Measurement and Geometry
Measurement and Geometry are related concepts that fall under what previous curricula called Shape & Space. Throughout K-7, the big ideas all share the foundational concept of the ability to describe, measure, and compare spatial relationships. This key concept is a critical part of numeracy as our learners develop spatial sense.
In Primary grades students identify, describe, build, and sort 2-D shapes and 3-D objects by exploring attributes and recognizing similarities and differences. As they go through the Intermediate grades students learn to classify shapes by their attributes, including learning vocabulary relevant to each type of shape or object. Our visible world is full of shapes and objects that our learners experience every day.
Many of these geometrical concepts then connect to number concepts through exploring measurement. Over K-7 students measure and compare length, area, volume, capacity, mass, time, and angles. Students begin developing the concepts by measuring common attributes through comparison. They then learn to appreciate the value of direct measurement, at first using non-standard units and then standard metric units. Indirect measurements are figured out by using direct measurements, for example, using dimensions to determine an area.
Beginning in Grade 4 with symmetry, students also develop spatial sense with transformations. In Grades 5-7 students identify and construct transformations using slides (translations), flips (reflections), and turns (rotations).
As students explore measurement and geometry, there are many opportunities to connect to students’ lives, community, culture, and place. With these experiences we are honouring the following First Peoples Principle of Learning: Learning is holistic, reflexive, reflective, experiential, and relational (focused on connectedness, on reciprocal relationships, and a sense of place).
As we learn about key concepts in measurement and geometry, we will also be developing many curricular competencies. Three that we have chosen to focus on in our designing of lesson ideas are:
- Estimate reasonably
- Visualize to explore mathematical concepts
- Use mathematical vocabulary and language to contribute to mathematical discussions
Although these three curricular competencies have been highlighted, there will be many opportunities to develop many curricular competencies during the investigation of measurement and geometry.
Learning Story for Grade 7
Measurement and Geometry
Prior to Grade 7, students have explored a variety of attributes and measurements of 2-D shapes and 3-D objects. In Grade 7, they build on their understanding by learning about the geometry of circles.
Students extend their understanding of how mathematicians use formulas as shortcuts to finding area, as they explore finding the area and circumference of circles. They also learn that multiplying the area of the base of a cylinder or rectangular prism by the height will give the volume of the object. Although students at this age should develop competency with formulas, it is an understanding of what the parts of the formulas mean and how they work that should be the basis for understanding.
Students learned about transforming shapes (slides, rotations, reflections) in Grade 5, and in Grade 6 they combine these transformations and situate them on a coordinate grid. In Grade 7, students perform and describe sequences of transformations in all 4 quadrants of the Cartesian Plane.
Grade 7 is an important year for solidifying concepts and connections introduced in the earlier grades, as they will need to have a strong understanding of them to support their learning about surface area and the Pythagorean Theorem in Grade 8. In grade 8, students will also expand their ability to construct objects and their understanding of nets using 3D figures.
Key Concepts
Circumference and Areas of Circles
Students learn to construct circles given the radius, diameter, area, or circumference and learn about the relationship of these measurements and the formulas for finding area and perimeter of a circle.
Volume of rectangular prisms and cylinders
Students add to their understanding of formulas for area of shapes by multiplying area by height to find the volume.
Cartesian coordinates and combinations of transformations
Students perform and describe sequences of translations, rotations and reflections in all 4 quadrants of the Cartesian Plane.
Key Measurement and Geometry Concept 1: Circumference and Areas of Circles
Overview
Students have explored finding area and perimeter of various shapes in previous grades. In grade 7, they are introduced to Pi and its relationship to finding the circumference and area of a circle. They explore creating circles given measurements of the diameter, radius, circumference or area and they learn to use standard formulas for finding the volume of cylinders and rectangular prisms.
