- Use the properties of operations, and the relationships between addition, subtraction, multiplication, and division, to solve problems involving whole numbers, including those requiring more than one operation, and check calculations.

- Explain the importance of various sampling techniques for collecting a sample of data that is representative of a population.
- Determine the mean and the median and identify the mode(s), if any, for various data sets involving whole numbers, and explain what each of these measures indicates about the data.

- Students investigate two different methods of extrapolation to plan a party for the entire school.
- Students should be comfortable with basic arithmetic operations (addition, multiplication, and division).

- Book:
*Moira’s Birthday*by Robert Munsch- Read Aloud Available: Video coming soon.

- Pencil crayons or markers — a variety of colours
- Appendix A (Pizza Sheet)
- Appendix B (Pizza Chart)
- Cookie sheets or cardboard/posterboard — optional

Introduction:

- In preparation for the lesson:
- Cut out pizza slices from pizza sheets (Appendix A), aiming for 2 pizza slices per student.
- Print out multiple copies of the pizza chart (Appendix B) so that there are enough rows. Alternatively, display the chart on the projector and add more rows if needed.
- Check the number of classes in the school and number of students enrolled in the school prior to beginning the lesson.

- Before reading the book, consider asking students whether they have helped plan a party. Potential discussion questions include:
*How many people were invited to the party?**What type of food was served at the party?**How much food do you estimate was needed for the party?*(e.g.,*Can you estimate how many slices of pizza were needed for all the party guests?*)

- Read
*Moira’s Birthday*to the class.

Part 1 (Arts & Crafts):

- Students will plan a party for the class in order to collect data.
- Students will decide how many pizza slices they would like to eat at the party. Write this number beside their name on the chart.
- Remind students to be reasonable in their estimates!

- Hand out pizza slices to each student following the numbers indicated on the chart.
- Students will decorate their slices using pencil crayons or markers.
- When students have finished decorating their pizza slices, they work together as a class to create whole pizzas from the slices (like putting together pieces of a puzzle).
- Teachers can use cookie sheets or large pieces of cardboard to contain each pizza.

- Students will count the number of pizzas required for their class party. The teacher can use blank pizza slices to indicate leftover slices if the last pizza is incomplete.
- For example, the class may end up with eight whole pizzas, with two blank slices on the eighth pizza.

- Ask students to count the total number of students in the class and add up the total number of slices required for the class party.

Part 2 (Estimation Using Multiplication Factor):

- Discuss the difference between a population and a sample.
- A population is the entire set of items from which you analyze data.
- A sample is a smaller set of items that effectively represents larger population.

- Introduce students to the concept of “extrapolation” and its relation to estimation.
*Recall that estimation is a powerful tool for planning, as it can help you quickly determine a numerical quantity that is close to the answer.**Similarly, extrapolation is a type of estimation that involves estimating a value outside a set of data. If we collect data from a sample, we can extrapolate to make conclusions about the larger population. However, this is only possible if the sample accurately represents the larger population.*

- Students use data from their class party (the sample) to plan a party for the entire school (the population). They will extrapolate the number of pizzas required for the school using two different methods: the multiplication factor and the mean.
- Students will complete a set of challenges to extrapolate the number of pizza slices required for the school using the multiplication factor. The following questions act as prompts for each challenge, along with hints to guide students through the procedures.
*Can you estimate how many classes are in our school?*- Students should explain how they arrived at their estimates.
- After students have shared their answers, share the actual number with the class.

*Can you use the number of pizzas required for our class party to estimate the number of pizzas required for the school party?*- Hint: If there are 10 classes in the school, that means 10 times more pizza is required.
- Solution: Multiply the number of pizzas required for the class party by the number of classes in the school.

Sample Calculations (highlighted values indicate information that the students will already have)

Number of Classes in School | 16 classes |

Number of Pizzas for Class | 4 pizzas |

Number of Pizzas for School | 4 pizzas per class × 16 classes = 64 pizzas |

Part 3 (Estimation Using Mean):

- Now, students will complete a set of challenges to extrapolate the number of pizza slices required for the school using the mean. The following questions act as prompts for each challenge, along with hints to guide students through the procedures.
*Can you estimate how many students are enrolled in our school?*- Students should explain how they arrived at their estimates.
- After students have shared their answers, share the actual number with the class.

*Can you use the pizza chart from earlier to calculate the average (mean) number of pizza slices consumed by each person at the class party?*- Solution: The mean equals the total number of pizza slices divided by the total number of students in the class.

*Do you have any strategies for how to use the mean of the smaller data set to estimate the number of pizza slices required for the school party?*- Hint: Consider the mean and the total number of students in the school.
- Solution: Students will multiply the mean by the total number of students in the school to estimate the total number of pizza slices.

*How can you use the number of pizza slices consumed by the school to calculate the number of pizzas required for the party?*- Hint: Consider the real-life context of the problem when rounding to a whole number.
- Solution: Divide the total number of pizza slices by the number of slices in one pizza to determine the total number of pizzas required. Round your answer up to the nearest whole number.

Number of Pizza Slices Eaten by Class | 30 slices |

Number of Students in Class | 22 students |

Mean | 30 slices ÷ 22 people = 1.36 slices per student |

Number of Students in School | 207 students |

Number of Pizza Slices for School | 1.36 slices per student × 207 students = 281 slices |

Number of Pizzas for School | 281 slices ÷ 8 slices per pizza = 35.125 ≈ 36 pizzas |

Conclusion:

- Discuss the calculations with students. Potential questions include:
*Which method resulted in less pizzas? Why?**What are the assumptions being made in each method? How do those assumptions impact the calculations?**What are some possible advantages and disadvantages for each of the methods?**Are there other methods of determining the number of pizzas required for the party?**Why is it important to reduce the number of pizzas ordered?*

- Finish the lesson by discussing the theme of “excess” within the story. Potential questions include:
*How could Moira have applied math to her birthday party planning?**Did our sample (class) accurately represent the population (school)? How could we improve our sample set to be more representative of the population*

- Are students able to understand the real-life context of mathematical procedures involving multiplication and division?
- What do students notice about the two different methods of extrapolation?
- Do students recognize the differences between samples and populations?

- Challenge students to determine the percentage leftover pizza slices using each method (multiplication factor and mean).
**Bonus challenges using the multiplication factor:***How can you estimate the number of leftover slices at the school party?*- Solution: Multiply the leftover slices at the class party by the number of classes to determine the number of leftover slices at the school party.

*How many slices are in one pizza? Can you calculate the total number of pizza slices ordered for the school party?*- Solution: Multiply the number of slices in one pizza by the number of pizzas to determine the total number of pizza slices.

*Can you find the percentage of leftover slices at the school birthday party?*- Solution: Divide the leftover pizza slices by the total number of pizza slices to determine the percentage of leftover slices.

**Bonus challenges using the mean:***Can you calculate the total number of pizza slices from all the pizzas, and the number of leftover slices?*- Solution: Multiply the number of slices in one pizza by the total number of pizzas from the last step. Subtract the number of pizza slices consumed from the total number of pizza slices to get the number of leftover pizza slices.

*Can you calculate the percentage of leftover slices? How does the percentage compare to leftover slices in the first method?*- Solution: Divide the leftover pizza slices by the total number of pizza slices to determine the percentage of leftover slices. It is likely that the percentage of leftover slices will be smaller when they use the mean.

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