BBC Bitesize - KS2 Maths - Addition and subtraction - inverse operations
Work on the understanding of the inverse relationship between addition and subtraction in an explicit, symbolic context with small numbers finds some evidence. -Recognise and use the inverse relationship between addition and subtraction and use this to check calculations and solve missing number problems. Number?omarcafini.infoions. PowerPoint. (McREL) Students understand the inverse relationship between addition and subtraction.
As we discuss the skill, I ask questions throughout the lesson. By doing this, it allows the students to think about the answer instead of me just giving them a lot of information. I feel that students learn better when they come up with things on their own. For this particular lesson, I will begin by reviewing place value and how we use it to add and subtract. When we add or subtract, we line our numbers up according to place value.
The numbers in the ones place should line up under each other, the tens place, etc. Some numbers need to be regrouped. When we regroup, we put the number in the ones place at the bottom, and regroup the number in the tens place at the top of the tens place. For example, with the number 15, the 5 would go in the ones place and the 1 would be regrouped to the tens place because it is valued at 10 3.
When we subtract, we may have to regroup as well. If the top number is smaller than the bottom number, we need to take 1 away from the place to the left and add it to the place to the right. For example, 17 - 8. The 7 is smaller than the 8, so we take one away from the tens place that becomes a zerothen we add that one to the ones place.
We now have a Because we have two numbers, we can take those 2 numbers and make a math problem. How does using the inverse operation help find missing numbers? I find that collaborative learning is vital to the success of students. Students learn from each other by justifying their answers and critiquing the reasoning of others MP3. As they work to solve problems, I will keep in mind that problems are easier for children to solve when the action or the relationships in the problems are as clear as possible.
This will be something that I check for as I create sets of word problems for my students. It is evident that the largest group of these problems includes word problems one through seven in the appendix because they are all Join problems in which the change is unknown. A second dimension of this set is problems that do not require regrouping to find the answer.
Problem numbers one, three, and six in the appendix are representative of this group. A final similar set of this group is problems that require regrouping to find the answer. These types of problems include numbers two, four, and seven in the appendix. As I created several of the problems in this word problem set, I realize that these problems are difficult for my students to solve as addition problems.
I realize that I have to find alternate ways for my students to "set up" the problems. For example, if I use problem number one to demonstrate, it will look like this: If I had students organize the problem differently, I believe that my students would see things much differently See below. When students are able to look at the problem in this manner, it becomes easier for them to see that in the ones column, six plus a number equals eight.
They are also able to see that in the tens column, two plus a number equals three. If they are not able to mentally use what they have learned about adding to ten, then they can use the Joining To method as described in Children's Mathematics: In this method, a child would use six counters to show the number in the ones column and "count on" from six in order to find how many they need to get to eight. In this case, the child would add two more counters to the six.
This would help the student to know that six plus two is eight. They would do the same for the number two in the tens column.
Join Problems with Start Unknown The third type of Join problem is created when the start minuend of the problem is unknown. Several of these types of problems are listed below. Todd had some hotdogs. Will gave him 12 more hotdogs. Now Todd has 27 hotdogs. How many hotdogs did Todd start with? Kelsey some pies on Monday. She baked 12 pies on Wednesday. She ended up with 21 pies in all. How many pies did she bake on Monday? The dimensions of this problem include the overall dimension of being problems that all have the first addend, or the start, missing.
Another dimension includes the problems that call for students to utilize regrouping. Examples of problems in this dimension include numbers one, three, and five. Problem numbers two and four are examples of problems that do not require regrouping to solve.
These types of problems will likely cause students the same trouble as those in the "Join with Change Unknown" section of this unit.
I would use this opportunity to use the prior knowledge that the students have acquired with setting the problems up differently, utilizing their knowledge of place value, and adding to ten to solve these in a way similar to the previous ones. Using problem number four as an example, I would initially like to see the students set up problem number three in the following way: It would look like this: If necessary, I would encourage them to use the Joining To method in order to solve the problem.
