3 read strategy for problem solving

3 Reads Strategy for Successful Problem Solving in Math

3 Reads strategy for problem solving with poster on clipboard and math manipulatives around it

Word Problems are often the hardest part of our math instruction. They can visually overwhelm students. They often contain extraneous information or multiple steps for completion. Students often struggle to persevere through complex problems. But, ultimately, it is through these complex problems that we are able to truly see our students’ understanding of math concepts and proficiency with the standards. Our students are expected to persevere through solving them, to demonstrate understanding, and to use a variety of strategies. I detail my experience with difficulties with story problems in my post Why Your Students Struggle with Word Problems . I use a modified 3 Reads Strategy in my classroom to help students make sense of complex word problems during our Word Problem of the Day . I connect it to the Close Reading we do during E/LA.

We have to read the problem closely to truly understand what is being asked of us as mathematicians.

The 3 Reads Strategy is a series of steps that helps students make sense of word problems. It’s focused on understanding the context. There are a variety of interpretations of the protocol. I have found my students have been increasingly successful following the 3 Read protocol daily. We do it during our Word Problem of the Day routine so we practice nearly every single day. At the beginning of the year, I walk my students through the 3 reads and we talk about the steps with each read. As the weeks go on, my scaffolding decreases as I expect students to apply the same steps independently. I often, especially with more complex word problems, do the first reading orally to provide access for all students. Here are the steps we take during our 3 Reads Routine.

3 Reads Strategy for Word Problems

1st read: read for gist.

The purpose of the first read is to get the gist of the word problem. Students should be able to answer what the problem is about; the context . Students should be able to retell, in their own words, what is happening in the word problem.

2nd Read: Read for the Unknown

The second read is focused on the unknown ; what is being solved for. Identifying the unknown helps students identify important and necessary information for solving during the third read. This helps them parse extraneous information out. It also helps students ensure they’re solving for what is actually being asked. I have my students underline important information in the question and also write a sentence frame for the solution.

3rd Read: Read for Quantities

In this read, students identify the quantities and relevant units. During this read, I have students circle the numbers and underline the key words (most often the units) for solving. It’s important to note that I do not mean keywords that are typically words relating to operations such as more. In this read, we focus on what is known ; the information given. With the unknown already being identified. Students then write an equation or expression to solve. They may also draw a picture if it’s helpful understanding the steps needed for solving.

Make a Plan

The last step in the 3 read protocol is to make a plan for solving. Now that students have identified what is being asked, the information that’s given to them, and what they are solving for, the last step is to actually solve. That may include modeling the problem with base ten blocks. It may also include using the standard algorithm to solve. Whatever strategy students feel they need, they do.

After students have worked through the problem, we share solutions and strategies. The focus is on so much more than a correct solution! I have students show their work and explain their thinking. Through our conversation we may critique someone else’s work to identify their mistake. We may share a variety of strategies for solving the equation. We may compare equations or expressions that were written for the problem. Because I’m walking around while students are working independently, I’m able to give on the spot support to some kids, while also identifying things I want to highlight for the group. This routine, and our steps after, go through so many of the Standards for Mathematical Practice!

It’s great to have a 3 Reads anchor chart or poster for student reference. A co-created anchor chart constructed with students while solving a complex problem would be great! I also have free black & white 3 Read Strategy posters that are perfect for printing on colored Astrobrights paper and made into a bulletin board. There’s also a 1-page 3 Reads Math Routine Poster that’s designed for student use.

3 Reads Strategy posters with the 4 steps on a bulletin board

Download the 3 Reads Routine posters here .

After students work to solve, we go over the problem. I scaffold the routine at the beginning of the year so we do each piece together. But as the weeks go on, I expect my students to become more and more independent in using the 3 Reads strategy. If they aren’t doing it independently when we’re working together, they’re not going to be using it independently in their work. Much of the power of our work comes from the discourse around the problem AND how students solved it. Depending on the problem, there can be value in focusing on the context and unknown. For others, the computation and strategies for solving may be the focus. I vary what our math discourse looks like. Some days, students talk with a partner as they work. Others, they talk with a partner after. Many times, it’s a whole class discussion.

