# 4 Equal Parts Of A Rectangle

(quad means four, lateral means side).

Cut out the square at the bottom of this page. Cut the square in half. Shade one half using your pencil. Rearrange the halves to create a new rectangle with no gaps or overlaps. Cut each equal part in half. Rearrange the new equal shares to create different polygons. Draw one of your new polygons from Part (d) below. By (date), when given (10) picture problems that show (2) shapes (circle, rectangle) and a number of equal parts, (name) will correctly draw lines to divide each shape into the. Appropriate amount of equal parts and circle the term that corresponds to that amount of parts (i.e. Halves, fourths, quarters), answering (8/10) problems correctly in (4 out of 5) activities. Since this line crosses the left and right edges of the whole rectangle, the four small triangles shown above (bounded between the purple and darker grey lines) have equal area, so the left rectangle is divided 1:4 and the right rectangle divided 2:3. Overall, the rectangle has been divided into four parts with ratio 1:2:3:4.

A Quadrilateral has four-sides, it is 2-dimensional (a flat shape), closed (the lines join up), and has straight sides.

## Try it Yourself

(Also see this on Interactive Quadrilaterals) ## Properties

• four sides (edges)
• four vertices (corners)
• interior angles that add to 360 degrees:

Try drawing a quadrilateral, and measure the angles. They should add to 360°

There are special types of quadrilateral:

Some types are also included in the definition of other types! For example a square, rhombus and rectangle are also parallelograms. See below for more details.

Let us look at each type in turn:

## The Rectangle

the little squares in each corner mean 'right angle'

A rectangle is a four-sided shape where every angle is a right angle (90°).

Also opposite sides are parallel and of equal length.

## The Square

the little squares in each corner mean 'right angle'

A square has equal sides (marked 's') and every angle is a right angle (90°)

Also opposite sides are parallel.

A square also fits the definition of a rectangle (all angles are 90°), and a rhombus (all sides are equal length).

## The Rhombus

A rhombus is a four-sided shape where all sides have equal length (marked 's').

Also opposite sides are parallel and opposite angles are equal.

Another interesting thing is that the diagonals (dashed lines) meet in the middle at a right angle. In other words they 'bisect' (cut in half) each other at right angles.

A rhombus is sometimes called a rhomb or a diamond.

## The Parallelogram

A parallelogram has opposite sides parallel and equal in length. Also opposite angles are equal (angles 'A' are the same, and angles 'B' are the same).

NOTE: Squares, Rectangles and Rhombuses are all Parallelograms!

### Example:

A parallelogram with:

• all sides equal and
• angles 'A' and 'B' as right angles

is a square!

## The Trapezoid (UK: Trapezium)

 Isosceles Trapezoid

A trapezoid(called a trapezium in the UK) has a pair of opposite sides parallel.

And a trapezium(called a trapezoid in the UK) is a quadrilateral with NO parallel sides:

 Trapezoid Trapezium In the US: a pair of parallel sides NO parallel sides In the UK: NO parallel sides a pair of parallel sides (the US and UK definitions are swapped over!)

An Isosceles trapezoid, as shown above, has left and right sides of equal length that join to the base at equal angles.

## The Kite

Hey, it looks like a kite (usually).

It has two pairs of sides:

Each pair is made of two equal-length sides that join up.

Also:

• the angles where the two pairs meet are equal.
• the diagonals, shown as dashed lines above, meet at a right angle.
• one of the diagonals bisects (cuts equally in half) the other.

... and that's it for the special quadrilaterals. The only regular (all sides equal and all angles equal) quadrilateral is a square. So all other quadrilaterals are irregular.

## The 'Family Tree' Chart

### Example: a square is also a rectangle.

So we include a square in the definition of a rectangle.

(We don't say 'Having all 90° angles makes it a rectangle except when all sides are equal then it is a square.')

This may seem odd, as in daily life we think of a square as not being a rectangle ... but in mathematics it is.

