This lesson's main goal is to help students become more fluent with basic number facts by employing various techniques. To make addition easier, teachers encourage students to break down numbers. Given that students are accustomed to the number 10, the "make ten" strategy will help them comprehend place value when adding two-digit numbers.
The objective of this exercise is to have students break down numbers into a systematic list of combinations. These combinations are useful for another strategy called "doubles plus or minus 1." Students typically pick up on doubles quickly and can use this knowledge to find sums that are either one more or one less than a double. They can also use this strategy to look at the pair of numbers in their combination list that are one away from the double, i.e., pairs with a difference of 2.
Part 1
"What are some methods we employ in addition and subtraction?" Allow students to reflect on the methods they already employ for addition and subtraction before having them write a list and discuss them.
"There are various methods we can employ to obtain the answers when we add and subtract. We will also benefit from using models, such as base-ten blocks and counters. We're going to put some of the techniques into practice so that we can consider addition and subtraction from various angles."
As they work to improve their fluency with number facts, students will be encouraged to use the "make ten" strategy and allowed to experiment with the "doubles plus" and "doubles minus" strategies in this lesson. Make an anchor chart that includes strategies for addition that students can consult as needed.
To ensure that students can easily view and comprehend the Double Ten-Frame page (M-2-3-1_Double Ten-Frame), set up the overhead projector to display ten on the left frame and four on the right frame (10 and 4). Employ the Base-Ten Dots sheet (M-2-3-1_Base-Ten Dots) if counters are not available.
Start by rapidly covering the overhead transparency after it has been visible for roughly five seconds. Don't allow students to count the counters or dots one at a time.
Ask, "How many dots were visible? How are you aware?" (There were 14 in total. I counted ten plus four, which makes fourteen. Alternately, I noticed a lot on one and four on the other. I'm not sure how many.)
Once more, display the frames and note that each one has ten squares. (The goal is to prove that there are ten spaces in every frame.)
"You are seeing the ten sets very rapidly. Let's try again now."
Show five and ten. Ask the same questions again. (There were 15 because one is 10 and the other is 5, and 10 plus 5 is 15.) I saw 15 because one was full and the other only had dots in one row, which is five. Ten and five is 15.
"How many would there be if both frames were filled?" (There would be 20.) This proves that 20 is the maximum.
"Let's take a closer look at one more." Put eight on one frame and two on the other to give the students a little more time to study before hiding them. Ask, "How many counters do we have? How are you aware?" (I counted eight, then two more. That makes ten.)
"Could you please come over and demonstrate your counting technique?" After pointing to each of the eight counters, the student touches the following two and counts from 9 to 10.
Accept any successful additions. Persist in inviting pupils to express and demonstrate their thought processes. Keep an eye out for more effective tactics.
"Who tried adding 8 and 2 differently?" (I got 10 because I moved the two counters from this side over here to make room for them and noticed two empty spaces on the side with 8. As a result, I filled this side up to ten. Alternatively, I moved the 2 over to make 10 because I knew that 8 plus 2 is 10.)
"You're relocating counters to the empty spaces to fill up the ten-frame. You're going to make ten. Here are a few more issues to examine."
Provide copies of the M-2-3-1_Double Ten-Frame worksheet and 20 counters to the students.
Establish the equation 8 + 4 = ____. Ask students to solve the puzzle using their counters and ten frames. Invite a few students to discuss their solutions and provide an explanation of their ideas. (I obtained 12 by combining 2 from 4 with 8 to create 10. I then had 2 remaining, making 10 plus 2 equal 12.)
"So, you employed the 'make ten' strategy. To get ten, you had to move counters from one frame to another. You had ten plus some left over. Give me a thumbs up if you also employ this tactic. Has anybody attempted using a different approach?" (I thought of 8 as 10. This gave me 12. Although I am aware that 10 plus 4 equals 14, I deducted 2 to arrive at 12.)
"In other words, you "made ten" in a different way by adding one number to make ten and then taking that number away."