At this stage of learning, students can be expected to be familiar with the use of formulas as shortcuts to finding area, perimeter/circumference and volume of shapes. The use of formulas should stem from a strong conceptual understanding of how formulas work and what each variable and constant within the formula means. This understanding is achieved by looking at the relationships between Pi, diameter, radius, area and circumference to develop the formula C = π x d
For example, instead of first memorising the formula, students can be taught to use a compass for drawing circles either in math or through an ADST or art project such as Islamic geometric design. The radius of the circle must be used to set a compass to the correct width in order to create the required circumference, so the relationship between circumference and radius can be experienced without anything more complicated than measuring with a ruler and a string. Students often hold their compasses incorrectly, leading to frustration. For a short tutorial on how to properly use a compass, see this youtube video. This activity is elaborated on in the week at a glance.
Once students understand the relationships for finding perimeter, they can then be taught to apply the formulas C= π x d and A=π x r^{2} to find the circumference or area given the radius or diameter of a circle.
Measurement & Geometry Foundations:
The following concepts and competencies are foundational in understanding circumference and areas of circles in grade 7:
- Basic properties of shapes
- Linear measurement
- Estimation strategies
- Understanding of perimeter and area
- Understanding of what a formula is and its parts (constants, variables, etc.)
- Multiplying and dividing a decimal number by another decimal number
Progression:
- Understands the meaning of radius, diameter, circumference and area
- Finds the radius and diameter of a circle
- Constructs circles with a given radius or diameter
- Finds relationships between radius, diameter, circumference, and area to develop C = π x d formula (Circumference = 3 x the diameter and a bit more)
- Constructs circles with a given area or circumference
- Applies formulas (A = π x r x r to find the area) given radius, diameter, area or circumference
Sample Week at a Glance:
This is intended as the introductory week to learning about circles in math. Monday’s lesson could be the basis for a deeper study into Islamic Geometry, which also connects to the Transformations week at a glance (WAAG). These lessons, therefore, could be interleaved with lessons from the WAAG in the transformations section or from other math connections to a class theme or project.
Focus: Explore the parts of a circle and how to construct them.
Part 1:
- Have students experiment for 3 minutes drawing circles with their compass
- Show this video: How to Use A Compass
Part 2:
- Give students 3 minutes to practise drawing circles.
- Show this IC Kids video from time (in minutes) 10:10-11:45 (1 min 35 sec)
- Give students some time to try to draw the pattern (you may need to replay the video or let students replay it on their devices.
- Start to add some math constraints to the circle drawing (define new vocabulary as needed):
- Draw a circle with a radius of 3 cm
- Draw a circle with a diameter of 6 cm
- Draw a circle with a perimeter of approximately 19 cm (provide students with string for measuring, but not the formula)
- Note students who are making connections to the relationships between radius, diameter and circumference or those that talk about Pi. You will call on these students in the consolidation discussion.
Part 3:
- Consolidation of learning discussion questions:
- What did you notice about the circles that had a diameter of 6 cm and a radius of 3 cm? They are similar.
- What does that tell us about the relationship? d=2r
- What strategies did you use to draw the last circle? Guess and check, measuring the previous circles and estimating, multiplying by 3.14
- Why would knowing the formula for circumference be helpful? It is more efficient and precise than measuring with string.
- What did you notice about the diameter of the last circle? It’s similar to the others.
- What does that tell us about the relationships of these measurements? They make up the formula, C = about 3d
Focus: Finding the circumference of a circle
Part 1 (whole class):
- Watch the first 8 minutes of A Brief History of Pi by Simon Clark or have students read this short Exploratorium article.
Part 2 (Partners): Original lesson plan and extensions
- Each group will need a ruler, string and several circular objects, such as cups, lids, bowls, etc.
- Students use the string to measure and record the diameter and circumference of each object:
Object Description | Diameter | Circumference | Circumference ——————– Diameter |
- While students are working, circulate to nudge thinking and monitor for students that struggle.
- Watch for students who are having difficulty measuring accurately, who have outliers or who are having difficulty with the calculations.
