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
| Overall Expectations | Specific Expectations |
| Numbers B2. Use knowledge of numbers and operations to solve mathematical problems encountered in everyday life. |
B2.2 Recall and demonstrate addition facts for numbers up to 10, and related subtraction facts.
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. |
Learning Goals
The purpose of this context is to help the student:
- bring two sets 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 such as those in The Effective Teaching Guide for Mathematics, K-3, Modelling and Algebra, Book 2: Equality Situations, Meaning of the Equality Symbol, pp. 36-48. 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:
|
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 tokens
- 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 wagon and that their bib represents the colour of a wagon.
Tell students that they must form trains made up of two different coloured wagons.
Invite a student pick students to form a train of two different coloured cars.
Instructions
The student chooses 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 the train comply with the directive?
Present the rules for trains with more than two cars:
- the cars must be aligned;
- the trains must all start with the same colour:
- cars of the same colour must follow each other (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 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? (for example, 1 blue car and 2 yellow cars)
- Does the train meet the three guidelines? If not, have the train re-form.
- 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. The selected students will sit next to the other trains.
Ask students the following questions:
- How many blue and yellow cars are there? (for example, 2 blue cars and 2 yellow cars)
- Does the train meet the three guidelines? If not, have the train re-form.
- Are there other ways to make up the train and still meet the guidelines? What are they?
Invite other students to form trains that represent the suggested ways (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 complies with the guidelines.
This train represents (2 + 2), but does not conform to the guidelines.
Ask students to show the train that conforms to the guidelines 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-coloured tokens.
Invite students to use the interlocking cubes or tiles or two-colored tokens 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.
Cut out the paper trains.
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 distributions 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 conform to the activity guidelines.
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 distributions.
Ask students about the relationships between the final quantity and the united quantities, and the representation (2 + 4 = 6), and the terms (2 and 4) and the sum (6).
Lead students to realize that 5-car trains can be useful in building 6+ car trains.
Examples of Success Criteria
The student:
- illustrates different distributions of cars to represent the same number;
- uses a strategy to find all possible distributions;
- 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 |
|
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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 corresponding to the sum of the two numbers 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 : Guide d’enseignement efficace des mathématiques de la 1re à la 3e année, p. 91-98.
Learning Situation 2: Ten in the Nest (Relationships)
Duration: 2 hours
| Overall Expectations | Specific Expectations |
| Operations B2. Use knowledge of numbers and operations to solve mathematical problems encountered in everyday life. |
B2.2 Recall and demonstrate addition facts for numbers up to 10, and related subtraction facts.
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. |
Learning Goals
The purpose of this learning situation is to allow students to:
- distinguish the relationships between numbers from 1 to 10;
- to represent in a concrete way and to know the addition facts that make 10;
- to 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 has practiced or practiced recognizing sets with more objects in them when the difference is visually obvious (for example, when there are 25 red chips and 5 blue chips). 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 taught in grade2 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:
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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)
- Appendix 1Rel.3 (Twelve in the Nest) (one copy per student)
- small mat or piece of cardboard
- large sheets of paper
- manipulatives (for example, buttons, blocks, tokens)
- 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 mime 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. Use the word pattern to describe what students are trying to say.
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 record the teams' findings.
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 members 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 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 this material?
- How did you solve the problem?
- How many distributions did you discover?
- Do you think you have found all possible distributions? Why or why not?
- How did you manage not to repeat the same distribution 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 the allocations they have made.
Encourage students to find other ways to divide the birds into the two nests if the work is incomplete.
Consolidation of Learning
Duration: approximately 40 minutes
Gather students and ask them to name the different number distributions 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 distribution 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 breakdowns.
Lead students to discover the patterns in the distribution 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 distributions 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 |
|
Ask students to find all the possible distributions of birds in three nests rather than in two nests. |
Follow-Up at Home
Twelve in the Nest
At home, the student can:
Divide twelve birds into two nests that they draws in a tree.
Give a copy of Appendix 1Rel.3 (Twelve in the Nest) to each student so that they can complete this activity with a family member.
Source : Guide d’enseignement efficace des mathématiques de la 1re à la 3e année, p. 109-114.