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E. Spatial Sense · Spatial Sense: Angles, Tessellations, Transformations, Scale, Pythagoras and Composite Measurement

Spatial Sense: Angles, Tessellations, Transformations, Scale, Pythagoras and Composite Measurement: Unit Review

A1 E1.1 E1.2 E1.3 E1.4 E2.1 E2.2 E2.3 E2.4About 40 minutes
Learning goals
  • I can use angle properties of parallel lines and polygons, and explain which shapes tessellate and the transformations in tessellations.
  • I can perform and predict translations, reflections, rotations and dilations on the Cartesian plane.
  • I can choose scales for models, and use scale drawings to find real lengths and areas.
  • I can use very large and very small metric units, the Pythagorean theorem, and formulas for composite shapes and objects.

Words to know

transversal
A line that crosses two or more lines.
(n − 2) × 180°
The sum of the interior angles of a polygon with n sides.
tessellation
A pattern of shapes with no gaps and no overlaps; the angles at each vertex add to 360°.
dilation
An enlargement or reduction from a centre by a scale factor k. Lengths × k, area × k².
scale
A ratio like 1 : 200 comparing drawing or model lengths with real lengths.
a² + b² = c²
The Pythagorean relationship for a right triangle with hypotenuse c.
composite
Made of two or more simpler shapes or objects.

Let's review!

In this unit you used angle properties of parallel lines and polygons, discovered which shapes tessellate and the transformations inside tessellations, transformed and dilated shapes on the Cartesian plane, built models from views at an appropriate scale, calculated real lengths and areas from scale drawings, worked with very large and very small metric units, described and used the Pythagorean relationship, and found the perimeter, area, surface area and volume of composite shapes and objects.

This review has a few questions from every lesson. Try each one, then press Show answer to check. Use π ≈ 3.14.

xxF: correspondingequalxxZ: alternateequal180° − xxC: co-interioradd to 180°
The letter shapes for parallel lines. F: corresponding angles are equal. Z: alternate angles are equal. C: co-interior angles add to 180°.

Angles Formed by Parallel Lines and a Transversal

1

Two parallel lines are cut by a transversal. One angle is 68°. What is its corresponding angle?

2

Two parallel lines are cut by a transversal. One co-interior angle is 103°. Find the other.

3

A transversal crosses two lines. The alternate angles are 58° and 62°. Are the lines parallel?

Interior and Exterior Angles of Polygons

1

Find the sum of the interior angles of an octagon.

2

Find each interior angle of a regular decagon.

3

Each exterior angle of a regular polygon is 20°. How many sides does it have?

4

A pentagon has angles of 100°, 105°, 110°, 115° and x. Find x.

Which Shapes Tessellate?

1

Why do regular hexagons tessellate on their own?

2

Will regular pentagons tessellate on their own?

3

A regular octagon and a square share a vertex with another regular octagon. Do the angles fit?

Transformations in Tessellations

1

What single transformation moves one brick in a brick wall onto the next brick in the same row?

2

What rotation around a vertex where three hexagons meet maps a hexagon tessellation onto itself?

3

A tile was made by cutting a piece from one side of a square and sliding it to the opposite side. What kind of tessellation does it make?

90°k = 2translationslidereflectionfliprotationturndilationenlarge or reduceimage is congruent (same size and shape)image is similar
The four transformations. Translations, reflections and rotations give a congruent image (same size and shape). A dilation gives a similar image. A reflection makes a mirror image; the others don't.

Reflections, Rotations and Combined Transformations

1

Reflect (4, −1) in the line y = x.

2

Rotate (5, 3) 90° clockwise about (2, 1).

3

Reflect a shape in x = 1 and then in x = 6. What single transformation does the same thing?

Dilations on the Cartesian Plane

1

Dilate (6, −9) by a scale factor of 1/3 from the origin.

2

Dilate (4, 5) by a scale factor of 3 from the centre (2, 1).

3

A shape with an area of 4 cm² is dilated by a scale factor of 5. What is the image's area?

Making Models from Views at an Appropriate Scale

1

Which view shows an object's depth and height?

2

A cottage is 9 m wide. How wide is a 1 : 50 model?

3

A canoe is 5 m long. A model must fit in a 40 cm box. Which scale fits: 1 : 10, 1 : 15 or 1 : 20?

Scale Drawings: Real Lengths, Real Areas and New Scales

1

A plan has a scale of 1 : 200. A room is 2.5 cm by 2 cm on the plan. Find its real area.

2

A map has a scale of 1 : 50 000. A lake is 6 cm long on the map. How long is it really?

3

A plan at 1 : 100 is redrawn at 1 : 25. What happens to the lengths and to the paper area?

