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 5
Measurement and Geometry
In Grade 1, students directly measure area using non-standard units that are non-uniform (e.g., handprints) or uniform (e.g., triangular pattern blocks). In Grade 3, students continue to directly measure area, this time using standard–and thus uniform–units (e.g., cm2). In Grade 5, students focus on area measurements of specific shapes: rectangles and squares. They investigate areas of rectangles using materials such as colour tiles, geoboards, and graph paper. Students use repeated reasoning to generalize the relationship between the area of a rectangle and its side lengths. They come to understand that the area of a rectangle is the product of its length and width. This means that students can determine the area of a rectangle abstractly (i.e., without measuring directly). Students use variables and formulas to express this relationship.
Key Concepts
Area measurement of squares and rectangles
Students continue to explore area measurements. They develop a formula to generalize the relationship between the area of a rectangle and its side lengths. Given two measurements (i.e., area, length, or width), they use this relationship to determine the third.
Relationships between area and perimeter
Students explore the relationship between area and perimeter. They come to understand that area and perimeter are related to but not dependent on each other: two 2-D shapes may have the same perimeter but different areas or the same area but different perimeters. They solve problems that involve minimizing or maximizing one measurement while holding the other fixed.
Single transformations
Students are introduced to single transformations that involve motion: translations (slides); reflections (flips); and rotations (turns). They determine the image of a shape after a slide, flip, or turn. They “go backwards”: given an original shape and its image, they identify and describe the transformation. They come to understand that these three types of transformations change location and direction, not size or shape.
Key Measurement and Geometry Concept 1: Area measurement of squares and rectangles
Overview
Students continue to develop conceptual understanding of area as the two-dimensional space that a shape covers (or the two-dimensional space that it takes to cover a shape). They make connections between the area of a rectangle and the array meaning of multiplication. They look for and generalize the relationship between the lengths, widths, and areas of rectangles: . Noticing this relationship means that students can also “go backwards” (i.e., calculate the width by dividing the area by the length or the length by dividing the area by the width). They come to realize that the square is a special case: so . In Grade 6, students will build on this concept; they will make connections between the areas of triangles, parallelograms, and trapezoids to that of rectangles.
Measurement Foundations:
The following concepts and competencies are foundational in supporting understanding of area measurements of squares and rectangles in Grade 5:
- direct measurement of area using non-standard (e.g., filling in the outline of a 2-D shape with pattern blocks) and standard (e.g., covering a 2-D surface of an object with centimetre blocks) units of measure
- familiarity with 1 cm2 and 1 m2 as metric units of area measurements
- understanding of fundamental meanings of multiplication, namely:
- repeated addition (i.e., repeated iterations of rows or columns)
- equal groups (i.e., thinking about rows or columns as having equal length)
- arrays (i.e., multiplying the number of rows and the number of columns answers “How many?”)
Progression:
- explore area measurements of squares and rectangles concretely, pictorially, and symbolically:
- use reasoning (i.e., patterns and logic) to generalize relationships between the side lengths of squares and rectangles and their areas
- express these relationships using variables (i.e., and )
- determine missing measurements
Sample Week at a Glance:
Before. Display a “partial array.” For example:
Ask “How many?”
Encourage students to attend to the row and column structure of arrays: to determine the total number of circles, multiply the number of rows, 5, by the number of circles in each row, 6 (or multiply the number of columns, 6, by the number of circles in each column, 5). The purpose of the “paint splatter” is to discourage counting.
Find more partial arrays, if needed, at: https://chrishunter.ca/2017/10/24/paint-splatter-arrays/.
Display a rectangle created with square tiles. For example:
Remind students that area is the space that a shape covers; it tells you the number of squares altogether in the rectangle.
During. Have students:
- build some different-sized rectangles using colour tiles (concretely)
- draw them using grid paper (pictorially), and
- record the lengths, widths, and areas in a table (symbolically)
A suggested sequence:
- length = 5 units; width = 3 units; Area = ? square units
- length = 6 units; width = 4 units; Area = ? square units
- length = 7 units; width = 5 units; Area = ? square units
- length = 6 units; width = 3 units; Area = ? square units
- length = 5 units; width = 4 units; Area = ? square units
- length = 8 units; width = 6 units; Area = ? square units
- length = 4 units; width = ? units; Area = 12 square units
- length = ? units; width = 3 units; Area = 21 square units
Have students work in pairs or groups of three. Ask “How can you find the area of a rectangle?” Look and listen for strategies that involve repeated addition (e.g., ) and multiplication (e.g., ). Nudge students toward multiplication. Ask “Why does it make sense to multiply the length and width?” Possible explanation: “The length tells us the number of rows of squares. The width tells us the number of squares in each row. So, multiplying length and width tells us the number of squares altogether.”
