La vue d'ensemble

At a Glance...

The Guides for Effective Instruction in Mathematics Revised

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B1. Number Sense

Demonstrate an understanding of numbers and make connections to the way numbers are used in everyday life.

Learning Situation 1: The Magic of Numbers (Counting)

Duration: approximately 2 hours

Overall Expectation Specific Expectations
B1 Demonstrate an understanding of numbers and make connections to the way numbers are used in everyday life.

 B1.1 Read, represent, compose, and decompose whole numbers up to and including 200, using a variety of tools and strategies, and describe various ways they are used in everyday life.

B1.3 Estimate the number of objects in collections of up to 200 and verify their estimates by counting.

B1.4 Count to 200, including by 20s, 25s, and 50s, using a variety of tools and strategies.

Learning Goals

The purpose of this learning situation is to allow the student to:

  • recognize regularities in numbers;
  • add mentally.
Learning Context Prerequisites
In Grade 2 , students further explore number patterns by counting by 1 and in intervals of 2, 5, 10, 25, and 50, up to 200. The use of grids, number charts, and the calculator allow students to explore counting patterns in a tactile and visual way to perform operations involving two-digit numbers.

In this learning situation, students will:

  • use one- and two-digit numbers in a variety of contexts;
  • count in increments of 2, 5, 10, 25, and 50 to 200;
  • use base-ten materials or other manipulatives (for example, interlocking cubes, centicubes, ten frames) to represent numbers to 200.

Materials

Main Activity

  • Appendix 2D.1 (Blank grid)
  • Appendix 2D.2 (Number grid)
  • Appendix 2D.3 (Guess the pattern!) (one copy per student)
  • number mat or blank number grid
  • manipulatives (for example, tokens)
  • magic wand or wizard hat (optional)
  • calculator

Additional Activity 1

  • Appendix 2D.2 (Number Grid) (one copy per team)
  • calculators (one per pair)
  • tokens
  • calculator on the computer

Additional Activity 2

Additional Activity 3

Mathematical Vocabulary

pattern, number grid, number mat, calculator

Before Learning (Warm-Up)

Duration: approximately 30 minutes

Place an enlarged copy of Appendix 2D.1 or a blank number mat on the floor, or project Appendix 2D.1.

Ask students to sit in front of the grid so that everyone has the same perspective.

Begin the Landing Position game by throwing a token onto the grid or mat. Ask students to guess the number in the box where the token fell and explain what they did to determine that number.

Ask students to communicate their thinking and justify their reasoning. For example, if the token fell on 32, a student might say, "I counted to 2, then I counted down in increments of 10, so 2, 12, 22 down to 32."

Vary the activity by saying a number and then asking a student to place an object on the corresponding square of the grid or move to that location on the mat.

Active Learning (Exploration)

Duration: approximately 45 minutes

Place a number mat or enlarged grid (Appendix 2D.1) on the floor and have students participate in the Number Magic game.

Clarify the various roles that students are to play.

  • The number wizard: The role of the number wizard is to choose a number and designate another student to become the apprentice in search of the number. The student may wear a special hat or use a magic wand.
  • The apprentice: The role of the apprentice is to discover the chosen number and to describe orally their movements on the grid to arrive at this number.
  • The verifier : The verifier's role is to record and verify, with the help of a calculator, the apprentice's movements on the mat or grid.

Designate three students to act out the roles.

Ask the number wizard to choose a number less than 50, and then ask the apprentice to go to the corresponding box.

Invite the number wizard to choose another number to add to the first.

Invite the apprentice to determine the sum of the two numbers and to move to the square corresponding to the total. The student moves one square at a time and describes to the class the steps of their moves.

Example

The number wizard chooses 23. The apprentice goes to the corresponding square. The magician decides that the number 34 must be added to 23. The apprentice can move down to square 33, which is a jump of 10, then to square 43, which is another 10, and to square 53 for a final jump of 10. The apprentice then moves one square at a time from 53 to 57, which is the destination. The verifier uses a felt-tip pen to trace the moves on a grid as the apprentice describes them.

