Grade 6

Measurement and Geometry

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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 6

Measurement and Geometry

Prior to Grade 6, students have explored a variety of attributes and measurements of 2-D shapes and 3-D objects, and in Grade 6 the learning continues across a wide variety of concepts.

Students extend their understanding of area to include triangles, parallelograms and trapezoids. They generalize to develop and apply formulas for these measures. In Grades 4 and 5 students have explored the perimeter of regular and irregular polygons, and in Grade 6 they explore the perimeter of complex shapes.

Grade 6 is the year when students learn about angles, both in terms of measurement (including constructing angles) and geometry (classifying angles). They use these angle classifications to classify triangles, as well as classify triangles using side lengths.

Though not addressed on this site, students also explore volume and capacity. Students have not explored capacity since Grade 3, and in Grade 6 they are introduced to volume as a measure, and connect it to capacity. In Grade 7 they will go onto exploring the volumes of rectangular prisms and cylinders.

Students learned about transforming shapes (slides, rotations, reflections) in Grade 5, and in Grade 6 they combine these transformations. They also situate transforming shapes on a coordinate grid using positive coordinates. In Grade 7 they will expand these ideas into all four quadrants of the coordinate grid.

Key Concepts

Measurement of 2-D Shapes

Students explore and generalize the areas of triangles, parallelograms, and trapezoids. They also explore the perimeter of complex shapes.

Explore

Angles and Triangles

Students estimate, measure, and construct angles up to 360°. Students classify angles and use these to classify triangles. Students also classify triangles using side lengths.

Explore

Transformations

Students combine different transformations of shapes on a coordinate grid (1st Quadrant only).

Explore

Key Measurement and Geometry Concept 1: Measurement of 2-D Shapes

Overview

In Grade 5, students would have explored the areas of rectangles and squares, including generalizing them with formulas. In Grade 6, they extend their understanding of area to include triangles, parallelograms, and trapezoids. As in Grade 5, the focus is more on the thinking that leads to generalization than it is on the formulas themselves.

The challenge with these shapes (triangles, parallelograms, trapezoids) is that they have at least one slanted side, so one cannot directly measure these shapes using square units. For each shape, students can first explore areas on a grid and estimate. They can count full squares, and then approximate the areas which are parts of a square. They may also combine two areas which together may make a square. For example, in this trapezoid, they can count 14 filled squares, then visually combine other parts to make 4 more filled squares.

To build towards generalizing these areas, students can relate these shapes to areas that they have already generalized. Starting with the parallelogram, they can relate its area to the area of a rectangle. So like a rectangle, the area of a parallelogram is base x height, or length x width.

The area of a triangle can be related to the area of a parallelogram. Rotating (half-turn) a copy of a triangle around one side creates a parallelogram. Because the parallelogram is made up of two of those triangles, each triangle has an area that is half of the parallelogram, so the area is half of base x height.

The area of a trapezoid can also be related to the area of a parallelogram. As done with a triangle, a copy of the trapezoid can be rotated (half-turn) to make a parallelogram. The base of the parallelogram is made up of the sum of the bases (top and bottom) of the trapezoid. So the area of the trapezoid is half of the area of the parallelogram, i.e., half of the height times the sum of the two bases.

In Grade 6, students explore the perimeter of complex shapes. Because the shapes are complex, there is no formula that is to be reached. There can still be some general ideas/strategies that can emerge. Complex shapes are shapes that are made up of other shapes. The elaborations mention using colour tiles, pattern blocks, or tangrams. They could also use a grid or geoboard to draw complex shapes. Be cautious using tangrams as some of the lengths are not rational numbers (in which case students could use a ruler to approximate lengths). Below are a couple of examples of complex shapes with their perimeter:

 

Measurement and Geometry Foundations:
  • Naming and classifying 2-D shapes
  • Meaning of area
  • Units of area (nonstandard and standard)
  • Using a grid or tiling to determine or estimate an area
  • Areas of squares and rectangles
  • Meaning of perimeter
  • Units of length (nonstandard and standard)
  • Estimating or measuring a length
Progression:
  • Estimating and determining areas of shapes on a grid.
  • Generalizing the area of a parallelogram by relating it to the area of a rectangle.
  • Generalizing the area of a triangle by relating it to the area of a parallelogram.
  • Generalizing the area of a trapezoid by relating it to the area of a parallelogram.
  • Throughout the progression they may solve problems involving areas of each type.
  • Determining the perimeter of complex shapes built by squares.
  • Determining the perimeter of complex shapes built by other shapes (e.g., pattern blocks).
Sample Week at a Glance:
Download as a PDF

This sample week starts at the beginning of this unit.

