As a student progresses through the developmental stages of mathematical reasoning, he will solve problems in different ways. The first approach that he will use is called a Direct Model. In this developmental approach, a student will represent each number in the problem with a concrete object (this may be a manipulative, such as a block, or a written representation of an object, such as a tally mark or a circle). Explicit modeling is beneficial for a student who understands the quantities and concepts underlying a math problem but is unfamiliar with a particular problem type. Direct models will help your student understand the action in the story and what he's trying to figure out, which will help him effectively identify and execute the procedure needed to solve the problem. Read on to learn about how to intervene to support your student's ability to use direct models to solve various problem types using addition, subtraction, multiplication, or division.
Math Story Problem Types. (n.d.). Retrieved from: http://www.teachertipster.com/CGI_problem_types.pdf
This page includes intervention strategies that you can use to support your students in direct modeling to solve join, separate, part-part-whole or compare problems. When using direct models of problems with single-digit numbers, students should be taught to represent each number with a single object (such as a counter, a cube, or a drawing of a circle). When direct-modeling problems with multi-digit numbers, students should start by representing smaller two-digit numbers with cubes and then learn to represent all multi-digit numbers using base-ten blocks (or pictures of these blocks), which represent 100s, 10s, and 1s (rather than counting out 100 single counters to represent 100). When direct modeling problems with fractions, students should be taught to represent numbers using fraction bars (or pictures of fraction bars).
Many of the examples below show students solving join, separate, part-part-whole, or compare problems with single-digit numbers, but the same strategies can also be applied to problems about multi-digit or fractional quantities. 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 use direct mathematical models for join, separate, or compare problem types, you should start by explicitly teaching the skill that you are about to show the student. This sounds like:
Activity A: Joining All (also known as Count All)
If your student is struggling to solve Join (Result Unknown) problems (4 + 6 =__ ) or Part-Part-Whole (Whole Unknown) problems (4 + 6 + __ ) , you can teach Joining All (Carpenter et al., 2015). This strategy teaches a child to use his fingers or objects to represent each of the addends, and then to count the total of both sets to find the solution.
This example refers to the following problem.
Jose had 4 teddy bears. His friends gave him 6 more teddies for his birthday. How many teddy bears did he have then?
Teacher: To solve this problem, I'm first going to count cubes. 1, 2, 3, 4. Now, I'm going to count 6 cubes. 1, 2, 3, 4, 5, 6. Now, I have my 4 cubes, and my 6 cubes. Now, I'm going to add them together, because I want to find the total. I'm going to push then all together, and then count them up: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10... He has 10 teddies total.
Joining All (or Counting All) in Action
In this video, Ama Awotwi explicitly models how to use a Count All strategy to join two quantities together. This is the most concrete strategy for solving a simple addition problem and therefore an appropriate strategy to model first. As you watch, consider: How does Ama's model support her student's execution of this strategy?
Activity B: Joining To
If your student is struggling to solve Join (Change Unknown) problems (14 + __ = 21 ), you can teach Joining To (Carpenter et al., 2015). This strategy teaches a child to find the number of objects added to the initial quantity to find the total.
Joining To in Action
This example refers to the following problem.
Jose had 14 teddy bears. How many more teddy bears does he need to get to have 21 teddy bears?
Teacher: To solve this problem, I'm first going to count the number of cubes, or teddy bears, that Jose already has. [Teacher counts out 14 cubes.] Now, I'm going to count additional cubes until I reach my goal number 21. So, remember, I'll start with 14, and count up, but I'll keep these cubes separate: 15, 16, 17, 18, 19, 20, 21. Now that I've reached my goal number, I'll count how many cubes I added to my original number, 14, to get to 21. 1, 2, 3, 4, 5, 6, 7. I added 7! So, that means Jose needs 7 more bears to get to 21.
