Grade 1: "Goody Bags for a Party"

(From: OAME )

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Content

Big Idea: Mathematical Modelling: making a model to predict the number of candies required in a candy bag.

Expectations:

Algebra

C2.1 identify quantities that can change and quantities that always remain the same in real-life contexts
C4. apply the process of mathematical modelling to represent analyse, make predictions, and provide insight into real-life situations.

Number Sense

B1. demonstrate an understanding of numbers and make connections to the way numbers are used in everyday life
- B1.1 read and represent whole numbers up to and including 50, and describe various ways they are used in everyday life

B2. use knowledge of numbers and operations to solve mathematical problems encountered in everyday life
- B2.1 use the properties of addition and subtraction, and the relationship between addition and subtraction, to solve problems and check calculations
- B2.4 use objects, diagrams, and equations to represent, describe, and solve situations involving addition and subtraction of whole numbers that add up to no more than 50

Data

D1. Data Literacy: manage, analyse, and use data to make convincing arguments and informed decisions, in various contexts drawn from real life
- D1.2 collect data through observations, experiments, and interviews to answer questions of interest that focus on a single piece of information; record the data using methods of their choice; and organize the data in tally tables
- D1.5 analyse different sets of data presented in various ways, including in tally tables, concrete graphs, and pictographs, by asking and answering questions about the data and drawing conclusions, then make convincing arguments and informed decisions

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 and build relationships and communicate effectively as they apply the mathematical processes of problem solving (develop, select, and apply problem-solving strategies) of communicating (express and understand mathematical thinking, and engage in mathematical arguments using everyday language, language resources as necessary, appropriate mathematical terminology, a variety of representations and mathematical conventions) and of representing (select from and create a variety of representations of mathematical ideas (e.g., representations involving physical models, pictures, numbers, variables, graphs), and apply them to solve problems) so they can make connections between math and everyday contexts to help them make informed judgements and decisions and work collaboratively on math problems - expressing their thinking, listening to the thinking of others, and practising inclusivity - and in that way fostering healthy relationships.

Learning Goals

Students are...

learning to understand how to solve everyday messy problems by seeking out important information.
learning to analyse mathematical situations.
learning to create mathematical models to solve problems by representing mathematical situations with objects, diagrams and equations.
learning to analyse and assess our mathematical models.

Success Criteria

I can...

determine important questions that need answering. I can determine missing information that is needed to solve the problem. I can show / explain my reasoning.
make assumptions about the situation and about missing information. I can describe what changes and what remains the same.
use representations, tools, technologies, strategies, facts and make assumptions to build my mathematical model. I can identify mathematical knowledge, concepts, and skills that might be involved in creating my model.
show / explain whether my mathematical model provides a solution to my problem. I can explain whether there are alternative models.

Materials

chart paper
A copy of the candy images for each group
markers
various manipulatives (ten-box frames, Rekenrek, tokens, cubes, sweets, ...)
Candy Bags
Various sweets, if possible
large sticky notes

For online/hybrid learning, small groups could be set up ahead of time. They could be facilitated via breakout rooms, depending on the different district conferencing tools used. In the breakout rooms, students could discuss to come up with questions and assumptions. They could use a virtual whiteboard to support their discussions, such as shared slides, Google Jamboard, Microsoft Whiteboard, Padlet depending on district tools, to see how different groups are collaborating. For virtual manipulatives, students can use the mathies website.

Vocabulary

addition
subtraction
groups

Pedagogy

Minds On

Project the image of the candy.

Ask students the following questions to get them thinking.

What do you see in this picture?

Allow sufficient time for students to reflect. Use the strategy (Think, Talk and Share)

Possible answers: I see candy. I see 9 different kinds of candy. I see heart-shaped candy. I see different shapes (e.g., square-based prisms, hearts, cylinders, etc.).

List their possible answers on chart paper, white board or an interactive board file.

What questions are you asking yourself?

Allow sufficient time for students to reflect. Use the strategy (Think, Talk and Share)

Possible answers: How many candies are there in total? How many of each type of candy are there? How many different colors are there?

List their possible answers on chart paper, white board or an interactive board file.

Opportunities for Differentiation

To Differentiate:

Continuously evaluate and adjust lesson content to meet student needs.
Parallel tasks: Have students work with a smaller number of candy bags.
Allow students to work individually if they wish.
Create quiet spaces where there are no distractions.
Have a variety of math manipulatives
(e.g., various types of candy) at the
students' disposition.