5 x 3=15
Select students to share their strategies.
After. Record their thinking. Use a table of values to organize this information.
Select students to share their strategies.
Rectangle | Length (units) | Width (units) | Area (square units) |
I | 5 | 3 | |
II | 6 | 4 | |
… | … | … | … |
VII | 4 | 12 | |
… | … | … | … |
Make connections between area and multiplication: side lengths are represented by factors and areas are represented by products. When students have generalized this relationship, introduce the formulas and .
Consolidate by encouraging students to “try on” this approach. For example:
- length = 6 units; width = 5 units; Area = ? square units
- length = 5 units; width = ? units; Area = 10 square units
- length = ? units; width = 4 units; Area = 28 square units
Before. Display the following Same But Different image:
Ask “What is the same? What’s different?” Possible responses:
- Both are arrays
- Both show sixteen objects
- The circles are arranged in two rows and eight columns; the squares are arranged in four rows and four columns
- The squares show area (i.e., they cover the space); the circles do not (i.e., there are spaces between)
- The are arranged like a rectangle; the squares are arranged as a larger square
Note that a square is, in fact, also a rectangle. In this lesson, students will explore squares as a special case of rectangles (i.e., ).
During. Give students the following open question: “A rectangle has an area of 36 square units. What could its dimensions be?” (Adapt, if needed. For example: 18 square units).
In pairs or groups of three, have students build, draw, and record as many rectangles that have an area of 36 square units as they can find. Challenge some students as needed: “Have you found all the possible rectangles? How do you know?”
Expect students to disagree about whether rectangles such as (4 rows; 9 columns) and (9 rows; 4 columns) “count” as different possibilities. If length and width refer to the longer and shorter sides, respectively, then they can be considered to be the same rectangle, just oriented differently.
Similarly, expect students to disagree about whether , a square, “counts” as a rectangle. It does, since a rectangle is defined as having four sides and four right angles!
After. Select students to share their possible solutions, sequencing from the common (i.e., , , ), to the overlooked (i.e., ), to the contentious (i.e., ). Introduce the formulas and as a special case of and . To consolidate, challenge students to build, draw, and record as many squares as they can.
Before. Launch Graham Fletcher’s 3-Act Math Task Cover the Floor. Show the video Act 1. Ask “What is the first question that comes to your mind?” Settle on “How many 100 flats (a/k/a “plates”) will it take to cover the yellow rectangle?” Have students determine a range of reasonable estimates: “What’s an estimate that is too low? Too high?”
During. Ask “What information do you need here?” Students might ask for the dimensions of the yellow rectangle. Show the Act 2 images. In small groups, have students figure out how many 100 flats it will take to cover the yellow rectangle. Invite students to retell: “What strategy did you use?” Ask “How else might you have solved the problem?” Listen and look for solutions that involve:
- working with length and width in 100 flats directly (i.e., )
- working with length and width in square tiles first (i.e., ; )
After. Select students to share their strategies. Make connections between two approaches. Emphasize that the smaller the unit, the larger the number of units (i.e., 12 tiles vs. 108 flats). Show the Act 3 “reveal” video.
Before. Display the following number talk image:
Ask “How many do you see? How do you see them?” Students might see as or . (Re-)introduce open arrays (or areas):
Display a second number talk image:
Again, show possible partitions:
During. Pose the following problem: “A rectangular field is 25 m by 32 m. Figure out the area of the field in as many ways as you can.”
Encourage students to represent their ideas using open areas.
After. Have students share their strategies. Possible partitions include:
Before. Ask the following Open Middle questions: “Place digits from 1 to 9 in the boxes below to make an area that is between 20 and 30 square units. Each digit can be used only once.”
The purpose of this prompt is to clarify expectations (i.e., although makes the “between” statement true, it breaks the “only once” rule).
During. Provide the following two parallel tasks:
Choice 1: Place the digits from 1 to 9 in the boxes below to make:
- the largest area [9 x 87=783]
- the area that is closest to 50 square units [3 x 17= 51]
Each digit can be used only once.