Activity C: Joining with Base-10 Understanding
If your student is struggling to solve multiple digit Join (Result Unknown) problems (15 + 33 =__ ), you can teach him to use Joining with Base-10 Understanding (Carpenter et al., 2015). This strategy teaches a child to add multi-digit numbers by using base ten blocks to represent larger quantities more efficiently.
Joining with Base-10 Understanding in Action
In the video below, Emily Art explicitly models how to solve addition problems by using base-10 blocks. As you watch, consider: How do the various colors of base-10 blocks help a student model a problem?
Activity D: Separating From
If your student is struggling to solve Separate (Result Unknown) problems (8 - 5 =__ ), teach Separating From (Carpenter et al., 2015). This strategy teaches a student to represent a large quantity and remove cubes to find a difference.
Separating From in Action
This example refers to the following problem.
Jesus had 8 toy cars. He gave 5 toy cars to Rosalee. How many toy cars does Jesus have left?
Teacher: I know the total number of toy cars, so I'll first count that many cubes. [Teacher counts 8 cubes.] Now, I know that some of these cars went away because Jesus gave some to Rosalee. I have to remove one cube for each toy car Jesus gave to Rosalee. 1, 2, 3, 4, 5. Now, I'll count my remaining blocks. 1, 2, 3. Jesus has 3 toy cars left.
Activity E: Separating To
If your student is struggling to solve Separate (Change Unknown) problems (8 -__ = 3 ), teach Separating To (Carpenter et al., 2015). This strategy teaches a student to represent a large quantity and remove cubes until he gets a number given in the problem.
Separating To in Action
In the video below, Emily Art works with a student to use this intervention. As you watch, consider: How does using blocks support a student's ability to carry out subtraction problems?
Activity F: Part-Part-Whole (Part Unknown)
If your student is struggling to solve Part-Part-Whole (Part Unknown) problems (7 - __ = 4), teach him to direct model these types of problems. Unlike Joining or Separating problems, these problems do not include action. However, the same strategies can be used to teach a student to model and solve the problem. In this intervention, the student represents the quantities in the problem and uses them to figure out the unknown part.
Part-Part-Whole (Part Unknown) in Action
This example refers to the following problem.
There are 5 kittens in Jodie's room. 3 of them are white and the rest are black. How many kittens are black?
Teacher: First, I'm going to count out the total number of kittens, which is 5. 1, 2, 3, 4, 5. Now, I'm going to move the kittens that are white. 1, 2, 3. Finally, I'll count the kittens that are left, which will be the ones that are black. 1, 2. My answer is 2! Two kittens are black!
Activity G: Matching
If your student is struggling to solve Compare (Difference Unknown) problems (5 - 2 = __ or 2 + __ = 5), teach Matching (Carpenter et al., 2015). This strategy teaches a student to construct a 1:1 correspondence between two sets until one is exhausted.
Matching in Action
This example refers to the following problem.
Ronald has 2 kittens. Lupita has 5 kittens. How many more kittens does Lupita have than Ronald?
Teacher: First, I'm going to count out the kittens that Ronald has. 1, 2. I'll put these in a row here. Now, I'll count out the kittens that Lupita has. 1, 2, 3, 4, 5. I'll put her kittens in her own row. Now, I'm going to put the rows next to each other. I'll put the 2 cubes here, and then line the 5 cubes up below the 2 cubes, matching as many cubes as I can by putting them next to each other. Now, I have to see which of Lupita's kittens don't have a match. 1, 2, 3... 3 don't match. That means, Lupita had 3 more kittens than Ronald!
Carpenter, T. P., Fennema, E., Franke, M.L., Levi, L., & Empson, S. B. (2015). Children's mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.
Forbringer, L., & Fuchs, W. (2014). RtI in Math: Evidence-Based Interventions for Struggling Students. Hoboken: Routledge Ltd.
