|Unit 2: Day 1: Linear and Quadratic Functions |MCT 4C | | |Learning Goals |Materials | |Minds On: 15 |Activate prior knowledge by reviewing features of linear and quadratic functions such as what the |PPT 2. 1. 1 | | |graphs look like, how could the graphs be described, and whether or not the graphs represent |BLM 2. . 1 | | |functions. |BLM 2. 1. 2 | | |Consolidate understanding of domain and range |BLM 2. 1. 3 | | |Learn end behaviour terminology and the definition of the leading coefficient |LCD projector | |Action: 20 | | | Consolidate:40 | | | |Total =75 min | | | | Assessment | |Opportunities | |Minds On… |Individuals or Pairs ( Activity | | | | | |Students complete BLM 2. 1. 1 using prior knowledge of linear and quadratic functions | | | | | | | | | | | |Curriculum Expectations/Observations/Checklist | | | | |Observe students as they complete BLM 2. 1. 1 and assess their prior knowledge, in particular what | | | | | |they recall about linear and quadratic functions. Use this information to determine the depth in
Use the graph and the equation to fill in the table relating to each graph. 1. [pic] |Domain | | |Range | | |Degree | | |Sign of Leading Coefficient | | |End Behaviour | | |Is the relation a function? | | . [pic] |Domain | | |Range | | |Degree | | |Sign of Leading Coefficient | | |End Behaviour | | |Is the relation a function? | | 2. 1. 3: Linear and Quadratic Functions † Practice (continued) 3. [pic] Domain | | |Range | | |Degree | | |Sign of Leading Coefficient | | |End Behaviour | | |Is the relation a function? | | 4. [pic] |Domain | | Range | | |Degree | | |Sign of Leading Coefficient | | |End Behaviour | | |Is the relation a function? | | 5. Is it possible to graph a line of the form [pic] that will not result in a function? Explain your reasoning. 6. Is it possible to graph a quadratic relation of the form [pic] that will not result in a function?
Explain your reasoning. |Unit 2: Day 2: A Higher Degree |MCT 4C | | |Learning Goals: |Materials | |Minds On: 10 |Investigate cubic and quartic functions and explain why they are functions. |BLM 2. 2. 1 | | |Graph the equations of cubic and quartic functions and investigate end behaviours, domain and range. |BLM 2. 2. 2 | |Describe end behaviours and the impact of the leading coefficient (positive and negative values). |BLM 2. 2. 3 | | | |Graphing calculators | |Action: 60 | | | |Consolidate: 5 | | | Total =75 min | | | | Assessment | |Opportunities | | |Minds On… |Pairs ( Activity | |Teacher should copy BLM | | | |Pairs are given a section of BLM 2. 2. 1 and take a few minutes to consider the relations. Each | |2. 2. and cut the page | | | |pair will have two functions from y = x, y = x2, y = x3, and y = x4 and must hypothesize about | |in half. There should | | | |how the relations will be the same and how they will be different, with reference to the following| |be one Venn diagram for | | | |items: | |each pair of students in| | | |effect of leading coefficient, end behaviour, degree, maximum number of x-intercepts, domain, | |the class. | | |range, whether or not the relations are functions | | | | | |Students will fill in the Venn Diagram in BLM 2. 2. 1 with their results. | | | | | | | | | | |Action! |Small Groups ( Jigsaw (Home Groups) | | | | |In groups of 4 , students should each select a different chart to complete on BLM 2. 2. 2. | | | | | | | | | | | |Small Groups ( Jigsaw (Expert Groups) | | | | | |Students form expert groups according to the chart that selected. Each group creates graphs for | | | | |each function in BLM 2. 2. 1 using a graphing calculator. Students make conclusions about the | | | | | |behaviour of cubic and quartic functions based on the investigations. | | | | | | | | | | | |Curriculum Expectations/Oral Questions/Rubric | | | | |Assess students as they complete BLM 2. 2. 1 in their expert groups on their understanding of the | | | | | |key components of functions. | | | | | | | | | | | |Small Groups ( Jigsaw (Home Groups) | | | | | |Students regroup too share their results.