Opportunities for Assessment

To assess our students based on this general expectation, we will need times when we can see who is able to look for relevant information and ask questions relevant to mathematical modeling. Whole class conversations may only be useful at certain times.

Assessment for Learning

Conversations:

Asking students questions to check their understanding of the problem.
What strategies did they use?
Ask questions to help students clarify their thinking.

Action!

Setting the stage for learning:

Soon, it will be Mrs. or Mr. _____ birthday. I wish we could organize a party to celebrate his or her birthday. Could you help me prepare bags of treats to give to our guests?

Understanding the problem

Information Gathering:

Ask students: “What information do we need to know? What do we already know?” They can share a few examples at this time. Fill in the hard facts that they remember and can list (that they learned from you). E.g., It is a birthday party. They need to prepare candy bags. We are making a plan to determine what consists of a good goodie bag.

Prepare the students to work in small groups to come up with possible questions and send them off to come up with a list of questions.

After an appropriate amount of time, bring the students together to share their responses. Record student responses on the K-N-L chart.

Here are some possible student responses:

How many kids are invited?
How many kids will be attending the party?
Do any of the kids have younger brothers or sisters that might tag along?
How many candy bags must you prepare?
How many candies should we purchase?
How many candies will there be in each bag?
How many of each type of candy will be added to each bag?
Which candy types are the best to put into the bags?
Do all children like candy?
Are there any children who have allergies or are diabetic?
How much money do we have to buy candies?
How much money does each bag of candy cost?
How much money do we have to spend?

Note : It is possible that some groups are working on different problems related to the same situation.

Analysing the situation

Providing Missing Information and Prioritizing Questions

By now you probably have a large collection of questions. Some of them generate relevant facts. Some of them will lead to useful assumptions, and some of them will lead to unimportant assumptions. We can see those differences. Children often don’t distinguish.

Meet back as a group to discuss which of the questions are the most helpful. Students determine which questions need answering.

What information is needed?

The teacher must answer these questions at this point. Provide them with some details necessary for decision-making at this point, without being too specific. For example, the teacher might tell how many kids were invited (which leaves open the possibility that there might be more or less in reality), while leaving out ideas of the other limitations.

You will also need to tell them the amount of candy in each store bought candy bag. This information has been screened and selected for you and placed into the lesson slide deck such as in the following example.

The bags come in packages of 2

There are 12 lollipops in a bag

You can fill in the (Know) section of the K-N-L chart now with some hard facts. E.g. “These are important questions that provide important information. I can answer some of them. Here is what I known. … .”

Assessing and Prioritizing Questions:

At this point it is likely important for the teacher to go through the remaining, unanswered questions with the class in a whole group to demonstrate how certain questions lead to useful assumptions and some lead to trivial assumptions. By asking students what information each question will yield, teachers can highlight and “star” questions which are productive and likely to yield useful assumptions and those which are non-productive and likely to lead nowhere.

Setting parameters

Making assumptions

Explain to students that sometimes we have to make decisions without all the facts, but that we can make some assumptions that are probably true and this can help us make a draft plan. The plan may not always work perfectly the first time, but we can always change a few things and improve it. That is called revising.

Tell students to get back into their groups and think about how many candy bags are needed and how many candies will each bag contain.

Help students think about their assumptions and make useful ones with questioning as you circulate among the groups. Here are some assumptions they might need to make, which you will need to listen for:

They might assume a typical party has 5-8 people.

They might assume that each bag should be the same or that they will want many kinds of candies.
They might think that too much candy is not a good thing.
They might think that there are too many different kinds of candy available so they will limit the choices.
They might think that they should make an extra bag of candy, in case a younger brother or sister decides to join the party.
They might think that they will not add any jelly beans (or other type of candy) cause they don’t like them or they contain too much sugar.
They might want to fill the bag completely.

Note that each group will have their own set of assumptions.

Reviewing Student Assumptions:

Have students come back together to share and analyze their assumptions. The teacher should consider having students record their ideas on sticky notes and posting them in the (Learned) column of the K-N-L chart.

As you facilitate a class discussion about these ideas, the teacher may have to step up and challenge the thinking behind the assumptions simply by asking questions such as, “Why do you think that?”, “What is the basis for your assumption?”, “How did you come to that conclusion?” “Is there another way to do this or another way to solve this problem?” Explain that when making assumptions, it is up to us to challenge our own assumptions, to think about them from other viewpoints.

