Dive into the world of numbers and equations as we unlock how to solve different types of word problems. By understanding the different structures of word problems, we are more prepared to teach students the various types, and therefore, our students will be more successful. Furthermore, here is a breakdown of the word problem structures along with strategies that do not involve teaching keywords to understand word problems.
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🧠💪Research shows a link between a growth mindset and math success. Kids with a growth mindset about their abilities perform better and are more engaged in the classroom.
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I am constantly looking for ways to help my students be more successful with word problems. I have introduced the 4 step problem-solving process, and we are now exploring the different structures. Yesterday’s lesson didn’t go well, though, because the students just kept wanting to add the 2 numbers. They weren’t thinking about what was happening in the problem. I decided to pull out the part-part-whole mats, use smaller numbers, and have students act out what was happening in the story. Eventually, we started to understand that these were missing addend types of word problems. Success!
What are Word Problems?
A word problem is a few sentences describing a ‘real-life’ scenario in which a problem needs to be solved through mathematical calculation.
A one-step equation is an algebraic equation you can solve in only one step. The common core lists several problem situations that are taught beginning in Kindergarten.
Word Problem Example
Paula has 16 tickets for the raffle. She has 8 fewer than Mark. How many tickets does Paula have?
Importance of Word Problems
Word problems challenge students to approach mathematics from a problem-solving perspective. They must interpret information, identify relevant data, and discern the appropriate operations needed to find a solution.
This process encourages a thoughtful examination of the problem at hand, promoting a mindset that values reasoning and logic. In a world that increasingly relies on data-driven decisions, the ability to navigate and solve word problems has never been more vital.
Types of Word Problems in Addition and Subtraction
1.OA.A.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions.
2.OA.A.1 Use addition and subtraction within 100 to solve one and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions.
Addition with Word Problems and Subtraction with Word Problems
These are the most basic word problems. Students and to and take from.
Missing Addend Word Problems
These word problems are a little more complex, and one “part” is missing. Students need to understand the information given and what they are trying to solve before tackling these types of word problems.
Addition Word Problems with Missing Addends
Teach Addition Word Problems with Missing Addends with this set of 20 word problems. These practices change unknown word problems within 20 using the problem-solving steps.
Word Problems with Unknown Addends
Teach Word Problems with unknown addends with this set of 20 word problems. These practices start unknown word problems within 10 using the problem-solving steps.
Start Unknown Word Problems Worksheets and Slides
Use these Start Unknown Word Problems Worksheet and Google Slides. Teach word problems with unknown addends with this set of 20 word problems. These practices start unknown word problems within 20 using the problem-solving steps.
Types of Word Problems in Multiplication and Division
3. OA. A. 3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
Students need to have many opportunities to use concrete materials to model the situations and identify the number of groups and the number of items in a group. Then, once students have an understanding, they can begin to draw pictures.
In 4th grade, students continue to solve multiplication and division word problems. In addition, they are introduced to comparison word problems.
4.OA.A.2 Multiply or divide to solve word problems involving comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
Teachers should provide a variety of multiplicative comparisons for students to model, describe, and solve. In addition, students should make connections between models (such as bar models) and written equations. Here are some examples of Multiplicative Comparison Problems.
Teaching Word Problems for Addition and Subtraction
1st Grade Word Problems
At the beginning of the first grade, review situations that were taught in kindergarten and increase the number to 20. Over time, the other situations will be introduced for students to model. Have students describe what is happening in the problem and make a plan for solving the problem. In addition, they should identify what they are looking for in each problem situation. For example, the number of apples.
Students solve problems using materials such as counters, five and ten frames, part-part-whole tables, and number lines to model various situations. In addition, they begin to use drawings to represent what is happening in the problem. Model writing equations that describe the student’s actions with the expectation that they will begin to use symbols to show their work when ready. Give students opportunities to say the equation orally and then in writing.
2nd Grade Word Problems
This standard extends the Kindergarten and Grade 1 one-step word problem standards. Students should continue to use linking cubes, place value blocks, ten frames, and counters to model problems. When solving problems, some students may be ready to move from the concrete stage to the representational stage.
Students may use materials or draw pictures to determine a strategy for solving a problem. As students work with concrete materials to solve problems, teachers should help students see the connection between pictorial representations and abstract (equations) representations.