- Prompt early finishers to graph their data to look for relationships
Part 3 (Whole Class):
- Consolidation discussion:
- What did you notice about the relationship between circumference and diameter? It is always about the same, it is close to 3 times diameter = circumference, Pi
- Choose students that you observed using good strategies as they worked or who made mistakes and discovered how to fix them to build towards an understanding that circumference is 3.14 or 3 and a bit more of the diameter.
- What does this tell us about the formula for circumference? C = D x 3.14, you have to multiply the diameter by Pi.
- What did you notice about the relationship between circumference and diameter? It is always about the same, it is close to 3 times diameter = circumference, Pi
Focus: Finding the area of a circle
Part 1(groups of 3):
Open Middle Question: Which circle is bigger: one with an area of 30 square units or one with a circumference of 30 units? How do you know?
- Give students 3-5 minutes of think/drawing time (provide both blank and grid paper) before discussing this question as a class
- Sample answers are included on the linked page
Part 2 (Partners): Youcubed activity: Finding Pi
- Circulate while students are working to support and nudge thinking.
- Note students that have ideas useful for the consolidation discussion in part 3
Part 3 (Class Discussion + Individual Journal):
- Consolidation discussion:
- How can we find the area of a circle? A= π x r^{2}
- What relationships are described in the formula A=π x r^{2}? Area is a little more than 3 times the radius times radius
- Math Journal:
- Use pictures, numbers and words to explain how Pi is related to the area of a circle?
Focus: Apply strategies for finding circumference
Part 1(Whole class): Math Talk – nrich: Circumferences and Diameter
- Project the prompt image for the class onto a white board.
- Depending on the class familiarity with line graphs, students may need talk time in small groups before the whole class discussion.
- Record students’ thinking on the white board.
Part 2 (Groups of 3): Illustrated Math Task
Part 3 (Whole class):
- Ask:
- Why might there be differences in the 2 tables? Measuring w string is difficult, measuring tools only allow a certain level of accuracy, drawn circles are not perfect.
- What does this activity tell us about working with formulas in real life? There is a margin of error.
Focus: Apply formulas for area and circumference
Part 1 (Whole Class):
- Project the image onto a white board (use the student view button in the link to the task.
- Ask: What do you notice? Units, groups of shapes, different colours, a set of 6, each shape has curved and straight lines…..
- Ask: I wonder how we might find the perimeter of the first shape? Measure the outside with string, estimate the curved sides, use formulas for area of a circle and area of a square…
- Ask: How would you start? Ask 2-3 students to explain different ways of starting the task: I would cut a long piece of string and get my group member to hold one end…; I would look at how many full squares there were and calculate the area and then calculate how many circles or parts of circles were included…
Part 2 (Groups of 3):
- Have each group stand with a non-permanent surface they can write on (whiteboard, window, Wipebook, etc.), as in Peter Liljedahl’s Thinking Classrooms
- Provide each group with a copy of the first (purple) image only.
- As groups solve, circulate to nudge thinking and collect information for the consolidation discussion.
- Provide subsequent images one at a time as groups finish each question. Order: purple, blue, orange, red, green, yellow
- Not all groups need to answer every option.
- Give hints or ask questions if students are stuck, but allow for productive struggle to occur, as long as it does not lead to frustration.
Part 3 (Whole class):
- Consolidation discussion: Start with the purple shape.
- Call students together and guide them through a discussion based on the examples you collected during the activity. Choose students who used different strategies.
- Ask: What challenges, if any, did you encounter?
- Ask: What might you do differently next time?
- It is not necessary to debrief all the shapes if there is not enough time.
- Focus on helping students realise that using formulas is more efficient, but that there may be many strategies that get you the right answer and that the most efficient strategy that they understand how to use is best for them.
Students will require practice measuring circles in a variety of problem solving contexts. Teachers can include explorations of circles and their properties through cross curricular projects throughout the year, building connections between math, ADST, art and science. Cross strand tasks in math are also useful for providing ongoing practice (spiralling) once the initial learning is complete.
Suggestions for Assessment
Success Criteria:
- Applies the formula for circumference correctly
- Applies the formula for area of a circle correctly
- Selects an appropriate strategy for solving problems involving area and circumference of circles.
