GEEM - Learning Situation

B2. Operations

Use knowledge of numbers and operations to solve mathematical problems encountered in everyday life.

Learning Situation 1: The Railway Station

Duration: 2 hrs 20 min

Learning Goals

The purpose of this context is to help the student:

bring two parts together to form a whole;
represent addition facts from 5 to 9 in a concrete way;
understand addition facts from 5 to 9.

Learning Context	Prerequisites
In Grade 1, students learn formal addition and subtraction. To develop number sense, students must understand the many applications of addition and subtraction in real-life situations. These operations are learned through the representation of problems using manipulatives or illustrations and drawings. It is preferable to let the student represent the problems in their own way rather than imposing a model. In other words, the process should not be the focus of the lesson, but rather a tool that the student uses to understand situations and problems in statement form. The addition (+), subtraction (-) and equal (=) signs are conventions that should be explored through activities. Some signs can have different meanings, which can become confusing for students; for example, the minus sign (-) usually means subtracting a quantity. This definition is limiting, however, as students will find in the junior and Intermediate Divisions that this sign can also mean something else as they explore negative integers in Grade 6. The concepts of decomposition and grouping develop in the early grades. Understanding the connection between these concepts helps in interpreting operations, number sentences, and equations. In representing decomposition and grouping problems, the use of concrete materials helps the student grasp the concept of quantity and can provide models during the solution of addition or subtraction problems.	In this learning situation, students will: understand the meaning of the addition (+), subtraction (-) and equal (=) signs; know the addition facts from 2 to 4.

Learning Context

Prerequisites

In Grade 1, students learn formal addition and subtraction.

To develop number sense, students must understand the many applications of addition and subtraction in real-life situations. These operations are learned through the representation of problems using manipulatives or illustrations and drawings. It is preferable to let the student represent the problems in their own way rather than imposing a model. In other words, the process should not be the focus of the lesson, but rather a tool that the student uses to understand situations and problems in statement form.

The addition (+), subtraction (-) and equal (=) signs are conventions that should be explored through activities. Some signs can have different meanings, which can become confusing for students; for example, the minus sign (-) usually means subtracting a quantity. This definition is limiting, however, as students will find in the junior and Intermediate Divisions that this sign can also mean something else as they explore negative integers in Grade 6.

The concepts of decomposition and grouping develop in the early grades. Understanding the connection between these concepts helps in interpreting operations, number sentences, and equations. In representing decomposition and grouping problems, the use of concrete materials helps the student grasp the concept of quantity and can provide models during the solution of addition or subtraction problems.

In this learning situation, students will:

understand the meaning of the addition (+), subtraction (-) and equal (=) signs;
know the addition facts from 2 to 4.

Materials

Appendix 1SO.1 (2 cm Grid Paper) (one copy per student)
Appendix 1SO.2 (Addition Game) (two copies per student)
blue and yellow bibs (one bib per student)
containers (one per team of two)
interlocking cubes of two different colours or two-coloured tiles or counters
scissors
crayons or markers
glue sticks (one per student)
construction paper (one sheet per student)
dice (2)

Mathematical Vocabulary

number sentence, addition facts, is equal to, pattern

Before Learning (Warm-Up)

Duration: approximately 40 minutes

Have students sit in a large circle.

Give half of the students a yellow bib and the other half a blue bib.

Explain that each person is a car, and their bib represents the colour of the car. The goal is to create a train composed of two different coloured cars.

Invite one student to make a train of two different coloured cars.

Instructions

The student should choose one student wearing a yellow bib and another wearing a blue bib to form the train. The train will be made up of a yellow car followed by a blue car or a blue car followed by a yellow car.

Ask students the following questions:

How many blue and yellow cars are there?
How many cars in total are on the train?
Does this train follow the rules for a train with two cars?

Present the rules for trains with more than two cars:

the cars must be in a row;
the trains must all start with the same colour:
cars of the same colour must stay together (for example, 4-car trains: 1 yellow car followed by 3 blue cars or 2 yellow cars followed by 2 blue cars or 3 blue cars followed by 1 yellow car).