There is an additional issue with this problem that allows me to incorporate the students' prior knowledge of fact families. The fact that the students need to start "counting on" with the bottom number will allow me to have the conversation with them that shows them more about the commutative property. It will be interesting to see if students will explore the possibilities of listing the addend on the bottom first in order to find the addend on the top.
Some students will need to review commutative basics such as: Separating - Class Two Separate problems are closely related to Join problems. Very similar to Join, Separate problems require an action that takes place over time.
The difference is that the initial quantity is decreased over time versus increased with Join problems. Again, very similar to Join problems, Separate problems involve three quantities. There is a starting quantity minuenda change quantity, or the amount taken away subtrahendand the result difference. Any of these quantities can be the unknown. Separate Problems with Result Unknown Todd cooks 27 hotdogs. Will took 14 of Todd's hotdogs.
How many hotdogs does Todd have left? Kelsey baked 20 pies. She gave 8 pies to her neighbor. The above word problem set include the overall dimension of being subtraction problems that require the students to find the result. Another dimension includes problem number one solely because it uses number words instead of numbers in the problem.
Problem numbers one and two are of the dimension that requires subtraction without regrouping. There is also a dimension that requires regrouping with subtraction and those include problems number three, four, and five. Separate Problems with Change Unknown Todd cooks 27 hotdogs.
Todd then gives some hotdogs to his neighbor Will. Todd is left with 13 hotdogs. How many hotdogs did Todd give to Will? Kelsey had 20 pies.
Addition and subtraction - inverse operations
She gave some pies to Ms. Kelsey was left with 12 pies, how many pies did she give to Ms. There is a dimension that includes both of the above word problems because they are subtraction problems that include a missing subtrahend. There is also another dimension of this problem that does not require regrouping in order to find the answer. Ironically enough, all five of the problems are a part of this category. I would encourage the students to find the answer to these types of word problems in a way that is very similar to the way in which we solve the Join with Change Unknown problems.
Initially, I would have the students set up the word problem as follows: They would also need to rely upon what they have learned about fact families. I would guide the students to arrange the problem vertically as follows: From there, if necessary, they would need to use the Join To method and figure out "five plus what number equals 7" or "what number plus five equals 7", depending on the direction that they chose to work.
Separate Problems with Start Unknown The last type of separation problem that students will need to understand is one that is missing the first number, or the minuend.
Some examples of those types of problems follow: He then gave Will12 hotdogs. Kelsey baked some pies.
She was left with 24 pies. The largest dimension of these problems includes all of them and that is the fact that they are all missing the starting number, or the minuend. Another dimension is when a problem requires regrouping and those are problems one and three. A third dimension includes problem numbers two, four, and five. This dimension includes the problems that do not require regrouping.
The last dimension includes problems numbers one and four. These problems utilize number words instead of numbers in the problems. As stated before, this is important for my students who experience literacy issues with the problem solving activities.
In other words, they are not reading on grade level and I want to incorporate some type of practice every chance I get. Cognitively Guided Instruction defines these types of problems as "static relationships among a particular set and its two disjoint separate subsets. Whole Unknown This type of problem gives the two disjointed parts and asks the problem solver to give the size of the whole.
A few of these problems are listed below. There are 18 girls in the class and 15 boys in the class. How many students are in the class? Four blue fish are in the tank with eighteen green fish. How many blue and green fish are in the tank? The overall dimension which includes all five of the aforementioned problems is that they are all requiring the problem solver to give the size of the whole.
The next dimension although in no particular order includes the problems where you are required to regroup. These problems include numbers one, two, three, and five. Another dimension does not require regrouping and problem number four is the sole problem in this dimension. A further look into the dimensions of these problems shows us a group of problems that use number words instead of numbers.
This dimension consists of problems number two and five. Part Unknown In these types of word problems, the students are given one of the parts along with the whole, and they are to find the size of the other part. Unlike the Join and Separate problems, it is not obviously clear to the students as to how they are to go about solving these problems. For example, if you look at number one below, you can either subtract 15 - 7 to solve the answer, or you can say, "7 plus a number equals 15" and this will cause you to use the count on or Join To strategies.