Note: This grid will be used during the consolidation stage to analyze the movements of the apprentices and to make a formative assessment of each student. If the same grid is used to record the movements of each team member, use different coloured markers.

Repeat the game until each team member has played all three roles.

Group students into teams of three. Provide each team with a copy of Appendix 2D.1 and tokens.

Observe students playing and note strategies used to determine numbers and add them up.

Record your observations in an anecdotal record or use a checklist to record strategies used by the student.

Consolidation of Learning

Duration : approximately 40 minutes

Share and discuss the various strategies used during the game.

Analyze, with the students, the moves recorded on the grid or the strategies used during the game.

Ask students the following questions:

  • What is the fastest way to determine the starting point?
  • What regularities did you notice?
  • What made the game easier?
  • How do you think this game would help someone add up two-digit numbers?
  • How would the game be different if the magician had asked you to subtract a number from the starting number?
  • Why did you decide to...?
  • How would you get to the box...?
  • How could we get to the box any faster?
  • Would there be a different way to get to the box...?

Examples of Success Criteria

The student:

  • uses an effective strategy for adding numbers;
  • uses patterns to determine numbers on the grid or number mat;
  • properly explains their movements.

Differentiated Instruction

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

To Facilitate the Task To Enrich the Task

  • Play the game using the number grid (Appendix 2D.2).
  • Ask students to mark with a pencil the moves made to reach the sum.
  • Suggest using a calculator to determine the sum or to check it.

  • Play the game by subtracting instead of adding.
  • Have the student explain their strategy in their math journal.

Follow-Up at Home

Guess the Pattern!

At home, students can play Guess the Pattern! with a family member.

Distribute a copy of Appendices 2D.2 and 2D.3 to each student.

Ask students to create patterns on the number grid by placing tokens on the grid.

Invite students to ask a family member to guess the pattern.

Extension 1: Secret Formulas

Group students into teams of two. Give each team a number grid (Appendix 2D.2).

Enter an operation in a calculator that involves repeating a number (for example, 5 + 5). Press the [=] key to show the first number in the pattern.

Ask each team to place a token on the square of the grid corresponding to the number.

Continue to press the [=] key as students add tokens to the grid to represent the new numbers that appear on the calculator until a student discovers the pattern and names it.

Repeat the game two or three times.

Distribute a calculator to each team or use the calculator on the computers. Have students play the game: one student uses the calculator and the other uses the number grid. The roles are reversed each time the solution to a game is found.

Modify the task by asking students to use two different numbers in the operation (for example, 2 + 5), which makes it more difficult to discover a pattern, or to subtract rather than add.

Extension 2: Missing Numbers

Group students into pairs. Give each student a copy of Appendix 2D.1.

Have one student from each team secretly cover four or five spaces on the grid with tokens.

Invite each partner to guess the numbers and explain how the student was able to figure them out (for example, It was the 4th number in the 3rd row, so I knew it was 34.)

Point out to students the strategies used to discover the missing numbers on the number grid.

Facilitate the task of struggling students by using the number grid (Appendix 2D.2).

Extension 3: Puzzles (reconstructing a number grid)

Make several copies of Appendix 2D.2.

Cut out the number grids in a variety of ways (for example, in strips, columns, or groups of 2, 4, or 5 squares…) and place the pieces of each grid in a plastic bag.

Distribute the bags to students and invite them to reconstruct the grids, an activity that serves to reinforce various concepts in numeration and number sense.

Source: Guide d'enseignement efficace des mathématiques de la 1re à la 3e année, p. 131-136.

Learning Situation 2: What is Your Estimation?

Duration: approximately 2 hours

Overall Expectation Specific Expectations
B1. Number Sense
Demonstrate an understanding of numbers and make connections to their use in daily life.

B1.1 Read, represent, compose, and decompose natural numbers from 0 to 200, using a variety of tools and strategies, in a variety of contexts, and describe how they are used in everyday life.

B1.3 Estimate the number of objects in collections of up to 200 and verify their estimates by counting.