Math Story Problem Types. (n.d.). Retrieved from: http://www.teachertipster.com/CGI_problem_types.pdf
Direct models can also be used for a student who is learning to solve problems that require them to multiply and divide single- or multi-digit numbers or fractions. This page includes intervention strategies that you can use to support your students in this area. 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 use direct models for multiplication and division problem types, you should start by explicitly teaching the skill that you are about to show the student. 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: Multiplication Modeling
If your student is struggling to solve multiplication problems (3 x 6 = __ ), teach Multiplication Modeling (Carpenter et al., 2015). This strategy teaches a student to solve multiplication problems by using tally marks or counters to model each group and then counting the total number of objects.
Multiplication Modeling in Action
This example refers to the following problem.
Dionne has 3 boxes of donuts. There are 6 donuts in each box. How many donuts does Dionne have in all?
Teacher: First, I'll count out one set of 6 cubes to represent one box of donuts: 1, 2, 3, 4, 5, 6. Now, I'll count another set: 1, 2, 3, 4, 5, 6. And, finally my last set: 1, 2, 3,4, 5, 6. Now, I've got my cubes that represent Dionne's 3 boxes of donuts. To find out how many donuts he has in all, I have to count up all the cubes. [Teacher counts each cube.] All in all, he has 18 donuts!
Activity B: Measurement Division
If your student is struggling to solve basic division problems (18 ÷ 6 = __ ), teach Measurement Division (Carpenter et al., 2015). This strategy teaches a student to solve division problems by using tally marks or counters to model the total amount and then separate the counters into groups by making an unknown number of piles of a known size.
Measurement Division in Action
This example refers to the following problem.
There are 18 donuts that need to be boxed. 6 donuts needs to go into each box. How many boxes does Dionne need for his donuts?
Teacher: First, I'll count 18 donuts. [Teacher counts 18 cubes.] Now, I'll count them into groups of six. Here is my first group. [Teacher counts out 6 cubes from the 18.] Here is my second group. [Teacher counts out 6 more into a second group.] Now, how many do I have left? 6, so that's my final group. I have 3 groups. That means that Dionne needs 3 boxes in order to fit his 18 donuts.
Activity C: Partitive Division
Another strategy to teach if your student is struggling to solve division problems (18 ÷ 3 =__) is called Partitive Division (Carpenter et al., 2015). This strategy teaches a student to solve division problems by using tally marks or counters to model the total amount and then "dealing" the total number of objects evenly among a known number of piles to figure out how many there will be in each pile.
Partitive Division in Action
This example refers to the following problem.
There are 18 donuts in boxes. There are 3 boxes total. How many donuts are in each box?
Teacher: First, I'll count 18 donuts. [Teacher counts 18 cubes.] Now, I'll divide them into three groups. [The teacher places cubes into 3 separate groups, dealing each cube until her pile is empty.] Now, I'll count up how many donuts are in each group: 6. There are 6 donuts in each box.
The following activities can be used for students who struggle to solve problems that require them to multiply and divide double-digit numbers.
Activity A: Multiplication with Base Ten Blocks
If your student is struggling to solve double-digit multiplication problems (16 x 3 = __ ), teach Multiplication with Base Ten Blocks (Carpenter et al., 2015). This strategy teaches a student to solve multiplication problems by using base ten blocks to model the quantity in each group and then counting the total number of objects.
Multiplication with Base Ten Blocks in Action
This example refers to the following problem.
Dionne has 3 boxes of donuts. There are 16 donuts in each box. How many donuts does Dionne have in all?
Teacher: First, I'll count out one set of 16 cubes to represent one box of donuts. To do this, I'll use one long block, because I know there are 10 cubes in each long, and then I'll use 6 ones: 10, 11, 12, 13, 14, 15, 16. Now, I'll count another set of 16 the same way: 10, 11, 12, 13, 14, 15, 16. And, finally my last set: 10, 11, 12, 13, 14, 15, 16. Now, I've got my cubes that represent Dionne's 3 boxes of donuts. To find out how many donuts he has in all, I have to count up all the cubes, by first counting my tens and then my ones. [Teacher counts: 10, 20, 30, 31, 32, 33...48.] All in all, he has 48 donuts!