Focus: Estimating areas of parallelograms, triangles, and trapezoids

Before:

  • Provide an image of a triangle on a grid or geoboard. Use 45° slants to make it simpler to count the area. For example,
  • Have students work in pairs or small groups. Ask them to determine the area of the triangle, and to see if there is more than one way they can determine the area.
  • For the example above, strategies that students may use include:
    • There are 6 half-squares so they make 3 whole squares. There are also 6 whole squares, so the total area is 9 square units.
    • You could take the left half of the triangle and put it on the right to make a 3 x 3 square to see it’s 9 square units.
  • Provide an image of a triangle with the top vertex moved over. Once again ask students to think about how they could determine its area. For example,
  • Some students may notice parts of a blue square that can be used to fill in white parts of a square.
  • Others may notice that they can look at each half of the triangle and notice each part is half of a rectangle. The total area is still 9 square units.
  • Summarize for the students that they have seen a number of ways to figure out the areas of shapes which are not a square or rectangle. Tell them these strategies will be helpful to them for the main activity.

During:

  • Display or provide a parallelogram, a triangle, and a trapezoid on a grid. For example.
  • Have students work in groups. Before they determine actual measures, ask them questions like:
    • What is the name of each shape?
    • Which do you think is largest?
    • Do any have the same area?
  • Ask them to use a variety of strategies to determine each area. They may wish to build each shape on a geoboard, or virtual geoboard.

After:

  • Note that a formula approach is not intended at this time. Those generalizations will be developed over the next 3 classes.
  • Have the class share their strategies for determining each area. General strategies include:
    • Partitioning each area into parts.
    • Visually slicing and moving some parts.
    • Noticing parts which are half of a rectangle.
  • For example, the image below shows each shape divided into parts which are rectangles or half of a rectangle:
  • Provide students some more parallelograms, triangles, and trapezoids on a grid, and ask them to use a strategy to determine each area.
  • Some students may wish to create their own shapes (even more complex ones) and determine their area.

Focus: Generalizing the area of a parallelogram

Before:

  • Do a Same but Different routine with two parallelograms on a grid that have the same base and height, but different slants. For example:
  • Things that students may notice that are the same:
    • Both are parallelograms.
    • Both have a base of 6.
    • Both have a height of 3 (students may need to be shown the meaning of a height of a parallelogram).
    • They have the same area (using reasoning from the prior class, they could determine that each area is 9 square units).
  • Things they may notice that are different:
    • The blue one slants to the right, and the green one slants to the left.
    • The green one slants more than the blue one.
  • Choose one of the parallelograms. Ask the class if there is a way to turn it into a rectangle.  For example, (and this idea may have already emerged in earlier class discussion):

During:

  • Provide several parallelograms without grids, and indicate the measures of the base and height of each. For example:
  • Have the class work in pairs or small groups. Ask them to figure out the area of each parallelogram. Encourage them to think about how they could change each parallelogram into a rectangle. Some students may benefit from drawing each parallelogram on a grid, or using a geoboard. Alternatively, if the parallelograms are printed, students could physically cut the parallelograms to make rectangles.

After:

  • Have students share their solutions and strategies. One way to visualize each area is to slice a triangle from one side and move it to the other side to make a rectangle. For example:
  • Ask the class what they notice about their area calculations. They should notice that just like a rectangle, the area of a parallelogram can be calculated using base x height.
  • An interesting idea that emerges from this generalization is that parallelograms with the same base and height dimensions will have the same area, even if it may not look like they do:
  • Provide students some more parallelograms, and ask them to use a strategy to determine each area. Some students may still need the support of using a grid or cutting out printed copies.

Focus: Generalizing the area of a triangle

Before:

  • Do a Which One Doesn’t Belong routine using triangles. In addition to other attributes, encourage them to use strategies to figure out the area of each. For example:
  • Sample responses:
    • Top left: Only one that slants to the left
    • Top right: Only one with an area of 8 (all others have an area of 6), only one with a height of 4
    • Bottom left: Only one with a base of 6 and a height of 2
    • Bottom right: Only one whose right side slants to the right

During:

  • Have students draw two identical triangles on grid paper. Make sure they use a straight edge.
  • Tell them to use a strategy to figure out the area of their triangle.
  • Have them cut out each triangle. Ask them to see if they can figure out how they could use the two triangles to make a parallelogram.
  • Ask them to reflect on what they notice.
  • Encourage them to try more triangles.