Learning Goals

The purpose of this learning situation is to allow the student to:

  • establish relationships between quantities of objects and numbers;
  • to make increasingly accurate estimations;
  • to establish benchmarks to facilitate estimations;
  • to explain their estimation strategies.
Learning Context Prerequisites
The student's ability to estimate is closely related to their understanding of quantity.

This skill helps the student use logic and reasoning in problem-solving situations in numeracy. The student who has difficulty estimating cannot evaluate the reasonableness of an answer. The student who automatically asserts that a large container holds 200 objects, regardless of whether the objects are small or large, has not fully understood estimation.

The use of benchmarks as an estimation strategy enables students to improve the accuracy of their estimations. A benchmark is a known and understood quantity. Knowing the quantity represented by a benchmark makes it possible to explore or estimate the number of objects included in a larger or smaller set. For example, students can more easily estimate the quantity of a large number of blocks by first counting 10. The benchmark of 10 blocks allows them to consider the larger set by estimating the number of groups of 10 in the set, and then estimating the total number of blocks in the set.

 In this learning situation, students will:

  • skip count by 2s, 5, 10s, 20s, 25, and 50s;
  • use the numbers 5 and 10 as benchmarks;
  • make estimations for small quantities of objects.

Materials

  • Appendix 2Q.1 (Example of an Estimation Table)
  • Appendix 2Q.2 (What is Your Estimate?) (one copy per student)
  • clear containers or reusable plastic bags (one per pair)
  • objects to fill containers or bags (for example, table tennis balls, marbles, golf balls, popcorn, dried beans)
  • sticky notes

Mathematical Vocabulary

estimation, near, double, half, benchmark

Before Learning (Warm-Up)

Duration : approximately 40 minutes

Place identical objects (for example, table tennis balls, marbles, golf balls, popcorn, dried beans), but vary the quantity, in three identical clear containers or plastic bags.

Show students the first container and circulate it around the classroom to familiarize themselves with the quantity and size of the objects.

Ask students to estimate the number of objects in the container and write the estimations on the board.

Empty the contents of the container and ask students to suggest ways to count the objects (for example, make groups of 2, 5, 10, 20, 25 or 50).

Count the objects and compare the total to students' estimations.

Ask students questions such as:

  • Was your estimation close to the actual number of items in the container?
  • What strategy could you use to get a more accurate estimation next time?

Show students the second container and ask them the following questions:

  • How many objects do you think are in this container?
  • Do you think it contains more or fewer objects than the first container?
  • How can the number of items in the first container help you estimate the number of items in this container?

Ask students to explain their reasoning.

Empty the second container of its contents and begin counting the objects.

Stop after counting 10 objects and ask students if they want to change their estimation based on the amount of 10 objects.

Ask students who changed their estimations to explain their reasoning.

Continue counting the objects in the second container and compare students' estimations to the actual quantity.

Ask students the following question: Why are the estimations more accurate this time?

Discuss with students the use of a benchmark to make estimations: knowing the number of objects in a small set can help determine the number of objects in a larger set.

Finally, show students the third container and ask them to identify strategies that might help them estimate the amount of objects in the container.

Encourage students to use the strategies mentioned to estimate the number of objects in the third container.

Clarify to students that the goal of estimation is to obtain a number that is close enough to the actual number, not necessarily to determine the exact number.

Project Appendix 2Q.1 on the interactive whiteboard.

Give students sticky notes and ask them to write their name and estimation on them and stick them in the appropriate box on the board.

Count the contents of the third container and discuss the results with students. Emphasize that estimations are more accurate when benchmarks are used.

Active Learning (Exploration)

Duration: approximately 45 minutes

Group students into pairs.

Provide each student with a copy of Appendix 2Q.2.

Give each team a plastic bag of small objects. Each bag should contain enough objects to make it impossible to visually count them.

Ask students to estimate and count the objects in the bag and have them write both numbers on the worksheet (Appendix 2Q.2).

Ask teams to switch bags with another team when they are finished.

Encourage students to use a strategy that uses a quantity as a benchmark before estimating the total number of items in the bag.

Circulate around the classroom and observe how students proceed.

