GEEM - Learning Situation

E1. Geometric and Spatial Reasoning

Describe and represent shape, location, ad movement by applying geometric properties and spatial relationships in order to navigate the world around them.

Learning Situation: Where Am I?

Total Duration: approximately 80 minutes

Summary

In this learning situation, students create their own tracking system and use it to clearly communicate the location of an object on a grid.

Overall Expectation

Specific Expectations

E1. Geometric and Spatial Reasoning

Describe and represent shape, location, and movement by applying geometric properties and spatial relationships in order to navigate the world around them.

E1.2 Plot and read coordinates in the first quadrant of a Cartesian plane, and describe the translations that move a point from one coordinate to another.

E1.3 Describe and perform translations and reflections on a grid, and predict the results of these transformations.

Learning Goals
The purpose of this learning situation is to have students: understand the concept of a tracking system; identify the key elements of a good tracking system; develop their ability to communicate instructions clearly and accurately.

Learning Goals

The purpose of this learning situation is to have students:

understand the concept of a tracking system;
identify the key elements of a good tracking system;
develop their ability to communicate instructions clearly and accurately.

Context/Prerequisites

Since Kindergarten, students have learned to locate objects or people in relation to reference points. Using terms such as in front of, behind, above, to the left, they have located them first in relation to themselves (for example, the desk is in front of me), and then in relation to objects around them (for example, the blackboard is to the left of the door). In Grade 4, students learn to use a coordinate system as a way of locating. In Grade 5, students further use the coordinate system by using variety of scales.

This learning situation does not require any specific mathematical knowledge. It requires students to describe the location of an object on a grid. The activity serves as a preparation for the use of a coordinate system, for example in a coding program. It allows students to identify the essential components of any coordinate system and to appreciate the importance of these systems in their daily lives.

Material

large sheet of paper
square cards (37)
blank sheets or grid paper (two per team)
pencils

Mathematical Vocabulary

reference point, grid, column, row, coordinate system

Before Learning (Warm-Up)

Duration: approximately 15 minutes

Ask students if they have ever had to give or follow instructions to a location and discuss as a class the difficulties they experienced.
Form teams of two students. Tell them that you want to provide instructions to the classroom for people who need to come to school (for example, parents, substitute teacher, community member). Ask the students to help you write the instructions. Stress the importance of providing clear and accurate instructions so that people who have never been to the school before can get to the classroom without difficulty.
When all teams have finished writing the instructions, ask them to exchange them with another team. Ask students to read the instructions they received and write on the sheet if they are clear and precise. If they are not, students should explain why. Each team then goes back to its copy and makes any necessary changes.
Invite a few teams to write their instructions on the board. As a class, discuss the merits of each. Point out that the more specific and precise the reference point, the easier it is to follow the instructions (for example, enter through the main entrance; turn right at the end of the corridor).

Active Learning (Exploration)

Duration: approximately 50 minutes

Note: Prepare a giant-sized hide-and-seek game by drawing a square grid of six rows and six columns (6×6) on a large sheet of paper. Cut out 37 congruent square cards that can be placed in the squares of the grid. On one of the squares, glue a picture on the back. (An alternative is to use playing cards. (An alternative is to use playing cards.)

Place the grid on the floor and arrange the 36 blank cards in the squares of the grid. Have students take a pencil and paper and sit on the floor around the grid with their partner. Tell students:

“I am going to ask two students to leave the classroom. When they leave, I am going to remove one of the cards on the grid and replace it with this card (show the card with the picture on the back). This will be the secret card! When the two students return, they will have to locate the secret card based on the instructions you write.
The challenge for each team is to write clear and precise instructions that will allow both students to locate the secret card on the first try, without any other help. Do not write your names on the sheet. When you are finished, I will bring in one person at a time and hand them one of the sheets. The student will need to successfully locate the secret card using only the instructions on the sheet. It will be important not to talk or react when the student is looking for the secret card, so as not to give them any new clues.”

Ask two students from a team to go into the hallway. Remove one of the cards from the grid and tell the students that this is where the secret card will be placed. Ask them to write down the instructions with their partner to find it.

Students will choose various strategies to indicate the location of the secret card. It is important not to interfere with their choice.