Activity B: Measurement Division with Base Ten Blocks
If your student is struggling to solve problems that require double-digit division (48 ÷ 12 =__), teach Measurement Division with Base Ten Blocks (Carpenter et al., 2015). This strategy teaches a student to solve division problems by breaking a total number of tens and ones apart to form an unknown number of groups of a known size.
Measurement Division with Base Ten Blocks in Action
This example refers to the following problem.
There are 48 donuts that need to be boxed. 12 donuts needs to go into each box. How many boxes does Dionne need for his donuts?
Teacher: First, I'll get out 48 blocks. I need 4 tens blocks, which is 40, and 8 ones. 10, 20, 30, 40, 41, 42, 43, 44, 45, 46, 47, 48. Now, I need to split these blocks into groups of 12. I'll put one ten block and two ones in this group to make 12. Now, I'll keep doing it until I have used up all of my blocks. It looks like I have just enough blocks to make 4 groups of 12. That means Dionne needs 4 boxes.
Activity C: Partitive Division with Base Ten Blocks
Another strategy to teach if your student is struggling to solve problems that require division of multi-digit numbers (48 ÷ 4 = __ ) is called Partitive Division with Base Ten Blocks (Carpenter et al., 2015). This strategy teaches a student to solve division problems by using base ten blocks to model a total number and then dealing this total into a known number of groups to identify how many will be in each group.
Partitive Division in Action
This example refers to the following problem.
There are 48 donuts in boxes. There are 4 boxes total. How many donuts are in each box?
Teacher: First, I'll count 48 donuts by using my base ten blocks. I need 4 tens blocks, which is 40, and 8 ones: 10, 20, 30, 40, 41, 42, 43, 44, 45, 46, 47, 48. Now, I'll divide them into four groups. [The teacher deals one ten block into each group, and then deals the remaining ones into each group until her pile is empty.] Now, I'll count up how many blocks are in each group. 12. There are 12 donuts in each box.
Carpenter, T. P., Fennema, E., Franke, M.L., Levi, L., & Empson, S. B. (2015). Children's mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.
Math Story Problem Types. (n.d.). Retrieved from: http://www.teachertipster.com/CGI_problem_types.pdf
Think about the following scenario, which takes place after a teacher has explicitly taught a student strategies for using direct models to solve problems. This example refers to the following problem.
Ronald has 2 kittens. Lupita has 5 kittens. How many more kittens does Lupita have than Ronald?
Teacher: "Show me how you can match cubes to solve this problem."
Student (putting the five blocks and the two blocks in a line next to each other): "One,
two, three, four, five, six, seven. She has 7 more kittens than Ronald!"
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 count again. When he is counting cubes, encourage the student to make sure he is using the correct strategy. | "Try again! Make sure you match each cube, so you can count how many more one person has than the other." |
| Medium Scaffold | Back it Up. If a student is struggling, back up your process. Ask the student to reread the problem and make sure he is following the correct process to solve the problem. | "I can see that you are stuck. Let's go back and review how to match the cubes." |
| 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 use the cubes to represent this problem." |
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! Modeling the use of manipulatives can take some time to do well, since students are managing two things at once: representing the problem through cubes, and then counting the cubes. Continue to practice such strategies to solidify the student's understanding. | "Let's try matching the cubes again. First, we'll count out 5 and place them here. Then, we'll count out 2 and place them below the 5 cubes we've laid out. Then, we'll count the difference." |
| Multiplying and Dividing Double-Digit Numbers | Trade in the Tens! If the student does not have an even number of base ten blocks (or needs to break the tens to find the solution), teach him how to trade in the base ten blocks for ten ones. If the student doesn't understand why this trade is even, line the blocks up next to each other and show how they are the same length. |
"If you can't evenly distribute your blocks, trade in your ten block for ten ones!" |