Students can be proficient without having to memorise the formulas. They can also use calculators to solve the equations, if it is application of the formula that is being assessed. Calculators should only be withheld if the teacher is specifically assessing the student’s ability to do computation, which is not the intention of this unit. Of course, it is possible to assess both in these lessons, but teachers must be clear with students about what is being assessed and share the criteria in advance.
Suggested Links and Resources
Some links can be embedded above, but include others here
● Samira Mian Islamic Geometry Website
● Make Math Moments Circle Lessons
- How To Use A Compass(Article)
- How To Use A Compass (Video)
- A Brief History of Pi (Video)
- A Brief History of Pi (Article)
Key Measurement and Geometry Concept 2: Volume of rectangular prisms and cylinders
Overview
Students first learn about volume in kindergarten through comparative measurement (What holds more?). In grade 3, students learn about standard units for measuring capacity (litre, millilitre) and they estimate and calculate capacity based on referents (If this cup holds 250mL, how much is in this pitcher?). At this stage, the focus is on understanding the concept of capacity, rather than on the use of formulas. In grade 6, students build objects concretely with cubes and calculate capacity and volume with the focus on the relationship between units. Finally, in grade 7, students learn the formula for volume by connecting it to their knowledge of area (V = area of base x height).
This Photo by Unknown Author is licensed under CC BY
Area is also a concept that students have been building throughout the elementary years, beginning with the introduction of the concept in grade 3. By grade 7, they can calculate the area of rectangles and have explored the relationships between area and perimeter, and this year they learn about the relationship between area and volume. In grade 7, students learn to use the formula for calculating the area of circles, which provides them the foundation for using the formula for calculating area of a cylinder.
Grade 7 is a year where the trajectory of concrete, pictorial, symbolic connections are solidified, so that students can access the efficiency of the symbolic formulas, while retaining their ability to visualise and represent mathematics in many ways. Although the ability to apply formulas in grade 7 is expected, students should not be rushed to this process by sacrificing sense-making. Fostering understanding of the meaning of the formulas they are using should always be the primary goal.
Measurement and Geometry Foundations:
Foundational, supporting concepts and related competencies that are needed to develop this grade level concept:
- Standard units for measuring capacity/volume
- Construction of 3D objects (base, sides, nets, etc)
- Area of a square/rectangle
- Understanding of capacity and it’s connection to volume
- Area of a circle
- Understanding of components of a formula (variables, constants, etc.)
Progression:
How this concept develops within the grade – where does it start? What are the learning stages?
- Estimates and finds volume concretely (using cubes, water, referents etc)
- Calculates area of the base of a rectangular prism (rectangle)
- Calculates area of the base of a cylinder (circle)
- Understands the relationship between area and volume
- Uses V = Area of base x height to calculate volume of rectangular prisms and cylinders.
Sample Week at a Glance
Since the learning in this week requires the ability to calculate the area of the base of a cylinder or prism, as part of the volume formula, this unit is best interleaved with, or introduced after, the grade 7 concepts in circle geometry.
Focus: Exploring volume of rectangular prisms.
Part 1(Whole class): Routine – Cube Conversations (Set 50)
Part 2 (Groups of 3): Tower Volumes
- Have groups of students build 5-6 rectangular prisms, with different dimensions, out of cubes and complete the table.
- In their groups have students consider how their models might help them determine the formula for volume of a rectangular prism.
- If students are struggling, ask them to relate to what they know about finding area. How is volume the same/different?
- Circulate as students work to nudge thinking and collect information for the consolidation discussion.
Volume | Base | Height |
Part 3: Consolidation of Learning
- Sequence the ideas that you heard students articulate during part 2, so that the sharing builds towards an understanding of the formula A = base area x height.
- Have students share what they discovered. Encourage them to use examples as they explain their thinking.
