enVision Math Grade 3 Topic 17-7: Problem Solving: Make and Test Generalizations | enVision 数学三年级 17-7：问题解决：提出并检验规律

📚 enVision Math Grade 3 Topic 17-7: Problem Solving: Make and Test Generalizations | enVision 数学三年级 17-7：问题解决：提出并检验规律

In this lesson, you will learn how to solve problems by making a generalization. A generalization is a rule or statement that works for many examples. You will also learn how to test a generalization to make sure it is true.

在本课中，你将学习如何通过提出规律来解决问题。规律（generalization）是指对许多例子都成立的规则或说法。你还将学习如何检验一个规律，确保它是正确的。

1. What Is a Generalization? | 什么是规律？

A generalization is a statement that is true for many different cases. For example, “The sum of two even numbers is always even” is a generalization. You can use patterns to make generalizations.

规律是对许多不同情况都成立的陈述。例如，“两个偶数的和一定是偶数”就是一个规律。你可以利用模式来提出规律。

When you make a generalization, you look at several examples and find something that is always the same. Then you write a rule that describes what you see.

当你提出规律时，先观察几个例子，找出总是相同的地方，然后写出一条描述你所见内容的规则。

2. Looking for Patterns | 寻找模式

Patterns help you make generalizations. Look at this list: 2 + 2 = 4, 4 + 4 = 8, 6 + 6 = 12, 8 + 8 = 16. What do you notice? Each sum is even. The pattern is that adding two equal even numbers gives an even number.

模式帮助你提出规律。看这一组算式：2 + 2 = 4，4 + 4 = 8，6 + 6 = 12，8 + 8 = 16。你发现了什么？每个和都是偶数。这个模式是：两个相等的偶数相加，结果是偶数。

In Topic 17, you have practiced identifying patterns in shapes, numbers, and operations. Now you will use those patterns to make a rule.

在主题 17 中，你已经练习了识别图形、数字和运算中的模式。现在你将利用这些模式来制定规则。

3. Making a Generalization | 提出规律

Follow these steps to make a generalization:

按照以下步骤提出规律：

Step 1: Look at several examples. Write down what you see.

第一步：观察几个例子，写下你所看到的内容。
Step 2: Find a pattern that is the same in all examples.

第二步：找出在所有例子中都相同的模式。
Step 3: Write a rule that describes the pattern.

第三步：写出描述该模式的规则。

Example: 5 × 2 = 10, 7 × 2 = 14, 9 × 2 = 18. The pattern is that multiplying an odd number by 2 gives an even number. Generalization: An odd number multiplied by 2 is always even.

例如：5 × 2 = 10，7 × 2 = 14，9 × 2 = 18。这个模式是：奇数乘以 2 得到偶数。规律：奇数乘以 2 的结果一定是偶数。

4. Testing a Generalization | 检验规律

After you make a generalization, you must test it. Testing means trying more examples to see if the rule still works. If it works, your generalization is probably true.

提出规律后，你必须检验它。检验意味着尝试更多例子，看看规则是否仍然成立。如果成立，你的规律很可能是正确的。

Use the table below to test the generalization: “The sum of two odd numbers is even.”

用下面的表格检验规律：“两个奇数的和是偶数。”

Example 例子	Sum 和	Even? 是偶数吗？
3 + 5	8	Yes 是
7 + 9	16	Yes 是
11 + 13	24	Yes 是

All three examples follow the rule. The generalization passes the test.

这三个例子都符合规则，因此该规律通过了检验。

5. Finding a Counterexample | 找出反例

A counterexample is one example that shows a generalization is not true. If you find even one counterexample, the generalization is false.

反例（counterexample）是指一个能证明规律不成立的例子。只要找到一个反例，这个规律就是错误的。

Suppose a student says: “All even numbers are divisible by 4.” Test it: 6 is even, but 6 ÷ 4 = 1 remainder 2, so 6 is not divisible by 4. The number 6 is a counterexample. So the generalization is false.

假设一个学生说：“所有偶数都能被 4 整除。”检验一下：6 是偶数，但 6 ÷ 4 = 1 余 2，所以 6 不能被 4 整除。数字 6 就是一个反例，因此这个规律是错误的。

When you test a generalization, always look for counterexamples. If you find one, change your rule or make a new one.

当你检验规律时，一定要寻找反例。如果找到一个反例，就修改你的规则或提出新的规则。

6. Using Math Vocabulary | 使用数学词汇

Here are important words for this lesson:

以下是本课的重要词汇：

Generalization: A rule that is true for many examples.

规律：对许多例子都成立的规则。
Counterexample: An example that shows a rule is not true.

反例：用来证明规则不成立的例子。
Pattern: Something that repeats or changes in a predictable way.

模式：以可预测的方式重复或变化的东西。
Test: To try more examples to check if a rule works.

检验：尝试更多例子来检查规则是否成立。

7. Practice: Make and Test | 练习：提出并检验

Try this problem: The table shows the number of sides and vertices for some shapes.

尝试这个问题：下表显示了一些图形的边数和顶点数。

Shape 图形	Sides 边数	Vertices 顶点数
Triangle 三角形	3	3
Rectangle 长方形	4	4
Pentagon 五边形	5	5

Make a generalization: For polygons, the number of sides equals the number of vertices. Test it with a hexagon, which has 6 sides. A hexagon has 6 vertices. The generalization works!

提出规律：对于多边形，边数等于顶点数。用六边形检验，它有 6 条边，六边形有 6 个顶点。这个规律成立！

Now test your own generalization: “Multiplying any number by 0 gives 0.” Try 7 × 0 = 0, 25 × 0 = 0, 103 × 0 = 0. It works every time.

现在检验你自己的规律：“任何数乘以 0 都等于 0。”尝试 7 × 0 = 0，25 × 0 = 0，103 × 0 = 0。每次都成立。

8. Real-World Connections | 与真实世界的联系

Generalizations are useful in everyday life. If you notice that every day at 3:00 PM your bus arrives, you can generalize: “The bus always comes at 3:00 PM.” But you should test it on different days to be sure.

规律在日常生活中很有用。如果你发现每天下午 3:00 公交车都会到站，你可以提出规律：“公交车总是在下午 3:00 到达。”但你需要在不同的日子检验它以确保正确。

In science, scientists make generalizations after many experiments. In math, you make generalizations after trying many examples. Testing helps you avoid mistakes.

在科学中，科学家在多次实验后提出规律。在数学中，你在尝试许多例子后提出规律。检验帮助你避免错误。

Remember: a good generalization must be tested. One counterexample can change everything.

记住：一个好的规律必须经过检验。一个反例就能改变一切。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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enVision Math Grade 3 Topic 17-7: Problem Solving: Make and Test Generalizations | enVision 数学三年级 17-7：问题解决：提出并检验规律