The Area Representation of Pythagorean Theorem

In Grade 8 here in Ontario, Pythagorean Theorem is introduced for the first time. It is pretty common for students to only see a² + b² = c² and they move on. This can be a problem for students since if they only see that formula, they can't get past the a's, b's & c's and often get them mixed up because they don't understand them (so many kids can recite a² + b² = c² proudly but that's where their expertise stops). But if you examine the expectations, you will see that really the focus is on the conceptual nature of the relationship. So we developed this activity to focus on the area representation of PT. The premiss is that students are given six sets of three numbers. The numbers come in the form of the side lengths of squares. Three of the sets are Pythagorean Triples the others are not (students are not told this). They then use the given squares to construct triangles (using the squares as the side lengths) and (hopefully) discover that right angled triangles have a special relationship with the areas of the squares.
A NEW ADDITION is an Explain Everything version. In this version students manipulate the squares right in the app.

• Gr8NS1.4 - determine the Pythagorean relationship, through investigation using a variety of tools (e.g., dynamic geometry software; paper and scissors; geoboard) and strategies; 
• MPM1D, MFM1P - relate the geometric representation of the Pythagorean theorem and the algebraic representation a² + b² = c² ;

• Print (on card stock preferably), laminate (optional) and cut out the squares. Note that the squares should be cut out as tightly to the edge as possible. Each group of 6 should have one set of cards. 
• Each group should have 1-2 pieces of chart paper
• Markers 
• whiteboards (optional)

1. Place students in groups of six (or in any group size that could be split in two).  Three students will construct triangles using squares with the following side lengths:     1) 5, 10, 12     2)  9, 10, 17   3) 12, 13, 15  (this group will create non right angled triangles - don't tell them this). The other three students will construct triangles using squares with side lengths:  4) 5, 12, 13   5) 6, 8, 10      6) 8, 15, 17 (this group will create right angled triangles - don't tell them this). Regardless, each set of students should trace the triangles and the squares that form them on their own chart paper (if they don't trace them, they won't have enough squares).
2. Ask the group of six if they notice anything different between triangles in groups 1, 2, 3 compared to groups 4, 5, 6 (hopefully they they will notice that in one set the triangles are right)
3. Ask students to find the area of each square and see if they can find any relationship in the squares in each of groups 4, 5 & 6 compared to groups 1, 2 & 3 ( you may need to steer some groups towards the sum of the areas with gentile questioning)
4. Discuss, as a group, what they discovered.
5. Give students a whiteboard and ask them to find the missing sides in triangles. A Smartboard file is attached with several more triangles.
6. As an extension students can investigate how general the area relationship is using this WebSketch.

Note: if using the Explain Everything version, all six groups are on the same file. So depending on how many iPads you have, you may group students differently. 

• Square Templates (pdf) (doc)
• Explain Everything (xpl)
• Pythagorean Relationship practice (not) (pdf) 
• Geometer's Sketchpad Area Relationship (WebSketch) (GSP file)

Did you use this activity? Do you have a way to make it better? If so tell us in the comment section. Thanks

Geometer's Sketchpad - Perfect Square Practice

When using the Geometer's Sketchpad it is often better to "start from sketch, not from scratch". That is, give students a premade sketch rather having them build something from nothing (as many textbooks would have you do). One of the advantages of doing this is that the bulk of the time spent on the software is actually doing math rather than building something. 

In this sketch students can practice recognizing perfect squares up to 144. It is a very simple sketch not meant to take much time but to just familiarize students with the first 12 perfect squares as well as to remind them that perfect squares can also be defined by physical squares.

• Gr7NS1.6 - represent perfect squares and square roots, using a variety of tools (e.g., geoboards, connecting cubes, grid paper);
• Gr8 - could be used as review or see our square root guesser sketch instead

• All that is needed is the electronic download (below)
• Note that this really works well on an iPad using the Sketchpad Explorer App (which is free)
• You can also use this on any web based computer (or Chromebook) with this Web sketch

This sketch randomly selects a number under 150 and asks students whether it is a perfect square. They can make a mental guess and check their answer. Or, before the check their answer,  if they want to test it out they can try to create a square that has area equal to the given number. Once done they can generate another random number and try again. The hope is that this will help them become familiar with the first 12 perfect squares. Watch this video to see a demonstration of how it works.

• PerfectSquares.gsp (iPad/V5)
• Web sketch here
• For more sketches like this go to my full page

Did you use this activity? Do you have a way to make it better? If so tell us in the comment section. Thanks

Christmas Review Activity

This is a Christmas review activity where students will answer exponent, powers of 10, square root and Pythagorean Theorem questions and collect presents on the Smartboard. 