Invite another student to form a three-car train and place it next to the first (2-colour) train. If the first train has a blue car at the beginning, the new train must also have a blue car at the beginning.

Ask students the following questions:

How many blue and yellow cars are there on this train? (for example, 1 blue car and 2 yellow cars)
Does the train follow the three rules? If not, have the students re-form the train.
Is there another way to form the train? (for example, 2 blue cars and 1 yellow car)
How many cars are on either train?

Invite another student to form a train of four cars, using the same rules. The selected students will sit next to the other trains.

Ask students the following questions:

How many blue and yellow cars are there on this train? (for example, 2 blue cars and 2 yellow cars)
Does the train follow the three rules? If not, have the students re-form the train.
Are there other ways to make up the train and still follow the rules? What are they?

Invite other students to form other possible trains that follow the rules (for example, 1 blue car and 3 yellow cars or 3 blue cars and 1 yellow car) and ask: How many cars are in each of these formations?

Introduce a four-car train built with interlocking cubes that conforms to the rules and a train that does not.

This train represents (2 + 2) and follow the rules.

This train represents (2 + 2), but does not follow the rules.

Ask students to show the train that follows the rules and explain why it does.

Active Learning (Exploration)

Duration: approximately 50 minutes

Form teams of two.

Give each team a container of two different coloured interlocking cubes or tiles or two-colour counters.

Invite students to use the interlocking cubes or tiles or two-color counters to build as many different 5-car trains as possible.

Bring students together once they have built several trains and give them a copy of Appendix 1SO.1 (2 cm grid paper).

Have students represent each of their trains on the grid paper using coloured pencils that match the colours of the cars.

Ask students to cut out the paper trains once they are done.

Give each student a sheet of construction paper.

Ask students to find a way to organize their trains, glue them together, and write a corresponding math sentence under each train (for example, if the train is 3 blue cars and 2 yellow cars, write: \(\ 3 + 2 = 5\)).

Circulate as students work and observe their process. Ask the following questions:

How would you describe your trains?
What can you do to find out if you have found all the possible combinations of cars?

The students' work could look like this:

\(\ 1 + 4 = 5\)

\(\ 2 + 3 = 5\)

\(\ 3 + 2 = 5\)

\(\ 4 + 1 = 5\)

Point out that the quantity of 5 interlocking cubes has been distributed in 4 different ways.

Note: Students may have built a train with 5 cars of the same colour, which is associated with the addition fact 5 + 0 = 5. It is important to point out that the train YBYBY also represents 2 + 3 = 5, but does not follow the rules for trains.

Consolidation of Learning

Duration: approximately 50 minutes

Invite each team to present the trains they have built.

Ask students the following questions:

What was difficult about building the trains?
What was easy?
What strategies did you use to build the trains?
How do you know if you have built all the trains you can?
What order did you follow to organize the trains?

Repeat the activity over the next few days, but increase the number of cars that make up the trains. Have them build trains of 6, 7, 8, and 9 cars by finding all possible combinations.

Ask students about the relationships between the whole, the parts, and the number sentence (2 + 4 = 6), the terms (2 and 4) and the sum (6).

Support students to realize that 5-car trains can be useful in building 6+ car trains.

Examples of Success Criteria

The student:

illustrates different combinations of cars to represent the same number;
uses a strategy to find all possible combinations;
writes number sentences correctly;
understands that the order of the terms does not change the sum;
identifies patterns;
explains their strategy and patterns using appropriate mathematical vocabulary.

Differentiated Instruction

The activity can be modified to meet the needs of the students.

To Facilitate the Task

To Enrich the Task

allow students to stay at the concrete stage and working with the interlocking cubes as long as necessary;
use the same small numbers repeatedly;
work one-on-one with students who are struggling and ask them questions to get them to think more deeply before acting.

ask students to build trains with a larger number of interlocking cubes (12 to 20);
have students build trains with interlocking cubes of three different colours.