Take a look at which students are employing the "make ten" tactic by trying a few more problems (8 + 3, 9 + 8, 7 + 9).
To set the stage for applying the "make ten" technique, use story problems. For instance, "Joshua possessed nine baseball cards." Fredrick gave him an additional six. What is Joshua's current baseball card count? Students are encouraged to add to the story's numbers and make ten.
Part 2: “Doubles Plus or Minus” Strategy
This section of the lesson will cover creating and utilizing doubles, as well as adding and subtracting.
Ask students (or groups) to create a combination of 11 using the Double Ten-Frame worksheet (M-2-3-1_Double Ten-Frame) and counters.
Describe their observations, write down the combinations, and assist them in making sure there are no more combinations than there are.
"Is there a method by which we could arrange these combinations?"
Students will recommend increasing by one and beginning at zero:
0 + 11, 1 + 10, 2 + 9, 3 + 8, 4 + 7, 5 + 6, 6 + 5, 7 + 4, 8 + 3, 9 + 2, 10 + 1, 11 + 0
Invite students to discuss what they noticed. To determine how many ways they could have thought of, some might count. While it's not the main objective of the lesson, this lets the students apply what they've learned. "There are 12 ways to get to 11," they might remark. "On this side, the numbers get smaller and bigger." "Every pair has a partner who is opposite them."
"Are all numbers paired with someone else? Are 7 + 4 and 4 + 7 equivalent?"
This raises the commutative property of addition and the idea that addition is an order-insensitive process. Even if the two outcomes are identical, the issues could differ.
Asking the same questions, repeat the procedure with fourteen counters. This will support both the commutative nature of addition and the combinations' flexibility and organization.
"Can we therefore conclude that we discovered pairs with the same numbers, albeit in the opposite order, using both numbers? Is there a related pair for every pair of numbers?" (There is only one pair in the 7 and 7. Two "7 and 7s" are not allowed.)
"You're accurate. You discovered two doubles. Is there a pair of doubles with our other number?" (There isn't a double pair for 11 because it's odd. Since 14 is even, it does.) To recognize this, students might require some guidance.
"Let's discuss doubles in more detail. They can assist us with addition and subtraction."
Under the "Math Facts" section, you can choose to teach your students the "Doubles Rap," which can be found at http://classrooms.tacoma.k12.wa.us/bpes/amichaels/homework.
Examine the double sums for the digits 1 through 10. Write the following on the board:
1 + 1 =___ 4 + 4 = ___ 7 + 7 = ___ 10 + 10 = ___
2 + 2 = ___ 5 + 5 = ___ 8 + 8 = ___
3 + 3 = ___ 6 + 6 = ___ 9 + 9 = ___
Ask students to share the sums after that.
Once each frame's seven squares have been filled in, present the double-ten-frame transparency before covering it. Find out how many squares were there. (There were 14 in total.)
"How are you aware of this?" (Because the amount in every frame was 7, and I know that 7 + 7 = 14.)
To each frame, add one counter.
"Now, what's the total?" (There are 15 since 14 had one added to it.)
Take out the fifteenth counter and remove the other counter.
"What's the current total?" (There are 13 instead of 14, as one was removed.)
"To find 7 + 8 and 7 + 6, we can either add or subtract one. This is because we know that 7 + 7 equals 14. I wonder if other doubles could use this matter." Put 8 and 9 in the transparency.
"How can the doubles strategy be used to find the total?" Invite students to explain various methods for calculating the sum. (I am aware that 8 + 8 equals 16, and 9 + 1 equals 17.)
"You employed the 'doubles plus' tactic, then. You increased it to 9 (8+1) by using the 8. You started with the double 8s and then added 1 more."
Put the equation (8 + 8) + 1 = 17 on the board.
"Did anyone take an alternative action?" (After realizing that 9 + 9 = 18, I had to deduct 1 because 8 is one less than 9, so I also got 17 after seeing the 9.)