Gr8 Number Sense

• express repeated multiplication using exponential notation
• represent whole numbers in expanded form using powers of ten
• multiply and divide decimal numbers by various powers of ten
• estimate, and verify using a calculator, the positive square roots of whole numbers, and distinguish between whole numbers that have whole-number square roots (i.e., perfect square numbers) and those that do not.

Gr8 Geometry and Spatial Sense

• determine the Pythagorean relationship, through investigation using a variety of tools
• solve problems involving right triangles geometrically, using the Pythagorean relationship

• three (or more) Christmas themed containers (find at a Dollar store)
• exponent, powers of 10, square root & Pythagorean Theorem Questions (copy on cardstock, laminate and cut) 
• solution handout
• Smartboard
• Smart Notebook file with score board
• whiteboard and markers (optional)
• Christmas decorations (optional)
• prizes for winning team (optional)

1. Cut out questions and place some in each of the containers.
2. Spread out containers on a table and add some Christmas decorations (optional).
   
3. Bring up the scoreboard on the Smartboard. (Could create your own scoreboard if a Smartboard is not available)
4. Place students is groups and give each student a whiteboard and marker.
5. Have each group choose a Christmas Tree from the scoreboard.
6. One student from the group will come up and choose a question from a container.  
   
   
7. They will bring it back to their group where all members will answer the question.
8. One person will then come and check their answer with the teacher.
9. The teacher will check off that the group has answered that question.  
10. The student will then drag a present or a Misfit toy under their tree on the Smartboard.  Questions with no candy canes are worth 1 present, questions with 2 candy canes are worth 2 presents and 3 candy canes are worth 3 presents.
11. Collect the question cards as students get them right.  When containers are empty, shuffle the cards and redistribute in containers.
12. The group who collects the most presents and/or Misfit toys will win.  

Note:  There are some special cards that students will find. Each group can have a chance to tap their tree on the Smartboard to play Christmas music.

The video, below, is only visible in the WECDSB domain. That is, only teachers in our school board can see the video if they are logged into their MyTools2Go accounts.

• Exponents, Square root, Pythagorean Theorem Christmas questions (pdf) (doc)
• Exponents,Square root, Pythagorean Theorem Christmas solutions (pdf) (doc)
• Christmas scoreboard (Smart Notebook file) (not)

Did you use this activity? Do you have a way to make it better? If so tell us in the comment section. Thanks

The 12 Days of Pascal's Triangle

This is a simple paper & pencil activity that is supposed to be a light exercise to bring in the Christmas season. We got the original idea for this from this site and modified it a bit. It is not particularly taxing but could be used when talking about Pascal's triangle or sequences and series in general (and of course if it's Christmas time). As it is the assignment is fairly simple but it could easily be extended to have students find the general term of both the series and sequences.

• MCR3U -  C1.5 determine, through investigation, recursive patterns in the Fibonacci sequence, in related sequences, and in Pascal’s triangle, and represent the patterns in a variety of ways; C2.2 determine the formula for the general term of an arithmetic sequence or geometric sequence, through investigation using a variety of tools and strategies, and apply the formula to calculate any term in a sequence; C2.2 determine the formula for the sum of an arithmetic or geometric series, through investigation using a variety of tools and strategies, and apply the formula to calculate the sum of a given number of consecutive terms 
• MDM4U - A2.4 make connections, through investigation, between combinations and Pascal’s triangle

• Just the hand out

• Hand out the sheet and let the Pascal/Christmas joy begin

• 12 Days of Pascal (with answers) (doc) (pdf)

Did you use this activity? Do you have a way to make it better? If so tell us in the comment section. Thanks

Geometer's Sketchpad - Square Root Number Line Guesser

When using the Geometer's Sketchpad it is often better to "start from sketch, not from scratch". That is, give students a premade sketch rather having them build something from nothing (as many textbooks would have you do). One of the advantages of doing this is that the bulk of the time spent on the software is actually doing math rather than building something. In this sketch students can practice their knowledge of estimating the square root of numbers up to 500. There are several levels of difficulty: perfect squares up to 100, perfect squares up to 500, square roots up to 100 and square roots up to 500. The intent was that this was built as a practice file for grade 8 students but grade 7 students could use it to practice perfect squares.

• Gr7NS - represent perfect squares and square roots, using a variety of tools
• Gr8NS - estimate, and verify using a calculator, the positive square roots of whole numbers, and distinguish between whole numbers that have whole-number square roots (i.e., perfect square numbers) and those that do not

• All that is needed is the electronic download (below)
• Note that this really works well on an iPad using the Sketchpad Explorer App (which is free)
• You can also use this on any web based computer (or Chromebook) with this Web sketch

Watch the video below to see how the sketch works

• SquareRootNumberlineGuesser.gsp (iPad/V5)
• Web sketch here
• For more sketches like this go to my full page

Did you use this activity? Do you have a way to make it better? If so tell us in the comment section. Thanks