Choice 2: Place the digits from 1 to 9 in the boxes below to make:
- the smallest area []
- the area that is closest to 500 square units []
Each digit can be used only once.
After. Have students justify their decisions (e.g., “We chose to place the largest/smallest digits in the tens place because…” or “We noticed that we couldn’t make 50 so we looked for ways to make numbers close to 50, like 48, and …”). Annotate solutions using the formula A = l x w.
Next, students will explore the relationship between area and perimeter of rectangles. See Key Measurement Concept 2.
Suggestions for Assessment
By the end of Grade 5, students can determine the area, length, or width of a rectangle given two of these three measurements. They can call upon their understanding of a fundamental meaning of multiplication (i.e., arrays) to explain the relationship between area and side lengths.
Suggested Links and Resources
Key Measurement and Geometry Concept 2: Relationships between area and perimeter
Overview
Students continue to “bump into” Key Measurement Concept 1, area measurement of squares and rectangles. They explore the relationship between area, the two-dimensional space that a shape covers, and perimeter, the one-dimensional distance around a shape. They notice that two rectangles may have the same perimeter but different areas or the same area but different perimeters. This can be a surprising discovery! Further, students solve problems, perhaps in context, that involve minimizing or maximizing one measurement while holding the other fixed. For example, students might find that, for any given area, the rectangle with the least perimeter is most like a square (or that, for any given perimeter, the rectangle with the greatest area is least like a square). In later grades, students will explore a similar relationship between volume and surface area.
Measurement and Geometry Foundations:
In addition to Key Measurement Concept 1 (and its foundations), the following concepts and competencies are foundational in supporting understanding of relationships between area and perimeter in Grade 5:
- direct measurement of length
- familiarity with metric units of length measurements
- understanding of perimeter as a combined linear measure
- direct measurement of perimeter (regular and irregular 2-D shapes)
- calculation of perimeter of a square and rectangle (e.g., “double the length, double the width, and then add” or “add the length and width and then double”)
Progression:
- explore relationships between area and perimeter concretely, pictorially, and symbolically
- understand that area and perimeter are related to but not dependent on each other: two 2-D shapes may have the same perimeter but different areas or the same area but different perimeters
- solve problems that involve minimizing or maximizing area or perimeter while holding the other fixed
Sample Week at a Glance
This week builds on Key Measurement Concept 1, area measurements of rectangles and squares.
Before. Provide pattern blocks. If students are unfamiliar with pattern blocks, ask “What do you notice?” Record students’ observations. For example:
- all sides are the same length… except the base of the red trapezoid
- the length of the base of the red trapezoid is twice that of the others
- two green triangles cover one blue rhombus; three blue rhombi cover one yellow hexagon; etc.
- the orange square and tan rhombus do not naturally cover the other shapes
- twelve tan rhombi can make 360°
Establish the length of one side of the green triangle as 1 unit. Have students find the perimeter of each pattern block. For example:
During.
Pose the following problem: Using 4 yellow hexagons and 6 red trapezoids, find the greatest and least perimeter.
Establish that blocks must be placed together so that they share at least one common side:
In small groups, have students determine the perimeters of new shapes. Ask “How would you describe the shapes that have the smallest/largest perimeter?”
After.
Select one student to share their shape with a small perimeter. Invite other students to verify its perimeter.
P = 18 units
Have students examine this shape: “Think about your smallest-perimeter shape. What is the same?” Students will describe these shapes as closely packed together.
Repeat for an arrangement with a large perimeter.
P = 36 units
Students will describe these arrangements as stretched out.
Have students compare the shapes with the least and greatest perimeters. Ask “What is the same?” [A: The area! It always takes 4 yellow hexagons and 6 red trapezoids to cover the space.] Emphasize that shapes may have the same area but different perimeters.
Before. Provide colour tiles. Have students build the following rectangle:
Ask students to determine the perimeter, the distance around the shape. Anticipate counting (i.e., 1, 2, 3, … 12) and “add-around-the-outside” (i.e., 4 + 2 + 4 + 2 = 12) approaches:
Ask “How else could you have arrived at the same result?” Listen and look for “double then add” (i.e., and “add then double” (i.e., ) approaches.
During. Present the following task: The area of a rectangle is 20 square units. What could its perimeter be?
