# Mapping relationships

Purpose

The purpose of this unit of two lessons is to develop relational thinking through the exploration and expression of the relationship between two related number patterns.

Achievement Objectives
NA2-7: Generalise that whole numbers can be partitioned in many ways.
NA2-8: Find rules for the next member in a sequential pattern.
Specific Learning Outcomes
• Recognise situations in which there is a relationship between two number sets.
• Use a ‘mapping diagram’ to show a number relationship.
• Understand the connection between the coordinate systems of maps and graphs.
• Identify a situation in which there is a unique relationship between two data sets.
• Transfer mapped values onto a graph (an xy coordinate system).
• Explain a simple relationship graph with reference to mapped values.
Description of Mathematics

In mathematics, a function is the relationship between two number sets, often referred to as inputs and outputs. What characterises this relationship is that one ‘input’ is paired with one single ‘output’.

As students begin to explore functional relationships, they are introduced to ways of expressing these, including function diagrams (or machines), tables, graphs, and equations. This relationship can also be expressed as a ‘mapping’ diagram, which provides a visual image of the relationship between two number sets. This ‘thinking tool’, or visual expression of a relationship, is often introduced very early for students to show relationships between important elements in their lives: for example, their relationship to other family members.

Students may also encounter this expression of a relationship in a pictorial mind map of ideas, where the relationship between one idea and another is depicted with an arrow or similar ‘mapped’ connection.

In relationship diagrams such as this:

2→4
3→6
4→8
5→10

one value is matched with, or mapped onto, one other value. In exploring the number relationships in these diagrams, students are developing the understanding of a unique paired relationship between variables.

The mapping diagram is a clear visual image, or representation of the special relationship that exists between the pairs of numbers that the relationship arrow connects. As students look carefully at the whole set of mapped values, they must consider the relationship that is common to all pairs. They look for the ‘rule’ that explains what is happening between the numbers. They see how the variables in one set are related to the variables in the other set.

It is important for students to have opportunities to both interpret and create sets of mapped pairs. When students create their own diagrams for others to interpret, they must generate the values of the mapped pairs, thinking about the relationship that connects the numbers.

This exploration of mapped values is essential to a student’s understanding of coordinate pairs. As a student interprets and represents a situation on a relationship graph, they are applying this understanding. They locate uniquely paired numbers on the (xy) coordinate system to represent the relationship in another way. Students need opportunities to identify the connections between ‘mapped pairs’, ‘mapping’, coordinate systems and graphs that show relationships. These two lessons introduce these concepts.

Early multiplicative (Stage 6)

Activity

Session 1

SLOs:

• Recognise situations in which there is a relationship between two number sets.
• Use a ‘mapping diagram’ to show a number relationship.

Activity 1

1. Begin by showing the students a picture of a tree and a seedling. Explain that the class will be investigating a relationship, over time, between the tree and the seedling.

Explain that the tree was planted on Arbor Day 20 years ago and that the seedling, that is just 1 year old, was planted this year on the same day, 5 June, which is also called World Environment Day.
Write beneath the picture, the headings, Tree and Seedling, the ages of the trees, 20 and 1.
Ask, “How can we show what ages they will be each year as they get older?”
Accept all suggestions but agree that a list could be created to show this. Have students write in the values for several consecutive years, beginning like this:

Ask, “What is happening to the numbers?”
Accept suggestions, but agree that they are both increasing by one year each time and that the seedling is always 19 years younger than (or less than) the tree.
Draw in a mapping arrow, on which is written -19. Explain that we say that we are ‘mapping’ one value onto one other special value, or showing the relationship between the pairs of numbers in each number set.
2. Explore this relationship further by:
a. Writing in some larger values for the tree, such as 50, 63, 78. Have the children discuss and agree with a partner the relative ages of the ‘seedling’ (becoming tree, and share their strategies for calculating these.
b. Writing in some larger values for the seedling such as 79, 85, 101. Discuss.
c. Write in some incorrect values such as Tree 63 Seedling 42; T100 S79; T109 S80.
From the resulting discussion, highlight the fact that the -19 value is known as the common difference and that it is an unchanging amount, so these values are not correctly ‘mapped’.
Emphasise that when we use arrows to map values in this way, we are showing the relationship between the values in two sets of numbers.

Activity 2

1. Make paper and pencils available.
Have students work in pairs to create their own diagram to show the relationship between the height of the trees, similar to the diagram the class has created for the trees’ age. They are to use arrows to map one value onto its paired value.
Pose this scenario: The tree and the seedling grow taller each year by approximately the same amount, half a metre or 50 centimetres. This year the tree was 10 metres tall and the seedling was just half a metre tall.