"By applying the 'doubles minus' strategy, you obtained the same result. To get 9, you added 1 to the 8, added the double 9s, and then deducted 1."
Write 9 + 9 - 1 = 17.
"Let's practice some more." Until students can articulate how they are utilizing the "doubles" strategies, provide additional examples.
On the transparency, display 8 and 6.
"Is it possible to add these two numbers using the doubles strategy?" (Alright! One counter moved from the frame containing eight to the other frame gives us seven plus seven, or fourteen.)
8 + 6 and 6 + 8 were two of the combinations that the students created when they added up to 14. Bring up the combinations the students made, which all added up to 14, and demonstrate on the transparency how 7 + 7 can become 6 + 8 by moving one counter, and vice versa, to get the students to understand this.
"What additional number combinations would this tactic be effective with?"
Assign groups of students to discover related pairings, then present their results to the entire class. Allow them time to tinker and experiment with various numbers and combinations.
Students will see that by adding one to each of the following numbers, they can convert 9 + 7, 8 + 6, 7 + 5, 6 + 4, 5 + 3, 4 + 2, and 3 + 1 into doubles.
"What's the connection between these pairings?" (Each of the of them is two apart.)
As they apply the technique more, some students might discover that pairs with a difference of 4 can be adjusted by 2 each to create doubles, pairs with a difference of 6 can be adjusted by 3 each to create doubles, and so on, until they have listed every possible combination for an even number.
Informal assessments that inform instruction can be derived from student responses to discussions and small group work. For any of the aforementioned tasks, students can also complete Random Reporter. Use the "make ten" strategy, the "doubles plus" or "doubles minus" strategies, or ask the class to add various combinations, like 6 + 7. You may also ask students how they would add 7+9. This will highlight the fact that any tactic is appropriate as long as it helps the learner and provides the right response.
There may be an assessment available in the form of M-2-3-1_Lesson 1 Assessment on paper and pencil.
Extension:
Routine: Instant recall is the aim of learning fundamental math facts. Instant recall is built on the strategies that students are learning in this lesson. It makes sense to apply these facts daily as a result.
For example, you could use Random Reporter to give each group one problem, like "What is 8 + 5?" or "Give me three ways to make 15," and give them only ten or fifteen seconds to consult with one another. Alternatively, you could post simple math fact problems on the board at the beginning of the day (or during the math period).
The preparation of transparency with as many issues as groups would be another tactic. Since they don't know which group you will start with, each group is accountable for all issues.
Either way, the entire exercise ought to take no more than five minutes.
Small Group: Assign a random number of counters to each member of the group. Students should be asked to determine every path leading to their number of counters. See if the students utilize the counters, follow a methodical approach, or just count the counters and make a list of the combinations.
Utilizing multiple instances, guide students to draw the subsequent conclusions:
One more pair exists than the total number of counters in each pair.
Though they don't have a double, odd numbers come in handy when using the "doubles plus one" or "doubles minus one" strategies.
Even numbers (such as 6 + 6, 5 + 7, and 7 + 5) have a double and can also use the difference of two pairs.
Expansion: Students use multidigit addition in this exercise by applying their understanding of fundamental number facts. Some of the tactics from this lesson will also be put to use by the students.
Students see the same number of facts in various contexts as they complete the Extension worksheet (M-2-3-1_Extension and Key.doc). Students might consider 20 + 13 = 33 or (20 + 10) + 3 = 30 + 3 = 33 when solving a problem like 28 + 5. Students can think of 80 + 50 as 10 tens plus 3 tens, or they can see that 8 tens plus 5 tens equals 13 tens, or 130.
If more counters are needed, students are free to use base-ten blocks. Discuss their conclusions once they've finished the exercise.
Workstation: Using whatever resources they require, students solve the Fact Family worksheet (M-2-3-1_Fact Family). Students are supposed to start investigating how addition and subtraction relate to one another. Regarding fact families and addition and subtraction properties, they will begin to conclude.