Very Large and Very Small Metric Units

1

Convert 4.5 GB to megabytes.

2

Write 1 nanometre in metres as a power of 10.

3

Convert 3 000 µm to millimetres.

Just for fun

What do you call an angle that's really cute?

An acute angle.

The Pythagorean Relationship

1

Is 9, 40, 41 a right triangle?

2

The squares on the legs of a right triangle have areas of 49 and 576. How long is the hypotenuse?

3

Classify the triangle with sides 4, 6, 8.

Using the Pythagorean Theorem to Find Unknown Sides

1

Legs 12 and 16. Find the hypotenuse.

2

The hypotenuse is 26 and one leg is 10. Find the other leg.

3

Find the distance between (1, 1) and (4, 5).

Perimeter and Area of Composite Shapes

1

Find the area of a semicircle with a diameter of 12 cm.

2

A 10 m by 6 m rectangle has a semicircle on one 6 m end. Find the perimeter.

3

A circle with a radius of 4 cm is cut from a 12 cm by 10 cm rectangle. What area is left?

Surface Area and Volume of Composite Objects

1

A box 6 cm × 4 cm × 5 cm has a cylinder (radius 1 cm, height 3 cm) on top. Find the total volume.

2

Three 1 cm cubes are glued in a row. What is the surface area?

3

A pipe is 50 cm long with outer radius 3 cm and inner radius 2 cm. Find the volume of material.

How to use this review

  • Work through one section at a time and write your steps, not just the answers.
  • Put a check mark beside each question you got right on your own. Put a star beside any you needed help with.
  • If you starred two or more questions in a section, go back to that lesson and redo its "More practice" questions.
  • Finish with the quiz. For every quiz mistake, read the explanation and find the lesson it came from.
  • Mistakes are useful: they show exactly what to practise next.
Check your understanding

Quiz

1.

Two parallel lines are cut by a transversal. Alternate angles are:

2.

What is each interior angle of a regular pentagon?

3.

Which regular polygon tessellates on its own?

4.

Reflect (2, −5) in the line y = x.

5.

A shape is dilated by a scale factor of 3. Its area is multiplied by:

6.

At a scale of 1 : 50, a 3 m wall is how long in a model?

7.

A room is 3 cm by 2 cm on a 1 : 200 plan. What is its real area?

8.

1 Gm is how many metres?

9.

A right triangle has a hypotenuse of 15 and a leg of 9. What is the other leg?

10.

A 4 cm cube is glued on top of an 8 cm cube. What is the surface area?

For parents and teachers

Use mistakes to choose what to revisit. Each review section matches one lesson.

  • Uses the parallel-line rules when lines aren't parallel, or mixes up which pairs are equal and which add to 180°: revisit Angles Formed by Parallel Lines and a Transversal. Redo the lined-paper angle hunt.
  • Uses n × 180° for the angle sum, or divides by n for a polygon that isn't regular: revisit Interior and Exterior Angles of Polygons. Draw diagonals from one vertex and count the triangles.
  • Guesses whether shapes tessellate instead of adding the angles at a vertex: revisit Which Shapes Tessellate?.
  • Can't tell a rotation from a reflection in a pattern: revisit Transformations in Tessellations. Label the corners of a tile and check the order.
  • Uses an origin rule for a different centre, or does a sequence in the wrong order: revisit Reflections, Rotations and Combined Transformations. Use move, turn, move back with tracing paper.
  • Multiplies coordinates when the centre isn't the origin, or multiplies area by k instead of k²: revisit Dilations on the Cartesian Plane.
  • Forgets to convert metres to centimetres, or mixes up which view shows which dimensions: revisit Making Models from Views at an Appropriate Scale.
  • Multiplies a drawing area by the scale number only once: revisit Scale Drawings: Real Lengths, Real Areas and New Scales. Convert lengths first, then find the area.
  • Multiplies when they should divide when changing prefixes, or mixes up M and m: revisit Very Large and Very Small Metric Units. Ask: is the new unit bigger or smaller?
  • Adds the sides instead of their squares, or uses the wrong side as the hypotenuse: revisit The Pythagorean Relationship. Redo the squares-on-the-sides activity.
  • Forgets the final square root, or adds when finding a leg: revisit Using the Pythagorean Theorem to Find Unknown Sides.
  • Counts inside joining lines in the perimeter, or uses the diameter in the area formula: revisit Perimeter and Area of Composite Shapes.
  • Includes hidden faces in surface area, or forgets to subtract a hole: revisit Surface Area and Volume of Composite Objects. Build a composite object from boxes.

A few mistakes are normal. Praise the strategy your child explains and the habit of checking, not only the right answer.