Once a student has mastered Counting On/Back to solve a particular problem type, he is ready to solve that type of problem in a more flexible way. Deriving interventions require a student to break down numbers and recombine numbers using known facts. These strategies support a student's ability to visualize the quantity (without using manipulatives) so that he can solve the problem with mental math. Read on to learn about how to support your student's ability to derive to solve problems using addition, subtractions, multiplication, and division.
This page includes intervention strategies that you can use to support your student's ability to derive single and multi-digit addition and subtraction problems. As you read, consider which of these interventions best aligns with the particular problem type with which your student is having difficulty.
Explicit Instruction
If you are intervening to support your students' ability to derive numbers to solve story problems, you should start by explicitly teaching the skill. This sounds like:
The following activities can be used for students who struggle to solve problems that require them to add and subtract single-digit numbers.
Activty A: Break Apart to Make Ten
If your student struggles to solve any type of story problems involving single-digit addition or subtraction quickly and accurately, teach him to Break Apart to Make Ten.
The following example refers to the problem below.
Laurie had 9 cards. She got 4 more cards from her friend Jamel. How many cards does she have in all?
In the following image (Clements & Sarama, 2009), you will see the work of a student who is using this strategy to add 4 to 9. First, she breaks 4 into a part she knows she needs to add to 9 to make 10 (1), and the left-over part (3). She then combines 9 and 1 to get 10 and then combines 10 and 3 to get 13.
Break Apart to Make 10 in Action
Now, watch as Ama Awotwi introduces this strategy to an intervention student struggling with mental math.
As you watch, consider: What does this student need to know already about addition in order to use this strategy?
Activity B: Number Fact Strategies (Addition and Subtraction)
If your student struggles to solve story problems of any type that involve addition or subtraction problems quickly and accurately, teach him Number Fact Strategies (Carpenter et al., 2015).This strategy teaches a student to use facts that they already know to derive other facts.
Number Fact Strategies in Action
This example refers to the problem below.
Dionne has 4 boxes of donuts. Then, his friend James gives him another 11 boxes of donuts. How many boxes of donuts does Dionne have now?
Teacher: To solve this problem, I'm going to think of the facts I already know. I know that 10 has two fives in it, and then I still have one left over. So, I'm going to add 4 + 5 + 5 = 14, and then add my 1 to make 15. What might be another way I could solve this problem, using what I know?
Student: I know that adding by 10 is easy, so 4 + 10 is 14. Plus 1 more is 15. So, the answer would still be 15.
Teacher: Well done. You are relying on facts you know quickly to solve the problem.
The following activities can be used for students who struggle to solve problems that require them to add and subtract multi-digit numbers.
Activity A: Incrementing Invented Algorithms (JRU or PPW)
If your student is struggling to solve Join (Result Unknown) and Part-Part-Whole (Whole Unknown) problems, such as 27 + 35 = n, you can support his understanding by teaching him to solve by Incrementing Invented Algorithms (JRU or PPW) (Carpenter et al., 2015). In this strategy, a student successively adds on to a partial sum. For example, he might round the numbers to the nearest ten, add those together, and then add on the ones.
Incrementing Invented Algorithms (JRU or PPW) in Action
This example refers to the following problem (Carpenter et al., 2015).
There were 27 boys and 35 girls on the playground at recess. How many children were on the playground at recess?
Teacher: Solve this problem by starting with easy numbers, such as a number rounded to its nearest ten, and then adding on.
Student: Let's see. 20 and 30, that's 50, and 7 more is 57. Then, the 5. 57 and 3 is 60, and the 2 more from the 5 is 62. There were 62.
Activity B: Incrementing Invented Algorithms (JCU)
If your student is struggling to solve Join (Change Unknown) problems, such as 37 + __ = 61, you can support his understanding by teaching him to solve by Incrementing Invented Algorithms (JRU) (Carpenter et al., 2015). In this strategy, a student successively adds on to a partial sum. For example, he might add a few ones to get a number to a ten, and then add the other ones.
Incrementing Invented Algorithms in Action
This example refers to the following problem (Carpenter et al., 2015).
Yoshi has 37 dollars. How many more dollars does she have to earn to have 61 dollars to adopt a dog?
Teacher: Solve this problem by adding the easiest way you can until you get to 61.
