“When we add and subtract we can use different strategies to get the answers. Using models, like counters and base-ten blocks, will also help us. We are going to practice some of the strategies that will help us think about adding and subtracting in different ways.”
This lesson will encourage students to use the “make ten” strategy and will let them explore the “doubles plus” and “doubles minus” strategies as they work to develop fluency with number facts.
Prepare the Double Ten-Frame overhead to show ten on the left frame and four on the right frame (10 and 4) so that it will be visible to students and can be covered quickly (M-2-4-1_Double Ten-Frame.doc). If counters are not available, use the Base-Ten Dots sheet (M-2-4-1_Base-Ten Dots.doc).
Begin by displaying the transparency on the overhead (for about five seconds) and covering it quickly. Do not give students time to count the dots (or counters) one at a time.
Ask, “How many dots did you see? How do you know?”
Students may say, “There were 14. I saw ten and four more. That’s 14.” “I saw four on one and a lot on the other one. I don’t know how many.”
Show the frames again and point out that they both have ten squares each. The purpose is to establish that each frame has ten spaces.
“You are seeing the sets of ten very quickly. Now let’s try some more.”
Display ten and five. Repeat the questions.
Students might answer, “I saw 15, because one was full and the other only had dots in one row; that’s five. So ten and five is 15.” “There were 15 because one is 10 and the other one is 5, and 10 plus 5 is 15.”
“If both frames were full, how many would there be?”
“There would be 20.” This establishes that the maximum is 20.
“Let’s look at one more.” Display eight on one frame and two on the other, giving students a little more time to observe; then cover them up. Ask, “How many counters do we have? How do you know?”
“Ten, because I counted eight and then two more.”
“Would you come up and show us how you counted?” The student points to each of the eight counters, then touches the next two, and counts “9, 10.”
Accept all successful attempts to add. Continue asking for students to share and model their thinking. Watch for more efficient strategies.
“Who tried another way to add 8 and 2?”
“I got 10 because I saw two empty spaces on the side with 8, and the two counters from this side fit over here, so I moved them and filled this side up to make ten.”
“I did that too because I know that 8 plus 2 more is 10, so I moved the 2 over and made 10.”
“You are moving counters to the extra spaces to fill up the ten-frame. You are making ten. Let’s look at some more problems.”
Set up the problem 8 + 4 = ____. Have students use their ten-frames and counters to solve the problem. Ask some students to share their answers and explain their thinking. [IS.4 - All Students and Struggling Learners]
“I got 12 by moving 2 from the 4 and putting them with the 8 to make 10. Then I had 2 left, so I had 10 plus 2 equals 12.”
“So you used the ‘make ten’ strategy. You moved counters from one frame to the other to make ten. Then you had ten and some left over. Show me a thumbs up if you also used this strategy. Did anybody try another way?”
“I got 12 by thinking of 8 as 10. I know 10 + 4 is 14, but 10 is 2 more than 8, so I took away 2 to get 12.”
“So you ‘made ten’ in a different way by adding to one number to make it 10 and then subtracting that afterwards.”
Give students copies of the Double Ten-Frame worksheet (M-2-4-1_Double Ten-Frame.doc) and 20 counters. Try a few more problems (8 + 3, 9 + 8, 7 + 9) and walk around to see which students are modeling the “make ten” strategy.
Use story problems to create a context for using the “make ten” strategy. For example, “Joshua had 9 baseball cards. Fredrick gave him 6 more. How many baseball cards does Joshua have now?” The numbers in the story encourage students to make a ten and add on.
Part 2: “Doubles Plus or Minus” Strategy
This part of the lesson will explore making and using doubles and then adding or subtracting.
Using the Double Ten-Frame worksheet (M-2-4-1_Double Ten-Frame.doc) and counters, ask students (or groups) to make a combination of 11.
List the combinations on the board, discuss their observations, and help them to check for all possible combinations.
“Is there a way we can put these combinations in order?”
Students will suggest starting at zero and going up by one:
0 + 11, 1 + 10, 2 + 9, 3 + 8, 4 + 7. 5 + 6, 6 + 5, 7 + 4, 8 + 3, 9 + 2, 10 + 1, 11 + 0
Ask students to share their observations. Some may count to see how many ways they came up with. This is not the focus of this lesson, but it allows students to express what they have learned. They may say, “There are 12 ways to get to 11.” “The numbers get bigger on this side and smaller on this side.” “The pairs each have an opposite partner.”
“Does every pair of numbers have a partner? Is 4 + 7 the same as 7 + 4?”
This brings up the commutative property of addition and a discussion about the fact that order does not matter in addition. Although the result is the same, the problems may be different.
Repeat the process with 14 counters, asking similar questions. This will reinforce the flexibility and organization of the combinations, and the fact that addition is commutative.
“So can we say that with both numbers we found pairs that have the same numbers, just in reverse order? Does every pair of numbers have a related pair?”
“No, the 7 and 7 is only one pair. You can’t have two ‘7 and 7s.’”
“You’re right. You found a pair of doubles. Does our other number have a pair of doubles?” Since 11 is odd, it will not have a doubles pair. 14 is even, so it does. Students may need some prompting to recognize this.
“Let’s talk more about doubles. They can help us when we add or subtract.”
