Where we are. HOW do we apply the concepts of quadratics to better understand higher order polynomials?

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1 A. Lesson Context BIG PICTURE of this UNIT: How & why do we build NEW knowledge in Mathematics? What NEW IDEAS & NEW CONCEPTS can we now explore with specific references to POLYNOMIAL FUNCTIONS AND RATIONAL FUNCTIONS? How can we extend our knowledge of FUNCTIONS, given our BASIC understanding of Functions and quadratic functions? CONTEXT of this LESSON: Where we ve been In Unit 3, you worked with quadratic functions in vertex, standard, and factored forms. Where we are HOW do we apply the concepts of quadratics to better understand higher order polynomials? Where we are heading How do we extend our knowledge & skills of polynomials given what we know about quadratics functions? B. Lesson Objectives a. Introduce cubic functions through a 3D box building modeling investigation C. Fast Five (Skills Review/Preview Focus) THIS NEEDS TO BE FOCUSED DOWN I BELIEVE>>>> 1(a). Evaluate P(2) if P(x) = x 3 x 2 + 2x 3 3(a). Expand and simplify (x 2)(x + 2)(x 1). Verify your expansion using WolframAlpha. 1(b). Evaluate P(-½) if P(x) = -2x 3 + 2x 5 2(a). Expand and simplify (x + y) 3 3(b). Multiply and then simplify -3(x + 2)(x 1)(2x + 3). Verify your expansion using WolframAlpha. 2(b). Expand and simplify (x y) 3

2 5(a). The formula for the volume of cylinder is V = πr 2 h. If the height of a cylinder is twice its radius and its radius is 5 cm, determine the volume of this cylinder. 4(a). Use DESMOS/TI-84 to graph and sketch: y = x 3, y = x 3 2 and y = (x 2) 3. 4(b). Sketch a function that crosses the x-axis at x = 6. 4(c). Sketch a function with x-intercepts at x = 6 & x = -1 4(d). Sketch a function with zeroes at x -2, 3 and 4. 5(b). The formula for the volume of a sphere is V = 4/3πr 3. Determine the volume of a sphere if its radius is 4 cm. 6(a). Use DESMOS/TI-84 to graph and sketch the polynomial functions: (i) y = (x 1)(x + 2)(x 3) and (ii) y = -(x 2)(x + 1)(x + 4). 5(c). Determine the radius of a sphere if its volume is 200 cm 3. 6(b). Use WolframAlpha to factor x 3 x 2 14x Sketch the graph as well.

3 D. Making an Open-Topped Box Instructions Materials Needed - 2 A4 Pieces of Paper 8.5 x 11 cm - Pair of Scissors - Tape - Ruler measuring cm - Brain Phase 1: The Model First Box Step 1: Get a sheet of 8.5 x 11 cm coloured paper. Step 2: In one corner, draw a square measuring 2cm x 2cm. Step 3: Cut this square out. Step 4: Go to the other three corners and measure & cut identical 2cm x 2cm squares. Step 5: You should now have 4 flaps that you will fold over (in order to make the sides of a box) Step 6: Tape these flaps together to complete the box. Check in # 1( Need Mr. S s Initials Here) Step 7: Measure the height, length and width of your box. Step 8: Calculate the volume of this box. Step 9: You now have one ordered pair in our volume modeling investigation (2 cm, ish cm 3 ), where x will represent the dimensions of the corner you cut out and V will represent the volume of the resultant box. Step 10: Calculate the outer surface area of this box and create another ordered pair in our modeling investigation (2 cm, ish cm 2 ), where x will represent the dimensions of the corner you cut out and y will represent the surface area of the resultant box. Volume of Box w/ 2 x 2 cm square cut out of the corners: Surface Area of Box w/ 2 x 2 cm square cut out of corners: Dimensions of Square cut out (cm) Volume of Box (cm 3 ) Surface Area of Box (cm 2 ) 2 cm Check in # 2 (Need Mr. S s Initials Here)

4 Phase 2: Creating Class Data Now we will repeat this box construction, wherein every student chooses/is assigned a different sized square to cut out of each corner. Step 1: Construct your box, measure the three dimensions & calculate the volume and outer surface area. Step 2: Record your data on the google doc. Step 3: Construct two graphs, one showing the relationship between corner size and volume and the second graph showing the relationship between corner size and surface area. Have the two graphs electronically or do them by hand on another sheet of paper. Please label your axes. Check in # 3 (Need Mr. S s Initials Here) Step 4: Looking at your scatterplots, what type of function could we use to model our relationships in this investigation? Justify your choice(s). Check in # 4 (Need Mr. S s Initials Here) Step 5: Use your TI-84 (or EXCEL) to determine an equation for the curve of best fit. Volume Equation: V(x) = Surface Area Equation: S(x) = Evaluate the following and interpret. V(2.3) = S(1.4) = 1. Use your model to predict the size of the corner that you should cut out in order to optimize the volume of the box. 2. Determine the domain and range for your model, explaining WHY you ve decided upon your domain and range.

5 a. EXTENTION: Can you PREDICT what the equation for the model should be, simply given the construction instructions? b. EXTENTION: Boxes with reinforced sides to construct a box with reinforced sides, use your original box (2 cm corners cut) and make one adjustment on the four side flaps fold this flap TWICE (once at the 2 cm mark and a second time at the 1 cm side), so that your sides are now twice as thick. Again, determine the volume of this box. Then, as before, predict an equation for an equation modeling the relationship between corner size and volume. c. EXTENSION: Cylinder?? Data Table for Exploration Enter your data on this google doc