Teaching Addition and Subtraction Word Problems to Children
For the purpose of this curriculum unit, I will guide my students to use subtraction to solve these problems. As an aside, if students used an alternative addition method to solve this type of problem, I would not count it against them or tell them that they were wrong. Several of the problems in this category are identified below.
How many girls were playing baseball? Deedre has 28 marbles. How many marbles are green? The most obvious of the dimensions is that they are all giving the whole and one part and they are asking for the missing part.
There is a dimension that asks for students to regroup with the subtraction and those are question numbers one and four. A dimension that does not require regrouping with subtraction in the word problems are numbers two, three, and five. An added dimension to this set of word problems is a problem that contains extraneous information. This is problem number four. Students will need to realize that the eight bones that T. Where the Part-Part-Whole problems "involve relationships between quantities rather than a joining or a separating action", Compare problems "involve the comparison of two distinct, disjoint sets rather than the relationship between a set and its subsets.
The chart below describes them. Mark has 3 mice. Referent set Joy has 7 mice. Compared set Joy has 4 more mice than Mark. Difference Amount by which one set exceeds the other Very similar to the Part-Part-Whole problem set, it is difficult to absolutely specify how I would prefer my students to solve Compare problems.
As you will see in the set of problems that follow, students will be able to draw on their prior knowledge of Join To or counting on skills as well as Separate strategies in order to find the answers. Difference Unknown Some of the most common Compare problems are listed below.
Kelsey had 19 pies. Polite had 7 pies. How many more pies did Mrs. Kelsey have than Mrs. If Veronica has 17 sandwiches and Jill has 11 sandwiches, how many more sandwiches does Veronica have than Jill? In order to identify the dimensions of this problem, you have to realize the most evident dimension of all five of these problems. This dimension is that all of the questions are missing the difference. Another dimension is one that includes the problems that require regrouping.
The regrouping problems in this set are numbers three and five. An added dimension is the set of problems that do not require regrouping with subtraction and those problems are numbers one, two, and four. The last dimension for this set includes a problem with extraneous information.
The only problem in this category is problem number five. Compared Set Unknown These word problems are a little more difficult than the previous problems and call for students to pay more attention to the way in which the problems are worded. Although the students are comparing, these are not subtraction problems.
They are actually problems in which they have to add in order to solve. Samples of Compared Set Unknown problems are listed below.
Lacie had 9 marbles. Laila has 8 more marbles than Lacie. How many marbles does Laila have? Langston has 24 gumballs. Lamar has 19 more gumballs than Langston. How many gumballs does Lamar have?
One dimension in the problem set listed above requires students to regroup when they add. These problems are two, three, four, and five. There is only one problem in the dimension that does not require regrouping when adding and that is problem number one.
Another dimension that I may explore when I continue to create word problems for my student is increasing complexity. Problems number one and two are basic addition problems and should not be very difficult for students to solve.
Problem number one - although it is a two-digit plus one-digit problem - is simple because there is no regrouping required. Problem number two should be very easy to solve as well because it consists of two numbers that students are exposed to early on when they learn "adding doubles" and "adding doubles plus one.
In this set, problem number four is the only one that is in this dimension. Referent Unknown In this Compare problem category, students are at a point where they need to subtract to solve the problems. Examples of Referent Unknown problems are listed below. Joy has 9 toy cars. She has 3 more cars than Mike. How many toy cars does Mike have? Tanya has 27 pens. She has 16 more than Travis. How many pens does Travis have? As stated before, students must know that they have to subtract to solve these word problems.
One dimension of these problems is that they all require subtraction to find the answer. Another dimension includes the problems that require regrouping.
These problems are numbers three and five. An added dimension which include problems that do not require regrouping are numbers one, two, and four. Although the wording in the problems in the preceding "four classes" of problems are similar, the basic structure of each problem is unique. The structures of the problems are related to how the students solve the problems. This lesson can be completed in one to two days. The teacher will need to make sure that they create several examples of each class of word problems.
The first thing that the teacher will do is ensure that students are aware of the different types of word problems. Although you may not teach the problems using the vocabulary presented in this unit i. This can be accomplished in one of several ways. Teachers can choose to review sample problems written on chart paper. Another way can be to create a PowerPoint presentation. Other choices include putting individual problems on index cards for small groups to review or even putting the problems on a sheet for each student to have for himself or herself.
