Exploring Expanded Form using Scales

Transcription

1 1 Exploring Expanded Form using Scales Chepina Rumsey, Scott Schreiner, and EDEL 463 Section D* Mathematics Lesson for Grade 4 Approximately 60 Minutes Common Core State Standard (Content): 4.NBT.2 Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. Standard for Mathematical Practice: SMP 2 - Reason abstractly and quantitatively. Objectives: The students will be able to write multi-digit whole numbers in expanded form and explain the connection to standard form. Materials:! Balance scales One for each group of four! Recording sheet! Dice, one per pair! Paper base-ten blocks for whole group instruction! Base-ten blocks* *Since this lesson requires comparing the weights of numbers using base-ten blocks, make sure that the base-ten block set that you are using do have the 10:1 relationship in weight. Some sets are made of different materials so not all sets have the same proportional weights. Groupings: The students will be working in pairs and as a whole group. Engage: Launching the lesson In this section, we will begin by asking students to compare two quantities, where the ones place of the smaller quantity is larger than the ones place of the larger quantity. (29 and 57). By doing this, we want to connect to students prior knowledge about numbers and the base-ten system. Then we will ask them to compare 290 and 57. For both of these questions, we will ask them to justify their reasoning. They can use the paper base-10 blocks to represent the numbers on the board.

2 2 Guiding Questions: Which number is larger, 29 or 57? Can you justify why 57 is larger than 29? What do you know about base-ten? If 9 is larger than 7, why is 29 larger than 57? How would you show 29 and 290 with the base-ten blocks? How many tens and ones do 57 and 29 have? What changes when we shift the numbers over and put a zero in the ones place? How does that effect the value of the number? How does the value change? What is zero? -If zero has no value and 57 is larger than 29, why is 290 larger than 57? Describe, using base-ten blocks, how you would show that 57 is larger than 29. Why is it important that we look at the place values of a number that has multiple digits? How can you tell how one number compares to another? Why is it important to know how to compare two numbers? Transition to the Explore section: Use a scale to weigh the two sets of base-ten blocks that represent the numbers 29 and 57. Explore: Investigate, problem solve, hands-on exploration In this section, groups of students will use a balance scale and the base-ten blocks to model the equations and determine which quantity is larger. They will make a record on their organizing sheets. Let the students know that we are trying to go beyond the correct answer so that we are able to observe patterns in the numbers and make connections to expanded form. Every student will fill out his or her own sheets but will have to figure out how to share the role of measuring with the scales since there is only one per group. Note: The comparisons are fairly easy for fourth grade students and many students will put the <, >, and = sign in the space without using the scale. Encourage them by saying that today we are going beyond the correct answer, but instead are looking for patterns. Some students will want to draw a picture as a record for each comparison, let them know that it s okay but encourage them to think of other ways to make a record. The numbers that the students will compare involve a variety of interesting situations, including (1) two numbers with the same numeral in the ones place, (2) two numbers with the same numeral in the tens place, (3) two numbers with the tens and ones place switched, and (4) two three-digit numbers with the same numerals out of order. While walking around, the teacher will be looking for different ways of representing the number sentences that could be shown during the next portion of the lesson.

3 3 Guiding Questions: How would you describe why the scale tips one way or another? How do you know how many blocks to put on the scale? What is the relationship between the number of blocks and the weight? Can you predict what will happen with the scale with the numbers 46 and 81? Why? When you represent these on the balance, what tells you which quantity is larger? -How does the balance scale represent the larger number? Why did you use 5 long blocks instead of the 5 of the little cubes? Why couldn t you use the little cubes instead of the long rods? How will you make a record? What is the important information to record? Explain how you came to that conclusion and the reasoning that got you there. Explain: Share understanding, new concepts and skills introduced. In this section, we will come back together as a whole group in order to share our results and methods for making a record of the results. There will be a progression of recordings that vary in abstraction. For example, to record the quantity 23 a student could write: 2 and 3 2 tens and 3 ones 2 x 10 3 x 1 or (2x10) + (3x1) = 23