$math talk posters with colorful dialogue boxes with sentence stems on bulletin board display$

These Math Talk posters help remind students of how they can engage in the classroom discourse around our word problem. They’re the perfect supplement to the 3 Reads Strategy because they refer back to the steps students followed based on the context of the problem. You can get My Math Talk posters for free in your inbox by filling out the form below.

Sign up to receive your free Math Talk Sentence Stem posters! After confirming your email, you’ll receive the file and be signed up for my weekly newsletter.

Loading…

You have successfully joined our subscriber list.

We use our 3 Reads Strategy during our Word Problem of the Day routine. You can read more about it in the blog post linked below. If you want to take a closer look at my Word Problem of the Day Bundles for 1st, 2nd, and 3rd grades, I have them in my TpT store. Each bundle includes a free Back to School version that gives you a great look at the format of the problems.

Newsletter Sign Up

Signup for my weekly-ish newsletter. I send out exclusive freebies, tips and strategies for your classroom, and more!

Please Read!

You have successfully joined our subscriber list. Please look in your e-mail and spam folder for Tales from Outside the Classroom. Often, the confirmation email gets overlooked and you're night signed up until you confirm!

I am one of the teachers that are teaching Word Problems wrong! I’ve read through all you said and it makes sense. Will use your strategy as our term starts tomorrow and this term the focus is on Word problems. Thanks for sharing. Do appreciate it.

I hope it helps you and your kiddos!

LOOKING TO SUPPLEMENT?

▹ NEWSLETTER ◃

Hi! I’m Tessa!

I’ve spent the last 15 years teaching in 1st, 2nd, and 3rd grades, and working beside elementary classrooms as an instructional coach and resource support. I’m passionate about math , literacy , and finding ways to make teachers’ days easier . I share from my experiences both in and out of the elementary classroom. Read more About Me .

Let's keep in touch! Sign up for my newsletter!

You will receive a confirmation email shortly. After confirming, you will be officially subscribed.

Powerful online learning at your pace

The 3 Reads Protocol for Solving Word Problems

Raise your hand if you wish your students were more confident and successful in solving word problems. Right, that’s what I thought. And the answer probably doesn’t change much based on your grade level. Face it, word problems are just plain hard!

What DOESN’T WORK

Over the years, very well-intentioned teachers have developed strategies designed to help students solve word problems . Two such strategies that are still quite prevalent are “problem-solving” models and the use of keywords. The idea is that if you follow these steps and look for these keywords, you will be able to solve any word problem. Unfortunately, it’s just not that simple, and despite their widespread use, these strategies are not very effective.

If you look at the CUBES problem-solving model, reading the problem is not even one of the steps! And if you’re thinking, Well, of course students know to read the problem! you might want to watch this model in action. I have more often than not seen students just literally start circling numbers (and not even the labels that go with the numbers…) without ever having read the problem. And keywords are not reliable either. Some word problems have no keywords, and keywords in multi-step problems end up confusing students because of the mixed messages they send.

So can we just agree that something else is needed and put these “strategies” to rest? Students fail at solving word problems for one reason—they don’t understand what the problem is asking them to do. It’s a comprehension problem, so students need reading comprehension skills.

The 3 Reads Protocol

Let me first say that if you search the Internet for 3 Reads Protocol , you’ll find that there are slightly differing versions. What I’m about to describe is the version that I find to be particularly effective. Regardless of the version, we are reading the problem three different times and each reading has a different focus.

The 3 Reads Protocol is a guided learning experience. Students are presented with the problem in stages, and with each read the teacher asks probing questions. Looking at an example is probably the easiest way to understand the protocol, so let’s dive in.