Students use the information shared to complete all | | | | | |charts and questions. | | | | | | | | | | | |Mathematical Process Focus: | | | | | |Reasoning and Proving † Students use their reasoning skills to determine patterns related to | | | | | |properties of polynomials. | | | | | | | | | |Consolidate |Pairs( Activity | | | | |Debrief |Students work in the same pairs from the Minds On activity and compare their hypotheses with the | | | | | |actual results. They should make any necessary changes to their Venn Diagram. | | | | | | | | |Reflection |Home Activity | | | | |Complete a Frayer Model (BLM 2. 2. 3) for cubic and quartic functions. | | | 2. 2. 1: Comparing Functions by Degree ( 2. 2. 1: Comparing Functions by Degree 2. 2. 1: Comparing Functions by Degree 2. 2. 1: Comparing Functions by Degree 2. 2. 1: Comparing Functions by Degree ( 2. 2. 1: Comparing Functions by Degree 2. 2. 2: Investigation – Cubic and Quartic Functions Part 1 1. Look at each equation and state the value of the leading coefficient. Fill in the information in the column specified below. The first one is done for you. 2. Graph each of the following functions on a graphing calculator and sketch a copy of what you see on the given grids. Use the sketch to fill in the other columns in the table. Again, the first one is done for you. Equation |Leading coefficient |Graph |Number of |End behaviour | | | | |x-intercepts | | | | | | | | |y = x3 |1 | |1 |As x((, y(( and as | | | | |x(-(, y(-( | | | | | | | | | | | | | | | | | | | | | | | | |y = x3 † 2×2 | | | | | | | | | | | | | | | | | | | | | | | | | | | | |y = 2×3 † 3 | | | | | | | | | | | | | | | | | | | | | | | | | | | | |y = x3 + 3×2 † x † 3 | | | | | | | | | | |y = 3×3 † 9x | | | | | | | | | | | |y = 3×3 + x | | | | | | | | | | 2. 2. 2: Investigation – Cubic and Quartic Functions Part 1 (continued) Refer to the chart that you just completed on the previous page to answer questions 3 † 7. 3. What is true about the leading coefficient of all of the polynomials? 4. What is true about the degree of all of the polynomials? 5. What is true about the end behaviour of all of the polynomials? 6. What is the maximum number of x-intercepts for all of the polynomials? 7. Do the graphs of the relations represent functions?
Explain. 2. 2. 2: Investigation – Cubic and Quartic Functions Part 2 1. Look at each equation and state the value of the leading coefficient. Fill in the information in the column specified below. The first one is done for you. 2. Graph each of the following functions on a graphing calculator and sketch a copy of what you see on the given grids. Use the sketch to fill in the other columns in the table. Again, the first one is done for you. |Equation |Leading |Graph |Number of |End behaviour | |coefficient | |x-intercepts | | | | | | | | |y = -x3 |-1 | |1 |As x((, y(-( and as | | | | | |x(-(, y(( | | | | | | | | | | | | | | | | | | | | | | | | y = -x3 † 2×2 | | | | | | | | | | | | | | | | | | | | | | | | | | | | |y = -2×3 † 1 | | | | | | | | | | | | | | | | | | | | | | | | | | | | |y = -2×3 + x2 + 2x † 1 | | | | | | | | | | | y = -2×3 † 6x | | | | | | | | | | | |y = -2×3 + 4x | | | | | | | | | | | 2. 2. : Investigation – Cubic and Quartic Functions Part 2 (continued) Refer to the chart that you just completed on the previous page to answer questions 3 † 8. 3. What is true about the leading coefficient of all of the polynomials? 4. What is true about the degree of all of the polynomials? 5. What is true about the end behaviour of all of the polynomials? 6. What is the maximum number of x-intercepts for all of the polynomials? 7. Do the graphs of the relations represent functions? Explain. 8. What impact do the signs of the leading coefficients seem to have on the graphs from the first chart and the second chart? . 2. 2: Investigation – Cubic and Quartic Functions Part 3 1. Look at each equation and state the value of the leading coefficient. Fill in the information in the column specified below. The first one is done for you. 2. Graph each of the following functions on a graphing calculator and sketch a copy of what you see on the given grids. Use the sketch to fill in the other columns in the table. Again, the first one is done for you. |Equation |Leading |Graph |Number of |End behaviour | |coefficient | |x-intercepts | | | | | | | | |y = 3×4 |3 | |1 |As x((, y(( and as | | | | | |x(-(, y(( | | | | | | | | | | | | | | | | | | | | | | | | y = x4 † 3×3 | | | | | | | | | | | | | | | | | | | | | | | | | | | | |y = 2×4 + 4 | | | | | | | | | | | | | | | | | | | | | | | | | | | | |y = x4 – 5×2 + 4 | | | | | | | | | | | y = x4 † x3 † 3×2 + 3x | | | | | | | | | | | |y = x4 † 4x | | | | | | | | | | | 2. 2. : Investigation – Cubic and Quartic Functions Part 3 (continued) Refer to the chart that you just completed on the previous page to answer questions 3 † 7. 3. What is true about the leading coefficient of all of the polynomials? 4. What is true about the degree of all of the polynomials? 5. What is true about the end behaviour of all of the polynomials? 6. What is the maximum number of x-intercepts for all of the polynomials? 7. Do the graphs of the relations represent functions? Explain. 2. 2. 2: Investigation – Cubic and Quartic Functions Part 4 1. Look at each equation and state the value of the leading coefficient.