You may need to digress and establish the difference between a fact and an assumption, between a want and a need. e.g., You may want all a full bag of candies but you don’t need one.

As you work through the assumptions, use the same process as you did with questions - highlight and star useful assumptions to provide a visible anchor chart as a reference point.

Testing Popularity Assumptions:

An important component of this lesson is teaching children about opinion polling and data display. When they present their partial plan about how they would like to fill the candy bags, a key teacher question for each group is, “How do you know the kids will like these candies?” Here is another opportunity for a discussion about the difference between fact and opinion (assumptions). Teachers may need to press the class with comments and questions such as, “There are a lot of different types of candy to buy. What if they don’t like these candies? How can we be sure they will like the candies? Is there a way we can find out?” One hopes that students will come up with the idea of asking them. You will need to give them the constraint or parament of taking up class time. We can’t ask each child individually. Is there another way? (A survey or poll.) How toy companies make predictions about what toys will sell well? Do they go out on the street and ask children or their parents? You may have to introduce the ideal of a survey.

Explain how we can use a sample survey to make assumptions or predictions.

Creating a mathematical model

Making the plan

Students work in groups to create/develop a model of what they think a good bag of treats looks like based on the assumptions they have made.

Tell students that they also need to determine the amount of candy bags they will need to purchase in order to create their own party candy bags.

Reminder: Each group will have its own set of assumptions that will guide its final model.

Teacher Moves

Setting in situation:

Soon, it will be Mrs. or Mr. _____ birthday. I wish we could organize a party to celebrate his or her birthday. Could you help me prepare bags of treats to give to our guests?

Ask students:

What information do we need to know?
What do we already know?

As students are working in their groups, circulate among them asking questions.

Note to Teacher: The teacher is going to chart ALL student responses on a K-N-L chart (What we Know-what we Need to know-what we have Learned and know now), so you may want to have the students write/draw on large sticky notes to make sharing easier when you come back together in whole group, or find an alternative yet simple process for sharing of ideas.

Since students are the ones determining what is important, it is key the teacher includes ALL items. It is the students who pick what is helpful to them. Eventually in this lesson, each group will have to make their own set of assumptions that will guide their own final plans.

It is important for teachers to think ahead and list as many questions (and later assumptions) as possible so teachers know what to listen for. We have listed some possible ones for you.

If students are stumped, then try in small groups or whole group to teach and coach students to brainstorm questions.

Assessment as Learning

Observations: Observe students and their ability to explain the reasoning behind the choices they make. - Observe students and check how they express themselves and organize themselves during teamwork.

Conversations : Observe students and check how they express themselves and organize themselves in teamwork. Listen to conversations between students.

Encourage classroom and small group conversations that allow students to clearly express their thoughts and develop their thinking.

Production : Evaluate students' written productions.

Assessing and Prioritizing Questions:

Delicately, without discouraging students, it is important to have a class discussion about “productive” questions, and to help clarify and categorize them for students to examine whether they are going to lead to helpful assumptions.

Here are some examples of questions which are not productive. Ask questions such as those below to help students:

Will the candy fit in the bags?
How big are the candy bags?

In fact, it is important to understand that these questions are less relevant depending on the assumptions chosen. The children could also go on a different tangent and be more interested in volume and capacity by determining the size of the bags and how the candy can be put in the bags.

Setting parameters :

Making assumptions

Help students make assumptions.

Boys and girls, it is your job to think about what makes a good goodie bag.
Why don't we hand out a goodie bag full of candy the size of a grocery bag?
I want you to get back into your groups and think about what a good goodie bag should have.

Help students think about their assumptions (healthy eating, too much sugar makes us silly, we want more than 1 kind of candy, enough to take some home with us…)

What consists of a good candy bag? How do you know?

While students are collaborating to solve the problem, the teacher should circulate and ask open-ended questions to move the students` thoughts forward. (see the document The Art of Effective Questioning - https://www.coachingforchange.com/pub10.html ). The idea is to help them test their assumptions, to critically evaluate them.

What strategies are you using to make your assumptions? Why?
What strategies are you using to TEST (think about) your assumptions?
Are you encountering any barriers/challenges? How are you overcoming them?
Why did you choose these assumptions?
Can you think of any other ways this could be accomplished?
What would happen if you changed this assumption? Try it and walk me through the new scenario if you do it the new way?
Are these assumptions useful?

