics content focus from Ohio’s Mathematics Learning Standards • A focus SMP • Designed to Orchestrate Productive Mathematics Discussions (The 5 Practices)

  36. Math Content for our Classrooms Three checks must be made for the completion of lesson plans: Check 1) Consult with Sandy and/or Mary about the mathematics content of the lesson and explain to her its mathematical appropriateness. When the lesson is complete Sandy, our resident mathematician, will sign off on its content (SMC’s). Check 2) Consult with Sherry about the design of the lesson to promote mathematical discourse (5 Practices). Sherry must sign off on the lessons discourse elements. Check 3) Consult with Dr. Matney about the design of the lesson having a focus Standard for Mathematical Practice. Dr. Matney must sign off on the lessons mathematics proficiency elements (SMP’s) ?Questions about COMP Lesson Plans?

  37. Time of Reflection • On a sticky note tell us one thing you learned today. • Tell us one think you liked or one thing you are still struggling with.

  38. Stay Safe • Please help us put the room in proper order. • Please leave your name tents for next time.

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