Follow-Up at Home

Addition Game

At home, the student can play the addition game with a family member.

Provide two copies of Appendix 1SO.2 (Addition Game) to each student.

Explain the rules of the addition game:

Roll two dice.
Ask students to find the sum of the two dice.
Ask them to find on the game board the column that is labelled with the correct sum and then colour a square in that column.
Repeat these steps a few times.
Point out that the first person to colour all the boxes in a column wins the game.

Source: translated from Guide d’enseignement efficace des mathématiques de la 1^re à la 3^e année, Numération et sens du nombre, p. 91-98.

Learning Situation 2: Ten in the Nest (Relationships)

Duration: 2 hours

Learning Goals

The purpose of this learning situation is to allow students to:

distinguish the relationships between numbers from 1 to 10;
represent in a concrete way and to know the addition facts that make 10;
build the concepts of decomposition and grouping.

Learning Context	Prerequisites
It is necessary for the student to have a clear understanding of what the expressions more than or less than mean before moving on to more complex relationships between numbers. These are concepts that the student began developing before entering school. In kindergarten, the child practiced recognizing sets with more objects in them when the difference was visually obvious (for example, when there were 25 red counters and 5 blue counters). An understanding of the meaning of more than or less than prepares the student for the use of the symbols greater than (>) and less than (<), which are used in the junior grades to show relationships between numbers. It is important to note that the ability to distinguish relationships between numbers is the basis for developing number sense. In Grade 1, students will perceive the relationship of 1 or 2 more than and 1 or 2 less than when counting up or down from a given number. Other fundamental concepts to be grasped at this level are decomposition and grouping, that is, the relationship between sum and terms, which the student acquires by handling numbers and discovering, for example, that 7 can be decomposed as follows: 6 and 1, 5 and 2, or 4 and 3. The relationship between 5 and 10 and between these numbers and other numbers is also very important. 5 and 10 are said to be anchors. To use them as anchors, the student must recognize the relationship between 5 and 10 and the other numbers (for example, 7 is 2 more than 5 and 3 less than 10). A solid understanding of the relationship between each number between 1 and 10 and the anchors 5 and 10 is of great help when moving to larger numbers (for example, 27 is 2 more than 25 and 3 less than 30). When performing addition and subtraction, the student consistently uses this relationship.	In this learning situation, the student should be able to: count backwards from 10; understand the meanings of the addition (+), subtraction (-), and equal (=) signs; know the addition facts for numbers from 2 to 9.

Learning Context

Prerequisites

It is necessary for the student to have a clear understanding of what the expressions more than or less than mean before moving on to more complex relationships between numbers. These are concepts that the student began developing before entering school. In kindergarten, the child practiced recognizing sets with more objects in them when the difference was visually obvious (for example, when there were 25 red counters and 5 blue counters). An understanding of the meaning of more than or less than prepares the student for the use of the symbols greater than (>) and less than (<), which are used in the junior grades to show relationships between numbers.

It is important to note that the ability to distinguish relationships between numbers is the basis for developing number sense. In Grade 1, students will perceive the relationship of 1 or 2 more than and 1 or 2 less than when counting up or down from a given number. Other fundamental concepts to be grasped at this level are decomposition and grouping, that is, the relationship between sum and terms, which the student acquires by handling numbers and discovering, for example, that 7 can be decomposed as follows: 6 and 1, 5 and 2, or 4 and 3.

The relationship between 5 and 10 and between these numbers and other numbers is also very important. 5 and 10 are said to be anchors. To use them as anchors, the student must recognize the relationship between 5 and 10 and the other numbers (for example, 7 is 2 more than 5 and 3 less than 10). A solid understanding of the relationship between each number between 1 and 10 and the anchors 5 and 10 is of great help when moving to larger numbers (for example, 27 is 2 more than 25 and 3 less than 30). When performing addition and subtraction, the student consistently uses this relationship.

In this learning situation, the student should be able to:

count backwards from 10;
understand the meanings of the addition (+), subtraction (-), and equal (=) signs;
know the addition facts for numbers from 2 to 9.