Ask students questions to allow them to explain the strategies they used, for example:

  • What benchmark did you use to estimate the number of items in the bag?
  • What other strategies can you use to estimate the number of items in the bag?
  • How can estimating the number of items in one bag help in estimating the contents of another bag?

Consolidation of Learning

Duration: approximately 35 minutes

Bring students together for the discussion and ask them the following questions:

  • How did you estimate the number of items in the bags?
  • Was the quantity of items in some bags more difficult to estimate? Why or why not?
  • What strategies did you use to count the objects?
  • If the same bag contained larger items, would there have been more or fewer of them?
  • If the same bag contained smaller items, would there have been more or fewer of them?
  • What kinds of groupings can be made to estimate the quantity of objects?

Discuss with students opportunities for estimation outside of the classroom; for example, when making a recipe that calls for 50 caramels, what can be done to estimate whether a bag contains enough? Remind them to use 5 or 10 caramels as a benchmark to determine if the bag contains about 50 caramels.

Examples of Success Criteria

The student:

  • uses the appropriate benchmarks;
  • makes accurate estimations;
  • explains their estimation strategies;
  • uses strategies to count quickly;
  • modifies their estimation based on what the student knows after counting a certain quantity.

Differentiated Instruction

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

To Facilitate the Task To Enrich the Task

  • Have the student who is having difficulty work with another student who has mastered the skill of estimating.
  • Use a small number of objects at first and increase as the student becomes more skilled.
  • Always use the same objects and change only the quantity or change the colour.

  • Use different shaped containers.
  • Use smaller and smaller objects.
  • Increase the amount of items in the container.
  • Ask the student to prepare a container of objects to be estimated.

Source : Guide d’enseignement efficace des mathématiques de la 1re à la 3e année, p. 147-151.

Learning Situation 3: Mystery Bags

Total duration : approximately 3 hrs 30 min

Overall Expectation Specific Expectations

B1. Number Sense
Demonstrate an understanding of numbers and make connections to their use in daily life.

B1.1 Read, represent, compose, and decompose natural numbers from 0 to 200, using a variety of tools and strategies, in a variety of contexts, and describe the ways in which they are used in everyday life.

B1.2 Compare and order natural numbers up to 200 in a variety of contexts.

B1.3 Estimate the number of objects in collections of up to 200 and verify their estimates by counting.

B1.4 Count to 200, including by 20s, 25s, and 50s, using a variety of tools and strategies.

B1.5 Describe what makes an number even or odd.

Learning Goals

The purpose of this learning situation is to allow the student to:

  • decompose numbers into hundreds, tens, and ones up to 200;
  • to exchange ten tens for a hundred;
  • represent numbers using base ten and other materials;
  • describe the relationships between numbers.
Learning Context Prerequisites

By the beginning of Grade 2, the student is able to construct the concept of positional value. For example, the student may see the number 33 as representing 33 ones as well as 3 tens and 3 ones. The learning is further complicated by the fact that this number can be seen as equivalent to 2 tens and 13 ones. In addition, the student has addressed the regularity of addition by 10 in the decimal system and should be able to mentally add 10 to 33 without counting each one.

In Grade 2 , students will be expected to understand that in any number up to 200 :

  • groups of 10 and 100 objects must be perceived as a whole and represented by a number;
  • the quantitative value of a number is determined by its position in the number.

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

  • read and write the symbols of natural numbers up to 200;
  • skip count by 1s, 2s, 5s, 10s, 20s, 25, and 50s;
  • count backwards;
  • determine the number of hundreds, tens, and ones in numbers up to 200;
  • use a variety of counting strategies to add and subtract.