Focus: Exploring volume of cylinders
Part 1 routine: Same, but Different
r = 4cm; h = 10cm
Possible responses:
- Same: Their height and width, both 3D, both have a base, sides and a top
- Different: round vs straight sides, circular vs square/rectangular base
Part 2: Explore
Note: Students will need 3-5 cylinders per group of 3. These can be geometric solids or other cylindrical objects.
Instructions:
- Find the height and diameter of each cylinder.
- Build a rectangular prism that has the same height and width
- Calculate the volume of each prism?
- How might you calculate the volume of the related cylinder?
If students struggle with making connections, pause the activity and have this conversation as a group/
- Estimate and then calculate the volume of the cylinders? How do they relate to the prisms with the same dimensions?
Part 3: Consolidation
- Class Discussion: How is figuring out the volume of a cylinder like figuring out the volume of a prism? Find the area of the base and then multiply by the height
- Math Journal: Using pictures, numbers and words, explain how to find the volumes of rectangular prisms and cylinders.
Focus: Apply formula for area of a rectangular prism to solving problems
Part 1(Whole class): Routine – Cube Conversations (Set 54)
Part 2: 3 Act task – Soup Du Jour
Instructions are provided on the webpage
Part 3: Consolidation
Math Journal: Using pictures, numbers and words, show how you solved the Soup De Jour problem. Be sure to explain why you made the choices you did.
Focus: Apply formula for volume of a cylinder to solving problems
Part 1(Whole class math talk): (adapted from nrich)
A cylindrical aluminium can contains 330 ml of cola.
.
If the can’s diameter is 6 cm what is the can’s height?
If the can’s height was 10 cm what would the can’s diameter have to be?
- Project this problem onto a whiteboard
- Present the first question. Focus on recording student strategies for how they know what the height is, rather than just on the answer to the question.
Part 2: (Source)
- For grade 7s the focus for this task should be on the area of the base and the height of the square used for calculating the volume, rather than on calculating the entire surface area. Specific surface area to volume relationships are explored in later grades.
- Have students calculate the volume of several hypothetical cylinders to determine the largest volume that can be created from one piece of paper.
Part 3: (Whole class consolidation discussion)
- Have students consolidate their data for base area, height and volume of the cylinders they made into a chart on the board.
- Ask:
- What do you notice about the relationship between these measurements? Wider base makes a bigger difference than height, wider containers have more volume than tall and narrow containers, the length of the square has to be a bit more than 3 times the diameter of the circle you draw
- Can we determine volume if we only have one of these measurements (height or base)?
- Can we determine height and base area if we know the volume? Only if we have volume and one of the dimensions.
Focus: finding dimensions given the volume
Part 1: Cube Conversations (Set )
Part 2: Fish Tanks task from the Math For Love volume menu (scroll down page)
- Give students these constraints for the task with the fish tanks worksheet:
- At least one of your tanks must be a rectangular prism.
- At least one of your tanks must be a cylinder.
Part 3: Consolidation Routine – Always, Sometimes, Never (Can be played as a quiz-show format)
Possible statements:
- A cylinder has less volume than a rectangular prism of the same height. sometimes: depends on the width
- We need to know the height of an object to calculate it’s volume. always
- A rectangular prism with the dimensions 10cm x 3cm x 12cm will enclose a cylinder with the dimensions D = 10cm, H = 12cm. Never: the prism needs to be wider.
- The base of a cylinder is a circle. always
- A rectangular prism has less volume than a cylinder of the same height and width. Sometimes: if the length of the prism is shorter than the diameter of the cylinder.
Students may require further practice with exploring volume and applying the formulas in a variety of contexts. Teachers can keep this learning fresh throughout the year by looking for opportunities to incorporate volume explorations and calculations into other curricular areas, such as ADST, art and science. Cross strand rich math tasks that combine the skills for calculating volume with other mathematical concepts to solve problems are particularly good for building connections between strands and facilitating the ability for students to apply their understandings and skills.