Teachers should continue to provide students with experiences using all the problem structures. Eventually, move to two-step word problems with smaller numbers and both operations being the same. Then, use larger numbers and both addition and subtraction within the same problem.
To read more on 2 Step Word Problems….What’s the Best Proven Way to Teach Word Problems with Two Step Equations?
Using Ten Frames and Double Ten Frames to Solve Word Problems
A ten frame is a rectangle with ten equal spaces. It has five spaces on top and five on the bottom. You often find the ten frames positioned horizontally and filled with counters placed from left to right. A double ten frame is 2 ten frames put together to show up to 20.
Ten frames are a fantastic tool used in kindergarten and first grade to help build an understanding of counting, place value, adding, subtracting, and more. You use counters or math manipulatives to represent word problems on the frame. Check out this link to find a free Ten Frames Template and double tens frames printable. Ten Frames Free Printable.
Read more about using ten frames here…Why Using Ten Frames for Math to Develop Understanding is Effective?
Word Problems of Subtraction
Do your students need more practice with subtraction word problems within 20? These slides use subtraction with ten frames and counters to solve the word problem.
Word Problems with Subtraction to 100
Do you need more practice with subtraction word problems within 100? These slides have a two-digit subtraction word problem to be solved using the place value chart and movable place value blocks.
Addition Word Problems | Word Problems to 20
Solve addition, find the sum, and missing addend word problems using ten frames and counters.
Using Part-Part-Whole Mats to Understand Word Problem Structures
The part-part-whole model is a pictorial representation of a problem that helps students see the relationship between a “whole” and its components.
The part-part-whole mat shows that you know some information but still need to find additional information. The bar can be split into as many parts as you need, depending on the information you’ve been given. The whole or any of the parts can be missing.
Children using this model will see the relationship between the whole number and its parts, helping learners to make the connections between addition and subtraction.
This is another valuable tool in modeling word problems. Check out this link to find a free Part-Part-Whole Template. Part- Part-Whole Mat
You can read more about using the part-part-whole strategy and find some part part whole mat activities here…How to Use the Powerful Part-Part-Whole Strategy to Improve Number Sense.
I think these are a great tool to use when one of the “parts” is missing so students can see if they need to subtract or add the numbers.
Word Problems of Addition and Subtraction with a Part Whole Mat
Practice word problems in addition and subtraction. Students model the problem with counters and a Part-Part-Whole mat or ten frames to build conceptual understanding and model solving the word problem for addition and subtraction.
Acting it Out or Modeling
It is essential to use concrete materials to act out or model problems. For example, after reading a book or giving students a situation, give them materials to act out the problem.
For example, after reading The Doorbell Rang, you could provide students with plates and cookies to act out a multiplication or division situation.
Eventually, students can begin to draw pictures and write equations to represent the problems.
Problem Structures
Research indicates that providing students with explicit instruction of problem structures can increase their success in solving problems similar in structure. Those effects are generalizable and can be maintained over time.
Word problems can be categorized into several distinct types, each presenting unique challenges and requiring different problem-solving strategies. One common type is the comparison problem, where students must determine the difference between two quantities.
For example, if one person has 5 apples and another has 3, the problem may ask how many more apples the first person has.
Another prevalent category is the part-part-whole problem, which involves understanding how parts relate to a whole. For instance, if a school has 120 students, with 70 being girls and the rest boys, a typical question would be to find out how many boys are in the school.
These problems require students to visualize relationships and use addition or subtraction to find the missing component, thus enhancing their comprehension of whole numbers and their components.
Word Problem Types for K-2
This is a breakdown of the word problem structures taught in Kindergarten and 1st grade and then continued into 2nd grade.
- Add to- result unknown ( 2 + 3 = ?)
- Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now?
- Add to- change unknown (7 add some more equals 9: 7 + ? = 9)
- Two frogs were sitting on the grass. Some more frogs hopped there. Then there were five frogs. How many frogs hopped over to the first two? 2 + ? = 5
- Add to- start unknown (have some add 4 more equals 9: ? + 4= 9)
- Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before? ? + 3 =5
- Take from- result unknown ( 5- 2= ?)
- Five apples were on the table. I ate 2 apples. How many apples are on the table now?