Student: 37 plus 3 is 40, plus 20 is 60, then just one more is 61. I added 3, 20, and 1, that's 24; 24 dollars.
Activity C: Combining the Same Units (Addition)
Another strategy to help your student solve Join (Result Unknown) and Part-Part-Whole (Whole Unknown) problems, such as 27 + 35 = n, is called Combining the Same Units (Addition) (Carpenter et al., 2015). This intervention is best for a student who is able to use mental math to add single digit numbers in his head. This strategy teaches a student to combine the tens and ones separately before adding them together.
Combining the Same Units (Addition) in Action
This example refers to the following problem (Carpenter et al., 2015).
There were 27 boys and 35 girls on the playground at recess. How many children were on the playground at recess?
Teacher: Solve this problem by adding up the tens, and then the ones, and then adding them all together.
Student: Let's see. 20 and 30 makes 50. Then 7 and 5, that makes 12. So, I'll add that to 50. The answer is 62.
Activity D: Incrementing Invented Algorithms (Subtraction)
If your student is struggling to solve Separate (Result Unknown) problems, such as 73-55 = n, you can support his understanding by teaching him to solve by Incrementing Invented Algorithms (Subtraction) (Carpenter et al., 2015). In this strategy, a student successively subtracts from a complete sum. For example, he might round the numbers to the nearest ten, and then subtract the ones from those numbers.
Incrementing Invented Algorithms (Subtraction) in Action
This example refers to the following problem (Carpenter et al., 2015).
Gary had 73 dollars. He spent 57 dollars on a pet snake. How many dollars did Gary have left?
Teacher: Solve this problem by finding the easiest numbers to work with, such as rounding to the nearest ten.
Student: Let's see. 73 take away 50. That's 23. Now the 7. Take away 5. That's 18, and take away the other 2- that's 16. He had 16.
Activity E: Combining the Same Units (Subtraction)
Another strategy to help your student solve Separate (Result Unknown) and Part-Part-Whole (Whole Unknown) problems, such as 73 - 55 = n, is called Combining the Same Units (Subtraction) (Carpenter et al., 2015). This intervention is best for a student who is able to use mental math to subtract single digit numbers in his head. This strategy teaches a student to subtract the tens and ones separately before adding them together.
Combining the Same Units (Subtraction) in Action
This example refers to the following problem (Carpenter et al., 2015).
Gary had 73 dollars. He spent 57 dollars on a pet snake. How many dollars did Gary have left?
Teacher: Solve this problem by finding the easiest numbers to start with, such as rounding to the nearest 10, subtracting those numbers, and then doing the same for the ones.
Student: Let's see. 70 take away the 50, that's 20. Then 3 take away 7 is minus 4. So, I need to take away 4 from the 20. So that's 16.
This page includes intervention strategies that you can use to support your student's ability to derive single and multi-digit multiplication and division problems. As you read, consider which of these interventions best aligns with the particular problem type that your student is having difficulty with.
Explicit Instruction
If you are intervening to support your students' ability to derive numbers to solve story problems, you should start by explicitly teaching the skill. This sounds like:
The following activities can be used for students who struggle to solve problems that require them to multiply and divide single-digit numbers.
Activity A: Number Fact Strategies (Multiplication and Division)
If your student struggles to solve story problems that involve multiplication and division problems quickly and accurately, teach him Number Fact Strategies (Carpenter et al., 2015).This strategy teaches a student to use facts that he already knows to derive other facts.
Number Fact Strategies in Action
This example refers to the problem below.
Dionne has 4 boxes of donuts. There are 6 donuts in each box. How many donuts does Dionne have in all?
Teacher: To solve this problem, I'm going to think of the facts I already know. I know that 2 sixes is 12, and another 2 sixes must also be 12. So, 12 + 12 is 24. He must have 24 donuts. What might be another way I could solve this problem, using what I know?
Student: I know that 3 sixes is 18, and then I have to add another 6. So, the answer would still be 24.
Teacher: Thanks! Can anyone else tell me another way you solved the problem?
Student: I started thinking about one box of donuts, which was 4. Then, I doubled it to get 8. Then, I know I needed 3 more of these, so 3 eights is 24.
Teacher: Well done. You are all relying on facts you know quickly to solve the problem.
The following activities can be used for students who struggle to solve problems that require them to multiply and divide multi-digit numbers.
