E. Spatial Sense · Shapes, Maps and MeasurementAngles and Square Corners
E1.1About 30 minutes
Learning goals- I can find the angles in a shape.
- I can check whether an angle is a square corner, smaller, or bigger.
- I can compare two angles and tell which is bigger.
- I can sort shapes using more than one attribute (sides, side lengths and angles).
Words to know
angle
The amount of turn between two sides that meet at a vertex. A wide-open angle is bigger. A pointy, closed-up angle is smaller.
square corner
An angle that is exactly like the corner of a sheet of paper. It is also called a right angle.
right angle
Another name for a square corner.
attribute
Something we can notice about a shape, like its number of sides, its side lengths or its angles.
Open and close
Hold your arm out straight. Now bend it at the elbow, just a little. Bend it more. Now open it wide again.
Your arm is making angles! When it is almost closed, the angle is small. When it is wide open, the angle is big.
What is an angle?
Every vertex of a polygon has an angle. The angle tells how open the two sides are where they meet.
- A small angle is pointy and closed up.
- A big angle is wide open.
The size of an angle does not depend on how long the sides are. It only depends on how wide the sides open.
The square corner
The corner of a sheet of paper is a special angle called a square corner. Grown-ups also call it a right angle.
Square corners are everywhere: the corners of a door, a book, a window, a tile. In drawings, we show a square corner with a tiny square at the vertex.
Make a corner checker
Take any sheet of paper. Its corner is a square corner. Use it to check angles:
- If the angle fits exactly under the corner, it is a square corner.
- If the angle is inside the paper corner with space left over, it is smaller than a square corner.
- If the sides stick out past the paper corner, it is bigger than a square corner.
Worked exampleExample 1: Check the corners of a rectangle
How many square corners does a rectangle have?
- Put your paper corner on the first vertex. It fits exactly. That is 1.
- Check the next vertex. It fits. That is 2.
- Check the last two vertices. They fit too. That is 3, 4.
A rectangle has 4 square corners. A square has 4 square corners too.
Worked exampleExample 2: Compare two angles
A slice of pizza has a pointy tip. A stop sign has 8 corners. Which angle is bigger: the pizza tip or a stop sign corner?
- Check the pizza tip with your paper corner. It fits inside with space left over. It is smaller than a square corner.
- Check a stop sign corner (draw one, or use a picture). The sides stick out past the paper corner. It is bigger than a square corner.
- Smaller than a square corner is less than bigger than a square corner.
The stop sign corner is the bigger angle. It is more wide open.
Sorting with more than one attribute
An attribute is something you can notice about a shape. Now you know three:
- Number of sides (and vertices)
- Side lengths (Are any the same length?)
- Angles (Are there square corners? Are angles big or small?)
Sorting by two attributes at once is like a puzzle.
Worked exampleExample 3: Sort by two rules
Look at shapes A to E. Which shapes have 4 sides AND square corners? Which have square corners but NOT 4 sides?
- Find the shapes with square corners: A (square), C (one square corner), E (rectangle).
- Of those, which have 4 sides? A and E.
- C has a square corner but only 3 sides.
- B (hexagon) and D (triangle) have no square corners. D's angles are smaller than a square corner. B's angles are bigger.
4 sides and square corners: A and E. Square corner but not 4 sides: C.
| Has a square corner | No square corner |
|---|
| 4 sides | A, E | (none here) |
| Not 4 sides | C | B, D |
Just for funWhy was the wide-open angle always grumpy?
Because it was never right! (Remember, a square corner is also called a right angle.)
Hands-on activitySquare corner hunt
You need: a sheet of paper (your corner checker), a pencil, and a list.
- Walk around your home. Check 10 corners with your paper corner: a table, a cereal box, a picture frame, a cushion, a cutting board, a book...
- Make three columns on your list: smaller, square corner, bigger. Write each object in a column.
- Fold a paper circle (like a coffee filter or a traced plate) in half, then in half again. You made a square corner! Check it.