- Take from- change unknown ( 9 take away some equals 7: 9 – ?=7)
- Five apples were on the table. I ate some apples. Then there were three apples. How many apples did I eat? 5 – ? = 3
- Take from- start unknown ( have some take away 4 equals 3: ? – 4=3)
- Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before? ? – 2 = 3
- Put together/ take apart- addend unknown,total unknown or oth addends unknown
- Five pears are on the table. Three are red, and the rest are green. How many pears are green? 3 + ? = 5, 5-3 = ?
- Comparison- difference unknown, smaller unknown, bigger unknown
- Difference unknown: Larry has 5 apples. James has 8 apples. How many more apples does Larry have than James? 5 + ? = 8 8 – 5 = ?
- Bigger unknown: Larry has 4 fewer oranges than James. Larry has 6 oranges. How many oranges does James have?
- Smaller unknown: Larry has 6 fewer apples than James. James has 9 apples. How many apples does Larry have?
Link to Addition and Subtraction Problem Structure Table
Structures of Word Problems for 3rd- 5th Grade
- Equal Groups- Unknown Product (3 X 6= ?)
- There are 3 bags with 6 apples in each bag. How many apples are there in all?
- Equal groups- group size unknown ( 3 x ? = 18 and 18 / 3 ?)
- If 18 apples are shared equally into 3 bags, then how many apples will be in each bag?
- Equal groups – number of groups unknown ( ? X 6 = 18, and 18 /6 = ?)
- If 18 apples are to be packed 6 to a bag, then how many bags are needed?
- Arrays- unknown product
- There are 3 rows of apples with 6 apples in each row. How many apples are there?
- Arrays- group size unknown
- If 18 apples are arranged into 2 equal rows, how mamy apples will be in each row?
- Arrays- number of groups unknown
- If 18 apples are arranged into equal rows of 6 apples. How many rows will there be?
- Compare- unknown product
- A blue jacket costs $6. A red jacket costs 2 times as much as the blue jacket. How much does the red jacket cost?
- Compare- group size unknown
- A red jacket costs $12 that is 2 times as much as the blue jacket. How much does a blue jacket cost?
- Compare- number of groups unknown
- A red jacket costs $12 and a blue jackets costs $6. How many times as much does the red jacket cost as the blue jacket?
Teaching Word Problems using Multiplication and Division
Word problems put multiplication and division equations into context and help students understand what is happening with the numbers. Students need to start with acting out and then draw pictures and write equations for the word problems. There are several different standards involving multiplication and division word problems for elementary students.
Multiplication Arrays and Equal Groups
Understand Multiplication with Equal Groups or Arrays using digital counters.
Word Problems Division and Multiplication
Teach Word Problems, Division, and Multiplication using the counters to represent the word problems for multiplication and division.
Don’t forget to incorporate all the different Multiplication and Divison Problem Situations
Read about the math array definition and resources to teach it… What’s a Multiplication Arrays Math Definition and Fun Activities to Use?
Strategies for Solving Word Problems
To tackle word problems effectively, students can employ several strategies that simplify the process and enhance understanding. First and foremost, it is essential to read the problem carefully. This initial step allows students to understand the situation, identify key information, and understand what is being asked.
Once the problem is understood, the next strategy is to translate the words into mathematical expressions. This stage involves determining the operations needed to solve the problem. Students should ask themselves what the problem is asking for and what information they have.
Finally, students should check their work after arriving at an answer. This involves reviewing the calculations and ensuring that the solution makes sense in the context of the problem. If the answer seems unreasonable or does not align with the original question, students should revisit their steps and reassess their reasoning.
Making Sense of Problems
The first Mathematical Practice Standard of the Common Core State Standards for Math describes mathematically proficient students as those who can:
- Explain the meaning of a problem
- Plan a solution pathway rather than jumping to a solution
- Continually check for reasonableness and ask, “Does this make sense?”
These three skills are essential to solving math word problems successfully. But, how do we help students develop them?
Problem Solving Steps in Math
Students are more successful when using these 4 problem-solving steps in math. This helps them think about the problem and make a plan before solving it. Then, after solving the problem, they check if their answer makes sense.
Step 1: Understand
- Read and think about the problem.
- Circle the needed information in the problem.
- Underline the question in the problem.
- Write an answer stem sentence. There are ____ pages left to read.
Step 2: Plan
- Make a plan to solve the problem.
