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Minds on

MINDS ON

Sharing pictures electronically is something that happens frequently; it may be something that you do. The pictures may be posted on social media sites, viewed in online newspapers/magazines, or copied and pasted into written reports. Sometimes the picture may need to be resized to fit the available space.

What happens to the picture if the original aspect ratio (the ratio of the width to the height of an image or a screen) is not kept when resizing the picture? How can you tell that the picture is distorted?

Maintained or Distorted?

In the following interactive there are a number of versions of the same picture. Some have been distorted while others maintain the original aspect ratio. Drag all the pictures that maintain the same aspect ratio into the box. Aspect ratio is a special term that is used for screens and images that describes the ratio between the width and the height.

aspectRatio

Long Description

 

 

Action.

ACTION

The word ‘similar’ is a commonly used word. How do you use the word 'similar' when you are talking or writing?

This is the Portfolio icon. Similar

Use the title U1A3 Similarity. Jot down a few expressions that include the word ‘similar’ that you may use when talking to your friends.

Save to your Portfolio.

 

Many everyday words are used in mathematics. When used in mathematics, the words may have a very precise definition or they may be used in a very different way than they are used in everyday language. ‘Similar’ or ‘similarity’ are words that have a precise definition in mathematics. 

Exploring Mathematical Similarity -
Relationships within Similar Rectangles

In this investigation, you will explore the precise mathematical meaning of similarity. As you complete the following investigations with similar rectangles, think carefully about the mathematical process that you are using. This will help you to identify and name the processes that are used when working mathematically.

In the following interactive, we are going to investigate the following questions:

  • What is the relationship between the width and the height in similar rectangles?
  • What do you notice about the diagonals in the family of rectangles?
  • How would you make another rectangle that fits in with this group?
  • How can you use a graph to determine another rectangle that fits in with this group?
  • How does aspect ratio describe the relationship between the width and the height in similar rectangles?

withinRect

Long Description

 

This is the Portfolio icon. Relationships Within Similar Rectangles

Download and open the following document: Relationships Within Similar Rectangles.

Print the first page of the document and cut out the rectangles. Your task is to sort the rectangles into two groups, then complete the document.

The interactive above takes you through the various steps that you will need to follow, make sure you review the interactive as you complete your work.

Save your completed document into your Portfolio.

 

This is the Portfolio icon. Word Journal

Return to your expressions for the word similar.

You read earlier that many words have an “everyday meaning” and may have a very specific meaning ‘in math.’

To help keep track of the words that have double meanings, you are going to create a Word Journal, starting with the words Similar and Similarity.

Create a 3 column table with a number of rows.

In the top row, label the first column “Word”, the second column “Everyday meaning” and the third column “In math.”

An image of the table to be created that includes three columns with the heading word, everyday meaning and in math.

In the second row, the first column has the two words - Similar and Similarity.

Complete the second and third columns with your expressions and ideas about the word ‘similar’ using your current understanding of the relationship within similar rectangles.

Consider the angles in the rectangles and the side lengths.

Save Word Journal to your Portfolio.

 

Exploring Mathematical Similarity -
Relationships Between Similar Rectangles

The previous section looked at the relationship within similar rectangles; you saw that the ratio of width to height was maintained as long as the rectangle was not distorted.

Now you will make comparisons between similar rectangles.

You will use the similar rectangles that you used previously.

As you complete the work, continue consider the mathematical processes you are using during each stage of the investigation.

In the following interactive, we are going to investigate the following questions:

  • What is the relationship between the width and the perimeter of similar rectangles?
  • What is the relationship between the ordered pairs on the diagonals of similar rectangles?
  • What is the relationship between the area and the width of similar rectangles?
  • How does the scale factor connect to the graphs for perimeter, diagonal length, and area?

betweenRect

Long Description

 

Scale Factor

The word ‘factor’ is used in many topics in mathematics. When similarity is the topic being discussed, the proper term to describe the change in size of similar figures is scale factor.

In the work that we are completing, the scale factor is either 2 or 3 or 4 or 5. This is the value that each dimension of the figure is multiplied by to create an enlargement that maintains the same proportions and is not distorted. The square of the scale factor is needed to calculate the area of a similar rectangle so the associated values are 4, 9, 16, or 25.

This is the Portfolio icon. Relationship Between Similar Rectangles

Download and open the following document: Relationships Between Similar Rectangles.

Your task is to complete the tables that allow you to compare the rectangles in groups 2 and 3 of our investigation.

The interactive above takes you through the various steps that you will need to follow, make sure you review the interactive as you complete your work.

Save your completed document into your Portfolio.

 

This is the Portfolio icon. Word Journal

Return to your Word Journal and add to your ideas about the word ‘similar’ using your current understanding of the relationship within similar rectangles.

Save again to your Portfolio.

 
Consolidation

CONSOLIDATION

This is the discussion icon. Similarity

Similarity In this activity, you used rectangles to look at the relationships within and between side lengths, perimeter, diagonals and area in similar figures.

You determined that a scale factor is used to create a similar rectangle from an original one. Some relationships in similar rectangles are linear while others are non-linear (e.g. quadratic).

The different relationships were clearly visible on the graphs. The same relationships are true for all types of figures e.g. triangles, pentagons, hexagons etc.

Two figures are similar if their corresponding angles (two angles are “corresponding” if when the figures are positioned in the same manner, the angles are occupying the same relative position) are equal and the scale factor is the same for each set of corresponding side lengths.

Using the page, U1A3 Triangles: Determine which of the 3 triangles belong to the same family.

Support your choice of triangles using a variety of facts to link the triangles to the same family.

Explain why the other triangle(s) are not included in the family.

Check your answers.

 

 

Mathematical Processes

Five of the seven mathematical processes have been highlighted in the work that you have completed.

These five processes are what happens in our heads as we work on solving problems and communicating our understanding or questions that we have about the problem.

An image of the math processes which include problem solving, reflecting, reasoning and proving, connecting, representing, tools and computational strategies and communicating.


In the image above, you can see that Problem Solving and Communicating are at the top and bottom, and that the five other processes are between these.

We use the other five processes to work through problems and then communicate our understanding to our audience.

As we continue on our journey through mathematics, we will be checking in with the processes regularly.

The following interactive will ask you to match a series of statements that describe the various mathematical processes:

mathProc

Long Description

 

This is the Portfolio icon. Mathematical Processes Reflection

Throughout this activity, the process that you might be using at that moment was identified alongside the task.

The processes are interconnected and are used as you develop the knowledge, understand the concepts, and learn the new skills in a math course.

Choose two processes that will be your focus during the rest of this unit.

At the end of each activity, you will return to this reflection and identify where you identified the use of one of the processes.

 
 

Congratulations, you have completed Unit 1, Activity 3. You may move on to Unit 1, Activity 4.

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