# Is it the same or different? #WithMathICan

#### 1 Hook

TEACHER INSTRUCTIONS:

A. Show the students a picture of old shreddies (square) and new shreddies (diamond) and ask them to decide if they think they are the same or different. See attached photo. https://meanwhile.files.wordpress.com/2007/07/shreddies-ooh-02.jpg

Have a debate and give points for each correct statement. Take up their responses and discuss the following as they arise:

- angles
- sides
- shape
- size
- rotations

B. Relevance: present today’s overlying problem: *Our school is currently undergoing renovations. Who do you think cares if the rooms are the same or different? Why would they care? * (This could also be posed as a question in Google classroom to record responses). Possible responses include:

- architects for symmetry and design
- construction workers for material orders and ‘easy’ math (if they are all the same)
- school board for cost of materials
- teachers to ensure fairness, will fit all students
- teachers for furniture design - will rectangular furniture fit in a circular room, if they switch rooms will all of their things fit

Explain: Mathematicians use special terms to explain when something is the same or different and we are going to learn them today. We will also discover how mathematicians check to see if two shapes that look the same are in fact identical.

A. Go to the left side of the room if you think the two objects in the picture are the same or the right side of the room if you think they are different.

**What is the same? What is different? How do you know? How could you find out?**

After debate, back to your seats.

B. Relevance: Our school is currently undergoing renovations. Who do you think cares if the rooms are the same or different? Why would they care? Pair - share with a partner. Come together and discuss as a class.

#### 2 Access Prior Knowledge

TEACHER INSTRUCTIONS:

Explain to students that the lesson will be sent out as we go using Google classroom. Post the first question (see Q1 in student instructions). Students should choose one statement to explain in Explain Everything. By offering choice, each student should meet with success, while also feeling challenged. By having more than one option available, students that finish early can try the other questions while they wait. As the students are working, keep projecting different responses using airplay and using it to start discussions. When they are done, have them summarize their observations by posting their comments in Google classroom.

By including correct and incorrect responses, you will increase their understanding. Try to comment on geometry skills as well such as hatch marks, angles, use of a ruler etc.

Guiding questions:

- is this a ...?
- how could you check?
- is there an example when this is not true?
- are they the same? why? what attributes make them the same?
- are they different? why? what attributes make them different?

Q1: Pick one of the following statements and use explain everything to decide whether these statements are a) always true, b) sometimes true or c) never true. You must include a drawing to explain your answer and an explanation (oral or written) using the app. Post your final answers to Q1 in Google classroom.

**What is the same? What is different? How do you know? How could you find out?**

- A diamond is a square.
- A rhombus is a square
- A parallelogram is a rectangle
- A triangle is a parallelogram
- A square is a parallelogram
- A trapezoid is a quadrilateral
- Create your own example!

Still have time? Choose a second one to explain that is more difficult for you!

#### 3 Explore Congruent Polygons

TEACHER INSTRUCTIONS

Post question number two in Google classroom (see Q2 in student instructions). This question asks them to draw a triangle with specific attributes. Briefly explain the main features of geometry pad, but otherwise let your students struggle a bit and collaborate with each other in order to create a triangle using your specifications. Can they figure out how to label it? Can they name their vertices? Can they find the measure of all the sides and angles?

You will need the paid version of geometry pad if you want to cut and paste it into another app such as Explain Everything. Otherwise the free version will suffice. Once again, project students' images as they work (using airplay), and start conversations surrounding the theme: are they the same? are they different?

Focussing questions:

- are they identical? equal? exact? similar?
- why? why not?
- how could you check? (measure, print it out, trace it, transform it)
- what attributes are the same? what are different?
- how could you show that? (colour coding, labeling)
- discuss side length, angles, perimeter, area, shape, size

Once students have shared with a partner, have them answer the question in Google classroom by posting their comments. Students that are still struggling with the concept of congruence can refer to the comments or the virtual nerd tutorial to get more guidance.

Explain: Mathematicians use the term congruent to describe two shapes that are identical.

- We write: ABC ⩭ FGH
- We use hatch marks to show equal sides
- We use symbols to show equal angles

Relevance: re-visit construction problem. *The rooms are congruent. How does that help with the construction? *Possible answers include:

- multiply by # of rooms to get materials / resources
- now know same perimeter, area, shape, size - help with design, materials etc.

Q2: "Use geometry pad to create a triangle with side AB=5 units, angle A = 90 degrees and side AC = 4 units."

When you're done, compare with a friend who is also done. **What is the same? What is different? How do you know? How could you find out? **Post your answers to Q2 in Google classroom.

Still stuck? See this tutorial on virtual nerd for more info!

http://www.virtualnerd.com/middle-math/geometric-figures/congruent-simil...

Extension:

- Can you draw a congruent triangle that is rotated 45 degrees? reflected? translated? How do you know it is still congruent? Prove it!

#### 4 Explore Similar Polygons

TEACHER INSTRUCTIONS

Repeat the process but with question 3 (see Q3 in student instructions), which asks them to draw a similar triangle. Share students’ work as you go by projecting it using airplay and creating discussion. Another virtual nerd video is also posted for students that need more assistance.

Focussing questions:

- what do you need to do to do this? (multiply? add?)
- what will be the same?
- what will be different?
- consider angles, shape, area, perimeters, sides, size

Students should post their discoveries in Google classroom by commenting on the question. Others can add to previous comments as they go. Encourage everyone to participate.

Summarize the following after most students have had the opportunity to discover it for themselves: Mathematicians call polygons similar if they have the same shape, number of sides, angles, but different perimeter, area and size. How could we label this?

Relevance: Re-visit construction problem. *Some rooms are similar. Why is this useful to know?*

- can draw a map to scale, down-size, up-size, multiply or divide measurements of one room to get other, know shape is still the same but size will be different, still same angles for doors, windows etc.

Q3: Can you now draw a triangle that is similar but three times bigger in size?

Compare this with your first triangle. **What is the same? What is different? How do you know? How could you find out? **Post your answers to Q3 in Google classroom.

Still stuck? See this tutorial on virtual nerd for more info!

http://www.virtualnerd.com/middle-math/geometric-figures/congruent-simil...

Extension:

- Can you draw a similar triangle to your first one that is one third of the size? How? Prove it!
- How does the idea of similar relate to our study of ratios? Show me! Post your comments to Google classroom!

#### 5 Independent Practice

Now it is their turn to reflect on their learning and apply it to another shape. It is also your turn to gain feedback about the lesson as well as collect some formative assessment. Post the assignment to Google classroom (see Assignment in student instructions). The assignment asks them to create their polygons using geometry pad. They will then import their drawing into Explain Everything and use the Explain Everything app to answer the questions. If you do not have the paid version of geometry pad, you can have the students complete the assignment using grid paper instead.

Choice is built into the question in that they can choose what polygon to draw based on their comfort and ability. Same rules apply: feel free to airplay students work as they go and use it to provoke conversation. Encourage use of all the mathematical terms used during the class as well as proper labeling of their diagrams.

Have fun!

Assignment: Use geometry pad to draw and then explain everything to answer the following. Please insert your geometry pad drawings into your Explain Everything presentation.

Pick a shape other than a triangle to illustrate your responses. Lots of time? Consider a decagon! Not so much time? Perhaps a pentagon is best. Try to challenge yourself!

- How do you know if two polygons are congruent? Illustrate with an example of congruent and not congruent.
- How do you know if two polygons are similar? Illustrate with an example of similar and not similar.

Reflection: describe a real-life situation when it would be helpful to know if polygons are similar or congruent.

All done? Try this puzzle!