Using the chart below we can answer such questions as:

• Is a Square a type of Rectangle? (Yes)
• Is a Rectangle a type of Kite? (No)

Oh Yes! when two sides cross over, we call it a 'Complex' or 'Self-Intersecting' quadrilateral, like these:

They still have 4 sides, but two sides cross over.

## Polygon

A quadrilateral is a polygon. In fact it is a 4-sided polygon, just like a triangle is a 3-sided polygon, a pentagon is a 5-sided polygon, and so on.

## Play with Them

Now that you know the different types, you can play with the Interactive Quadrilaterals.

## Other Names

A quadrilateral can sometimes be called:

• a Quadrangle ('four angles'), so it sounds like 'triangle'
• a Tetragon ('four polygon'), so it sounds like 'pentagon', 'hexagon', etc. Alignments to Content Standards:1.G.A.3

Materials

• Paper cut outs of rectangles, circles, and squares

• Blank paper

Actions

Part One:

1. Give each pair of students a square and ask, â€œHow can you share the square equally so that you and your partner get the same size piece?â€ Â

2. Ask students to fold the paper to show how they could get two equal parts.

3. Call on student volunteers to share, asking the class as each example is displayed, â€œIs the paper shared equally? Â Will each person get the same size piece? Â How do you know?â€

4. Create a chart, showing some of the ways students folded the square to make two equal parts. Â Students may fold vertically, horizontally, or diagonally.

5. Tell students, â€œThere are two equal parts.â€ Â Ask students, â€œWhat can we call each part of the rectangle?â€ Â Elicit student thinking, building the understanding that each piece is one of two equal parts, or half of the rectangle. Â The standard calls for students to 'describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of.' Â Helping students develop this language is critical. Â If students don't come up with this language, it's important to introduce it. Â After introducing the terms 'half' and 'half of' in this case, adding labels to the chart will help anchor the language.

You may repeat this with other shapes (circle, rectangle) or increase the number of students sharing the paper shape from 2 to 4. If you repeat with a rectangle some students may a scissors as scaffolding to cut the rectangle instead of folding when they divide the rectangle diagonally.

Part Two:

1. Pose the problem to students: Â 'If you and three friends want to share a cake so that you each get the same amount, how can you much can each person have?' Â Ask questions to ensure students understand the context and problem being posed. Â Questions might include:

• What is the story about? • How many people are sharing the cake?
• What does it mean for each person to get the same amount?

Provide students with tools such as blank paper and/or paper shapes to solve the problem. Â The cake context was selected to allow students to explore with multiple shapes as cakes may be round, square, or rectangular in shape. Â

### 4 Equal Parts Of A Rectangle Graph

1. As students solve the problem, monitor their progress looking for students that partition a shape into four equal shares.

2. Conduct a share out, showing several different student solutions that show a shape partitioned into four equal parts. Â It may be helpful to consider the features of each solution being shared by posing questions that get at student thinking and the essential mathematical ideas.

The teacher might ask questions such as:

### 4 Equal Parts Of A Rectangle

• How many people are sharing the cake?

• What does the picture represent? Â What is the circle? Â What is the square? Â What is the rectangle?

• How is ___'s picture similar or different than ____'s?

• How much cakeÂ does each person get? Â

• Some people represented the cake with rectangles and others with squares or circles, did that change the amount of cake each person gets? Â Why or why not?

• What can we call these parts? Â How can we label them?

• Why do the parts need to be equal?

## IM Commentary

This purpose of this task is for students to understand how to partition shapes into equal pieces. Â This task starts students with concrete representations of the shapes that they can fold and cut, so that later they will understand more abstract representations like diagrams and symbols.Â Part one provides students with opportunities to manipulate paper shapes, folding them to create equal parts. Â Students start with a square so that vertical, horizontal, and diagonal folds all match the two sides exactly, this makes it easy to see what is meant by â€œequal parts in this case. Rectangles pose a greater challenge, because the diagonal fold does not match the two halves up exactly. The teacher should lead a discussion about how we know the two halves are equal by cutting them apart and showing they match up exactly (or providing students with scissors and allowing them to do this for themselves). The idea of â€œequal partsâ€ with shape is actually very subtle; in grade 1, students only look at congruent figures as equal. Later they can talk about equal area even when shapes arenâ€™t congruent; students begin to tackle these issues in Grade 3 and beyond.