After:

  • Have students share their triangles and what they discovered. For example:
    • The two triangles made a parallelogram with a base of 5 and a height of 2, so the area of the parallelogram is 10. There are two triangles, so each triangle has an area of 5.
  • Have a discussion about how they could generalize how to find the area of a triangle.
    • If the triangle is duplicated, you can make a parallelogram with the same base and height. So to get the area of the triangle, you get the area of the parallelogram (base times height) and divide by 2.
    • Alternatively this could be expressed as half of the base times the height.
  • Show a new triangle without a grid and ask them to find the area. For example:
  • Explore additional triangles. Even if students understand the formula/generalization, they may still struggle with less familiar situations, such as:
    • Height is outside the triangle
    • Base is not horizontal
    • Area is a decimal

Focus: Generalizing the area of a trapezoid

Before:

  • Show several shapes, including some trapezoids, and have a class discussion on which ones are trapezoids and which are not. For the ones which are not, have them explain why they are not a trapezoid. You may need to remind them of vocabulary such as quadrilateral and parallel. For example:
  • Sample responses include:
    • A, B, and F are trapezoids
    • C is not a trapezoid (or even a quadrilateral) because it has a curved side
    • D is not a trapezoid (or even a quadrilateral) because it only has 3 sides
    • E is a parallelogram. Some people define a trapezoid to have at least one set of parallel sides, so they would consider this a trapezoid. Others, including this author, prefer to define a trapezoid as having only one set of parallel sides.

During:

  • This activity is similar to the previous day, but this time they are doing it with trapezoids.
  • Have students draw two identical trapezoids on grid paper. Make sure they use a straight edge.
  • Tell them to use a strategy to figure out the area of their trapezoid (from Monday’s lesson).
  • Have them cut out each trapezoid. Ask them to see if they can figure out how they could use the two trapezoids to make a parallelogram.
  • Ask them to reflect on what they notice.
  • Encourage them to try more trapezoids.

After:

  • Have students share their trapezoids and what they discovered. Just as with the triangle, one can rotate the second trapezoid to make a parallelogram. For example:
    • The two trapezoids made a parallelogram with a base of 11 and a height of 4, so the area of the parallelogram is 44. There are two trapezoids, so each trapezoid has an area of 22.
  • Illustrate how it is the two parallel sides which connect to make the base of the parallelogram. Usually these parallel sides are referred to as the two bases of the trapezoid.
  • Have a discussion about how they could generalize how to find the area of a trapezoid.
    • If the trapezoid is duplicated, you can make a parallelogram with the same height. The base of the parallelogram is made up of the sum of the bases of the trapezoid. So to get the area of a trapezoid, add the two bases, then multiply the sum by the height. Then divide by 2.
    • Alternatively this could be expressed as half of the base times the sum of the heights.
  • Have students determine the area of some additional trapezoids.
  • To extend some students, ask them to build complex shapes made up of parallelograms, triangles, and/or trapezoids. They can make a shape and challenge a classmate to determine the area.

Focus: Perimeter of complex shapes

Before:

  • Ask the class what they can tell you about perimeter.
  • Show a complex shape constructed with colour tiles (or use a grid or geoboard). For example:
  • Ask them to determine the perimeter of this shape (18 units).
  • Tell them you are going to add one more square, and ask them to think about what the perimeter of the new shape would be, without counting at first.
  • Tell them they may now use a counting strategy to determine the perimeter. Many will be surprised to find out the perimeter has not changed. Ask them to think about and share why the perimeter didn’t change in this case.
    • Even though the new shape added 2 new sides, its 2 other sides covered 2 of the original sides, so there isn’t a change.
  • Ask them if there is a place they could add a new square to that would change the perimeter.
    • If you place a square so that it is only covering one side, then the perimeter will increase by 2 (3 new sides, minus 1 lost side).

During:

  • Provide students with pattern blocks (hexagons, trapezoids, rhombuses, and triangles only). Explain that each side can be considered one unit, except for the long side of the trapezoid which is two units.
  • Have them work in pairs or small groups.
  • Ask them to use pattern blocks to make a shape that has a perimeter of 18 units.
  • Ask them to think about whether they could add or remove any blocks from their shape, and still have a perimeter of 18 units.
  • If any finish early, encourage them to try to build other shapes, making them very different from their first shape.

After:

  • Have several students show their shape. Ask others to confirm that each shape has a perimeter of 18 units. For example, here are two different shapes made using Polypad.
  • Ask the class what strategy they used to build their shape.
    • Some may have just used a bunch of blocks at first, then made changes to get to 18 units.
    • Others may have built it one block at a time, keeping track of the perimeter along the way.
  • Ask if any were able to add or remove a block and still have a perimeter of 18 units.
  • Have them return to their shape. Without adding or removing any blocks, can they re-arrange the blocks to make a larger perimeter? smaller?
  • Provide other interesting problems to explore. For example:
    • Using 10 triangles only, what is the least perimeter shape you could build. What is the greatest perimeter? What if you used 10 hexagons instead?
  • Ask students to explore the perimeter of other shapes that they make. They can continue using pattern blocks, or change to colour tiles, cuisenaire rods, or grid paper.