(Note that this task is very similar to one described in Key Measurement Concept 1. In this task, students think about perimeter, not just length and width.)
In small groups, have students find as many possible perimeters as they can. Have students record their thinking using pictures, numbers, and words. Ask “How did you know which rectangles were possible (and which were impossible)?” and “How did you determine the perimeter?” Nudge students toward more efficient methods, if needed: “How else could you figure out the perimeter?”
Repeat: “What if the area is 24 square units? 25?”
After. Select students to share possible perimeters, one at a time and from greatest to least (i.e., ; ; ).
Ask “What do you notice?” Invite students to make their own conjectures (e.g., “The rectangle with the shortest perimeter is the one that looks most like a square” or “The closer together the side lengths, the smaller the perimeter”). Ask “Is this always true? What if the area is 35 square units?”
Before. Hold up 6 coffee stir sticks (or popsicle sticks or toothpicks). Build (under a document camera) or draw (on a whiteboard) the following rectangle:
Point out the difference between perimeter (i.e., the distance around the rectangle) and area (i.e., the space enclosed by the rectangle).
Show that determining area involves visualizing squares:
During. Pose the following “real-world” problem: Your class is building a garden for your school. You have 20 metres of lumber. What is the largest garden you could build? What is the smallest?
In small groups, have students model and solve the problem. Provide concrete materials for students to represent the context (e.g., one toothpick = 1 m of lumber). Encourage students to draw each garden that they find:
After. Select one student to share their garden with the largest area. Ask “What do you notice?” (“Its length is much greater than its width.”) Repeat for the garden with the smallest area. (“Its a square!”) Help students make connections between maximum and minimum area and perimeters: “How is this problem similar to and different from yesterday’s colour tile task?” Possible responses:
- Yesterday, the area stayed the same, 20 square units; today the perimeter stayed the same, 20 units.
- A lot of different rectangles with the same area or perimeter were possible!
- For rectangles with the same area, the square (or “square-ish”) rectangle has the smallest perimeter.
For rectangles with the same perimeter, the square (or “square-ish”) rectangle has the largest area.
Before. Display the following open question:
Which two rectangles are most alike?
Possible responses:
- Left and Centre since both have an area of 24 square units
- Centre and Right since both have a perimeter of 22 square units
- L and R since both have a side length of 4 units
- C and R since both have a side length that is odd
- C and R since they are both taller than they are wide
During. In small groups, have students solve several “Mystery Rectangle” problems. For example, “A rectangle has an area of 7 cm2. It has a perimeter of 16 cm. What could its dimension be? Sketch the rectangle.”
A suggested “thin-sliced” sequence:
- Area = 7 cm2; Perimeter = 16 cm; length = ?; width = ?
- Area = 12 cm2; Perimeter = 16 cm; length = ?; width = ?
- Area = 30 cm2; Perimeter = 22 cm; length = ?; width = ?
- Area = 24 cm2; Perimeter = 22 cm; length = ?; width = ?
- Area = 24 cm2; Perimeter = 28 cm; length = ?; width = ?
- Area = 36 cm2; Perimeter = 30 cm; length = ?; width = ?
- Area = 36 cm2; Perimeter = 24 cm; length = ?; width = ?
- Area = 64 cm2; Perimeter = 32 cm; length = ?; width = ?
Make colour tiles and graph paper available; students might decide that one–or both–of these tools are helpful. Look and listen for strategies that involve factors (e.g., “If the rectangle has an area of 30 cm2, then its dimensions might be 1 and 30, 2 and 15, 3 and 10, or 5 and 6…”) or sums (e.g., “If the rectangle has a perimeter of 22 cm, then its length and width add to 11. The dimensions might be 1 and 10, 2 and 9, …, or 5 and 6…”).
After. Select students to share their solutions, as above. To check students’ understanding, have them create and solve a “Mystery Rectangle” problem of their own.
Before. Draw a rectangle having a width of 5 cm and a length of 8 cm. Have students list characteristics of the rectangle. For example:
- the area is 40 cm2
- the perimeter is 26 cm
Press students to find more characteristics: “What else?” For example:
- the area is greater than the perimeter
- the area is a multiple of 5
- the side lengths are odd and even
- the area and perimeter are both even
- the length is 3 cm more than the width
During. Display the following set of characteristics:
https://docs.google.com/document/d/1PNgyKfPCb4iKXQtEGy1XaA72tzRlz1V6bt5RK5vw8ss/edit
In small groups, have students build rectangles that have these characteristics. Challenge them to build as few rectangles as possible that satisfy each constraint at least once. Observe groups. Ask students to justify their mathematical decisions. Ask:
- “Which constraints pair nicely?”