To begin the 3 Reads Protocol, the teacher presents the students with a problem, and the class reads the problem together. Probably the easiest way to do this is with a PowerPoint or Google Slides file. Notice that with the first read, there are no numbers and no question. We just want the students to understand what the story is about and make a mental picture. Without numbers, students have to focus on the meaning of the words! After reading the problem together, the teacher asks what the story is about and calls on students for responses. Don’t be surprised if the responses are very general at first ( girls, flowers, etc.). Ask for additional details, if necessary. Ideally, for this problem, you’d like the students to offer the names of the girls and the types of flowers.

For the second read, the problem is again presented to the students, but this time it includes the numbers. Read the problem again whole class. The questions you will ask now are all related to the numbers in the story. Our goal is for the students to understand that it’s not just 10, it’s 10 daisies . Students might also offer relationships—e.g., Natassja picked more daisies than Ayriale.

Finally, with the third read, students are asked to generate questions that could be answered using the information in the problem. Even though the problem looks just like it did for the second read, don’t skip the reading part! Some problems won’t lend themselves to very many different questions. I like to use this problem as an example because many different questions can be generated. Why? Because there are lots of different numbers in the problem. Here’s a sampling of questions that could be asked. I’m sure you can think of many others.

How many flowers did Ayriale pick?
How many flowers did Ayriale and Natassja pick?
Which girl picked more flowers? How many more?
How many daisies did the girls pick?

That’s the protocol in a nutshell! Once a question or questions have been generated, you can have students go on to solve the problem.

Frequently asked questions

1. When students are solving word problems independently, do I ask them to ignore the numbers and the question?

No! That would be pretty much impossible for them to do. By routinely solving problems using the 3 Reads Protocol with either the whole class or in small groups, you are helping students develop good reading habits that will transfer to their independent work. When they are working independently, the idea is that they will automatically think about the context, identify what the numbers mean within that context, and better understand what the question is asking them to find.

2. Where do I find problems for the 3 Reads Protocol?

I’m sure you can find some that have already been prepared, but it’s super easy to make your own! Just set up a PowerPoint or Google Slides file and format it however you like. Maybe you want a colorful border or a particular font. Use problems that you already have from your resources—textbooks, supplemental books, etc. You’ll need two slides for each problem. On the first slide, type the problem from your resource, leaving out the numbers and the question. On the second slide, add in the numbers. Use a nice big font so students can easily read the problem when projected on your interactive whiteboard. That’s all there is to it!

If you have other questions, add them in the comments below, and I’ll add them to the FAQ. I’d also love to hear how the 3 Reads Protocol is working out in your classroom!

Finding Factors and Common Factors

A skill that 5th-grade kiddos have trouble with is finding factors. There’s a couple of reasons for that. First, if kids aren’t strong with their multiplication facts, finding factors is…

Part/Whole Relationships

It seems that lately we hear more and more about the importance of number bonds, part-whole relationships, and composing and decomposing numbers. As I spend more time in primary classrooms,…

Number of the Day Routine

I’m always looking for a good number of the day routine, and I found one on my own campus! One of my sweet 3rd-grade teachers gave me the idea for…

Fractions on a Number Line with Cuisenaire Rods

A couple of days ago, I blogged about equivalent fractions, and I promised a follow-up post about fractions on the number line. This post contains affiliate links, which simply means…

The Ten Wand

Whether the beginning of the year or somewhere in the middle, it’s never too late to establish routines for your class. To that end, I want to offer another daily…

First, I have a quiz for you. If you speak and write Hindi, you can skip this quiz. For the rest of you, look at the symbols below. Can you…

16 Comments

fantastic thoughts! I LOVE this approach. I’ve used a resource from Lucky Little Learners that is called Numberless Word Problems. This 3 reads protocol will work perfectly with that resource!

Yes! It does include elements of numberless word problems, which I was first introduced to by Brian Bushart .

I appreciate your thoughts on problem solving however the use of CUBES has been totally effective for the past three years with my students who never attempt word problems. You stated that not circling the “labels”, I suppose you mean units connected to the problem. I use this strategy daily and we circle both numbers and units. I’ve made it my own. Let me also reiterate that this strategy works exceptionally well with students who have low reading abilities too. I spend 5 weeks on CUBES and we slowly work our way into the Three Read Protocol which also helps the “non-reader”

Thank you for sharing your experiences! Since you transition to Three Reads, I’m curious why you wouldn’t start with it. I’d also be interested to know what grade level you teach.