Reviewing Student Assumptions:

It is important for teachers to think ahead and list as many student assumptions as possible so teachers know what to listen for. Some possible ones are listed for you below.

It is also important to have a class discussion about assumptions as you did with questions to help students clarify and categorize assumptions as useful and to evaluate whether their parameters are hard limitations (e.g., a budget), a reasonable assumption (e.g., an informal poll, well none of my friends like a certain type of candy), or wishful thinking (e.g., “We need to fill the bags completely. There are 10 people that were invited so we need 10 bags of candy.”)

Testing Popularity Assumptions:

Conduct a class survey. Display the data on a simple pictograph on chart paper such as in the example below. It thus becomes an anchor chart (reference) for the children.

Consolidation Of Learning

Analyse and Assess the Model

Use the Aquarium strategy to present their solution:

Invite 2 groups to settle in the aquarium (delimit an area in the classroom). These students will become the "fish.”

Invite all of the other students (the “cats”) to sit outside the aquarium area. These students will observe and listen to the “fish” presentations. The cats cannot speak at this time. They can only listen to the two presentations.

Group A presents their work and the strategies used to group B. Group B can pose questions to group A about their strategies.

Group B then presents their work and the strategies to group A.

When the two groups have finished presenting their strategies, the other students "cats" can now ask questions.

After the presentations,the teacher can ask the group :

Do these two examples consist of a good candy bag? How do we know?

Consider how you would want students to assess the effectiveness of their own models. Asks other students or groups :

Does your model represent a good candy bag? How do you know?
Are there alternative models?

In this step, the student must also be able to verify how his or her model would fit if changes were made to the variables. For example, how do the following variables affect my model :

change in bag size
unavailability of a certain kind of candy
adding or removing friends
change of budget
guest who is diabetic or has an allergy

The teacher can voluntarily introduce some of these changes at the end and invite students to discuss what changes they should make to their model.

The teacher can proceed by asking questions that will lead the mathematical conversation. (The Art of Effective Questioning)

What questions did you ask yourself during your work?
Why did you choose this strategy to solve the problem?
Which assumptions do you think were most productive? Why?
Are there groups that have used different strategies or made different assumptions?
What was the most difficult part of the task?
How did you feel about doing this task? Why?
… and the most important question, “What assumptions would you change? How would you change your plan?”

Invite students to make connections to everyday events.

Can you relate to other life events?
Can you think of another situation where these strategies would come in handy?

Discuss together the “winning strategies” that were used:

Communicating clearly
Good listening
Precise word choice
Thoughtful assumptions

Assessment

Assessments as, for and of learning are embedded throughout the lesson plan.

SEL Self-Assessments (English) and Teacher Rubric

Extension

Many Grade One students might struggle with determining what information is important given a real problem. The following next steps are related to these areas of the mathematical modelling process, and ideas about how to explicitly teach those skills informally and/or formally.

Tell students :

I have been invited to a friend's birthday party and I was asked to bring cupcakes. The problem is that I’m not sure how many and what kind of cupcakes I should bring.

Ask students :

What information would help me determine how many and what kind of cupcakes I should bring? What information is required?

Students’ possible answers :

How many people are attending the party?
Does anyone have food restrictions?
Do people prefer chocolat, vanilla or other flavoured cupcakes?
Will there be any other types of deserts?
How many cupcakes will each person eat?
Will everyone be eating cupcakes?
Who else is invited to the party?
How old are the invitees?

Write all of the questions on the board.

Ask students :

Amongst all of the questions presented, which questions will provide the information needed to solve this specific problem. Why is this question important? How will it be useful or how can it help me?
Which questions are not important? Why?

Students need to be able to justify their reasoning.

If possible, plan to review this mathematical modeling task several times during the year, allowing students to make connections between different domains, different problems with similar underlying structures, different tools and different strategies.

Technology

For online/hybrid learning, small groups could be set up ahead of time. They could be facilitated via breakout rooms, depending on the different district conferencing tools used. In the breakout rooms, students could discuss to come up with questions and assumptions. They could use a virtual whiteboard to support their discussions, such as shared Google slides, Google Jamboard, Microsoft Whiteboard, Padlet, to see how different groups are collaborating.

For virtual manipulatives, students can use the mathies website.

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