This concludes the learning for this unit.

Suggestions for Assessment

By the end of Grade 6, students should be able to apply a formula (generalized strategy) to determine the areas of parallelograms, triangles, and trapezoids. They should also be able to determine the perimeter of a complex shape that they are given or that they build themselves.

Suggested Links and Resources

Key Measurement and Geometry Concept 2: Angles and Triangles

Overview

As students have explored shapes in earlier grades, they would have encountered angles, mostly as corners of 2D polygons. Grade 6 is the year that students explore angles in depth. An angle is the measure of rotation around a point (called the vertex). It is usually represented using two rays.

In Grade 5 students learned about rotations using the language of turns. In Grade 6, students learn classifying language. A quarter-turn is a right angle. A half-turn is a straight angle. In Grades 6 to 11, angles are measured in degrees, and there are 360° in a full turn, so a right angle is 90° and a straight angle is 180°. In addition to right and straight, students learn to classify angles as acute, obtuse, or reflex, as shown below:

Using referents of familiar angles (e.g., 45°, 90°, 180°), students learn to estimate angles. They then learn to use a protractor to measure angles. For reflex angles, students may use a 360° protractor, or use a regular protractor and some reasoning. Finally they use a protractor to construct angles of a given measure.

 

Students explore angles in polygons. Beginning by discovering that the sum of angles in a triangle is 180°, students can figure out the sum of angles in other polygons by figuring out how many triangles make up that polygon. Generally, however many sides a polygon has, there will be 2 fewer triangles. For example, an octagon (8-sided polygon) can be divided into 6 triangles. So the sum of angles in an octagon is 180 x 6 = 1080°.

 

Returning to the classifications of angles, angles are also used as one way to classify triangles.

Triangles can also be classified depending upon how many sides of the triangle are congruent (same length).

Because triangles can be classified in two ways, these ways can be combined. For example, one can imagine an isosceles right triangle.

Measurement and Geometry Foundations:
  • Naming polygons (triangle, quadrilateral, pentagon, etc)
  • Rotations/turns
  • Multiplicative number relationships (half, quarter)
Progression:
  • Meaning of an angle, and vocabulary involving an angle (ray, vertex)
  • Classifying angles (right, straight, then acute, obtuse, and reflex)
  • Estimating angle measures
  • Measuring angles using a protractor
  • Measuring angles using reasoning
  • Constructing angles
  • Determining the sum of angles in a triangle, then other polygons
  • Classifying triangles by angles
  • Classifying triangles by side lengths
Sample Week at a Glance:
Download as a PDF

This sample week starts at the beginning of this unit.

Focus: Classifying angles

Before:

  • Show a rotation of an image. Start with the original image and a rotation point, then draw its rotated image. For example:
  • Explain that the amount of rotation is measured by an angle. Add to the drawing to show the parts of the angle representation, and label with vocabulary.
  • You may also want to do a physical rotation of an object to illustrate the action of rotation. For example, hold arms together then open them apart, or stand in place, then rotate.
  • Show two new illustrations of rotations to introduce the language of a right angle (it’s okay if students think about this like an L), and a straight These words will be helpful for the next activity. For the right angle, show how a square is used to indicate it.

During:

  • This activity can be done as a whole class, in small groups, or in pairs.
  • Show an image or picture that has many angles in it, including angles of different types. For example:
  • Ask the class to identify the angles that they see. In the image above, the polygon has 9 angles in its interior.
  • Ask the class to sort these angles using sorting rules that they develop.

After:

  • Ask the class what they noticed, and how they decided upon a set of sorting rules. Choose the set of rules (or provide if not groups come up with it) that matches the convention for classifying angles:
    • Angles less than a right angle: B, H
    • Right angles: A, E
    • Angles bigger than a right angle but less than a straight angle: D, F, I
    • Angles bigger than a straight angle: C, G
  • Provide names for these sorting rules: acute, obtuse, right, reflex. You may wish to colour code and label the diagram.
  • Students may struggle to understand a straight angle (it doesn’t look like an angle) and a reflex angle (it looks acute or obtuse). Remind students that angles are measures of rotations, so it makes sense to rotate something a half-turn, or more than a half-turn.
  • You may decide to introduce angle measures to bring numerical definitions to the types of angles (e.g., an obtuse angle is between 90° and 180°), or wait until the next class.
  • Have students go around the classroom, outside, or at home, to identify angles of different types. They can record what they find in a journal, or take photos and annotate them to put in a portfolio.