Here are some examples of strategies they could use:

Draw a grid similar to the one on the floor and place an X in the appropriate square.
Indicate the location of the card according to a move on the grid (for example, three squares to the right and four squares up).
Indicate the location of the card according to the position of the rows and columns it is found in (for example, 3rd column, 4th row).
Indicate the location of the secret card according to cardinal points (for example, three squares to the east and four squares to the north).

Note: The challenge for students is to think of a reference point from which to indicate the location of the secret card. Many students will tend to give instructions based on their location around the grid, but will not write it down (for example, “the secret card is in the 3rd column, 4th row”). When the student tries to locate the card, they will not know which side of the grid to be on and may not be successful.

Circulate among the students and observe the clues proposed by the different teams and decide which instructions will be given to the waiting students. In order to stimulate discussion, it is preferable to choose two sets of clues that do not have the same degree of clarity and precision. When the students have finished, collect all the sheets.

Before bringing in the first person, copy the instructions on the first selected sheet of paper on the board for all students to see. Remind students that they should not to talk or react when the student tries to locate the card. Ask the student to enter, read the instructions aloud, and ask that person to follow them to locate the secret card. When the student has turned over the card they believe to be the correct one, write the second set of instructions on the board. Bring in the other student and follow the same procedure.

When both students have completed the activity, discuss with the whole class why the students were successful or unsuccessful in finding the secret card. Ask them questions such as:

Why was [student's name] unable to find the secret card?
What do you think was not clear enough in the instructions?
What would have helped [student's name] find the card the first time?
In order to make the written instructions on the board clearer and more complete, what elements could be added, clarified or modified?
How could you clearly indicate where, in relation to the grid, [student's name] should start? (If students suggest using a student's location as a reference point, ask them if that would work for someone who just came into the classroom and does not know our names).
Are there other reference points in the classroom that can be used?

Emphasize the importance of finding a fixed point of reference (for example, window, door) from which to indicate unequivocally the starting location in relation to the grid. Also emphasize the importance of the person trying to find the secret card reading the instructions carefully and following them exactly.

To ensure that students understand how to give clear and accurate instructions, repeat the activity with two more students. Change the location of the secret card and repeat the activity. Tell them, “This time I'm going to draw two sheets of instructions at random and I'd like them to be clear and accurate enough to allow both students to locate the secret card without difficulty.” Challenge them to think of different ways to give the instructions.

Consolidation of Learning

Duration: approximately 15 minutes

After repeating the activity, lead a math conversation with students who are still around the grid. Ask questions such as:

Do you find that the instructions were clearer on the second try? Why?
What did you add or improve when you wrote your instructions the second time?
In your opinion, what are the elements that make instructions clear?
What did you use as a reference point?
When you read the instructions, what helped you find the secret card? What were you missing?

The purpose of this discussion is to make students aware that when writing instructions for another person to find an object or place, it is important to imagine yourself in that person's location and how they might interpret what is written. Take this further and discuss the advantages of defining a reference point on the grid itself (for example, a ruler placed on one side of the grid). To do this, turn the grid a quarter turn and ask them if it is still possible to find the secret map using the same instructions. This is usually not possible if the instructions are written with a reference point outside of the grid.

Note: This discussion can be used as a starting point to explore the coordinate system on a road map. Point out to students that a road map is gridded, like the grid on the floor. To locate a city on the map, all that is needed is a good indication of the square in which it is located. Discuss with students how the numbers and letters associated with the columns and rows serve as reference points on the map and make it easy for anyone to locate a city. A discussion can also be generated about other location systems that students are familiar with (for example, map at the entrance to a mall, seat numbering system in a room, postal code, pagination of a book).

Differentiated Instruction

To Facilitate the Task	To Enrich the Task
Suggest that students show the location of the secret card using a grid drawing on a grid; Give students partial wording of instructions (for example, “stand facing the windows and look at the grid; starting from the left corner at the bottom of the grid, count…”).	Have students write down the instructions to find the secret card in two different instructions; Have students design a scavenger hunt (for example, instructions lead to a first card on the back of which there are instructions to a second card, and so on, until the secret card is found).

Follow-Up at Home

At home, students can hide an object and write down instructions to find it. They then give these instructions to a family member to find the hidden object. Once the object is found, the roles are reversed. In each case, discuss the clarity and accuracy of the instructions given.

Source: translated from Guide d’enseignement efficace des mathématiques, de la 4^e à la 6^e année, Géométrie et sens de l'espace, Fascicule 2, p. 51-57.