In this lesson students are led to make connections between addition and subtraction. They find the difference between two numbers (how far apart they are) by counting up from the smaller number. Students use strategies that were developed in Lesson 2, but apply them to find sums and differences. Students then work with fact families to further develop the connections between addition and subtraction.
Say, “In our everyday lives we use addition and subtraction to answer questions and solve problems. What are some times throughout the day that you use addition or subtraction? I’m going to share some stories that have problems for us to solve, and we’re going to choose strategies to help us.”
The purpose of this lesson is to provide opportunities for students to construct their understanding of addition and subtraction. They need to see the connection between addition and subtraction and to understand that the “comparison” and “removal” strategies, which they applied concretely in Lesson 2 and 3, can both involve subtraction. Catherine Twomey Fosnot and Maarten Dolk in their book Young Mathematicians at Work say, “Traditionally, teachers have often told learners that subtraction means ‘take away.’ This is a superficial, trivialized notion of subtraction, if not erroneous. It is just as erroneous to say subtraction means ‘difference’” (p. 90).
The numbers in this problem were chosen carefully. Students may not see the relationship between the first and second problems. Guide their discussions, accepting all attempts and reinforcing any demonstrations of part-whole relationships.
Read the story problem (you may also want to write it for all to see and refer to).
“Ms. Clover is planting a flower garden. She bought a package of 100 seeds. So far, she has planted 76 seeds. How many more does she need to plant to use all of her seeds?”
“Her neighbor, June, is planting flowers, too. She also bought a package of 100 seeds. June has only planted 24 seeds. How many more seeds does June need to plant to use all of her seeds?”
Give students paper to record their work as they attempt to solve the problems. Allow adequate time for students to come up with solutions; then begin a class discussion. Along with introducing the lesson, these problems will also serve as a review of the strategies students were exposed to in Lesson 2.
“Who would like to share what they did on the first part of our problem, finding how many more seeds Ms. Clover needs to plant?” (I said she needs to plant 24 more seeds.)
“How did you get your answer?” (I counted up by tens from 76 to 96 and that was 20. Then I counted by ones from 96 to 100 and that was 4. So I added 20 and 4 and that’s 24.)
“You used a ‘counting on’ strategy by counting up from 76 by tens and ones. Did someone try another way?” (I looked at the number line and jumped by tens from 76 to 96 and then jumped by ones to 100. That was 24.)
“The number line is a good tool to help us compare numbers and find ‘how many more’ or ‘how many less.’ Did anyone try a subtraction strategy?” (I subtracted 100−70 which is 30 and then 30−6 = 24.)
“I see, you broke apart the second number but left the first number whole. That works!”
“Let’s talk about the second part of our story. Ms. Clover’s neighbor, June, also starts with a package of 100 seeds. She has only planted 24 seeds. How many more seeds does June need to plant to use all of her seeds? Who would like to share his/her work?” (I know that June planted 24 seeds. First I counted by tens: 34, 44, 54, 64, 74, 84, 94. Then I counted by ones: 95, 96, 97, 98, 99, 100. To get to 100, I counted 7 tens and 6 ones, which makes 76.)
“Good thinking. Did anyone use another strategy?” (I started with 100 and took away 20 to get 80. Then I started with the 80, a decade number, and took away 4 more to get 76, because
10 – 4 is 6.)
“So you subtracted 100 − 24 and also got 76.” (Hey, Ms. Clover planted 76 seeds and had to plant 24 more. June planted 24 seeds and had to plant 76 more! They both used the same numbers.)
Some students may start seeing the relationships between the numbers; 100 − 24 = 76 and
100 − 76 = 24; so 76 + 24 = 100 and 24 + 76 is also 100. This is the part-part-whole relationship that we want students to develop. The connection between addition and subtraction will also be developed with more practice. Some students may understand that if 76 + 24 = 100, then
100 − 24 = 76. Allowing students to share their thinking will help classmates grow in their development of these concepts.