- “Which constraints cannot be paired?”
- “Is it possible to build 4 rectangles? 3? 2?”
After. Select students to share their sets of rectangles. Ask “What decisions did you make?” Multiple strategies and solutions are possible. For example, one group of students might explain that they began with the constraint “One side length is double the other side length” (E) and built a rectangle with a length of 20 cm and a width of 10 cm. Then, they might explain that they noticed that this rectangle satisfies four additional constraints: “The area is more than 50 cm” (A); “The perimeter is a multiple of 3” (B); “The area and perimeter are both even numbers” (C); and “The perimeter is more than 24 cm” (F). Next, they might explain that they tackled “Both side lengths are odd numbers” (D) and, with an eye on “The area is a multiple of 3” (H), built a rectangle with a width of 3 cm and a length of 5. They might explain that this rectangle also satisfies “The perimeter is bigger than the area” (G). Repeat, pointing out constraints that pair nicely.
Ask “Is it possible to build one rectangle? How do you know?” Students can reason that it is not possible. For example, students might explain that “Both side lengths are odd numbers” (D) cannot be paired with “One side length is double the other side length” (E) since a double is always even. They might also explain that “Both side lengths are odd numbers” (D) cannot be paired with (part of) “The area and perimeter are both even numbers” (C) since .
Students might make some surprising discoveries! For example, if one side length is double the other side length, then the perimeter is always a multiple of three:
Students can continue to revisit Key Measurement Concepts 1 & 2 throughout the year. For example, the following “Would You Rather Math” prompts could be used as “warm-ups” to ignite short discussions about content learned earlier in the year:
- https://www.wouldyourathermath.com/would-you-rather-77/
- https://www.wouldyourathermath.com/cheezits/
- https://www.wouldyourathermath.com/perimeter/
https://www.wouldyourathermath.com/birthday-cake/
So, too, could “Alike and Different” prompts:
Alike & Different: Which One Doesn’t Belong? & More
Suggestions for Assessment
By the end of Grade 5, students can determine different rectangles that have the same area or perimeter. They can communicate strategies to determine which rectangles are possible (and which are not). They can use reasoning to predict which of these rectangles will have the greatest or least perimeter or area; they can calculate to test their predictions. They can determine the side lengths of a rectangle given its area and perimeter (or its perimeter given its area and a side length or its area given its perimeter and a side length).
Suggested Links and Resources
Key Measurement and Geometry Concept 3: Single transformations
Overview
Students are introduced to single transformations that involve motion: translations (slides), reflections (flips), and rotations (turns). Students slide shapes horizontally, vertically, and both horizontally and vertically. They move every point on an original shape in the same way to form its image. Students reflect shapes in lines that can be on, inside, or outside the shape. They keep the distance from the reflection line to corresponding points on the original and image the same. Students rotate shapes, clockwise or counter-clockwise, in increments of quarter turns. Like reflections lines, rotation centres can be on, inside, or outside the shape. The location of the rotation centre, alongside the size and direction of the turn, affects the location of the image. Throughout, students work with concrete materials and, later, simple grids. They use mirrors (or MIRAs) and tracing paper (or transparency sheets) to develop understanding. Students use both familiar terms (i.e., “slide,” “flip,” “turn”) and mathematical terms (i.e., “translation,” “reflection,” “rotation”) to talk about these three motions. They use these terms to describe transformations, moving from simple to ambiguous (e.g., transformations that can be described as more than one type). In Grade 5, students come to understand that these three types of transformations change location and direction, not size or shape. (Dilations, a type of transformation that does change size, are introduced in <>.) In Grades 6 and 7, students extend their understanding of transformations, from single transformations to combinations of transformations and from simple grids (or no grid at all) to coordinate grids. In later grades, students transform functions rather than shapes; in doing so, they make necessary connections between algebra and geometry.
