In this lesson students are first asked to compare two numbers. The activity is set up as a game, with randomness to increase student interest. This starting exercise reviews and reinforces the understanding of place value, which is key in the main portion of the activity. As students do the Goal activity, they see that they can compose and decompose numbers, focusing on the tens separately from the ones, and in either order. They also see that ones can combine to become tens, and tens can be broken into ones.
Student thinking is reinforced as different strategies are validated. The strategies become internalized (and ready to be applied to addition and subtraction problems) as students extend their thinking to computing mentally.
Say, “We use base-ten blocks in many ways. We can use them to build different numbers and to help us add or subtract numbers. We can also use them to compare numbers and decide which is larger. As we look at some numbers using the base-ten blocks, think about how you can tell which number is larger.”
This lesson will help students see the connection between addition and subtraction as they explore how far apart numbers are, beginning with a brief review of comparing numbers.
Display the Tens and Ones transparency on the overhead (M-2-3-3_Tens and Ones.doc). Show 3 tens blocks and 5 ones blocks in the place-value section. Write the number 48 in the small square. Ask a student to name the number in the square and tell whether it is greater than or less than the number of tens and ones blocks. Write “>” or “<” between the square and the place-value chart. (48 > 35)
Ask the student to explain how s/he knew. The student will likely say, “48 has 4 tens and there are only 3 tens on the chart. So I know 48 is more than 35.”
Repeat the activity a few more times, changing the blocks and using different two-digit numerals. Be sure to show some numerals where the tens place is the same (such as 49 and 42), and the ones place is the same (such as 14 and 34). Include sets that demonstrate both > and <.
Once students understand this activity, move on to this next activity. Replace the Tens and Ones transparency with the Goal transparency on the overhead projector (see M-2-3-3_Goal in the Resources folder). In this activity one student chooses a number from 1 to 99, while another student chooses a number from 1 to 99 to build with base-ten blocks. The class will compare the two numbers.
Leaving the projector off, have a student come up to the projector and write a number between 1 and 99 in the square. This number is the “goal.”
Cover the square to hide the number, and have a different student use tens and ones blocks to represent a number between 1 and 99. (If you see that the numbers are the same, have the first student change the number in the square.) Turn the projector on so the whole class can see the number and the blocks.
Tell students that it is their job to make the set of blocks equal to the number in the square, the goal. They must decide if they need to add or subtract blocks. Guide their discussion to model ideas for the rest of the class, and have them change the blocks to make the collection match the number.
For example, if the number in the square is 23 and the number represented by tens and ones blocks is 56, students may have this discussion:
“56 is more than 23. We need to take some blocks away.”
“We have too many tens, so let’s take three off; that leaves two tens, which is 20.”
“Okay, so we have 26 but we should have 23. 6 minus 3 is 3, so I’m taking 3 ones off.”
“There. That’s 23 blocks and that’s the same as the 23 in the square.”
Now have students write a number sentence to show their work. (We started with 56 blocks and took away 3 tens. So that’s 56 − 30 = 26. Then we had too many ones so we subtracted 3 ones. 56 − 30 = 26 and 26 − 3 = 23.)
Ask questions about their numbers to help them realize the relationship between their actions (adding or subtracting blocks) and the number sentence they write.
“Which number shows how many blocks you started with?” (56)
“Which number shows your goal?” (23)
“Which symbol shows if you added or subtracted?” (The minus sign.)
“Which number shows how many tens and ones you subtracted?” (We have two numbers, the 30 and the 3.)
“What is another name for 30 and 3?” (OK, that’s 33.)
“That’s right. You took 3 tens and 3 ones off the mat. So you subtracted 33. Is there another way to write your number sentence?” (We could write 56 − 33 = 23.) Allow students time to write this new number sentence as a model sentence.
Students may use numbers that require them to compose or decompose tens blocks. If the goal number is 43 and the blocks represent 19, students will have 9 ones from 19 and 3 ones from 43. They will want to compose 10 of the ones blocks into 1 tens block.
Choose another pair of students and repeat the activity, starting with a new goal number and a new set of blocks. It is important to have the initial discussion as a whole class so that all the different strategies students use are validated. There are several ways that students may compose or decompose the numbers (some of the steps could be done mentally):
- Add ones, add tens: In the simplest case, such as going from 24 to 58, a student could add 4 ones, then add 3 tens, making 34 added.
- Add to make ten, add tens, add ones: In a variation of the above problem, a student might add 6 to 24 to get to 3 tens. Then 2 tens can be added to get to 50, and 8 ones to get to 58, for a total of 34.
- Add tens, subtract ones: To go from 24 to 52, a student might add 3 tens (30), then subtract 2, leaving 28 added.
- Add ones, add tens, subtract: In the same example above, a student might add 2 to 52 so the ones match 24. Then adding 3 tens (30) makes them equal. However, the 2 needs to be subtracted now, leaving 28 added.
- Subtract ones, subtract tens: To go from 58 to 24 a student could subtract 4 ones, then 3 tens.
- Add ones, subtract tens: To go from 52 to 24, a student might add 2 to the ones, making it 54. Then 3 tens can be taken away. 28 is taken away altogether (+2 − 30, or 30 − 2).
- Subtract tens, add ones: This is the same as the above strategy, but students could choose to start with the tens. To go from 52 to 24, take away 3 tens, and then add 2 ones, for a total of 28 subtracted.
There are other ways students may think about composing and decomposing (and all are valid if they make sense to the student and are mathematically correct), but these are the main ones you will encounter.
Monitoring student responses during discussions and small-group work can be used as informal assessments to guide instruction.
Random Reporter can be used to continue the activity above. Choose the starting number of blocks and the goal number, and show them on the overhead. Each student group discusses how they got to the goal, and all strategies are considered. (Did any group do it differently?)
Groups could propose problems to the class. This would begin to reinforce the connection between addition and subtraction, as well as introducing the language to use. If a group started with 24 blocks and added 37 for a total of 61, they would ask the class: “We started with 24 and our goal was 61. How much was added?”
A paper-and-pencil assessment may be used (M-2-3-3_Lesson 3 Assessment and KEY.doc) to assess students’ progress.
Extension:
Use the activities and strategies listed below to meet the needs of your students during the year.
- Routine: As students become more comfortable using the different strategies, have them do the same activities mentally. Continue to use Random Reporter to encourage groups to discuss their strategies.
This could be a way to start the day, period, or activity: Simply write the starting number and goal number on the board, and give groups time to discuss how they would solve the problem. There is no need to do more than one problem a day. The benefit of the activity is the discussion in the group and with the class about how the problem was solved. Having this brief discussion and practice daily will greatly help students’ number sense and flexibility with numbers.
- Small Group: For students who need opportunities for additional learning, more practice and small-group instruction will be beneficial. Students can begin by writing several two-digit numbers on their sheet of paper. They can take turns calling off one of their numbers to establish the number and the goal. The group can build the number together, using the base-ten blocks, and then discuss how they will get to the goal. Students can keep track of their work on the sheet of paper.
Students may also need some support regarding which strategy to use. As students work, ask questions that will suggest certain strategies. For instance, in going from 24 to 89, ask them how many ones they need to add or subtract so that the ones are the same for both numbers.
- Expansion: Students who are ready can do the same activity as above, but using three-digit numbers. They will see that the same principles and strategies still apply. In fact, they may ask to do four-digit numbers as they begin to realize the basis of our number system.
