Geometer's Sketchpad - Practice the Pythagorean Theorem

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). 
In this activity students can download a GSP sketch that allows them to practice determining the hypotenuse (on the first page) or a leg (on the second page). The sketch will generate an infinite number of questions and give a full solution for each.
Note: Although Pythagorean Theorem is introduced in grade 8, it is only supposed to be relating more to the area model so these practice problems may not fit that. 

• MPM1D, MFM1P: D, C2.2 - solve problems using the Pythagorean theorem, as required in applications
• MPM2D, MFM2P: C,A2.2 - determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios and the Pythagorean theorem;  

• All that is needed are the electronic downloads (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 to use the sketch. The first page can be used for determining the hypotenuse and the second page can be used for determining a leg. Both will randomly generate an infinite number of problems.

• Pythagoras_Generator.gsp 
• 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

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

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

Geometer's Sketchpad - Practice Distance Between Points

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). In this activity students can download a GSP sketch that allows them to practice determining the distance between two points (this part could also be used to check answers) and then to be quizzed with randomly generated sets of points.

• MPM2D - develop the formula for the length of a line segment, and use this formula to solve problems (e.g., determine the lengths of the line segments joining the midpoints of the sides of a triangle, given the coordinates of the vertices of the triangle, and verify using dynamic geometry software);

• All that is needed are the electronic downloads
• 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 to use the sketch. The first page can be used for discovery or for checking problems and the second page can be used for quizzing students as it will generate an infinite number of random points to find the distance between.

• DistancePractice.gsp (iPad/V5) (V4)
• 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