Materials

Main Activity

  • reusable plastic bags
  • mystery bags containing identical objects (one per team of two)
  • Appendix 2Rep.1 (Mystery Bag) (one copy per pair)
  • Appendix 2Rep.2 (Example of an Organizational Chart)
  • Appendix 2Rep.7 (Anecdotal Record)
  • Appendix 2Rep.8 (Let's Play Detective) (one copy per student)
  • large sheets of paper
  • felt-tip pens

In Each Centre

  • sheets of paper or math journal
  • calculators (optional)

Centre 1: The Helpers

  • Appendix 2Rep.3 (Aids) (one copy per student)
  • manipulatives: interlocking cubes, centicubes (small cubes of 1cm3), tokens

Centre 2: The Countdown

  • Appendix 2Rep.4 (Place Value Mat) (one copy per pair)
  • base ten materials: unit cubes, rods, and flats
  • sets of number cards from 1 to 9 (several)

Centre 3: Mystery Numbers

  • interlocking cubes
  • cards (one per student)
  • paper bags (one per student)
  • stapler

Centre 4: The Number Wheel

Mathematical Vocabulary

hundreds, tens, ones, represent, value, place value, group, grouping, exchange

Before Learning (Warm-Up)

Duration : approximately 40 minutes

The Mystery Bags

Prepare enough mystery bags for each team to have one. Mystery bags should contain between 50 and 200 objects (for example, 148 pattern blocks, 63 tokens or 142 sticks). Ensure that each bag contains only identical objects. Using a variety of objects that vary in number and size provides students with different experiences.

Begin the activity by telling students that several mystery bags containing a variety of different objects have been left in the classroom and that they need to determine the quantity of objects in each bag.

Group students into teams of two. Give each team a copy of Appendix 2Rep.1.

First ask students to:

  • estimate the quantity of objects contained in the bag;
  • explain how they arrived at this estimation.

Invite students to determine the exact number of objects in their bag using the counting strategy of their choice (for example, students can count objects by 1 or in increments of 2, 5, 10, 20, 25, and 50).

Circulate around the classroom as students work and observe the counting strategies that teams use.

Assist students who are struggling by asking questions such as:

  • What strategy did you use to count the objects?
  • How many objects are there?
  • How can you prove it?
  • Is there another way to count objects? What is it?
  • Is there an easier way to count objects? What is it?

Bring students together to share their findings.

Help teams share their counting strategies with the class.

Ask each team:

  • to identify the objects in the bag;
  • to indicate the estimation made;
  • to explain how the team made their estimation;
  • identify the counting strategy used;
  • to specify the actual number of objects in the bag;
  • to explain why the team is satisfied or not with their estimation.

Highlight effective counting strategies that students have used (for example, counting by 1 is not effective because it is too long and easy to make mistakes) and list them on a large sheet of paper for reference in future activities.

Active Learning (Exploration)

Duration: approximately 2 hours

Organize four learning centres in advance.

Explain to students that they will be doing four different activities involving positional value.

Group students into four teams.

Using Appendix 2Rep.2 as a guide, prepare a chart showing the activities and rotation schedule for the teams.

Organize the materials needed for each activity and place them in separate bins. Label each bin with the activity name and number.

Prepare a checklist for each activity indicating the criteria by which students will be assessed. Attach each checklist to a clipboard and place it in the corresponding centre. Use Appendix 2Rep.7 as a guide.

Delineate the area of the classroom reserved for each activity, by grouping desks or reserving a table or corner.

Prepare a chart with students indicating the criteria for each activity.

Introduce students to the activities one at a time: state the name of the activity, explain the work to be done, show the materials, and model the activity as needed.

Tell students that they will be observed while working in a specific centre to assess their work in a formative manner.

Accompany students to the centre being evaluated and use the list of established criteria to evaluate each team member.

Note: It is important that students understand the numerical value of the ones units, tens rods, and hundreds flats that are part of the base ten materials before undertaking the activities that require these materials. If they do not, have them use interlocking cubes or centicubes to complete the activities.

Centre 1: The Helpers

Give each student a copy of Appendix 2Rep.3 and read the problem aloud.

Mrs. Cadieux needs additional students to help her organize a party. Make a list of students from the class that she could choose to help her. Write the names of the students in the box provided. Assume that Mrs. Smith already has 125 students helping her. How many will there be if she decides to choose all the students you suggested? Write down on a piece of paper the counting strategy you used to determine the number of students who will help organize the party.