Through a class discussion, students will understand that shapes can be partitioned in many different ways and that there are conventions for naming those parts. The task intentionally begins with partitioning a shape into two equal parts to build on students intuitive knowledge and experience working with halves. Â Students may be familiar with the term half but think of it as two parts rather without understanding that the parts must be equal in some sense. Â Developing student language is critical component of this task, the standard calls for students to 'describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of.' Â

Part one can be extended to provide students with opportunities to partition additional shapes and increase the partitioning to create four equal shares. Â Extending the activity to include partitioning four equal shares can create opportunities for students to understand that decomposing into more equal shares results in smaller parts. Â

Part two provides students with a context, sharing cake, to further build on the ideas above. Â Students will see there are many ways to represent and partition shapes. Â However, the parts must be equal. Â Students will reason abstractly and quantitatively as they create equal shares in and out of context.

## Solution

Part One: Â Students may fold the rectangle vertically, horizontally, or diagonally to show two equal parts.

Â

Part Two: Â Students may draw a circle (or other shape) to represent the cake and then partition the shape into four equal parts. Â Itâ€™s important for students to understand that the parts must be equal so that everyone gets a fair share. Â Students may label each part one of four equal parts, fourths, or one out of four equal parts.

Â

## Equal Shares

Materials

• Paper cut outs of rectangles, circles, and squares

• Blank paper

Actions

Part One:

1. Give each pair of students a square and ask, â€œHow can you share the square equally so that you and your partner get the same size piece?â€ Â

2. Ask students to fold the paper to show how they could get two equal parts.

3. Call on student volunteers to share, asking the class as each example is displayed, â€œIs the paper shared equally? Â Will each person get the same size piece? Â How do you know?â€

4. Create a chart, showing some of the ways students folded the square to make two equal parts. Â Students may fold vertically, horizontally, or diagonally.

5. Tell students, â€œThere are two equal parts.â€ Â Ask students, â€œWhat can we call each part of the rectangle?â€ Â Elicit student thinking, building the understanding that each piece is one of two equal parts, or half of the rectangle. Â The standard calls for students to 'describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of.' Â Helping students develop this language is critical. Â If students don't come up with this language, it's important to introduce it. Â After introducing the terms 'half' and 'half of' in this case, adding labels to the chart will help anchor the language.

You may repeat this with other shapes (circle, rectangle) or increase the number of students sharing the paper shape from 2 to 4. If you repeat with a rectangle some students may a scissors as scaffolding to cut the rectangle instead of folding when they divide the rectangle diagonally.

Part Two:

1. Pose the problem to students: Â 'If you and three friends want to share a cake so that you each get the same amount, how can you much can each person have?' Â Ask questions to ensure students understand the context and problem being posed. Â Questions might include:

• What is the story about?
• How many people are sharing the cake?
• What does it mean for each person to get the same amount?

Provide students with tools such as blank paper and/or paper shapes to solve the problem. Â The cake context was selected to allow students to explore with multiple shapes as cakes may be round, square, or rectangular in shape. Â

1. As students solve the problem, monitor their progress looking for students that partition a shape into four equal shares.

2. Conduct a share out, showing several different student solutions that show a shape partitioned into four equal parts. Â It may be helpful to consider the features of each solution being shared by posing questions that get at student thinking and the essential mathematical ideas.

The teacher might ask questions such as:

• How many people are sharing the cake?

### Divide A Rectangle Into Thirds

• What does the picture represent? Â What is the circle? Â What is the square? Â What is the rectangle?

• How is ___'s picture similar or different than ____'s?

• How much cakeÂ does each person get? Â

• Some people represented the cake with rectangles and others with squares or circles, did that change the amount of cake each person gets? Â Why or why not?

• What can we call these parts? Â How can we label them?

• Why do the parts need to be equal?