Review the doubles sums for the numbers 1 through 10. Write on the board:
1 + 1 =___ 6 + 6 = ___
2 + 2 = ___ 7 + 7 = ___
3 + 3 = ___ 8 + 8 = ___
4 + 4 = ___ 9 + 9 = ___
5 + 5 = ___ 10 + 10 = ___
Then call on students to give the sums.
Display the Double Ten-Frame transparency with seven squares filled in each frame; then cover it. Ask how many squares there were.
“There were 14.”
“How do you know?”
“Because each frame had the same amount: 7, and I know 7 + 7 = 14.”
Add one counter to either frame.
“What is the total now?”
“There are 15, because one was added to 14.”
Remove the 15th counter and take an additional counter away.
“What is the total now?”
“There are 13 because one was taken away from 14.”
“So, if we know 7 + 7 is 14, then we have a strategy to find 7 + 8 and 7 + 6: add one, or subtract one. I wonder if this would work for other doubles.” Display 8 and 9 on the transparency.
“How can I find the total using the doubles strategy?”
Call on students to describe different ways to find the sum.
“I know that 8 + 8 is 16, and 9 is 1 more, so that’s 17.”
“So you used the ‘doubles plus’ strategy. You used the 8 and made it 9 (8+1). You added the double 8s first and then added 1 more.” Write the problem on the board, (8 + 8) + 1 = 17.
“Did anyone do something different?”
“I saw the 9 and thought that 9 + 9 is 18, but I had to take 1 away because 8 is one smaller than 9, so I got 17 too.”
“You got the same answer using the ‘doubles minus’ strategy. You added 1 to the 8 to get ‘9,’ added the double 9s, and then subtracted 1.”
Write (9 +9) −1 = 17.
“Let’s practice some more.” Give more examples until students are able to explain how they are using the “doubles” strategies.
Show 8 and 6 on the transparency.
“Can we use the double strategy to add these two numbers?”
“Yes! If we move one counter from the frame with 8 to the other frame, we have 7 + 7, which is 14.”
When students made combinations of 14, 8 + 6 and 6 + 8 were two of the combinations. Elicit this from students by reminding them of the combinations they made (all of which equaled 14), and showing on the transparency that 7 + 7 can be changed to 6 + 8 by moving one counter, and vice versa.
“What other number combinations would this strategy work for?”
Have students work in groups to find similar combinations and report their findings to the class. Give them time to explore and investigate different numbers and combinations.
Students will see that 9 + 7, 8 + 6, 7 + 5, 6 + 4 (as well as 5 + 3, 4 + 2, and 3 + 1) can all be converted to doubles by adjusting each number by 1.
“What do these combinations have in common?”
“They all are two apart.”
Some students may take the strategy further and realize that pairs with a difference of 4 can be adjusted by 2 each to make doubles, pairs with a difference of 6 can be adjusted by 3 each to make doubles, and so on, eventually listing all combinations for an even number.
Student responses during discussions and small group work can be used as informal assessments to guide instruction. Students can do Random Reporter for any of the above activities. Ask the class to add different combinations such as 6 + 7 using the “make ten” strategy or the “doubles plus” or “doubles minus” strategies. You could also ask students what strategy they would use to add 7 + 9. This will help emphasize that any strategy is valid, if it works for that student and gives the correct answer.
A paper-and-pencil assessment may be given (M-2-4-1_Lesson 1 Assessment.doc).
As students start the day (or the Math period), there can be basic number fact problems on the board for students to answer, or you can use Random Reporter and give each group one problem, such as “What is 8 + 5?” or “Give me three ways to make 15,” allowing group members only 10 or 15 seconds to consult each other.
Another strategy would be to have a transparency prepared with as many problems as there are groups. Each group is responsible for all problems because they don’t know which group you will start with.
In either case, the whole activity should be quick, perhaps five minutes.
Small Group: Give each person in the group a random number of counters. Ask students to find all of the ways to get their number (of counters). Observe whether students use the counters, work systematically, or simply count the counters and list the combinations.
Use several examples, and lead students to the following conclusions:
The number of pairs is one more than their number of counters.
Odd numbers have no double, but are handy for the “doubles plus 1” or “doubles minus 1” strategies.
Even numbers have a double, and include the difference of two pairs
(i.e., 6 + 6, 5 + 7, 7 + 5).
- Students will complete the worksheet, (see M-2-4-1_Extension in the Resources folder), seeing the same number facts in different situations. For a problem such as 28 + 5, students may think of 20 + 13 = 33, or (20 + 10) + 3 = 30 + 3 = 33. For 80 + 50, students will see that 8 tens + 5 tens is 13 tens, or 130, or they may think of it as 10 tens + 3 tens.
Students may use the base-ten blocks (or other counters) as needed. After they have completed the activity, discuss what conclusions they have come to.
The focus of this lesson is to develop fluency with basic number facts, using different strategies. Students are encouraged to decompose numbers in order to add them more easily. The “make ten” strategy is useful because students are comfortable with the number 10, and because it will be useful for understanding place value in adding two-digit numbers.
Students are asked to break down numbers into a systematic list of combinations. These combinations will be useful for another strategy: “doubles plus or minus 1.” Students usually learn doubles quickly and can use this knowledge to find sums that are one more or one less than a double. Students extend the strategy further to look at the pair of numbers in their combination list that is one away from the double: pairs with a difference of 2. In each of these activities, students are given the opportunity to practice independently and explore on their own to draw their own conclusions.