Chapter 1: Understanding Multi-Digit Place Value

Transcription

1 Chapter 1: Understanding Multi-Digit Place Value Student friendly standard: I can determine that a digit in one place represents ten times what it represents in the place to its right. (UW.4.M.NBT.A.01) In this lesson we will answer these questions: -What is base ten? -How can you use it to divide? KEY VOCABULARY Base Ten: A number system based on ten. Digit: A symbol used to show a number. Place Value: The value of a digit depending on its place in a number. Whole Number: A positive number that has no pieces with it Problem of the Week Samantha s math teacher has given her the three sets of numbers below and asked that she explain WHY this pattern works. Help Samantha with this discovery. Using what you know about place value, determine the pattern and justify why it works. 8, 80, 800, 8,000 2,000, 20,000, 200, , 7,000, 70,000 To demonstrate your understanding, create a pattern using the number 4. Be sure to justify why each number has a zero at the end.

2 Lesson 1.1: Base Ten Basics (UW.4.M.NBT.A.01) Lesson Objective: The students will be able to understand what a digit represents in a given place value. Video Lesson: Base Ten Basics Your parents gave you a choice for your birthday. You could either choose to have 76 one dollar bills, 11 ten dollar bills or one hundred dollar bill. Which would you choose? Why? (Answer at end) When you say 53, what does the five represent? How about the three? The answer can be found be looking to the digits place value. The five is worth five tens because it is located in the tens place. The three is worth three ones because it is located in the ones place. Hundreds Tens Ones We can say that we have 5 groups of ten (5 * 10) and 3 groups of one (3*1) for a total of 53 ones.

3 What would 7 groups of ten look like? In the space below (or a separate sheet of paper) draw out 7 groups of ten. Hundreds Tens Ones It should look similar to what we have below. You would have seven groups of ten. What multiplication problem can be written to represent this?. Hundreds Tens Ones Because there are seven groups of ten ones, we know that 7 tens or 70 (7 * 10 = 70). So which amount did you go with at the beginning? Was it the 76 ones, 11 tens, or 1 hundred? The 76 ones are worth 76 ones. 11 tens can be broken apart into 10 tens and one ten for a total of 110 ones. 10 tens are the same as 100 ones, so which one would you choose.

4 Practice: 1. 3 * 10 = 30 what does the three in the product (answer) of this problem represent? 2. If the number 2 is multiplied by 10, what will the 2 digit represent in the product? 3. How is the 4 similar and different in the numbers 44 and 14? 4. If the number 6 is multiplied by 10, what will the new 6 digit represent?

5 Lesson 1.2: Hundreds and Beyond Lesson Objective: The students will be able to explain a digits place value in relation to others up to the hundreds place. Your weekend has been kidnapped! You are working in the garden with your brother. Your parents are willing to pay you 15 dollars to pick weeds for an hour. They said if you picked all the weeds they would pay you ten times that amount. Your brother says, I am not going to try to get them all. 25 dollars isn t worth all that work. You would be happy with 25 dollars, but something is telling you that is not what they meant. Foraging Dandelions, by the Weed, CC BY-NC-SA Yesterday we compared numbers in the ones place to those in the tens. We concluded that numbers in the tens place were ten times bigger than those in the ones. For example, the problem 3 * 10 = 30 helps to show this. Three is in the tens place so that means three groups of ten. The same is true when dealing with bigger numbers. Write down the number four thousand, two hundred twenty-four in the space below., Let s compare the place value of the tens and hundreds. Both of those digits are two. We have two tens and two hundreds. How many groups of two tens will it take to make the two hundreds (in other words, what times 20 will give us 200)? Use place value blocks to help you. How many of these Would give us those

6 So how many was it? Let s look below. There are five groups of two tens in each hundreds block. There are two hundreds blocks so it will take ten of the groups. Another way to look at it is the hundreds place is ten time the tens place. If we took the number 45 and multiplied it by 10, what should the new four s value be? It started off being worth four tens. When we multiply four tens by ten, we get forty tens, or 400 ones. 5 ones times 10 gives us 5 tens or 50 ones. 40 * 10 = * 10 = = 450 You decided to pull all of the weeds. How much money should you have made? Your brother just added ten dollars to get twenty five, but if you realized that it would be 15 * 10 =.