Materials

Appendix 1Rel.1 (Ten in the Nest) (one copy per student)
Appendix 1Rel.2 (Ten in the Nest, Problem Solving) (one copy per team of ten)
small mat or piece of cardboard
large sheets of paper
manipulatives (for example, buttons, blocks, counters)
drawing representing two nests

Mathematical Vocabulary

add, subtract, group, divide, decompose, pattern, in all

Before Learning (Warm-Up)

Duration: approximately 30 minutes

Present the nursery rhyme Ten in the Nest (similar to Ten in the Bed) (Appendix 1Rel.1).

Have ten students sit in a tight group on a small mat or piece of cardboard (mat or cardboard just large enough for all ten students to sit on). The mat or cardboard represents the nest and each student represents a bird.

Assign a number from 1 to 10 to each student. Have the first student play the role of the little bird who says, "Move over, move over." Throughout the rhyme, the bird that falls out of the nest is the one with the number corresponding to the number of birds in the nest.

Read the first two verses of the rhyme and ask the students sitting on the carpet to act out the scene.

Ask other students:

How many birds were in the nest?
How many have fallen?
How many birds are left in the nest?

Read the rest of the rhyme in the same way.

Represent what happens as the rhyme unfolds.

Example

Number of Birds in the Nest	Number of Fallen Birds
10	0
9	1
8	2
...	...

Ask students to say what they observe. Allow time for them to express themselves in their own words and introduce the word pattern when appropriate.

Active Learning (Exploration)

Duration: approximately 50 minutes

Form two teams of ten students and tell them that they are representing birds. If there are more than twenty students in the class, designate students to be team recorders.

Propose the following problem:

The bird family in the nursery rhyme Ten in the Nest decides that they need another nest. After building a new nest, they look for different ways to divide the family between the two nests.

Ask a few students to take turns explaining the problem in their own words. Help them to express it by asking them the following questions:

What do you already know?
What do you want to know?

Discuss with students the various possible strategies for solving the problem.

Ask both teams to find all possible ways to divide themselves into the two nests, either by simulating the situation or using manipulatives.

Give each team or team recorder a copy of Appendix 1Rel.2 (Ten in the Nest - Problem Solving) and ask them to record their solutions.

Circulate and observe the approaches used by the students. Ask questions such as:

Why did you choose these manipulatives?
How did you solve the problem?
How many combinations did you discover?
Do you think you have found all possible combinations? Why or why not?
How did you manage to not repeat the same combinations twice?
How is this problem similar to what we did when we recited the rhyme?

Meet with the teams when they are finished to verify their solutions.

Encourage students to find other ways to divide the birds into the two nests if their work is incomplete.

Consolidation of Learning

Duration: approximately 40 minutes

Gather students and ask them to name the different number combinations used to place the birds in the two nests.

Ask them to explain the strategies they used to solve the problem.

Draw attention to new strategies employed by some students.

Ensure that the strategy proposed by a student is consistent with the work the student has done. Accept the answer "10 birds in one nest and 0 in the other" as a possible combination of family members. Ask students to explain this answer.

Return to the chart used in the warm-up and ask students to identify other possible combinations.

Support students in discovering the patterns in the combinations of birds using the relationship of one more or one less bird in the nest.

Examples of Success Criteria

The student:

uses an appropriate strategy to solve the problem;
draws or represents in a concrete way as many combinations as possible that make 10;
recognizes patterns;
explains how the problem was solved using words, numbers or drawings.

Differentiated Instruction

The activity can be modified to meet the needs of the students.

To Facilitate the Task

To Enrich the Task

Give cubes and several drawings of two nests to the student, ask them to divide the cubes into the nests and write the combination under the nests. Leave the cubes in all the nests from beginning to end so the student can see the combinations already found;
ask to place the nests in an order that shows the existing pattern (+ 1 or - 1).

Ask students to find all the possible combinations of birds in three nests rather than in two nests.