Fill in the information in the column specified below. The first one is done for you. 2. Graph each of the following functions on a graphing calculator and sketch a copy of what you see on the given grids. Use the sketch to fill in the other columns in the table. Again, the first one is done for you. |Equation |Leading |Graph |Number of |End behaviour | | |coefficient | |x-intercepts | | | | | | | |y = -5×4 |-5 | |1 |As x((, y(-( and as | | | | | |x(-(, y(-( | | | | | | | | | | | | | | | | | | | | | | | | |y = -x4 † 3×3 | | | | | | | | | | | | | | | | | | | | | | | | | | | | y = -2×4 + 4 | | | | | | | | | | | | | | | | | | | | | | | | | | | | |y = -x4 + x3 + 4×2 † 4x | | | | | | | | | | | |y = -x4 + 4×2 | | | | | | | | | | |y = -x4 +5×2 – 4 | | | | | | | | | | | 2. 2. 2: Investigation – Cubic and Quartic Functions Part 4 (continued) Refer to the chart that you just completed on the previous page to answer questions † 9. 3. What is true about the degree of all of the polynomials? 4. What is true about the leading coefficient of all of the polynomials? 5. What is true about the end behaviour of all of the polynomials? 6. What is the maximum number of x-intercepts for all of the polynomials? 7. Do the graphs of the relations represent functions? Explain. 8. What impact do the signs of the leading coefficients seem to have on the graphs in the third and fourth chart? 9. In general, what is the relationship between the degree of a polynomial and the maximum number of x-intercepts for the polynomial? 2. . 3: Frayer Model – Cubic and Quartic Functions |Unit 2: Day 3: Cubic and Quartic Functions |MCT 4C | | |Learning Goals: |Materials | |Minds On: 20 |Investigate cubic and quartic functions. |PPT 2. 3. 1 | | |Consolidate results from the previous activity to reinforce end behaviours, domain, range, sign of the|LCD Projector | |leading coefficient, and maximum number of zeros for cubic and quartic functions. |BLM 2. 3. 1 | | | |BLM 2. 3. 2 | | | |BLM 2. 3. 3 | |Action: 30 | | | Consolidate:25 | | | |Total=75 min | | | | Assessment | |Opportunities | |Minds On… |Pairs ( Activity | |This activity will give | | | |Students complete BLM 2. 3. 1. | |the students an | | | | | |opportunity to share | | | |Whole Class ( Discussion | |their findings from the | | | |Discuss student responses to BLM 2. 3. 1. Do not ask for support for answers as students will have | |work they completed the | | |an opportunity to revisit their answers after the Action portion of the lesson. | |day before. You will be| | | | | |able to assess what is | | | | | |to be focused on today, | | | | | |based on students’ | | | | |demonstrated | | | | | |understanding of a | | | | | |concept. | | | | | | | | | | | | | | | | | | | | | | | | | | | | |You may want to have | | | | |students fill in | | | | | |portions of the BLM | | | | | |2. 3. 2 before the slide | | | | | |appears in the | | | | |presentation. | | | | | | | | | | | | | | | | | |You may want to ask “Who| | | | |answered ‘yes’ to | | | | | |question one? ” and then| | | | | |ask “Why? ”. Repeat the | | | | | |process for the rest of | | | | |the questions. | | | | | | | | |Action! |Whole Class ( Demonstration | | | | | |The teacher should reiterate the topics mentioned in the previous lesson using the PowerPoint | | | | | |presentation PPT 2. 3. | | | | | | | | | | | |Individual ( Note Making | | | | | |During the PowerPoint presentation, PPT 2. 3. 1, students should make notes on their copy of BLM | | | | |2. 3. 2 | | | | | | | | | | | |Mathematical Process Focus: | | | | | |Reflecting † Reflect on findings from the previous lesson to the PowerPoint shown. | | | | | | | | | |Consolidate |Pairs ( Activity | | | | |Debrief |Each pair from the Minds On portion of this lesson will revisit BLM 2. 3. 1 and make any necessary | | | | | |changes and/or additions. | | | | | | | | | | |Whole Class ( Assessment/Discussion | | | | | |Discuss BLM 2. 3. 1 by asking for examples that support a true statement and non-examples that | | | | | |support a false statement. | | | | | | | | | | |Mathematical Process/Discussion/Mental Note | | | | | |As students share their answers and provide examples to support their thinking, assess reasoning | | | | | |skills. | | | | | | | | |Concept Pratice |Home Activity or Further Classroom Consolidation | | | |Reflection |In your own words, describe/define the following: | | | | |Effect of leading coefficient | | | |End behaviour | | | | |Degree | | | | |Maximum number of x-intercepts | | | | |Domain | | | |Range | | | | |Are the relations functions? | | | | |Complete BLM 2. 