Suggestions for Assessment
Success Criteria:
- Reasonably estimates the volume of rectangular prisms and cylinders
- Describes the relationship between volume and area
- Applies the formula V=L x W x H to find the volume of a rectangular prism to solve problems
- Applies the formula V=π x r^{2}x H to find the volume of a cylinder to solve problems
- Justifies the reasonableness of measurements and calculations
Students can be proficient without having to memorise the formulas, as long as they understand how to apply it. They can also use calculators to solve the equations if it is application of the formula that is being assessed. Calculators should only be withheld if the teacher is specifically assessing the student’s ability to do computation, which is not the intention of this unit. Of course, it is possible to assess both in these lessons, but teachers must be clear with students about what is being assessed and share the criteria in advance.
Suggested Links and Resources
Key Measurement and Geometry Concept 3: Cartesian coordinates and combinations of transformations
Overview
Students have been learning about transformations since grade 5. By Grade 7 they know how to perform and describe translations(slide), rotations(turn) and reflections(flip) on a coordinate grid. In grade 7, students learn to perform and identify sequences of transformations in all four quadrants of the Cartesian Plane.
Introducing the Cartesian Plane at this time may or may not be necessary, depending on whether this topic is being taught before or after linear relations and graphing are introduced, as these topics also require students to work in all 4 quadrants for the first time in grade 7.
Students in grade 7 perform only sequences of rigid transformations, where the shapes remain congruent. It is important that they learn to do this accurately, by understanding how the coordinates work to position objects on the grid. This ability is the basis for further learning in high school about the movement and manipulation of shapes, such as with non-rigid transformations (eg: dilations). Working with concrete shapes and cut-outs can support this important development of the visualisation abilities important to geometry and to mathematics in general.
Measurement and Geometry Foundations:
Foundational, supporting concepts and related competencies that are needed to develop this grade level concept:
- Spatial awareness (visualisation)
- Graphing coordinates on a grid
- How to describe a translation, rotation and reflection
- Performing transformations in sequence
Progression:
- Describes single rigid transformations on a coordinate grid
- Plot coordinates on all 4 quadrants of the Cartesian Plane
- Perform single transformations in all 4 quadrants
- Describe a series of transformations in all 4 quadrants
- Perform a given series of transformations on the Cartesian Plane to solve problems and create specific designs
Sample Week at a Glance
While these lessons work as a series for a complete week, there is intersection in the learning with many other areas of Math, including linear relationships (patterns) and graphing (data and probability). Therefore, those strands can be interleaved (spiralled) if desired or Monday’s lesson may be used as a lesson in a larger lesson sequence that includes both line graphing and transformations. Likewise, the Tessellations activity can be integrated with Art and Social Studies to create a cross curricular study of Islamic geometry. This unit would also include the lesson about Islamic Geometry in the week at a glance for circles.
Focus: Four quadrants of Cartesian plane
Math = Love – Where’s Polygon math routine(Instructions)
- Display a blank coordinate plane for students to see.
- Have the coordinates of a specific polygon drawn on another, secret, paper
- Have students offer coordinate pairs
- Colour code their guesses:
- Red = outside the shape
- Yellow = inside the shape
- Blue = the perimeter of the shape
- Green = vertices of the shape
- Continue until all 4 vertices are found.
Part 1 (Whole class): Play Where’s Polygon as a class.
- Begin by drawing only in quadrant 1. Students are familiar with how to plot on a grid from previous grade
- Teacher draws the hidden polygon
- Students collectively give coordinates to try
- Once they understand the process on the grid, then introduce the 4 quadrant Cartesian plane and use the whole plane when choosing where to place the shape.
Part 2 (Groups of 3):
- Students take turns being the person that draws the hidden polygon.
- Circulate during this time to support students who are struggling and collect data for which students might have something helpful to contribute to the consolidation of learning discussion.
- Note students using strategies to make finding the polygon easier.
- Note students who make common mistakes, such as reversing the order of the coordinates, and then fix them. They can explain what they learned in consolidation as well.
Part 3 (Whole class): Consolidation of learning discussion
- Sequence the students who are sharing ideas to build understanding of how shapes are described on the Cartesian plane.
