📚 Composing and Decomposing 2-D Shapes | 拼搭与拆分平面图形
In this lesson, you will learn how to put small shapes together to make bigger shapes. This is called composing. You will also learn how to break big shapes into smaller parts, which is called decomposing. These skills help you understand geometry and prepare for learning about area.
在本课中，你将学习如何把小图形拼在一起组成更大的图形，这叫做“拼搭”（composing）。你还将学习如何把大图形拆成更小的部分，这叫做“拆分”（decomposing）。这些技能能帮助你理解几何，并为将来学习面积打下基础。
1. Essential Shape Words | 图形基本词语
Before we compose shapes, we need to know the names and parts of common 2-D shapes. A 2-D shape is flat, like a drawing on paper. Every shape has sides and vertices. A vertex is a corner where two sides meet. Two or more vertices are called vertices.
在拼搭图形之前，我们需要认识常见平面图形的名称和组成部分。平面图形是平的，就像画在纸上的图形。每个图形都有边和顶点。顶点是两条边相交形成的角，一个顶点用英文说就是 vertex，两个或更多顶点用复数形式 vertices。
| Shape 图形 | Sides 边数 | Vertices 顶点数 |
| Triangle 三角形 | 3 | 3 |
| Square 正方形 | 4 | 4 |
| Rectangle 长方形 | 4 | 4 |
| Trapezoid 梯形 | 4 | 4 |
| Rhombus 菱形 | 4 | 4 |
| Hexagon 六边形 | 6 | 6 |
When you count the sides of a shape, count each straight edge once. When you count vertices, count each corner once. A circle has no straight sides and no vertices.
数图形的边时，每条直边只数一次。数顶点时，每个角只数一次。圆没有直边，也没有顶点。
2. What Does It Mean to Compose Shapes? | 什么是拼搭图形？
To compose a shape means to join two or more smaller shapes together to make one larger shape. The small shapes act like puzzle pieces. When we place them side by side, we create a brand new shape.
拼搭图形就是把两个或更多的小图形拼在一起，组成一个更大的图形。这些小图形就像拼图碎片一样。当我们把它们并排放好时，就能创造出一个全新的图形。
□ + □ → ▭
For example, two squares of the same size can make a rectangle. The rectangle has the same area as the two squares combined.
例如，两个同样大小的正方形可以拼成一个长方形。这个长方形的面积等于两个正方形面积之和。
When we compose shapes, we usually want no gaps and no overlapping. The pieces should fit together perfectly, like tiles on a floor.
拼搭图形时，我们通常希望没有空隙，也没有重叠。所有碎片应当严丝合缝地拼在一起，就像铺地板时拼接瓷砖一样。
3. Combining Triangles to Make New Shapes | 用三角形拼出新图形
Triangles are very useful for composing shapes. Two identical triangles can be joined along a side to create a rectangle, a parallelogram, or a larger triangle.
三角形是拼搭图形的得力工具。两个完全相同的三角形沿一条边拼在一起，可以组成长方形、平行四边形或更大的三角形。
2 triangles → 1 rectangle
If you place two right triangles so their slanted sides touch, you make a rectangle. If you slide them a little, the same two triangles can make a parallelogram.
如果把两个直角三角形斜边相对拼在一起，就能拼出长方形。如果把其中一个稍微平移，同样两个三角形又能拼成平行四边形。
Four small identical triangles can also make one larger triangle. Put one triangle at the top, then place three triangles below it to form the next row.
四个完全相同的小三角形还可以拼成一个大三角形。把一个小三角形放在最上面，再在下面放三个小三角形，排成一行，就能得到一个大三角形。
4 triangles → 1 large triangle
4. Building Rectangles and Squares | 拼出长方形和正方形
You can build a rectangle by putting small squares in rows and columns. For example, place 4 squares in a row, then place another 4 squares in a second row below them. This makes a rectangle that is 2 rows by 4 columns.
你可以把小正方形按行和列排起来拼成长方形。例如，先放一排 4 个正方形，再在下面放第二排 4 个正方形，这样就拼成了一个 2 行 × 4 列的长方形。
rows × columns = total squares
If the rows and columns have the same number of squares, the shape you build is a square. For example, 3 rows of 3 squares make a larger square.
如果行数和列数相同，那么拼出的图形就是正方形。例如，3 行 × 3 列就拼成一个更大的正方形。
A square is actually a special kind of rectangle because it has 4 right angles and 4 equal sides. Remember that when you describe the shapes you build.
正方形其实是一种特殊的长方形，因为它有 4 个直角，而且 4 条边都相等。描述你拼出的图形时，要记住这一点。
5. Making Hexagons with Pattern Blocks | 用图案块拼出六边形
Pattern blocks are special classroom shapes. They include triangles, rhombuses, trapezoids, and hexagons. You can compose a hexagon in many different ways.
图案块是课堂上常用的特殊图形，包括三角形、菱形、梯形和六边形。你可以用多种不同的方法拼出一个六边形。
6 triangles → 1 hexagon
A regular hexagon can be made from six identical equilateral triangles. Arrange them around a central point, like the slices of a pie.
一个正六边形可以用六个完全相同的等边三角形拼成。把它们围绕一个中心点摆放，就像切开的披萨饼一样。
| Ways to Make a Hexagon | 拼六边形的方法 |
| 6 triangles | 6 个三角形 |
| 3 rhombuses | 3 个菱形 |
| 2 trapezoids | 2 个梯形 |
| 1 trapezoid + 3 triangles | 1 个梯形 + 3 个三角形 |
There is often more than one correct way to compose the same shape. Try to find as many different ways as you can.