- Bonus: open a pair of scissors or a laptop a little, then more. Say whether the angle is getting bigger or smaller.
Math talk: "Which angle was the smallest one you found? How do you know?"
Your turn
1What is another name for a square corner?
2How many square corners does a square have?
3Open a door a little. Then open it wide. When is the angle bigger?
When the door is open wide. The more it opens, the bigger the angle.
4A triangle has 3 pointy corners. Each corner fits inside a paper corner with room to spare. Are they smaller or bigger than a square corner?
Smaller than a square corner.
5Ben draws an angle with very long sides. Ava draws an angle with short sides but much more open. Whose angle is bigger?
Ava's. The size of an angle is how open it is, not how long the sides are.
6Name 2 things at home that have square corners.
Answers will vary. Examples: a book, a door, a window, a cereal box, a sheet of paper, a phone.
More practice: warm-up
1Where do you find angles in a shape: at the sides or at the vertices?
At the vertices (corners), where two sides meet.
2What tool can you use to check for a square corner?
The corner of a sheet of paper (or a book, or a card).
3Does a rectangle have square corners?
Yes, all 4 corners are square corners.
4Does a circle have any angles?
No. A circle has no sides and no vertices, so it has no angles.
5A pair of scissors opens wider and wider. Is the angle getting bigger or smaller?
More practice: on your own
1A shape has 4 sides, but none of its corners are square corners. Is it a rectangle? Explain.
No. A rectangle must have 4 square corners. This shape is a quadrilateral, but not a rectangle (it could be a parallelogram).
2A regular hexagon (like a honeycomb cell) has angles bigger than a square corner. A stop sign's angles are even bigger. Put these in order from smallest to biggest: stop sign corner, square corner, hexagon corner.
square corner, hexagon corner, stop sign corner. The more sides a shape like this has, the more open its corners are.
3Can a triangle have a square corner?
Yes. Fold a sheet of paper from one corner to the opposite corner and cut. Each half is a triangle with one square corner.
4Sort these by the rule "has 4 square corners": book cover, slice of pizza, stop sign, door.
Has 4 square corners: book cover, door. Does not: slice of pizza, stop sign.
5Mia says, "All shapes with 4 sides have square corners." Is she right? Explain.
No. A parallelogram (a pushed-over rectangle) has 4 sides, but two of its angles are smaller than a square corner and two are bigger.
Challenge
1Can a triangle have 2 square corners? Try to draw one. What happens?
No. If two corners are square, the two sides going up from them stand straight up side by side and never meet. You can't close the shape, so it can't be a triangle.
2I have 4 sides that are all the same length, but my angles are not square corners. Am I a square? What could I be?
Not a square (a square must have square corners). You are a rhombus, a quadrilateral with 4 equal sides that is "pushed over". (Calling it a quadrilateral is also correct.)
3Make a sorting rule that puts a square and a rectangle in the same group, but keeps a triangle with a square corner out.
Answers will vary. Example: "Shapes with 4 square corners", or "Shapes with 4 sides". A triangle has only 3 sides and only 1 square corner.
Quiz
1. What is a square corner also called?
2. You check an angle with a paper corner. The sides stick out past the paper. The angle is...
3. How many square corners does a rectangle have?
4. What makes an angle bigger?
5. Which object has a corner that is smaller than a square corner?
6. Which shape has 4 sides but no square corners?
Big idea: An angle is the amount of "opening" between two sides. Grade 2 children compare angles to a square corner (right angle) using a paper corner. They do not measure angles in degrees yet.
Common difficulties:
- Thinking longer sides make a bigger angle. Draw two angles with the same opening, one with long arms and one with short arms. Check both with a paper corner.
- Holding the corner checker wrongly. Line one edge of the paper up with one side of the angle, with the paper's corner right on the vertex.
- Thinking only squares and rectangles have square corners. Some triangles and other shapes do too.
How to help: Ask your child to open a book, a laptop or a door "to a square corner", then "to less than" and "more than" a square corner. Make sorting games with paper shapes: "Find all the shapes with 4 sides and square corners."