Focus: Measuring Angles

Before:

  • Ask the class if they have any ideas about how angles are measured.
    • Some may have familiarity with 90° being a right angle
    • Some may be aware of 360° being a full circle from skateboarding or other sports
  • Tell the class that one way to measure angles is to give a number for a full rotation, namely 360°. So if 360° is a full rotation, explore other benchmark angle measurements visually. For example:
  • Provide several angles and ask the class (in pairs) to use the benchmarks to estimate the measure of each angle. Be sure to include at least one acute, one obtuse, and two reflex angles (one larger and one smaller than 270°).

During:

  • Demonstrate how to use a protractor.
  • There are several things to be mindful of when measuring an angle:
    • Align the vertex with the bottom centre of the protractor
    • Align one of the rays with the 0° line
    • Decide which range of numbers to read depending on if the angle is acute or obtuse.
    • Recognize which numbers the angle is in between. For example, the angle above is 2° away from 40°. The measure is 42°, not 38°.
  • Have the class work in pairs to measure each angle that they had previously estimated.

After:

  • Have the class share their measurements. For the reflex angles, they need to use some reasoning (unless they are using a 360° protractor). For example:
  • If the angle were acute, it would be 42°. As a reflex angle, we can calculate its measurement as 360° – 42° = 318°.
  • Provide some other angles for students to measure as practice, and/or they may measure the angles that they had collected the previous class. They could also search for images of angles in real-life contexts (e.g., slanted roofs, ramps) and measure those angles.
  • In addition to getting angle measures using a protractor, students can also determine angle measures using reasoning. For example:

Focus: Constructing Angles

Before:

  • Discuss the following problem:
    • Noah was asked to construct an angle of 114°. His construction is below. Is he correct? If not, what advice would you give him?
  • Beyond just saying that Noah is incorrect because he constructed an acute angle and 114° is obtuse, students should recognize why he constructed his angle this way. He correctly identified 114° on the protractor, but he should have chosen the other place that 114° appears on the protractor so that the angle would be obtuse.

During:

  • Model for the class how to construct an angle. Key steps are:
    • Draw a ray.
    • Align the centre of the protractor with the vertex of the ray.
    • Align the base-line (where 0° is) with the ray.
    • Identify where the desired measurement is on the protractor, keeping in mind whether the angle is acute or obtuse.
    • Make a mark at that measurement value.
    • Remove the protractor and draw the other ray.
    • For a reflex angle, they’ll need to do some reasoning first unless they are using a 360° protractor.
  • Provide the class several angle measurements (of different types, including reflex) and ask them to work in pairs to construct them.

After:

  • Have a few students share their angle constructions. Discuss any challenges they had.

Have students construct a few more angles of their choice. They can give their angles to a partner to measure and check.

Focus: Sum of Angles in Polygons

Before:

  • Have students explore interior angles in pattern blocks. Rather than use protractors, they can put blocks together and use reasoning.
    • For example, use the blocks to make a benchmark angle:
    • Use angles from one block to determine angles in another block:
  • Ask students to add up the angles for each pattern block and share what they noticed.
    • Triangle: 180°
    • Quadrilaterals (red, blue, brown, orange): 360°
    • Hexagon: 720°
  • They may notice these are all multiples of 180°, but it’s okay if they don’t. That understanding will emerge in today’s lesson.

During:

  • Ask students to construct a triangle and find its angle sum by doing the following:
    • Draw the base of the triangle. They should make it at least 5 cm so that the protractor can be used.
    • Choose two acute angles (their choice). Construct one of the angles at one end of the base, and the other angle at the other end.
    • Measure the third angle.
    • Determine the sum of the three angles.
  • They’ll notice that everyone got the same sum of 180° (some may be off by a degree or two due to the accuracy of their constructions).
  • So for any triangle, the sum of the interior angles is 180°. To highlight this generalization, you may wish to show this triangle created in Desmos. The sum is shown on the left. You can drag each vertex around to make several different triangles. The sum is always the same.

  • An alternative exploration is to create any triangle, cut out each vertex, and line them up to see the angles add up to a straight angle.
  • Show them a quadrilateral with its angle measurements, or use this Desmos version.
  • Ask them to add the angles. What did they notice? (The sum is 360°)
  • Ask why it makes sense that this is the same as the sum of two triangles. (A quadrilateral can be made into two triangles)
  • In general, the sum of the angles in any polygon can be determined by dividing the polygon into triangles. Have students work in pairs.
    • Draw different polygons of their choice.
    • Divide each polygon into triangles to determine the sum of the interior angles of each polygon.
    • Ask them to keep track of their results in a table:

After:

  • Ask for students to share what they found, starting with an example of a pentagon, then hexagon, then other polygons.
  • As examples are shared, fill in a table. For example:
  • Ask the class if they notice a way to generalize how to determine the sum of angles in a polygon.
    • The number of triangles is always two less than the number of sides. So for a polygon with n sides, the sum of its interior angles will be (n –2) x 180°.
  • If none of the student examples were concave polygons (one with a reflex angle), draw a concave polygon and ask if they think the rule would apply for the sum of the interior angles. If they apply the same strategy of using triangles, they’ll see that the rule does apply.
  • Have students explore other problems involving angles in polygons. For example:
    • Applying the rule to determine the sum of interior angles of various polygons.
    • Given the sum of the interior angles of a polygon, how many sides are there?
    • Determining the measure of every angle of a regular polygon (all sides equal)
    • Determining the value of a missing angle in a polygon if all of the other interior angles are given.