Activity A: Invented Algorithms (Multiplication)
If your student is struggling to solve multi-digit multiplication problems, help support his understanding by teaching him how to solve these problems by using Invented Algorithms (Multiplication) (Carpenter, 2015). In this strategy, the teacher helps encourage the student to use tens to find the answer. For example, a student might multiply the tens and then the ones. This strategy can be used only after a student has developed a conceptual understanding of how to use base-ten blocks. When solving multi-digit multiplication and division problems, you may encourage your student to use manipulatives if he needs additional support. According to Carpenter et al. (2005), this reliance of manipulatives does not mean that the student has regressed; it only suggests that he may not yet understand the problem and is seeking out additional tools to help him conceptualize the problem. With time, the use of manipulatives should fade as a student extends his ability to use invented algorithms to solve multi-digit multiplication and division problems.
Invented Algorithms (Multiplication) in Action
This example refers to the problem (Carpenter et al., 2015) below.
The school bought 16 boxes of markers. There are 24 markers in each box. How many markers are there altogether?
Teacher: Solve this problem by first multiplying by tens.
Student: 10 twenties is 200. 6 twenties is 120. 6 fours is 24. So, 200 plus 120 is 220. 220 plus 24 is 244. So, there are 244 in all.
Activity B: Invented Algorithms (Measurement and Partitive Division)
If your student is struggling to solve multi-digit measurement and partitive division problems, help support his understanding by teaching him how to solve these problems by using Invented Algorithms (Measurement and Partitive Division) (Carpenter, 2015). In this strategy, the student uses everything he knows about addition, subtraction, multiplication, and division to solve problems.
Invented Algorithms (Measurement and Partitive Division) in Action
This example refers to the problem (Carpenter et al., 2015) below.
The class baked 84 cookies. We want to put them into boxes to sell at the school bake sale. If we put 12 cookies into each box, how many boxes can we fill?
Teacher: Solve this problem by using everything that you know about addition, subtraction, multiplication, and division to solve the problem.
Student: 12 and 12 is 24, so that's 2 boxes. 24 and 24 is 48, that's 4 boxes. If I doubled 48, it would be more than 84, so I added just 1 more box instead. 48 and 12 is 60, 5 boxes, Then I added 2 more boxes and got 84, so it was just 7 boxes.
Think about the following scenario, which takes place after a teacher has explicitly taught a student strategies for deriving numbers. This example refers to the problem below.
Dionne has 4 boxes of donuts. There are 6 donuts in each box. How many donuts does Dionne have in all?
Teacher: "How can you solve the problem using what you know?"
Student: "Well, I know that 4 + 4 is 8. And, 8 times 6 is 48, so he must have 48!"
In such a case, what might you do?
When you are planning your lessons, you should anticipate that your student will make errors throughout. Here are a series of prompts that you can use to respond to errors. Keep in mind that all students are different, and that students might respond better to some types of feedback than to others.
| Level of Support | Description of Scaffold | Script |
|---|---|---|
| Smallest Scaffold | Try Again. Ask the student to try the problem again. | "Try again! See if you can think through how to use what you already know to help you figure out how many donuts Dionne has. |
| Medium Scaffold | Back it Up. If a student is struggling, back up your process. | "What are you trying to figure out? What's the equation that you need to solve? (4x6 = ?). So, what other facts do you know that will help you figure out what 4 x 6 is?" |
| Highest Scaffold | Model. If the student continues to struggle, model the process for him. | "I can see that you are stuck. I'll show you how I solve this problem using what I know. Then, we can practice using what you know to solve it a different way. I know that 2x6 = 12. So, 4 x 6 = 12 + 12 = 24." |
If your student struggles to meet your objective, there are various techniques that you might try in order to adjust the activity so as best to meet your student's needs.
| Activity | Description of Strategy | Script |
|---|---|---|
| All Activities | Practice, practice, practice! One reason that deriving can be difficult is that it can be hard to keep track of the additional steps that are needed to find the answer. For example, in the above problem, the student had accounted for 8 out of the 24 donuts total, but didn't understand the process for finding the rest. Offering continued practice with deriving numbers, and paying special attention to what is already accounted for and what next steps need to be taken is essential. | "You've identified that 4 + 4 is 8. Now, how many more 4s do you need to figure out how many Donuts Dionne has? What do we need to do next to solve this problem?" |