Measurement and Geometry Foundations:
The following concepts and competencies are foundational in supporting understanding of transformations in Grade 5:
- regular and irregular polygons:
- identify, describe, and create polygons (i.e., triangles, quadrilaterals, pentagons)
- line symmetry:
- understand that a shape has line symmetry if one half of the shape is a mirror image–or reflection–of the other half
- determine (i.e., test 2-D shapes for) line symmetry and construct (i.e., build or draw) 2-D shapes with line symmetry
- time:
- differentiate clockwise from counter-clockwise
- fractions:
- represent ¼, ½, and ¾ of one whole
Progression:
- draw the image of a shape after a translation, first horizontal OR vertical, then horizontal AND vertical
- identify and describe the translation given a shape and its image
- draw the image of a shape after a reflection, first on a horizontal or vertical reflection line that is on, then outside, then inside the shape
- identify and describe the reflection given a shape and its image
- draw the image of a shape after a rotation, first at the centre (and without a simple grid), then at a point on the shape (and with a simple grid)
- identify and describe a rotation given a shape and its image
- determine whether a transformation is a translation, reflection, or rotation
Sample Week at a Glance
This sample week starts at the beginning of this unit.
In pairs, have students solve spatial puzzles that involve placing puzzle pieces (e.g., tans, pentominoes) inside outlines of shapes. For example:
https://polypad.amplify.com/tangram
https://polypad.amplify.com/lesson/pentomino-zoo
Listen for students to describe these “moves” of puzzle pieces (e.g., tans, pentominoes) as slides, flips, and turns.
Before. In the game Rush Hour, you slide blocking vehicles in their lanes until the path is clear for the red car to escape. Vehicles can only move forward and backward, not sideways. Show a player’s first move:
https://www.thinkfun.com/wp-content/uploads/2015/09/RushH-5000-IN02.pdf
The orange car slid one space up. The puzzle can be solved with three more moves. Challenge students to figure out this sequence. Call on students to describe each move. Note that a move includes both direction and distance. Compare students’ solutions to the posted solution. The green truck slides left two spaces:
The blue truck slides down two spaces:
The red car slides right 5 (or more) spaces:
Today, students will explore “slides” of shapes. Mathematicians use the word “translation” to describe a slide. Draw a triangle on a simple grid:
Unlike Rush Hour vehicles, shapes can slide in all four directions. Ask students to predict the location of the triangle after the following slide/translation: 1 space left, 4 spaces up. Reveal the new location:
Introduce the word “image.” Draw one or more arrows to illustrate the translation:
During. Draw a pentomino “F-pentomino” on a simple grid:
For reference:
Ask students to predict the location of the pentomino after the following slide/translation: 3 spaces right, 2 spaces down. Reveal the original pentomino alongside its image:
Make pentominoes available to students, if possible. Have students draw a pentomino of their choice: the 13 different shapes provide different levels of challenge. Ask students to predict, then draw, the location of the image after the following translations:
- 3 spaces right
- 5 spaces down
- 4 spaces right, 1 space up
- 2 spaces left, 4 spaces down
Observe (Circulate). Ask “How did you know how to draw the image?” Some might focus on moving each of the five squares (or their vertices) one-by-one; others might focus on moving one key square (or one of its vertices) and then draw the pentomino appropriately from there.
After. Select students to share some of their solutions. Sequence in increasing complexity (i.e., from sliding the I-pentomino 3 spaces right to sliding the W-pentomino 2 spaces left and 4 spaces down).
Consolidate. Ask “What stays the same?” [A: shape, size, direction] and “What changes?” [A: location]. Point out that each square (or vertex) moves the same distance.
Before. Today, students will explore “flips” or “reflections” of shapes. Have students fold a piece of blank paper in half:
Have students draw a quadrilateral on the left side:
Ask students to predict the image after a flip over the fold line.
Have students use a bingo dabber to mark each vertex. Fold and press.
Ask “What do you notice?”:
- The image is “backwards,” like in a mirror
- Distances from the reflection line stay the same
During. Have students draw a polygon on a simple grid. Ask students to predict, then draw, the location of the image after a reflection in:
- a horizontal line outside the polygon
- a vertical line inside the polygon
- a horizontal or vertical line on the polygon
- a diagonal line (extension)
For example:
horizontal; outside
vertical; inside
horizontal; on
Make MIRAs available. Encourage students to use this tool to determine (or confirm) the image. (You might decide to have students first explore reflections using mirrors and pattern blocks in a separate lesson.)
Observe (Circulate). Ask “How did you know how to draw the image?”
After. Select students to share some of their solutions.