Focus: Classifying Triangles

Before:

  • Explain to the class that they will be looking at two different ways to classify triangles.
  • How do the angles make these triangles different from each other?
    • The first only has acute angles → acute triangle
    • The second has a right angle → right triangle
    • The third has an obtuse angle → obtuse triangle
  • How do the sides make these triangles different from each other? (Explain that the marks indicate equal sides)
    • The first has all three sides equal → equilateral triangle
    • The second has two sides equal → isosceles triangle
    • The third has no equal sides → scalene triangle

During:

  • Provide, or ask students to create a table as follows:
  • Explain that any triangle can be classified in two ways: by angles or by sides.
  • Tell students (in pairs or small groups) to draw a sketch of a triangle in each box, if one is possible. For example, in the top left box, they would sketch a triangle that is acute and scalene.

After:

  • Have a class discussion about which triangles were possible, and what each triangle looks like. For example:
    • An equilateral triangle has to be acute since all of its angles are 60°.
  • Provide students a few triangles to classify in two ways. They could include measurements rather than markings.
  • Explore some problems involving classifying triangles. For example:
    • A triangle has one angle that is 32° and another angle that is 58°. What kind of triangle is it? (it’s a right triangle, and with further reasoning they may also identify it as scalene).
      • A similar question could be asked so that the third angle ends up being obtuse.
    • An isosceles triangle has a perimeter of 20 cm. What could it’s side lengths be? (Even with whole numbers, there are still many possible, but not as many as students may think. If the equal sides are 5 cm or less, the triangle cannot be made because the third side would be too long).

This concludes the learning for this unit.

Suggestions for Assessment

By the end of Grade 6, students should be able to classify angles both visually and by given measurements. They should be able to measure angles by estimating using benchmarks, using a protractor, or reasoning from known angles. Students should be able to construct any whole number angle measurement up to 360°. Students should be able to solve problems involving angle measurements, including the sum of interior angles in polygons. Finally, students should be able to classify triangles by angles and by sides.

Suggested Links and Resources

Key Measurement and Geometry Concept 3: Transformations

Overview

In Grade 5, students learned single transformations: slide/translation, flip/reflection, and turn/rotation. In Grade 6 there are two further developments that occur together:

  • Combining transformations (e.g., a reflection, followed by a translation)
  • Mapping transformations in the first quadrant of a Cartesian plane (coordinate grid)

Students need to learn how the Cartesian plane (named after Descartes) describes the position of points on a grid. Each point has a horizontal position (x-coordinate) and a vertical position (y-coordinate). Together, these coordinates form what is called an ordered pair. They also learn the language of axes, and the origin at (0, 0). For example, for the ordered pair (6, 4):

Students in Grade 6 plot shapes on the grid using coordinates (ordered pairs), perform transformations, then identify the coordinates of the image (transformed shape). Because the Cartesian plane is new to them, it makes sense first to graph single transformations before graphing combinations of transformations. For example, here’s a translation of a parallelogram 5 to the left and 2 up:

Notice that every coordinate has changed in the same way. The x-coordinates of the image are all 5 less than the x-coordinates of the original shape. Similarly for the y-coordinates. Though not required, we could describe this transformation symbolically as (x – 5, y + 2).

Below are the other two transformation types:

Note that a reflection requires a line (usually horizontal or vertical) to reflect in, and a rotation requires a point to rotate around, a direction (clockwise or counter-clockwise) and an angle (quarter-turn, half-turn, or using degrees like 90° or 180°).

Students should then do combinations of two transformations, and finally using all three transformations. Below is a combination of all 3. The intermediate images are included for clarity.

First there was a quarter-turn rotation counter-clockwise about the lower left corner, then a reflection in the line through y = 6, then a translation 3 to the left and 1 up. Students should be able to perform these transformations when given the description. They should also be able to look at an image and figure out which transformations led to it. There are usually several different combinations that can lead to the same image.

There are many designs and examples of art, particularly in Indigenous art, that make use of transformations.

