First, students are asked to compare two numbers in this lesson. To pique students' interest, the task is organized as a game with random elements. Place value is a fundamental concept in the main part of the activity, and this first exercise reviews and reinforces it. Students can compose and decompose numbers while working on the Goal activity. They can focus on the tens independently of the ones and in any order. They also realize that tens can be divided into ones and that ones can be combined to form tens.
Different strategies are validated, which reinforces student thinking. Students' thinking extends to mental computation as the strategies are internalized and prepared to be applied to addition and subtraction problems.
Say, "There are numerous applications for base-ten blocks. They help us add and subtract numbers and can be used to construct various numbers. They also allow us to compare numbers and determine which is bigger. Consider how you can determine which number is larger as we examine some numbers using base-ten blocks."
This lesson starts with a quick review of comparing numbers and will help students understand the relationship between addition and subtraction as they investigate how far apart numbers are.
Show the Tens and Ones transparency (M-2-3-3_Tens and Ones) on the overhead. In the place-value section, display 3 blocks representing tens and 5 blocks representing ones. In the tiny square, write the number 48. Have a student identify the number in the square and indicate if it exceeds or falls short of the total number of blocks consisting of ones and tens. Put a space character ">" or "<" between the place-value chart and the square. (48 > 35)
Ask the student to explain how he or she knew. "48 has four tens, but there are only three tens on the chart," the student will most likely say. "Thus, I am aware that 48 is greater than 35."
A few more times, repeat the exercise using different two-digit numerals and different blocks. Make sure you display some numbers with the same tens and ones, like 49 and 42 and 14 and 34, respectively. Incorporate sets illustrating both > and <.
Go on to the following exercise once students have mastered this one. Use the Goal transparency (refer to M-2-3-3_Goal in the Resources folder) in place of the Tens and Ones transparency on the overhead projector. In this activity, two students each select a number between 1 and 99, and one student uses base-ten blocks to build that number. We'll compare the two numbers in class.
Have a student approach the projector and write a number in the square between 1 and 99 while the projector is off. This amount is the "target."
Have a different student represent a number between 1 and 99 using tens and ones blocks after covering the square to conceal the number. (If the first student changes the number in the square and you notice that the numbers are the same.) Activate the projector so that the blocks and the number are visible to the entire class.
Inform the students that the objective is to match the number in the square with the set of blocks. They have to choose whether to add or remove blocks. Ask them to adjust the blocks so that the collection matches the number, and lead their discussion to serve as an example for the rest of the class.
If the number in the square is 23, and the number represented by the blocks of tens and ones is 56, for example, students could discuss the following:
"56 is greater than 23. We have to remove a few blocks."
"Let's take 3 out because we have too many tens; that leaves 2 tens, or 20."
"Alright, so while we have 26, we really ought to have 23. 6 minus 3 equals 3, so I'm subtracting three ones."
"That's right. That's 23 blocks, which corresponds to the 23 in the square."
To demonstrate their work, have the students now write a number sentence. (We deducted three tens from the original 56 blocks. That comes to 56 - 30 = 26. After that, we deducted 3 because we had too many. 26 - 3 = 23 and 56 - 30 = 26.)
Ask questions about their numbers to help them understand the connection between their actions (adding or subtracting blocks) and the number sentence they write.
"Which number indicates the initial number of blocks you had?" (56)
"Which number represents your objective?" (23)
"Which symbol indicates addition or subtraction?" (The minus sign.)
"Which number demonstrates the number of ones and tens you subtracted?" (We have the 30 and the 3 and two numbers.)
"What other name is there for thirty and three?" (Okay, that makes 33.)
"That's accurate. You removed three ones and three tens from the mat. Hence, you deducted 33. Is there a different way you would write your numeric sentence?" (We could also write 56 - 33 = 23.) Give students some time to write a model sentence using this new number phrase.
Students are allowed to use numbers that call for them to combine or separate blocks of ten. If the blocks represent 19 and the goal number is 43, the students will have 9 from 19 and 3 from 43. They will want to combine ten of the one blocks to make one tens block.
Select a different pair of students and carry out the exercise again, this time using a different goal number and blocks. The class must have this initial conversation to validate the various strategies that the students employ. Students can compose or decompose the numbers in a variety of ways (some steps can be completed mentally):
Add ones, add tens: In the most basic scenario, such as increasing from 24 to 58, a student could add 4 ones and 3 tens, for a total of 34 added.
Add to make ten, add tens, add-ones: In an alternate version of the above problem, a student could add 6 to 24 to get to 3 tens. Next, add 2 tens to get 50, and add 8 ones to get 58, for a total of 34.
Add tens, deduct ones: A student could add three tens (30) and deduct two to go from 24 to 52.
Add ones, add tens, and subtract: Using the previous example, a student could add 2 to 52 to make the ones equal 24. Then, they are equalized by adding three tens (30). But now we must deduct the 2 to leave 28 added.
Subtract ones, then subtract tens: To get from 58 to 24, a student could subtract four ones, and then three tens.
Add ones, deduce tens: To reduce 52 to 24, you could add 2 to the ones to get from 52 to 24, or 54. Then 3 tens can be taken away. A total of 28 is removed (+2 − 30, or 30 − 2).
Add ones, subtract tens: This method is the same as the one above, but students can decide to start with the tens. To get from 52 to 24, subtract 3 tens and then add 2 ones for a total of 28 subtractions.
These are the main conceptions of composition and decomposition that students will come up with; there are others, too, and they are all legitimate if they make sense to the student and are accurate mathematically.
Informal assessments can be used to inform instruction by keeping an eye on students' responses during discussions and small-group work.
To carry out the previous task, utilize the Random Reporter. Determine the desired number of blocks to start with and the end number, then display them above. Every student group explains how they arrived at the objective, taking into account every tactic. (Did any group approach it differently?)
Groups could present issues to the class. This would introduce the terminology and start to reinforce the relationship between addition and subtraction. A group might ask the class, "We started with 24 and our goal was 61; how much was included?" if they began with 24 blocks and added 37 for a total of 61.
Student's progress can be evaluated using the M-2-3-3_Lesson 3 Assessment and KEY, paper-and-pencil assessment.
Extension:
To fulfill your students' needs throughout the year, employ the techniques and activities provided below.
Routine: Have students mentally complete the same tasks as they grow more accustomed to using the various strategies. To encourage groups to talk about their strategies, keep using Random Reporter.
This could be used to begin the day, the workday, or an activity: All you have to do is write the goal and starting numbers on the board and allow groups of people to talk through potential solutions. It is not necessary to work on more than one problem each day. The discussion of the problem-solving process in the group and with the class is the activity's advantage. Students' number sense and flexibility with numbers will be substantially improved by having this quick conversation and practice every day.
Small Group: Extra practice and instruction in small groups will be helpful for students who require chances for further learning. Students can start by writing multiple two-digit numbers on their paper. Once the number and the objective are established, they can alternately call off one of their numbers. Using base-ten blocks, the group can construct the number collectively and then talk about how they plan to reach the target. On the paper sheet, students can record their work.
When deciding which tactic to employ, students might also require assistance. As students work, ask questions that will lead to the development of specific strategies. To ensure that the ones are the same for both numbers, ask them how many ones they need to add or subtract when going from 24 to 89, for example.
Expansion: If students are prepared, they can complete the previous task using three-digit numbers. They will observe that the same tactics and tenets remain relevant. When they get a better understanding of the foundation of our number system, they might even ask to practice four-digit numbers.