4 4 Students may also use the abbreviations H, T, and O for hundreds, tens, and ones. If students write 2T and 3O for two tens and three ones, you can highlight the fact that this is a good idea, but that the O could look like a zero, therefore confusing 3O with the number 30. Some key words to introduce in this section are standard form and expanded form if they haven t already learned them. We will begin by discussing the second comparison question (23 and 32) on the worksheet they completed for the Explore section and have different students share their representations. Our goal is for the students to see the expanded form shown in the far right. Students will draw a variety of representations, but our goal is to make connections between whichever methods they use, the base-ten blocks, standard form, and expanded form. As students are presenting their ideas, record them on the board in the progression that is shown above. Read all of the representations, highlighting 2 tens and 3 ones for 23 and how they are represented on the board. Guiding Questions: Does anyone have another way to make a record of the blocks you used to compare 23 and 32? What similarities and differences do you see in the various records? Is it possible that all of these ways are correct? What do you know about the number 23? How many different ways can you write it? How do we know that these different ways all show 23? What is similar and different about the ways of writing it? What is the relationship between expanded form and standard form? Explain the difference between expanded form and standard form. Can we always break a number down to expanded form? Then we will compare the quantities 47 and 52. We will show both numbers in various forms, working to the expanded form that we want the students to be able to understand. Is the following statement true or false (why, why not?): 47=40+7 Are these statement also true: 40=4x10 and 7=7x1? = / \ (4 x10) + (7 x 1)

5 5 Elaborate: Further activity allowing the students to put new knowledge into practice In this section the students will work in pairs to go back to the original recording sheet and try to write the quantities using expanded form. An example of expanded form will be left on the board for the students to refer to, but as we are walking around we can pose the following guiding questions: Guiding questions as students are working: How many tens are in the number? How do you show two tens in expanded form? How do you represent zero tens or ones in expanded form? (For 301, which may pose some difficulty for the students) What does the 3 represent? -What does the 0 represent? -What does the 1 represent? How is your original record similar to the example of expanded form on the board? How could you explain expanded form? Why are you multiplying and then adding when writing in expanded form? Does the order matter? Evaluate: Assess student understanding and lesson effectiveness To evaluate the objective we will have the students play a game in pairs and then write the numbers that they roll in expanded form as an exit slip. To play the game, each student takes turns rolling the die and placing a single number at a time in the spaces that make up a three-digit number. Then the students will compare the numbers and individually write both numbers in expanded form. Because they are using the die, there will be no places that have a zero. Take turns rolling the die and deciding where to put the numbers so that you can have the largest three-digit number. Once you write a number in the space, you cannot move it. When you have filled in the spaces, put a <, >, or = in the to make the statement true. Then, individually write both numbers in expanded form in the space below.

6 6 Resources: None used *2013 EDEL 463 Section D students from Kansas State University: Allison Fehr Megan Graham Whitney McGee Alexandra Myers Cortney Nagel Kaitlin Odlenhuis Cat Parr Paige Strecker Sarah Swenson Danielle Thomas Olivia Van Hook Cassandra Wood Katie Wootten Lauren Wormington Kayla Yank Kelsey Youngworth Fourth Grade Classroom Teachers: Andrea Fields and Brooke Snyder

7 With your groups use the scales to compare the sets of quantities listed below. In the space between the numbers use the symbols <, >, or =. Below the number sentence, make a record of your work with the scales

8 Take turns rolling the die and deciding where to put the numbers so that you can have the largest three-digit number. Once you write a number in the space, you cannot move it. When you have filled in the spaces, put a <, >, or = in the to make the statement true. Then, individually write both numbers in expanded form in the space below. Take turns rolling the die and deciding where to put the numbers so that you can have the largest three-digit number. Once you write a number in the space, you cannot move it. When you have filled in the spaces, put a <, >, or = in the to make the statement true. Then, individually write both numbers in expanded form in the space below.