 

Measurement and Geometry Foundations:
  • Triangles and quadrilaterals of various types, and other polygons
  • Single transformations: slide/translation, flip/reflection, and turn/rotation
Progression:
  • Understanding plotting points on a Cartesian plane, and related vocabulary (axes, origin, ordered pair)
  • Plotting single transformations on a Cartesian plane
  • Plotting a combination of two transformations on a Cartesian plane
  • Plotting a combination of three or more transformations on a Cartesian plane
  • Identifying which transformations led to an image (could be integrated with above)
  • Identifying and/or applying transformations in art and design
  • Extending: reflections about a horizontal line, rotations about a point that is not on the shape
Sample Week at a Glance
Download as a PDF

This sample week starts at the beginning of this unit.

Focus: The Cartesian Plane

Before:

  • Provide a map such as the example below. It is intended to provide a familiar situation that deals with location and direction.
  • Ask the class to describe where each location is (separately) relative to home. For example, the library is 3 blocks to the right, and 4 blocks up. Settle on a convention of going horizontal first, and then vertical.
  • Add detail to this map to turn it into a Cartesian plane (coordinate grid), providing vocabulary to the students as you go.
  • Explain that just as they could describe a position by the number of blocks in each direction, in a Cartesian plane we use coordinates to describe the positions of points – first with the x-coordinate, and then the y-coordinate. Together, these coordinates form an ordered pair, and are represented using the numbers within brackets. For example, the library is at the point (3, 4).

During:

  • Have students play a game similar to Battleship™ with a partner. The difference is that coordinate points will be used (i.e., (4, 5) instead of F4, and points will make up the ships). Or play a different point-target type of game. Below is what a game board may look like for one’s ships:

After:

  • The purpose of this game is to practice what they learned about plotting points on a coordinate grid. Watch for students who switch x and y as this is the most likely mistake when first learning about coordinates.

Focus: Plotting Single Transformations in the Cartesian Plane

Before:

  • Show a shape with three different transformations. For example:
  • Ask the class to describe each transformation. For example:
    • A is a slide (or translation) of 3 to the left and 4 up
    • B is a quarter-turn clockwise turn (or rotation) about the lower left point
    • C is a flip (or reflection) about the horizontal line through y = 6
  • Ask them where the original point (4, 2) moves to in each image.
    • (1, 6) on A, (6, 4) on B, and (4, 10) on C
    • Note that identifying where a point has moved is called mapping
  • The purpose of this Before activity is to review what they learned about transformations in Grade 5. You may need to have them work through some more examples.

During:

  • Provide grid paper or a ready-made coordinate grid that is at least 15 by 15 (a large grid provides more flexibility).
  • Have students work in pairs or small groups. Ask them to plot the point (5, 6). Tell them to make a shape that includes this point. They will then be performing 3 separate transformations (not combined):
    • Translation 3 to the right and 4 down
    • Reflection in the vertical line through x = 9
    • Half-turn rotation about the point (5, 6)
  • As students are working on their graphs, circulate and support as needed. Some students may need to start with a different shape if one of their transformed images ends up outside of the grid, or they could extend their grid.
  • Sometimes a transformed image may overlap with the original shape. Though it may be ‘messier’ to look at, it is not an issue.

After:

  • Have a few groups share their shape and the transformations. Ask them to explain how they knew where to map the shape with each transformation.
    • Some may focus on the key points of their shape.
    • Others may focus on mapping one point (5, 6), and then draw the shape appropriately from there.
  • Confirm that the point (5, 6) was mapped to the same place no matter the original shape. For example:
  • Possible situations:
    • A reflection could be the same as a full-turn rotation
  • Have the class practice some more single transformations on a coordinate grid. You could provide some specifically, and/or students could create their own shapes and transformations.
  • Some students may appreciate some more challenging tasks. For example:
    • Given a mapping, what could be the transformation? For example, if the original shape has a point (7, 3) and it maps to (7,7), which transformations could have led to that mapping?
    • Do rotations about a point that is not on the original shape.
    • Do reflections about a diagonal line.

Focus: Plotting Combinations of Two Transformations in the Cartesian Plane

Before:

  • Show the class a shape that has had two transformations. Indicate a reflecting line. For example:
  • Do a think-pair-share about which two transformations led to where the image ended up.
  • There are two possible combinations:
    • Reflect in the line through x = 6, then translate 1 to the right and 4 down, or
    • Translate 1 to the left and 4 down, then reflect in the line through x = 6
  • This is an example that shows that when performing a combination of transformations, the order in which they are done may make a difference.
  • It helps to sketch in the intermediate transformations. The image below shows the intermediate stage for each option.
  • Show how a combination of transformations can also be done point-by-point. For example, consider the combination that has a reflection first, then the translation:
    • (6,7) stays where it is after the reflection. Then it moves 1 to the right and 4 down to become (7, 3).
    • (1, 9) moves to (11,9) after the reflection. Then with the translation it moves to (12, 5).