7 Practice 1. How is the number 2 in 425 different from the number two in 245? 2. Explain why 3 * 10 = What is true about 135? a. The place value of the 3 is 10 times the 5. b. The place value of the 1 is 10 times the 5. c. The place value of the 5 is 10 times the If 34 is multiplied by 10, what will the new 3 represent?

8 Lesson 1.3: Base Ten Patterns (UW.4.M.NBT.A.01) Lesson Objective: The students will identify and explain a pattern using base ten concepts. Video Lesson: multiplying by powers of ten Your favorite game, Stuffed Animals vs Aliens just came out and you cannot wait to play it. After playing for an hour you started to notice something. Round one had 4 aliens, round two had 40 aliens, and round three had 400 aliens. When getting ready for round four your friend asked if you were ready to take on 4,000 aliens. How did he know? When multiplying a number by powers of ten you can see a pattern. 6 * 10 will give you 60, 60 * 10 will give you 600. The pattern looks like this: 6, 60, 600. If you look at each individually it makes sense. 6 ones times ten gives you 60 ones or 6 tens. 6 tens times ten gives you 60 tens or 6 hundreds (600). 6 hundreds times ten gives you 60 hundreds or 6 thousands (6,000). Look at the pattern below: 5, 50, 500, 5,000 5 * = 50, 50 * 10 = 500, 500 * = 5,000 What is true about this pattern? If you said each number is ten times the number before it then you would be right. What is true about this pattern? 5,000, 500, 50, 5

9 In the first pattern we had numbers that were increasing (getting bigger). Now we have ones that are decreasing (getting smaller). To get from 5 to 50 we multiplied by ten. Now we are going from 50 to 5. We need to divide. Take our 5 tens from above and break them into ten groups. How many did you get in each group? Our number sentence would be = 5. (Five tens put in 10 groups puts five in each group). In our pattern of 5,000, 500, 50, 5 we are dividing by ten. Another way to say this is each number is one tenth of the number to its left. This same principal can be applied to multiples of ten as well. Look at the following pattern: 3 * 4 = 12 four groups of three are 12 (3+3+3=12) 30 * 4 = 120 four groups of thirty are 120 ( = 120) 300 * 4 = 1,200 four groups of 300 are 1,200 ( = 1,200) So how did your friend know 4,000 aliens would be coming? Let s look at the pattern the video game was following. Level 1 Level 2 Level 3 Level ? 4, 40, 400, 4,000. As you can see from above, each number is times the number to the left of it.

10 Practice: 1. 6,, 600, 6,000 Explain this pattern: 2. 7,000, 700, 70, Explain this pattern: 3. 5 * 6 = * 6 = * 6 = 4. How is 5 in the number 578 similar and different to the 5 in 758? (review) The next equation would be:

11 Lesson 1.4: Understanding Place Value in Numbers (UW.4.M.NBT.A.01) Lesson Objective: The students will compare values of digits in different place values. Video Lesson: Your local radio came to your school to announce that your friend Matt won their Guess the Jelly Beans contest. Because of this Matt now has to choose an amount of money using the following digits: 1, 2, 3, 4. Matt tells you that he just won one thousand dollars! While he is happy you tell him he could have much more. How? When we look at numbers, the place value tells us a lot of the story. Take the number 3,987. Even though the digit 3 is normally less than 9, its value will be more because of where it is. In fact the place value of the 3 (thousands place) is worth 10 times that of the nine (hundreds place. Look at the numbers below: 7,216 What can be said of the Place Value of the 2 and 1? The 2 is in the place. The is in the tens place. The place value of the 2 is times more than the place value of the 1.

12 The 2 is in the hundreds place. The 1 is in the tens place. The place value of the 2 is 10 times more than the place value of the 1. Let s try another: What can be said of the place value of the 1 and 8? 1,874 When comparing numbers, understanding their values can help in deciding which are bigger. Let s take Matt s decision for a second. He thought he had to get $1,234. Since he can arrange these any way he wants, even one move of 2,134 will be bigger. It is easy to see if we break them apart into expanded form. 1,234 = 1, ,134 = Just looking at the thousands place value will tell us that is bigger because 2,000 is bigger than 1,000. What should Matt have picked in order to get the most money? Since each place value is worth ten times the one to the right, Matt should get:

13 Practice: 1. In the number 3,549, what can be said about the place value of the 5 and the 4? 2. In the number 7,128, what can be said about the place value of the 2 and the 8? 3. Justify in words why this problem is true: (review) 25 * 10 = How is 3 in the number 301 similar and different to the 3 in 103? (review)