Consolidate. Ask “What stays the same?” [A: shape, size] and “What changes?” [A: location, direction].
To check for understanding, display the following:
Ask students to describe the transformation that would result in each image:
A: translation of 3 spaces left
B: reflection in a vertical line (outside the shape)
C: reflection in a horizontal line (on the shape)
D: translation of 3 spaces right, 4 spaces down
Point out that A & D can be visualized as sliding the original pentomino since the direction is the same. In B & C, the direction is different. They can be visualized as picking up the pentomino and flipping it.
Before. Display a design that involves translations, reflections, and rotations. For example:
(The shapes are labeled A to F to assist communication.) Ask “What transformations do you see?” Students will find and describe translations (e.g., sliding F down and to the right results in C), reflections (e.g., flipping F over a vertical line results in B), or both (e.g., sliding A down or flipping A over a horizontal line both result in D).
Ask students to describe how they might transform A so that it covers C. Some might spot a slanted reflection line; others might spot a “turn”:
Today, students will explore turns, or “rotations.” Display a large, simple shape or picture. For example:
Have students predict and describe what the picture will look like after a quarter turn. Ambiguity (i.e., finger pointing right, thumb pointing up vs. finger pointing left, thumb pointing down) is intentional. To make an accurate prediction, the direction of the turn must be known. Discuss the term clockwise and counterclockwise.
Repeat, this time with clearer directions (i.e., “¾ turn counter-clockwise,” “½ turn clockwise”).
During. Provide plastic pentominoes and one-inch graph paper. Print your own, if needed. Pick a pentomino. Ask students to predict its image after a rotation at a vertex. Have students use the provided tools to test their predictions. Show–and record–the rotation under a document camera or on a whiteboard. Repeat. Mix it up: ask about ¼, ½, and ¾ turns, both clockwise and counterclockwise. For example:
Have students explore rotations of different pentominoes (or simple shapes). Invite students to play with different rotation centres. For example:
Observe (Circulate). Ask questions to help students reflect on their thinking: “What was the most challenging part?” etc.
After. Consolidate. Ask “What did you notice?” or “How are a ¼ turn clockwise and a ¾ turn counter-clockwise the same? How are they different?” [A: image is the same; motions are different]
Before. Provide each student with two red trapezoid pattern blocks–an original and image. Have students predict (i.e., visualize) the direction (not location) of the image after a:
- translation
- reflection
- rotation
Have students test their predictions on their pattern blocks.
Discuss. Record all possibilities:
During. Display the following:
The blue flag is the original and the red flag is its image. Ask “How can you transform the blue flag to cover–or “capture”–the red flag?” Discuss. Ask “How can you improve the description of the transformation?” if needed (i.e., “slide/translate the blue flag 2 spaces right and 3 spaces down” > “slide/translate the blue flag down and to the right” > “slide/translate the blue flag”). (You might consider this small group conversation to be a rehearsal for the whole-class discussion.)
In small groups, provide students with different “Capture the Flag!” challenges. For example:
Create new challenges (or have students create their own):
Observe. Move about the room to check for understanding. Ask “How did you figure out the transformation?” Press for precision if appropriate.
After. Select groups to share their solution to a challenge. Sequence in increasing difficulty–think “beginner” to “expert.” Together, solve a new challenge created by a student!
Next, students might explore “ambiguous” transformations in which the image can be the result of more than one transformation. For example:
Students will continue to explore transformations by noticing and naming transformations in the world around them. Please visit the Indigenous Connections page of this site as many of the resources listed there provide ideas for learning about translations, reflections, and rotations in First Nations art.
Suggestions for Assessment
By the end of Grade 5, students can perform translations, reflections, and rotations on shapes. They can accurately (i.e., precise location on a simple grid, correct direction) draw the image of a shape after a transformation; they can identify and describe a transformation given a shape and its image. Students can explain how each transformation affects a shape–what stays the same and what changes.
Suggested Links and Resources
https://mathequalslove.net/printable-tangram-pieces-pdf/
https://mathequalslove.net/tangram-challenge-binder/
https://polypad.amplify.com/p#tangramhttps://polypad.amplify.com/tangram
https://polypad.amplify.com/lesson/building-tangram-puzzles
https://mathequalslove.net/printable-one-inch-pentominoes/
https://mathequalslove.net/category/puzzles/movable-pieces/pentominoes-puzzles/
https://polypad.amplify.com/p#polyominoes