3. 3 | | | 2. 3. 1: True or False? Read each of the following statements and circle True or False below. |A degree of four is the highest degree that a cubic function can have. |True |False | A cubic function has to have at least one x-intercept. |True |False | |The leading coefficient for the function 2×3 † 5×2 + 10x + 3 is 2×3. |True |False | |The domain and range for all quartic functions will never be restricted. |True |False | |The domain and range for all cubic functions will never be restricted. |True |False | |A quartic function can have three x-intercepts. |True |False | |A cubic function can have four x-intercepts. |True |False | |A quartic function can resemble a quadratic function when graphed. |True |False | Sometimes a quartic relation is not a function. |True |False | |As x((, y(-( and x(-(, y(-( means the graph is starting on the left in quadrant 3 and ending on the right in |True |False | |quadrant 4. | | | |The leading coefficient does not influence the graph of a quartic function. |True |False | |The x-intercepts do not change when the graph is reflected on the x-axis. |True |False | |The function y = (x † 3)2 (x + 1) would have two x-intercepts. |True |False | |The function y = (x † 2)(x † 2)(x † 2) would have three x-intercepts. True |False | |The function y = (x + 4)2(x † 4)2 would create a “W” shape. |True |False | |The end behaviour for the function y = -x4 + 2×3 † x2 + 3x † 10 would be as x((, y(( and x(-(, y((. |True |False | 2. 3. 2: PowerPoint Presentation SLIDE 5: [pic] SLIDE 6: [pic] 2. 3. 2: PowerPoint Presentation (continued) SLIDE 7: [pic] SLIDE 8: [pic] 2. 3. 2: PowerPoint Presentation (continued) SLIDE 9: [pic] SLIDE 10: [pic] 2. 3. 2: PowerPoint Presentation (continued) SLIDE 11: [pic] SLIDE 12: [pic] 2. 3. 2: PowerPoint Presentation (continued) SLIDE 13: [pic] 2. 3. : Properties of Cubic and Quartic Functions 1. Based on the graphs given, complete the chart. | |Sign of Leading |Number of |End Behaviour |Domain |Range |Type of | | |Coefficient |x-intercepts | | | |Function | | | | | | | |Cubic or | | | | | | | |Quartic? |[pic] | | | | | | | |[pic] | | | | | | | |[pic] | | | | | | | |Unit 2: Day 10: Using Polynomial Functions to Model Real Life Data |MCT 4C | |Learning Goals |Materials | |Minds On: 10 |Activate prior knowledge by reviewing features of the graphing calculator such as regression analysis,|BLM 2. 10. 1 | | |if needed. |BLM 2. 10. 2 | | |Apply knowledge of polynomial functions to a set of data to determine an appropriate model for the |Graphing calculators | | |data. |Overhead panel or | | |TI-Smartview, if | | | |available | |Action: 60 | | | |Consolidate:5 | | | Total =75 min | | | | Assessment | |Opportunities | | |Minds On… |Pairs ( Activity | |This activity is only | | |Students complete BLM 2. 10. 1 to review how to plot data and use the graphing calculator for | |needed if students have | | | |regression analysis. | |not recently used the | | | | | |graphing calculators for| | | | | |regression analysis. | | | | | | | | | | |If time is a concern, | | | | | |have students complete | | | | | |all of the parts that | | | | | |require the use of the | | | | |graphing calculator | | | | | |during class and the | | | | | |conclusions (#7) can be | | | | | |done for homework or in | | | | |class the following day. | | | | | | | | | | | | | | |Action! |Individuals or Pairs( Activity | | | | |Students complete BLM 2. 10. 2 using the graphing calculators. | | | | | | | | | | | |Note: It is very difficult to find data that fits a cubic or quartic function. The given data | | | | | |does not really fit a quartic model outside of the given data. The website | | | | | |calculator. maconstate. du/cubic_modeling_lesson/index. html has a set of data that fits a cubic | | | | | |model nicely. However, the statistics are US based. Teachers may decide to use this data instead| | | | | |of the given data. | | | | | | | | | | | |Mathematical Process/Observation/Checklist | | | | |Assess students’ ability to reflect and connect as they complete the task. | | | | | | | | | | | |Mathematical Process Focus: | | | | | |Reflecting – Reflect on understanding of regression to complete the journal