# Patterns and the Coordinate Plane: Ordered Pairs Initials

## Patterns and the Coordinate Plane Played Out

This activity (Ordered Pairs Initials) will provide students with practice in graphing ordered pairs and seeing how patterns in the ordered pairs play out on the graph.  Students are usually pretty good at graphing ordered pairs.  It takes a class period or two to get the X and Y axis straight, but I find that students struggle in seeing how patterns in the ordered pairs appear on the graph and, often, need extra practice with that concept.

To make sense of the concept of the scale factor while providing students with more practice using the axes and coordinate pairs, I lead students in a little mathematical art project in which they draw their initials in block letters on graph paper and use multiplication to enlarge the letters.  This enables students to see how patterns in the ordered pairs are transferred to the visual representation on the graph.

## Set up for the Lesson: Ordered Pairs Initials

• Each student needs their math notebook or a piece of paper
• 2 pieces of graph paper for each student- any size
• One copy of Ordered Pairs Initials worksheet
• Markers, crayons or colored pencils

Here are the links to the Worksheets:

## Launch the Lesson

Introduce the activity in a fun way by telling students that they will be graffiti artists in this lesson. Depending on the size of the space where graffiti artists want to tag their initials, they need to be able to make their letters bigger or smaller.  But how do they make sure their tag looks the same no matter the size they are making it?

Tell the students that you are going to experiment by “tagging” a triangle in two different sizes. Tell them using a graph will help you pay attention to exactly how much the triangle has to grow.

### Have a set of ordered pairs on the board, such as:

(5,5) (5,10) and (15,5)

The students can help you graph the ordered pairs on the board.  I like to use a projected piece of graph paper for this.  You could also use magnetic graph paper.  If you have to hand draw your graph, that can also work, just make sure you have enough space to graph the ordered pairs when doubled. Review the graph vocabulary before you get going.  The x-axis is the horizontal line in the graph  The y-axis is the vertical line in the graph.  The x coordinate is the first number in the pair. And the y coordinate is the second number in the pair.

Call students up to graph each point.  Have them explain how they know where to place the ordered pair. As they label the points, have students talk out what they are doing to the class, saying: 5 is the x coordinate, so I count out 5 spaces to the right on the x axis. 10 is the y coordinate so then I count 10 spaces up the y axis. Draw lines to connect the points.  You should have a scalene triangle.

Ask the students how they can make a triangle with the same proportions, but larger? Have them turn to their neighbor and explain their thinking.  How could they do it? What would you need to do to the ordered pairs?  Are there other ways to achieve this? How would you make it smaller?

Call on students for ideas on how to create a proportionally larger triangle.  They should come up with the idea that you multiply the X and Y coordinate by the same number to make it proportionally larger.  And you divide the X and Y coordinate to make it proportionally smaller.  If students aren’t connecting to the idea of multiplying the coordinate pairs, you could draw the larger triangle, have students jot down the new set of ordered pairs and compare to the first set to determine what changed about the numbers and what operation could be used to represent that change.

Try it by multiplying each of the coordinate pairs by two.  You should end up with (10, 10) (10, 20) and (30,10).  Graph that triangle.  Discuss how the triangles are proportionally the same, but different sizes.

So graffiti artists have to pay attention to proportion! Using coordinate pairs on the graph can help ensure that each point of a shape or a graffiti tag grows proportionally so the bigger version looks just like the original, only larger!

## Lesson Instructions: Ordered Pairs Initials

Hand out graph paper to each student.   They are going to be graffiti artists now. They will tag their initials on the graph paper in block letters.  Have the students place a dot at each of the vertices.  On the Ordered Pairs Initials worksheet, have them record the ordered pairs of the vertices.  For letters that have round parts, like  J, have them create a boxy letter J so it is made with all lines and no curves.

Students should then trade papers with a partner who should double check that their ordered pairs correspond correctly to the vertices of each letter on the graph.

Students should choose a number to multiply by. In order to keep the proportion of the letters the same and maintain the correct form,  students need to multiple all of the ordered pairs by that same number in order to enlarge their letters. Depending on the starting size of the letters, you may want to guide students to choose a number under 5 to ensure it still fits on the graph.  After multiplying each of the ordered pairs by their chosen number, they will find the new values of the ordered pairs and record them in the last column of the Ordered Pairs Initials worksheet.

They should then use these new ordered pairs to create their initials on a new piece of graph paper. If the shape or form of the letters doesn’t match the original, they should work with a partner to check their multiplication across all the pairs.

When all the graphing has been completed and the teacher has checked the students’ work, students can add color to their graffiti projects. Encourage students to be creative and neat and to consider using both sizes of initials to create their own unique graffiti tag.

I encourage you to have the students write a short paragraph explaining how they created their initials.  Have them explain the process of multiplication and how multiplying those numbers makes them all larger in the same proportion, thereby producing an enlarging effect when graphing the new points for the letters.

## Reflecting on the activity: Ordered Pairs Initials

Have the students share their work with others.  They can hang their work on a bulletin board or in the classroom upon completion.  As they reflect on their own work, you can ask the questions:

• Why are the enlarged versions different from each other?
• Why are some enlarged versions larger than others?
• If you multiply the original ordered pairs by the same number, how does that affect the overall size of the new ordered pairs and the height and width of the new initials?
• How does this change if we were working on a 3-dimensional plane?
• What happened to the shape of the letter if one coordinate was multiplied incorrectly?

## Extensions/Next Steps

This activity is a great entry into scale drawing in hobbies, art and architecture.  Models are created by using proportions to either enlarge or reduce the size of an object.  Whether you are into miniature trains, towns or building model airplanes, this applies to the work you do in building, painting and setting up displays. In this case though, you are using division to create miniatures of the real thing.

Talking about models gives you the chance to address what happens when you work on a three dimensional plane.  Science fiction often includes evil villains with the ability to shrink or enlarge objects or people.  Could this really happen? What would need to have to happen on for that to be possible?

You can then ask them to reproduce the initials project, this time dividing the original ordered pairs by a number to see what happens to their initials.

## Resources

https://mathsnacks.com/scale-ella.html

https://wisconsin.pbslearningmedia.org/resource/mket.math.rp.miniland/scale-models-in-the-real-world/

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