After about ten to fifteen minutes of reviewing various problems, teachers will present the students with at least ten word problems that are comprised of a mixture of samples from each class. The best grouping for this activity will be small, heterogeneous groups consisting of no more than four children each.
The students will decide what operation they will use to solve the problem. Along with identifying the operation, students will write the number sentence that they will create to solve. These are the only things that they will do. Although this activity is seemingly very simple, I believe that this is a crucial activity because this will help the teacher to learn which students have mastered the concepts and those who have not. An alternative to this activity could be to discuss several problems of one type, and then introduce a new type while using a strategic set of questions for students.
When introducing a second set, teachers may ask, "How is this problem different from the problems we have been looking at? Questions to ask could ask students which problems are related to which, how they are similar, and how they are different. It is not important for students to learn the whole taxonomy. What is important is that they learn to read the problem carefully and think about what it says.
The comparison activities are to help them make distinctions in language. As stated before, students will work in small, heterogeneous groups for this activity. The teacher will circulate among the groups listening to discussion. This is how you will be able to assess whether or not individual students understand the previous lessons that have been describing the methods for solving word problems.
Of course, teachers will use the results of this activity to decide whether to reteach or continue to expand the word problems lessons.
This activity can be completed in one to two days. The teacher will need to make sure that there are several examples of each class of word problems that could be taken from the appendix.
Once again, teachers will need a variety of word problems from the four classes of problems presented in this curriculum unit. The way that teachers choose to present the problems should be in a manner that best fits their personalities and the class' needs and learning styles. Teachers will arrange students in pairs for this activity.
The name of this activity explains exactly what the students will be doing. After more exposure to a variety of word problems, teachers will use this opportunity to verify that students are indeed becoming more proficient at solving word problems. While in their pairs, students will decide whether they will use addition or subtraction. This will satisfy the "what" of this activity.
After specifying the operation, students will show the number sentence that they will use to solve the word problem. This is the "how" of the activity. In order to satisfy, the "why" portion of the activity, the students will explain to their partners why they choose to solve the problems in that manner. It is during this time that the teacher will listen for reasoning that will help them to solve various problems consistently.
There will be times when students will be able to "guess" the correct answer, but the discussion piece will clarify whether or not they really understand what they are doing. This activity can be completed in one to three days. In order to complete this activity, students will need to be placed in small heterogeneous groups of no more than four students. They will need paper and pencils to complete the activities and then chart paper and markers or crayons to display their word problems and share them with the class.
This activity will be completed at a point when the teacher is confident that the majority of students are able to complete any of the word problems within the four classes of addition and subtraction word problems. Students will work in their groups to create four different word problems.
Students can decide whether or not they will use addition or subtraction as long as they use different types of word problems. For example, they can create two addition word problems as long as they are not both "join problems with start unknown". After students have created the problems in their groups, the teacher should review their work. Once students have followed the guidelines, the teacher should give them approval to create a poster that they will share with the class.
Once all groups have created their problems, had them approved, and rewritten them on chart paper, the class should go through each chart, discuss, and solve the problems. Implementing District Standards Georgia Performance Standards This curriculum unit satisfies several sections of the Georgia Performance Standards for second grade students.
The second Number and Operations domain states that "students will build fluency with multi-digit addition and subtraction". The elements of the standard include correctly adding and subtracting two whole numbers up to three digits with regrouping, understanding and using the inverse relationship between addition and subtraction to solve problems and check solutions.
Two additional elements of importance in this unit include students using mental math such as benchmark numbers to solve problems and using basic properties of addition commutative, associative, and identity to simplify problems.
In addition to the Numbers and Operations domain, this curriculum satisfies several elements that fall under the Process Standards that are required for the second grade. Some of these standards include building new mathematical knowledge through problem solving, applying and adapting a variety of appropriate strategies to solve problems, recognizing and using connections among mathematical ideas, and understanding how mathematical ideas interconnect and build on one another to produce a coherent whole.
