Big Idea: Mathematical Modelling

Situation: Developing a mathematical model that will help us prepare breakfast for the school while keeping in mind different factors.

Expectations:

Algebra

• C4. apply the process of mathematical modelling to represent, analyse, make predictions, and provide insight into real-life situations

Number Sense

• B2. use knowledge of numbers and operations to solve mathematical problems encountered in everyday life

  • B2.1 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

Data

• D1. manage, analyse, and use data to make convincing arguments and informed decisions, in various contexts drawn from real life

  • D1.2 collect data from different primary and secondary sources to answer questions of interest that involve comparing two or more sets of data, and organize the data in frequency tables and stem-and-leaf plots

  • D1.3 select from among a variety of graphs, including multiple-bar graphs, the type of graph best suited to represent various sets of data; display the data in the graphs with proper sources, titles, and labels, and appropriate scales; and justify their choice of graphs

Social Emotional Learning (SEL) Skills in Mathematics and the Mathematical Processes

• A1. Throughout this grade, in order to promote a positive identity as a math learner, to foster well-being and the ability to learn, build resilience, and thrive, students will apply, to the best of their ability, a variety of social-emotional learning skills to support their use of the mathematical processes and their learning in connection with the expectations in the other five strands of the mathematics curriculum.

In this lesson to the best of their ability students will learn to think critically and creatively as they apply the mathematical processes reasoning and proving (develop and apply reasoning skills (e.g., classification, recognition of relationships, use of counter-examples) to justify thinking, make and investigate conjectures, and construct and defend arguments) and selecting tools and strategies (select and use a variety of concrete, visual, and electronic learning tools and appropriate strategies to investigate mathematical ideas and to solve problems), so they can work collaboratively on math problems – expressing their thinking, listening to the thinking of others, and practising inclusivity – and in that way fostering healthy relationships.

Present and explain the challenge to your students. It can be written on the board or even have the principal participate and join in to start the discussion.

Challenge: The principal is challenging us, Grade 4 students. We have to prepare the breakfast orders for a week. How are we going to achieve this challenge? What questions should we ask ourselves? What information is needed to resolve our problem?

Asking good questions:

In order to begin math discussions, it will be important to make sure that students ask relevant questions that will help them find necessary and important information.

Here are some examples of questions students could formulate:

• Have you ever made a grocery list?

• What does a breakfast grocery list look like?

• What are the restrictions when the people in charge of the school breakfast make purchases?

• What is our budget?

In pairs, have students discuss the relevant questions to gather the necessary information. (Think-Talk-Share activity). The teacher circulates to ensure that the questions are related to the initial challenge. Students can record their questions on a chart paper or in a virtual removable sticky note app (eg, Padlet, Jamboard, Microsoft OneNote, Post It Plus).

Virtual learners can use virtual breakout rooms to discuss.

Opportunities for Assessment

Note the type of questioning each student has. Is the student able to ask an open question? A question that demonstrates knowledge of the subject? A question that will gather information that will be useful?

Opportunities for Differentiation

Provide some sample questions so that students can benefit from examples.

It is important to come back in a large group. Facilitate a discussion to bring out all the questions that have been raised in small groups while keeping the initial challenge in mind.

Use the recording tool to display questions raised by small groups.

Afterwards, students can return to small groups to discuss further. Some groups will have thought of other questions and these can be added. Students need to be more critical in selecting their questions.

They can also discuss the relevance of the issues raised.

Use virtual breakout rooms and the recording tool to select questions.

Do they have enough information about the problem?

The table can be divided into 3 categories: important questions, those we are not sure and those we can let go. Students will be able to use the same virtual sticky note app to continue this activity.

Bring the student back together for a whole class discussion. Ask each group of students to share their questions to determine what information is most useful and necessary to answer the challenge by explaining their choices.

During this group discussion, it is important to put the new information (questions) in the notes.

Virtual learners can use a virtual whiteboard to add information in the notes.

Opportunities for Assessment

• The teacher can determine which student is able to determine the information that is needed to answer the challenge question.

• A checklist can be used to remember the questions you want to bring up during the whole class discussion.

• The teacher can determine which student is able to ask a good question. The student will be able to collect the necessary information in order to answer the initial challenge. Students can also draw on their previous knowledge in order to be able to respond effectively.

• The teacher can determine which student is able to analyze and justify the information that will be relevant to answer the question of the initial challenge.

Opportunities for Differentiation

• Provide sample assumptions for students who have a little more difficulty with this concept.

• Students who have more difficulty with these concepts could use sample tables and diagrams.

• Guided practice with some students would be appropriate.

Opportunities for Assessment

• Do they recognize an assumption? Can they explain what is an assumption and if it is relevant?

Depending on the mathematical situation, they may have several different opportunities to assess. Here are a few:

• collecting data in a table;

• creating diagrams;

• estimate the cost for breakfast foods.

Students can now proceed to the step of creating a mathematical model.

Teacher Moves:

The models can be very diverse. Here are some possible examples of mathematical models students might create:

• A grocery list of food to buy as well as the total amount they have calculated is needed of each.

• A menu for a few days with the necessary quantity.

• A list of necessary foods, now we have to do the shopping.

• A list of necessary food for the week with the prices.

Virtual Learners: Encourage students to use a shared document on the cloud to create their model.

The mathematical models students create should be based on the assumed number of students participating in the breakfast program, and assumptions of how much students might eat. For example, students, through research or experience, might learn a box of cereal contains about 7 bowls. This information, along with assumptions about how many students might participate, types of cereals students might like… will help them make decisions about the problem. If any assumption changes, their model should help them adjust the decisions they might make.

Tell them it is time to evaluate their models in their small groups. They will revisit the assumptions they made and consider if those assumptions still seem reasonable/valid.

Have them decide if their model addresses the new situations. How might they improve it ?

Here is where the teacher discusses assumptions -- what they are, how they influence our work, why they need to be considered. Might some of the assumptions they made when creating their rating scale need to be reconsidered?

Student groups will decide if their model has worked well, and what they might change.

Students can present their findings to the class. They should include what they saw as successes and failures of their model and what they are planning to change.

Teacher Moves:

As the students analyse and assess their models, reflect on using questions such as:

• Are the assumptions that we made reasonable ?

• Would our model need to change if the situation changes ?

• What information should you include to present your models clearly ?

• What mathematical language can you use when you present your model to share your ideas clearly? (Consider developing a list of related vocabulary such as area, surface area, mass, capacity.)

Encourage discussion about the strengths and challenges of each model.

It is important to review this lesson by taking into consideration the first two steps of mathematical modeling:

• We understand the problem.

• We know what questions we need to answer.

• We have identified the information that is needed.

• We have analyzed the problem.

• We know which assumptions are good and which we need to check.

• We discussed the different mathematical ways to meet our challenge.

• We want to collect data to meet our challenge.

Students must come together and make decisions based on mathematical reasoning and what they will actually do.

Groups share their decisions along with their assumptions that led to those decisions, so the class can agree on what actions to do (what to purchase, or how much to make in the morning).

Students could make the order or actually prepare the food in the quantities they have suggested.

Think about how students will collect data to see if the model is successful.