for the home activity | | | | |Connecting † Students connect prior content to new terminology introduced | | | | | | | | | | |Consolidate |Whole Class( Discussion | | | | |Debrief |Discuss the assignment, once it has been handed in (if it is being assessed), to determine which | | | | |model is the better fit. Also discuss the limitations that students found in the model and their | | | | | |confidence in using the model to predict values that are outside of the data range. Explain that | | | | | |when given a set of real-life data, there are two major considerations when determining a best-fit| | | | | |model for the data. The first consideration is whether or not the model is a good mathematical | | | | |fit. This entails looking at the graph to see if the model is close to the data points and | | | | | |checking the value of R2. The second consideration should be whether the fit seems reasonable | | | | | |based on what the data is representing. This entails looking at what happens to the function and | | | | | |determining if it makes sense that the data would continue to increase where the function is | | | | |increasing and to decrease where the function is decreasing. | | | | | | | | | | | | | | | |Journal |Home Activity | | | |Reflect on the activity and complete the following statements: | | | | |The hardest part of the assignment was… | | | | |I think I could do better if … | | | 2. 10. 1: Modelling Data Using a Graphing Calculator Objective: To plot the following data and model the data using an appropriate polynomial function. x |-2 |-1 |0 |1 |2 |3 | |y |-43 |-10 |-1 |2 |1 |15 | Clearing Lists When drawing a scatter plot, it is advisable to first clear all the lists † this will delete any data that is already in the lists. To do this, press (, ( and choose option 4: ClrAllLists and press Enter. [pic] [pic] Entering Data The data must be entered in lists before it can be graphed. To do this, enter the x-values in [pic] and the y-values in [pic].
Press STAT and then select 1:Edit. Enter the x-values in [pic]. Now press ( so the cursor is in [pic] and enter the y-values. 2. 10. 1: Modelling Data Using a Graphing Calculator (continued) Plotting Data To plot the data, it is necessary to turn the plotting feature of the calculator on so that it will graph the data in the lists. The lists to be used must also be specified. Remember that the x-values have been entered in[pic] and the y-values have been entered in[pic]. Press (, ( and choose 1: Plot1. Move the cursor to On and press ( so that Plot1 is on. Note that the Xlist is [pic] and the Ylist is [pic]. pic] [pic] Adjusting Viewing Window It is important to let the calculator know what values for x and y need to be plotted. For our data, the x-values range from -2 to 3 and the y-values range from -43 to 17. Press ( and set the x- and y-values to include these ranges. [pic] Now press ( and the scatter plot should appear. There were six points to be plotted so make sure that 6 points are visible in the graph. If this is not the case, adjust the viewing window. (The arrows on the graph indicate the six points. ) [pic] 2. 10. 1: Modelling Data Using a Graphing Calculator (continued) Regression Analysis
Now it is necessary to determine what polynomial function would be a good model for the data. Since the data goes from Q3 to Q1 and has 2 ‘turnaround’ points, it would be logical to say that a cubic function would be a good model for the data. [pic] Now, to have the calculator create a cubic function that fits the data. Press (, then ( to select the CALC menu, and then select 6:CubicReg for cubic regression. Press (, (, (, (, (, (, ( and (. Your screen should look like the screen below on the far right. This will perform the regression on the data in [pic] and [pic] and store the equation of the cubic function in [pic].
The screen to the left shows the general cubic function and the values for a, b, c, and d. It also shows a value for [pic]. This is called the coefficient of determination. This value is useful to determine if the function is a good fit to the data † the closer this value is to 1, the better the function fits the data! Since [pic] is about 0. 99898, this shows that the function is a great fit. If your calculator doesn’t show [pic], refer to the next page. 2. 10. 1: Modelling Data Using a Graphing Calculator (continued) If your calculator doesn’t show [