Have students use the manipulatives and solve the problem.

Centre 2: The Countdown

Explain to students that they will play a game in which they will subtract from 100 (a hundreds flat) to 0. The first team to reach 0 will be the winning team.

Inform students that they will work in pairs and play one group against the other.

Provide each group with a copy of Appendix 2Rep.4, base ten materials, and several decks of number cards from 1 to 9.

Invite students to place the cards between the two groups, face against the table. Invite each pair to place a hundreds flat on their place value mat before beginning the game.

Have each student take a card and work with their partner to add up the numbers on it.

Invite students to subtract this number from their hundreds flat by exchanging the flat for tens rods and the rods for ones units so that they can represent the new number on their place value mat.

Example

One group randomly draws cards 2 and 6. Students add these two numbers and get 8, then exchange their hundreds flat for tens rods. Students should notice that it is still impossible to remove 8 from a rod. Therefore, they need to exchange one rod for ten units. Students can now remove the 8 units, giving them 92 on their place value mat.

Have students place the cards they have picked up, face down on the table, next to the deck when the group is finished.

Invite students to shuffle the cards and reuse them if none of the groups have reached zero.

Centre 3: Mystery Numbers

Ask students to:

  • choose a mystery number from 100 to 200, then represent it using base ten materials (hundreds flat, tens rods and ones units);
  • write at least three clues on an index card to help the class detectives discover the mystery number.

Example

If a student chooses 142, the clues may be as follows:

  • I have 7 items in my bag.
  • The number is made up of an odd number and two even numbers.
  • The tens digit is higher than the ones digit.
  • The tens digit is 1 more than 3.

Invite students to put the flat, rods, and units representing the mystery number into a paper bag and staple the clues to the bag.

Ask students to exchange the bag with a partner. Each student should try to figure out the other's mystery number before the bag is presented to the class, to see if the clues are clear.

Centre 4: The Number Wheel

Use the template in Appendix 2Rep.5 to prepare spinners on cardboard. Laminate them and add arrows or paper clips to them before putting them in the bin.

Provide each student with a copy of Appendix 2Rep.4.

Have students take turns spinning the arrow on the spinner and, depending on where it stops, they receive a unit cube or rod and place this material on their place value mat.

Invite students to continue spinning the arrow and tell students to add the appropriate base ten material to their place value mat depending on the number the arrow stops on.

Ask students to exchange ones for tens when the time comes and to say the new number each time.

Continue the game until one student reaches 100 and exchanges their 10 tens for a hundred. Continue until the student reaches 200, which means the student can trade 20 tens for two hundreds.

Consolidation of Learning

Duration : approximately 40 minutes

Bring students back together when all groups have completed the four activities.

Ask them to share their discoveries and challenges and suggest tips to make the games easier.

Ask students questions to help them express themselves more accurately about the place value, such as:

  • What did you learn about the usefulness of grouping when it comes to counting?
  • What have you grouped into tens?
  • What have you grouped into hundreds?
  • How did the manipulatives help you?
  • What grouping strategies did you use to facilitate the activity?
  • How did trading ones for tens help you determine the total?
  • How did trading tens for hundreds help you determine the total?

Provide each student with a copy of Appendix 2Rep.6 and invite them to select a few sentences to complete to record their thoughts in their math journal. This can be done individually or in groups of two or four.

Examples of Success Criteria

The student:

  • exchanges ones for tens at the appropriate time;
  • names the new numbers;
  • represents numbers using manipulatives;
  • describes their work in their math journal 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

  • Have the struggling student work with a partner who can help.
  • Accompany the student and ask questions at each stage of the discussion to guide their thinking.

Ask the student to create their own set of place values based on what they have done.

Follow-Up at Home

Let's Play Detective

At home, the student can play detective to discover numbers from given clues.

Play the detective game (Appendix 2Rep.8) a few times to familiarize students with the rules of the game.

Give each student a copy of Appendix 2Rep.8 so that they can play the game with a family member.

Source: Guide d'enseignement efficace des mathématiques de la 1re à la 3e année, p. 163-173.