Math word problems demystified. Georgia Department of Education n. Georgia department of education, early intervention program EIP guidelines,Retrieved July 2,from Georgia Department of Education website: K-2 mathematics georgia performance standards.
Retrieved July 3,from http: A book for grownups about children's mathematics. They sold three popsicles this afternoon. How many popsicles were sold altogether?
Todd cooks 20 hotdogs. Will cooks 10 hotdogs. Kelsey baked 26 pies last week. She baked 13 more pies this week. Veronica made 3 sandwiches for the picnic. Jill made 25 sandwiches for the picnic. How many sandwiches did the girls make together? Jason can make 35 cones in five minutes. Erin can make 7 cones in five minutes. How many cones can they make in five minutes together? Jeremy used 35 different crayons to make his drawing. Later, he used 56 more crayons. How many crayons did Jeremy use altogether?
Andy had 19 videotapes. He bought 24 more at the store. How many videotapes does Andy have now? Toni has 17 blue beads for her bracelet. Her mother gave her 19 green beads. How many beads does Toni have in all? Leon put 37 cans of fly soup in the cart and Ken put 13 cans of cream spider soup in the cart.
How many cans of soup went into the cart? How much money have you saved in the last two months? Join Problems with Change Unknown Todd cooks 26 hotdogs. If they end up with 40 hotdogs altogether, how many hotdogs did Will cook? How many pies did she bake this week if she ended up with a total of 20 pies? Veronica made 13 sandwiches for the picnic.
Jill made some sandwiches for the picnic. If they ended up with a total of 28 sandwiches, how many sandwiches did Jill make? Jeremy bought 18 crayons. How many more crayons will he have to buy in order to have a total of 37 crayons? Jeremy bought 14 crayons.
His mother gave him some crayons. If Jeremy ended up with 32 crayons, how many crayons did his mother give him? Andy has 23 videotapes. He buys some more. If Andy ends up with 48 videotapes total, how many videotapes did he buy? Toni has 17 blue beads. Her mother gave her some more beads. How many beads did her mother give her if Toni ended up with 30 beads? Leon put 37 cans of fly soup in the cart and Ken put some more cans of cream spider soup in the cart.
If they ended up with 50 total cans of soup altogether, how many cans of soup went into the cart?
You saved some more money this month. The ice cream store sold twenty-five popsicles this morning. They sold some more popsicles in the afternoon. If the ice cream store sold a total of thirty-eight popsicles, how many did they sell in the afternoon?
Join Problems with Start Unknown Todd had some hotdogs. Kelsey baked some pies on Monday. Tuesday she took a break from baking, but she baked 12 pies on Wednesday. She ended up with 21 pies. Veronica made some sandwiches for the picnic.
Jill made 14 sandwiches for the picnic. Veronica and Jill had a total of 28 sandwiches. How many sandwiches did Veronica make? Jeremy bought some crayons. His mother gave him 27 crayons.
Jeremy ended up with a total of 40 crayons. How many crayons did Jeremy buy? Andy has some videotapes. He bought 15 more at the store. If Andy ends up with 48 videotapes, how many tapes did Andy begin with? Toni has some beads. Her friend Zora has 19 beads. Together the girls have 37 beads.
How many beads does Toni have? Joseph had some stamps in his collection. He bought 42 more. Now he has 60 stamps in his collection. How many stamps did he have to begin with? The museum sold some tickets on Monday and 72 on Wednesday. If a total of 87 tickets were sold on Monday and Wednesday altogether, how many tickets were sold on Monday? Some people that got on the bus in Atlanta.
If there are now 28 people on the bus altogether, how many people got on the bus in Atlanta? Separate Problems with Result Unknown The ice cream store had twenty-five popsicles this morning. How many popsicles were left? Todd cooked 27 hotdogs. Jill had 35 sandwiches and gave 19 of them away. How many sandwiches did Jill have left?
Recognise and use the inverse relationship - New
Katie had 23 stickers. She gave 14 stickers to Drew. How many stickers did Katie have left? Andy has 48 videotapes and he sells 29 of them.