- What strategy will you use? (draw a picture, work backward, look for a pattern, create a table, bar model)
- What math tools do you need?
Step 3: Do
- Solve the problem.
- Show your work to solve the problem.
Step 4: Check
- Answer the question and check your work.
- Write the answer into the answer stem.
- Does your answer make sense?
- Check your work using a different strategy.
Tips for Mastering Word Problems
Mastering word problems is a skill that develops over time through consistent practice and the application of effective strategies. One of the best tips for improvement is to practice regularly with a variety of word problems.
This diversity exposes students to different types of challenges and helps them recognize patterns and common techniques for solving similar problems. Regular practice builds confidence and familiarity, making it easier to approach new problems with a positive mindset.
Collaborating in Solving Word Problems
Another helpful approach is to work collaboratively with peers or seek guidance from teachers. Discussing problems with others allows students to see different perspectives and problem-solving methods.
I,n addition, group work can foster an environment where students feel comfortable sharing their thought processes, asking questions, and learning from one another’s mistakes. This collaborative effort not only enhances understanding but also reinforces the idea that problem-solving can be a collective endeavor.
Real-World Connections & Math Literature
Using real-world situations or situations from literature books will help students visualize what is happening in the word problems. Then, students can act out the problem or use counters to model what is happening.
Word problems are not just academic exercises; they reflect real-world scenarios that require mathematical reasoning. Understanding their applications in everyday life makes learning math more relevant and engaging.
Tools and Resources for Practicing Word Problems
- Mastering Missing Addend Word Problems: Unlocking the Secrets to Problem-Solving Efficiency
- How to Unlock the Secrets of Teaching Word Problems Using Addition
- How to Successfully and Easily Teach Word Problems in Multiplication and Division
- How to Get Students to SHOW Their Reasoning
- ABCYA.com: Molly Add and Subtracts from 20
Overcoming Challenges in Solving Word Problems
Many students encounter challenges when faced with word problems, often feeling intimidated by the complexity of the language or the mathematical concepts involved. One common hurdle is the difficulty in understanding the wording of the problem.
In addition, ambiguous language or complex phrasing can confuse students, making it challenging to identify the core question. To overcome this, students should practice rephrasing problems in their own words, breaking down the sentences to distill the essential information.
Another challenge is the fear of making mistakes. Students may hesitate to attempt word problems due to a lack of confidence in their mathematical abilities. Addressing this fear requires fostering a growth mindset, where mistakes are seen as valuable learning opportunities rather than failures.
Encouraging students to embrace trial and error and to view problem-solving as a process rather than a destination can significantly boost their confidence and willingness to engage with challenging problems.
Read more about fostering a growth mindset…. Unlocking the Power of Growth: How to Help Students Develop a Growth Mindset
Not Using Word Problem Keywords
When students rely on keywords rather than reading and thinking, their understanding is hindered and may prompt them to use the wrong operation. The use of math keywords focuses on looking at the words in isolation and not in the context of the problem.
“Mathematics is about reasoning and making sense of situations” (Van de Walle & Lovin, 2006, p. 70); therefore, students should analyze the structure of the problems in the context, not just dissect them for keywords.
When students begin to view problem situations in this way, they can identify the bigger picture and make connections between problem situations and the solution strategy required to solve them.
Consider the following problem: Julie left $9 on the table. Her brother left $6 on the table. How much money was left on the table? Use of the word “left” might indicate to some that the solution to this problem is obtained with subtraction; however, this is an addition situation because two quantities are being joined together.
Teaching the Types of Word Problems in Mathematics
Mastering math word problems is an essential skill that not only enhances mathematical understanding but also equips students with critical life skills.
By understanding how to apply math in everyday situations, learners can appreciate its value and become more motivated to engage with the mathematics.
☀Download this BONUS Guide with everything you need to cultivate a positive classroom community a improve mindset.
➕Includes definitions, lesson ideas, mindset surveys for students and teachers, and printable
posters.
🧠💪Research shows a link between a growth mindset and math success. Kids with a growth mindset about their abilities perform better and are more engaged in the classroom.
👉Includes everything you need to start cultivating a more positive math classroom and students who love math.
Download and get started today! Click here to download the Bonus Mindset Guide & Survey.
With the right tools, resources, and mindset, students can transform their understanding of word problems. Begin teaching the different types of word problems and see success!
