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Investigations

Investigations. For 6 th , 7 th , and 8 th grades. Project commitment. Sign up for what level you plan to go for as we begin investigation.

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Investigations

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  1. Investigations For 6th, 7th, and 8th grades

  2. Project commitment • Sign up for what level you plan to go for as we begin investigation. • You are also expected to complete a word wall project for each investigation unit. Comparing and Scaling is 6th grade. Stretching and Shrinking is 7th grade. Thinking with Mathematical Models is 8th grade. Label your word wall on the back with the appropriate unit that your word belongs to with your S.A.N.D.

  3. 8th grade Investigation 2 Opener Did you know…? • Ever thought of this for a job? • Highway construction workers make about $35-40,000 per year.

  4. 8th grade Launch 2.1: “Linear Models” Launch Look at table for 2.1. Do you see a pattern in these data? Students need to make a scatter plot for this data. Do you think connecting the first and last points in a data set will always give a line that fits a pattern? Can you think of a situation in which that line would be a good representative of the overall relationship? Will any line that has an equal number of points above and below be a good model? • Goals: Fit a line to data that shows a linear trend. Write an equation for a line based on a graph of the line. Use mathematical models to answer questions about linear relationships.

  5. 8th grade Continue with 2.1 Launch …. Does the linear model in the book have the same slope as our models that we drew? How would predictions made from the two models differ? • How can we find the equation for our line? • Can we use our line or equation to predict the costs for other bridges? How? • MATHEMATICAL MODEL: used to describe the relationship of the line to the data pattern; helps us make predictions about the data

  6. 8th grade Continue with 2.1 Launch …. What information does the model give that the four points alone do not? What questions could answer using the model? What information do you need to write an equation for the line? • How do the cost estimators decided where to draw the line? • Do you think the line is a reasonable model for this data?

  7. 8th grade 2.1 Explore ….. “Manhatten Bridge” What does the y-intercept tell you about bridge length and painting cost? Do problem A. Then go to next slide. • What does the slope mean in terms of bridge length and painting cost?

  8. 8th grade Three Musketeers: go by ABC order Final Summary What directions would you give someone to help him/her draw a mathematical model? • Person One

  9. 8th grade 2.1 Summarize Final Summary Once you have drawn the line, how do you find the slope and y-intercept? • Person Two

  10. 8th grade 2.1 Summarize Final Summary How do the slope and y-intercept help you find the equation for the line? • Person Three

  11. 8th grade 2.1 Summarize Final Summary What directions would you give someone to help him/her draw a mathematical model? Once you have drawn the line, how do you find the slope and y-intercept? How do the slope and y-intercept help you find the equation for the line? • After problem A, who used the line to find approximate answers for parts (2) and (3)? Show us how you found the answers. • Who used the equations to find the exact answers for parts (2) and (3)? Show us how you found the answers.

  12. 8th grade Final Snapshot for this section 2.1 Final Summary If you have to make predictions, from a linear model, which is more helpful, the equation or the line? What are the advantages of having a linear model for your data set, rather than just a set of individual data points?

  13. What is a function? • A relation… relationship • Input – output • If it is a one, it cannot have a two and a three… input can only have one arrow in the model • There’s a correspondence between 1 and 2 • Constant rate of change • A rule • A relation where each input can only have one output • For every input there is exactly one output and there is sometimes a constant rate of change • For every input there is exactly one output; if there is more than one output per input then it is not a function… this is just a relation

  14. What is a function? • When you are talking about slope is there a function involved? Give examples of why are why not.

  15. What is a function? • When you are talking about slope is there a function involved? Give examples of why are why not. • Using your bodies in groups of five, make a human model that shows how to find the y-intercept.

  16. 8th grade Launch 2.2: “Equations for Linear Relationships” Launch What do you think self-service car wash owner consider when they decide how much to charge a customer? Sudzo Wash and Wash charges customers 75 cents per minute to wash a car. Write an equation that relates the total charge c to the amount of t in minutes. • Goal: Practice effective strategies for writing linear equations from verbal, numerical, or graphical information.

  17. 8th grade Launch 2.2 Launch U-Wash-It charges $10 for each car. The business owners estimate that it costs them 60 cents per minute to provide soap, water, and vacuums for a car. Write an equation for the profit p U-Wash-It earns if a customer spends t minutes washing a car. • Pat’s Power Wash charges two dollars per car to cover the cost of cleaning supplies, plus 49 cents per minute for the use of water sprayers and vacuums. Write an equation for the total charge for any car-wash time. Again, how do we represent symbolically the car-wash cost? Time?

  18. 8th grade Check for Understanding Launch 2.2 Continued Homework: Individual Posters Created for the Three Car Washes as Advertisement; one group figure out how to graph the three equations on one graph for display (12,13,11,17,8) • Go to edmodo.com for 8th grade and student number 10 type in these questions for students to respond to about today’s investigation: • Explain what the number and variable in each equation represent. • What questions can your equations help you answer?