During:

  • This activity carries out a similar way as the day before.
  • Provide grid paper or two ready-made coordinate grids that are each at least 15 by 15 (a large grid provides more flexibility).
  • Have students work in pairs or small groups. Ask them to plot the point (7, 4). Tell them to make a shape that includes this point. They will then be performing a different combination of transformations on each grid.
    • Rotate a quarter-turn clockwise around the point (7, 4), then translate 2 to the left and 3 up
    • Rotate a half-turn around the point (7, 4), then reflect in a line through x = 4
  • As students are working on their graphs, circulate and support as needed.
  • Doing the rotation second is more challenging since the rotation point is no longer on the shape. For those who would like the challenge, ask them to perform each combination in the reverse order on the same grid. For example:
    • Translate 2 to the left and 3 up, then rotate a quarter-turn clockwise around the point (7, 4)
    • Reflect in the line through x = 4, then rotate a half-turn around the point (7, 4).

After:

  • Have a few groups share their shape and the transformations. Ask them to explain how they knew where to map the shape with each transformation.
    • Some may focus on the key points of their shape (i.e., the point-by-point strategy from the Before).
    • Others may focus on mapping one point (7, 4), and then draw the shape appropriately from there.
  • Confirm that the point (7, 4) was mapped to the same place no matter the original shape. For example (this also shows C and D):
  • Have the class practice some more combinations of transformations on a coordinate grid. You could provide some specifically, and/or students could create their own shapes and transformations. Be sure to include these combinations:
    • A reflection then a translation
    • A translation then a reflection
    • A rotation then a translation
    • A rotation then a reflection
  • Some students may appreciate some more challenging tasks. For example:
    • Given a mapping, what could be the transformation?
    • Do a combination that does the rotation second about a point that is on the original shape.
    • Do reflections about a diagonal line.

Focus: Plotting Combinations of Multiple Transformations in the Cartesian Plane

Before:

  • This activity is modelled with the class so that they can do it on their own after.
  • Start with a shape on a coordinate grid.
  • Roll a 6-sided die. Perform a transformation based upon which number is rolled:
    • 1: Translation
    • 2: Reflection in a vertical line
    • 3: Reflection in a horizontal line
    • 4: Quarter-turn clockwise rotation
    • 5: Quarter-turn counter-clockwise rotation
    • 6: Half-turn rotation
  • After each roll, decide as a class the specific transformation:
    • Translation: how far in each direction?
    • Reflection: about which line?
    • Rotation: around which point? (it’s recommended to choose a point on the shape).
  • Do at least 4 rolls until the class is comfortable about how the activity is played.
  • Here’s an example involving 5 rolls (generated randomly on Polypad). So you can see the order, the colours of the transformations correspond to the colours of the dice.

During:

  • Have students work in small groups. Give each group a 6-sided die and grid paper or a prepared coordinate grid.
  • Remind them that the dice determines which transformation they use, and they can make a choice after each rolls about the specifics of each transformation.

After:

  • Have students share their graphs with another group or two.
  • Have a class discussion with some general questions about the activity. For example:
    • How did you decide which shape to use?
    • Which transformations were easier?
    • Which transformations were more challenging?
    • How did you decide the specific transformations to make?
    • What other things did you notice or wonder about?
  • After discussion, consider adapting the activity to be more like a game. The objective of the game is to arrive back to the original shape. How many rolls did it take them? Which rolls were they hoping for? What is the fewest number of rolls possible? They could play a few times. Note that the fewest number of rolls would be two, if they rolled a double 1, 2, 3, or 6 in their first two rolls.

Focus: Applying Transformations to Art and Design

Before:

  • Show students a few examples of art or design that makes use of translations, reflections, and/or rotations. Tesselations (from Escher and others) are great examples.
  • Ask them to notice which transformations were used, and if there could be different ways to describe them.
  • Symmetry and other transformations are an important part of Indigenous art. Explore and connect with your local Indigenous community to see examples of their art, and how it is important to them. You may also wish to explore the links in the Indigenous Connections page of this website.

During:

  • Have students create art or design that integrate transformations. With the focus of being a combination of transformations, a grid would not be necessary to use, but it could be done on a grid as well.
  • They could also make use of tools such as using Polypad.

After:

  • Have students share their art/designs. Rather than students explaining their own work, have peers identify and discuss them.

This concludes the learning for this unit.

Suggestions for Assessment

By the end of this unit, students should be able to understand combinations of two or more transformations (translations, reflections, and rotations), and how to represent these transformations on a Cartesian plane (coordinate grid) using positive coordinates (i.e., first quadrant). They can show their understanding by identifying, describing, and creating these combinations of transformations.

Suggested Links and Resources

Combining Transformations (nRich, which also has many other transformation problems)

Elementary

Coast Metro Math Project

Math Project Home