2. Have student pairs create and share their diagrams, discuss and resolve any differences. Emphasise that the arrow is telling us the relationship between the first set of numbers and the second. It is a rule that tells us how to find the second number if we are given the first number. When we are adding or subtracting, the amount is known as the common difference (in this case - 9.5m).

Activity 3

1. Have the students create a problem for another pair to solve. They should choose their own common difference.
Pose this scenario: As the trees grow taller, they also get wider, both by approximately the same amount.
Discuss whether they would be as wide as they are tall. Agree, no, probably not.
Students should create their own relationship diagram showing at least five number values for both the tree and the seedling, they should not include the common difference in ‘mapping arrow’.
For example: In the first year, the tree is 5 metres wide and the seedling is just 25cm wide.
2. Have them share their diagrams with another pair who is required to insert in pencil the arrows showing the relationship between the pairs of variables. (eg. –4m75cm or +4.75)
Use at least one diagram generated by the students and as a class, together answer the questions: Can there be more than one value for the seedling in any of the paired situations? (Answer: No, the number pairs are unique.)

Activity 4

Make available Attachment 1 for students to complete individually as they complete Activity 3 (above).

Activity 5

Conclude the lesson by reviewing the relationship information data that have been created. Have students explain what they have learned about mapping one value onto another and record their comments on the class chart/modeling book.

Session 2

SLOs:

• Understand the connection between mapping values and the coordinate systems of maps and graphs.
• Identify a situation in which there is a unique relationship between two data sets.
• Transfer mapped values onto a graph (an xy coordinate system).
• Explain a simple relationship graph with reference to mapped values.

Activity 1

Begin by reviewing the record of student learning from Session 1, Activity 4. Revisit the context of mapping the values for the tree onto values for the seedling.

Activity 2

1. Display a diagram with mapped values.
Pose the question: Why do we use the word ‘mapping’ when we talk about the relationship between one value and one other value? Elicit the importance of unique pairs.
Point out that when we use a map we also use unique pairs (the coordinate system).
Display a simple NZ map with grid reference.
Have several students demonstrate how they would describe the location of a specific place (eg. Wellington) on a NZ map.

Discuss the pairs of numbers (coordinates) that allow us to give a precise location.
Coordinates are a set of values that show an exact position. On a map, a pair of numbers show where a specific point is located:
Identify several more locations and record the coordinates.

2. Explain that a graph is also known as a coordinate system.
Display a blank graph grid and mark a point on the grid.

Number the axes and read the location of the data point. (7,4).
Point out that for consistency we give the numbers in a particular order (horizonatal axis first).
We are ’mapping’ paired values.
Use the data from a mapping diagram from Attachment 1. Together have students plot these values onto the graph, carefully discussing the process, including the order in which the number pairs are graphed (horizonatal axis first).

Activity 3

1. Explain to the students that they are going to make a ‘mapping puzzle’ for other class members to solve.
Show the students one of the pictures on Attachment 2.
For example: the frangipani flowers.
Together create a mapping diagram for the petal pattern for one flower, two flowers, three flowers, etc.

Point out that this time the rule is x 5, multiplication, and we call this the common ratio.
(Students do not need to remember this.)
Together create a relationship graph from the data. Highlight the importance of considering how they will number the vertical (y) axis.

Make the connection to ‘mapping’ pairs and coordinates on a map.
(Together look at the line of the data points. Point out that the graph shows that the relationship between the points is the same each time because the relationship rule, x 5, is the same for all pairs of numbers. This concept will be developed in later units of work and in later student modules).

2. Make available to each student pair, one piece of card, (approximately 10cm x 10cm), the pictures in Attachment 2. Photos for Mapping, and one sheet of graph paper.
Student pairs are to select a picture and create a mapping diagram for multiples of the item in picture. For example:

They can see one building with three windows. What if there were two buildings, three buildings, four buildings, etc.?
When their mapping diagram is complete, they should consider how they will number the vertical (y) axis on their graph. Have them complete their graph.
Have 3 separate trays available for students to place each of their items in when they are complete.
Scramble the content of each tray.

3. When all work is complete, randomly distribute a picture, a diagram or a graph to each student in the class. Have one student, who has a picture, hold this up for others to see. The other students must check to see if they have the matching mapping diagram and graph. The two students who have these, must show them and explain the relationship between the representations. The student with the picture confirms if they are correct.
Continue till all pictures have been matched with their mapping diagrams and graphs.
This is an important opportunity for students to articulate their understanding of the lesson content: mapping relationships and transferring data pairs onto a graph.

4. Retain these work samples as a matching task to be independently completed by students in small groups or individually.

Activity 4

Conclude the session by discussing and recording on the class chart what students have learned about mapping values and about graphing these relationships between two data sets.

Attachments