  19. 8th grade Check for Understanding Revisited 2.2 Explore and Monitor Your Work with Questions Do Questions A-F and ask: How can you check that the equations you write are correct? What information do you need to write an equation of a line? How are these values related to the equation y=mx+b? Next page… • Zoom to edmodo.com for 8th grade and see student responses.

  20. 8th grade 2.2 Explore: Review of slope 2.2 Review of y-intercept How can you find the y-intercept? • How can you find the slope of a line?

  21. 8th grade 2.2 Summarize 2.2 Summarize 1. a description of the relationship in words? 2. two or more (x,y) values or a table of (x,y) values? 3. a graph showing points with coordinates? Stop here for the day. • Students need to share for each part of question F. • F.) Suppose you want to write an equation of the form y=mx +b to represent a linear relationship. What is your strategy if you are given…?

  22. 8th grade 2.2 Summarize BIG IDEAS Do you know the answers? How can you find the slope if you are given the coordinates of two or more points on a graph? How can you find the slpe if you are given a table of (x,y) data values? How can you find the y-intercept from a table? How can you find the y-intercept from a graph? Check for Understanding at Edmodo.com coming soon! See next slide first. • What two values are important when writing a linear equation? • Which of these two is easier to find, in your opinion? • In an equation of the form, y= mx + b, what does each letter represent?

  23. 8th grade 2.2 Check for Understanding at Edmodo.com coming soon! 2.3: Tomorrow, wear (long) shorts for the launch of 2.3. * • Before you take this quiz on Edmodo.com, can you put equations like this into slope-intercept form? y= 4 - 3x y= -3 + x y= 6 – 4x y = 5.5 + 1/3x

  24. 8th grade 2.3 “Solving Linear Equations” Sandy’s Boat House rents canoes! • Goals: Develop skill in solving linear equations with approximation and exact reasoning methods. Write inequalities to represent “at most” situations.

  25. 8th grade 2.3 Launch Sandy’s Boat House rents canoes! • What does the 0.15 represent? • What does the 2.50 represent? Okay, you are applying for jobs at the boat house and the manager will check your ability to see if you can be trusted to calculate correct rental charges. (next slide)

  26. 8th grade 2.3 Launch: Quiet Think Time Can you share your strategies? Suppose you were applying for a job at Sandy’s. What strategies would you use to answer these questions? Sandy asks, • “What is the charge for renting a canoe for 30 minutes?” • “A customer is charged $8.50. How long did he use the canoe?” • “A customer has $10 to spend. How long can she use a canoe?”

  27. 8th grade 2.3 Explore * Work on problems A-E. Share strategies and solutions with details in your Math Tab. Remember, your presentation should include multiple strategies for solving problems as you arrive at your solutions.

  28. 8th grade 2.3 Summarize * If you have an equation and you know the value of one variable, either x or y, how can you use a graph to estimate the value of the other variable? How can you use a table to find the value of the other variable? How can you use an equation to find the value of the other variable?

  29. 8th grade Launch 2.4 “Intersecting Linear Models” Movie multiplex vs. amusement park • Goal: Use equations to represent questions about problem situations and interpret the solutions in the context of the problem • Do NOT take out your math investigation yet. Think about the fun you would have at these sites!

  30. 8th grade 2.4 Launch: think about first Chance of rain? • How might the probability of rain affect attendance at the amusement park? • How might the probability of rain affect attendance at the movie theater? • Now, take out your investigation packet. • Look at page 32, do your ideas match the trend in the data?

  31. 8th grade 2.4 Explore Discuss symbols and the language here. What does the “A subscript B” mean? What does “A subscript G” mean? Math has its own language. Just like learning a foreign language like German, one must make sense of the symbols and words in math too. • What would a plot of these data look like? • What do you need to know to find the equation of a line?

  32. 8th grade 2.4 Summarize Go to next slide for the BIG IDEA. • Students should display strategies using the document camera for part B.

  33. 8th grade 2.4 Summarize Systems of Equations! What question does this equation represent? How could you estimate the solution using a table? Using a graph? How could you find the solution symbolically? Does Mrs. Schlechter know how to graph it on a TI-73 yet? • Big Fun = Get Reel • A subscript B = A subscript G • What is the actual value for the above equation?

  34. 8th grade • Student presentations for this investigation shared after two class session spent on final preparations to share. Spruce up your presentations with color, drawings, neatness, etc. • Presentation Days: compare work in your Math Tab to the presentations presented by student groups. Be prepared to critique their argument for solving the problems. Teacher assesses with rubric and checks for reliable valid strategies and solutions. • Summative Assessment Piece • Reflections shared in Reflection Journal • Check our vocabulary and fill in definitions. • Check word wall progress.

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