同一个图形往往不止一种拼法。尽量找出尽可能多的不同拼法。
6. Decomposing Shapes | 拆分图形
Decomposing is the opposite of composing. When we decompose a shape, we split it into smaller shapes. We can cut along straight lines, fold, or simply imagine the parts.
拆分是拼搭的逆过程。拆分图形时，我们把一个图形分割成更小的图形。我们可以沿直线剪开、折叠，或者直接想象它的各个部分。
1 rectangle → 2 triangles
If you draw a diagonal line across a rectangle, you cut it into two triangles. The diagonal is a straight line from one corner to the opposite corner.
如果在长方形上画一条对角线，就能把它分成两个三角形。对角线是连接相对两个角的直线。
A rectangle can also be split into two squares if its length is exactly twice its width. Draw a line down the middle to see the two squares.
如果长方形的长正好是宽的两倍，那么它还可以分成两个正方形。在中间画一条竖线，就能看到两个正方形。
The same large shape can be decomposed in different ways. For example, a hexagon can be split into six triangles or into two trapezoids.
同一个大图形可以有不同的拆分方式。例如，一个六边形可以拆成六个三角形，也可以拆成两个梯形。
7. Hidden Shapes in Real Life | 生活中的隐藏图形
Shapes are everywhere around us. Finding them is a great way to practice composing and decomposing. Look closely and you will be amazed.
图形在我们身边无处不在。寻找生活中的图形是练习拼搭与拆分的好方法。仔细观察，你会感到非常惊奇。
- Honeycombs are made of many small hexagons composed together.
蜂巢是由许多小六边形拼在一起组成的。 - Floor tiles are often squares or rectangles that fit together with no gaps.
地板砖通常是正方形或长方形，它们拼在一起时没有空隙。 - A window frame can be decomposed into four smaller rectangles.
一个窗框可以拆分成四个更小的长方形。 - Quilt patterns use triangles and rhombuses to make beautiful larger shapes.
拼布图案用三角形和菱形拼出漂亮的更大的图形。 - Traffic signs sometimes contain a small circle inside a larger rectangle.
交通标志有时在大长方形里面有一个小圆形。
When you look at a building or a picture, ask yourself which small shapes were put together to make it.
当你观察一座建筑或一幅图画时，可以问问自己：它是由哪些小图形拼成的？
8. Using Grid Paper | 用方格纸拼搭
Grid paper is very helpful for composing shapes. Each small square in the grid is a unit square. We can count the unit squares to describe the size of a composed shape.
方格纸对拼搭图形非常有帮助。方格纸上的每个小正方形都是一个“单位正方形”。我们可以通过数单位正方形的个数来描述拼出图形的“大小”。
2 × 4 = 8 unit squares
If a rectangle has 2 rows and 4 columns, it contains 8 unit squares. We write that as 2 × 4 = 8. This is an early idea of area.
如果一个长方形有 2 行 4 列，它就包含 8 个单位正方形。我们可以写成 2 × 4 = 8。这就是面积概念的雏形。
Use grid paper to draw shapes that cover a certain number of squares. For example, can you compose a shape that covers exactly 5 unit squares? Try different arrangements.
在方格纸上画一画，尝试拼出覆盖一定数量小正方形的图形。例如，你能拼出一个正好覆盖 5 个单位正方形的图形吗？试试不同的摆法。
Counting squares helps you compare shapes. A shape with 8 unit squares is larger than a shape with 5 unit squares, even if they look different.
数格子能帮助你比较图形。包含 8 个单位正方形的图形比包含 5 个单位正方形的图形更大，即使它们看起来形状不同。
9. Practice Problems | 练习
Try these problems on your own. Use your hands or draw pictures to help you think.
请你独立完成下面的练习。可以动手摆一摆，或者画图来帮助思考。
- How many identical triangles can completely cover one regular hexagon?
需要多少个完全相同的三角形才能正好铺满一个正六边形？
Answer: 6. 答案：6 个。 - Two identical squares can be joined to make which shape?
两个相同的正方形拼在一起能组成什么图形？
Answer: a rectangle. 答案：长方形。 - If you split a rectangle along its diagonal, which two shapes do you get?
如果沿对角线把一个长方形剪开，会得到哪两个图形？
Answer: two triangles. 答案：两个三角形。 - Use 3 rows of 3 squares. What large shape do you build?
用 3 行 × 3 列的小正方形拼图形，你拼出了什么大图形？
Answer: a larger square. 答案：一个更大的正方形。 - Can a shape made of 4 unit squares be a triangle? Draw it on grid paper.
一个由 4 个单位正方形组成的图形能拼成三角形吗？在方格纸上画一画。
Hint: Yes, by arranging the squares in steps. 提示：可以，把正方形排成阶梯状。
Check your answers by composing and decomposing the shapes with real pattern blocks or paper.
你可以用真实的图案块或纸片拼一拼、拆一拆，来检查你的答案。
10. Common Mistakes to Avoid | 常见错误
Even experienced students make mistakes when composing shapes. Here are some common ones to watch out for.
即使是熟练的学生，在拼搭图形时也会犯错误。下面是一些需要留意的常见错误。
- Turning a shape does not change it. A
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