“Let’s listen to more of our story: Ms. Clover is also planting a vegetable garden. She has 305 corn seeds but has only planted 28 seeds. How many seeds does she still have to plant?”
This problem essentially asks that students find “how many more,” which suggests adding on. But this problem does not make adding on easy because 305 is so much bigger than 28. Watch for students who use a removal strategy to subtract. By subtracting in “chunks,” students are practicing estimation strategies.
“Who will share an answer and tell us how you got it?” (If she has only planted 28 already, then she has a lot more to plant. So the number has to be big. I thought 28 and how many more make 305? I decided to start with the 305 and take away the part she is done with, the 28. So I have to subtract 305 − 28.)
“How did you subtract 305 − 28?” (First I took away 200 because I knew I needed to have 28 left, but 300 was too much to take away. So now I had 105 − 28 and I took away 80 to get to the 20s. But that was 25 so I took away 3 too many. So, I counted back 3 from 280.)
“So how many more corn seeds does Ms. Clover need to plant?” (277)
“Did anyone solve this problem differently?”
There will be students who use adding on strategies, such as: (Add 80 to 28 to get to 108; add 200 to get to 308; subtract 3 to get to 105; 80 + 200 − 3 = 277.)
As you listen to students share their work, make note of those who arrived at the wrong answers and listen to their explanations as well, perhaps in a one-on-one dialogue. Bring them together for guided practice in a small group.
“If Ms. Clover has 305 seeds and she has already planted 277, how many more seeds does she have left to plant?”
If students are seeing the connections, they will reply that she has 28 seeds to go, without having to calculate.
“How do you know?” (Because 277 + 28 = 305)
Monitoring student responses during discussions and small-group work can be used as informal assessments to guide instruction.
Random Reporter can also be used to continue the activity above. Choose two numbers and write them on the board. Each group makes up a simple story that involves these two numbers, and reads their story to the class.
Some groups will simply add or subtract the numbers, but some will add the numbers and use the sum as a third number: 24 and 37 are on the board (24 + 37 = 61). The group responds: “Mr. Smith has 61 marbles. If he gives 24 to his son, how many does he have left?”
Extension:
- Routine: There is a logical progression in the Routine activity in each lesson. Students start with daily practice of number facts and then move to daily practice of doing the Goal activity mentally. The next step is to have daily practice of a two-digit addition or subtraction problem, working toward students doing the problem mentally. Again, one problem a day is sufficient, as it is the discussion that is the key. Once the problem is solved, you could also ask students to give you one more number sentence using those three numbers (a member of the fact family).
- Small Group: Provide practice mentally estimating and computing with addition or subtraction. Decide which operation your group needs to work with.
Give students the Balloon Math worksheet (M-2-3-4_Balloon Math.doc) and ask them to “pop” two balloons, one from the A bunch and one from the B bunch by drawing an X on the balloon. They will then use the two numbers to write an addition or subtraction sentence to solve. Encourage students to use a number line or models such as base-ten blocks if needed and to write their actions in the extra space on the page. When they have successfully completed one problem, choose two more balloons and write a new number sentence. Continue in this way until all balloons are “popped.”
You can let students choose individually which balloons they want to pop, or they can take turns choosing which balloons everyone should pop, so the whole class is working on the same problem. Ask students to explain their thinking as they work.
- Expansion: Students will complete the Lost Families worksheet (M-2-3-4_Lost Families and KEY.doc). The goal is to find the numbers that go together and make a fact family. Students will have to apply problem-solving skills and their knowledge of basic number facts to identify the ones place for the sum, to reunite the families.
- Workstation: Students work in pairs. Each student draws two cards and lays them on the table to create a two-digit number. The student with the largest number chooses an operation: addition or subtraction. Each student then creates a number sentence using both two-digit numbers and the selected operation.
When both partners arrive at a solution, they check each other’s work. If only one student has the correct solution, s/he keeps all four cards. If both students are correct, they each keep two cards. If they both are wrong, the cards go to the bottom of the deck.
Play continues until five number sentences have been solved correctly. The winner is the student with the most cards.