- Ask: What is it important to know when plotting coordinates on the Cartesian plane? Plot x first, the origin is (0,0), once you have each vertex, you can draw the shape, etc…
Focus: Identifying combinations of transformations.
Part 1: Which One Doesn’t Belong routine
Project this image
Possible answers:
Top left: No negative y coordinates
Top right: Only one that can be solved in one move.
Bottom left: No positive x coordinates
Bottom right: Only one that crosses quadrants
Part 2: Desmos Activity: Transformation Golf: Rigid Motion
*Note: Individual or small group depending on the number of laptops/tablets available. The teacher can also project the activity slide by slide and the class can do the work on their own grid paper. This activity can also be done offline by printing out the pages from the previews on the webpage.
- Desmos Provides teacher notes within the slides (as in Powerpoint), which you can access in the preview.
Part 3: Math Journal
- Name 3 mistakes that are commonly made when making transformations on the Cartesian Plane. Describe the impact each mistake has on the accuracy of the transformation description.
Reversing coordinates, mislabelling images, moving the opposite direction (+ instead of -), etc…
Focus: Drawing shapes on the Cartesian Plane to given specifications.
Part 1(Whole Class): Routine – Where’s Polygon?
Part 2(Partners):
- Give students grid paper on which to draw a Cartesian Plane or a template
- Provide the following instructions:
- Draw a triangle on the grid with one vertex at (5, -1). Make sure the other 2 vertices are on coordinate points as well.
- Label the vertices with the coordinates.
- Translate the triangle twice and describe the translation each time, giving the new coordinates of the triangle.
- Perform a rotation and then a reflection of the original triangle.
- Describe the rotation and reflection you chose. Draw and label the coordinates of each image.
- Transform the original triangle so that one of the vertices is now on (-5, 1). Can you do it using 3 different transformations? Can you do it using only one kind of transformation? Can you do it in one move?
- Label each of your final images with coordinates for all 3 vertices.
- Part 3 (Groups of 4):
- Pair each group with another.
- Take turns explaining your strategy for performing multiple transformations (encourage students to be as specific as they can and use verbal or pictorial examples.)
- What is the same about all our strategies? What is different?
Note: This lesson will take 2 blocks of time to complete and may be expanded into a larger exploration of Islamic Geometry in Art and Social Studies.
- The activity is best done on individual laptops/tablets. If necessary, students can be partnered.
- The worksheet students need for the offline activity can be downloaded from the linked page.
- Teacher notes are included in the slide deck.
Part 1 (included in the Desmos slide-deck): Which One Doesn’t Belong
Part 2 (1st block): This block is online
- Students follow the prompts in the activity to learn about Islamic geometry and tessellations.
- Follow the “Teacher Moves” notes included
- Instruct students to skip screen 8 for this block.
Part 2 (2nd block): This block is offline
- Provide students with the worksheet for designing their own tile and complete the activity to create a tessellated design.
Part 3: Math/Art Journal Reflection
*Note: Students will have completed an initial reflection as part of the online Desmos activity (screens 9-11) in Block 1.
- How are transformations connected to Islamic Geometry?
- Describe how you used combinations of transformations in your design?
- Explain what you would do differently next time and why.
As mentioned previously, students could now engage in deeper learning about Islamic geometry or study other artists, such as Susan Point and M.C. Escher, who use transformations in their artwork. Students may need more practice to accurately transform shapes and describe the transformations on the Cartesian Plane, so more experiences like those in Wednesday’s lesson can be provided. This topic shares skills with line graphing (linear relationships), so the lessons in this section can be interleaved with or followed by that concept.
Suggestions for Assessment
Success Criteria:
- Plots the vertices of a shape in any quadrant of the Cartesian Plane (C.P) given a set of ordered pairs.
- Performs sequences of transformations with shapes on the C.P.
- Describes sequences of transformations to move a shape from one position to another.
- Correctly labels vertices of both the original shape and subsequent images.