I teach both 7th and 8th grade, and many times they have not had any experience with either. CUBEs is embedded with 3-read. I read the question. Students read the questions, we use cubes to understand quantities and the operations ( C and B) used to bring these quantities together. Then we underline the question to provide clarity and to determine what type of answer we are seeking. In our data, we found that our students do not attempt word problems on any formative assessments so we had to start with decreasing student fear of problem-solving and reading. When I used CUBES, I found that many students could answer the question however they had no idea what to do (operations) with the numbers due to a lack of experience,(key operational words) not knowledge ( they always wanted to add). Immediately, students would attempt word problems because CUBES allowed them to pull out what they needed, producing increased math confidence. As we continued with problem-solving, the 3 read protocol was introduced when solving multi-step word problems, student growth was tremendous!

Excellent post!

Thank you! This strategy has so much potential for teaching students to approach word problems in a much more meaningful way!

Hello! This strategy looks great and reaffirms how I’ve been teaching my son to read the problems through as many times as he needs to understand. Now I have the right questions to ask with each reading! Thank you! My question, are there still some strategies to go along with this protocol that you have found to be helpful to use in conjunction i.e. drawing a picture, making a chart, etc. Thanks so much!

I’m so glad this provides you with the missing link! Drawing a picture or diagram is another powerful strategy that goes really well with 3 Reads because it helps students visualize what’s going on in the problem.

Thank you for your post. The Three Read strategy is a gentle reminder of the power of cross-curricular reading strategies; coupling this annotation and graphic organizers will help students develop ways to process and solve simple to more complex multi-step word problems.

I am a math interventionist at an elementary school. We see students K-5 and realize how important it is for us to have a common process/language when it comes to word problems. Looking forward to learning more about this and helping guide our school, thank you!

I absolutely agree about using common language and strategies campus-wide! It’s so much more effective than students having to adjust to their teacher’s language/strategies every new year.

Thank you! I am going to try Three Reads Protocol for the first time next week with my 4th grade students. I am excited. I have had students read word problems all the way through before trying to solve the problem, however, I love the 3 Reads approach and the questions that go with each read. Thank you for breaking it down.

It’s a very powerful strategy! Good luck.

Many of the problems already include the question or questions. When first teaching the strategy, I can take the questions away and then we can compare to see if one of our questions matched the questions posed, but how to do facilitate the strategy when the question is there and you can’t take it away (thinking of assessments).

The idea is that when done regularly, students will develop comprehension skills that will transfer to solving “normal” word problems. They’ll know, for example, that reading it through the first time and ignoring the numbers helps them focus on what’s happening in the problem. You certainly want to help them make that connection, though. You might even show them a normal problem and explicitly model how to use 3 read strategies when the numbers and question are present.

Want to create a balanced schedule to fit EVERYTHING into your math block?

Join me for a FREE , Guided Math workshop to discover how to easily incorporate Guided Math into your current schedule!

3 Reads Strategy to Problem Solve

Problem solving strategies in math are not always as easy as they seem. Many times, students read a mathematical problem and have no idea what to do from there. I’m often asked how to teach word problems. Word problems especially tend to give students the most trouble because they oftentimes require multiple steps in order to solve. They also require reading and comprehension skills. Although there are multiple problem solving strategies in math, one of my favorite ways to break down these steps is to use the 3 reads strategy for problem solving. Read on to learn how it works!

How to Implement the 3 Reads Strategy

In order for the 3 reads strategy to be successful, it’s important that you introduce it to your students by explaining the various steps and the goal of each step. Just like the strategy says, let your students know that you will be reading the math problem 3 times. Each time you read the word problem, you will be looking for different pieces of information each time. This is a much different approach than other problem solving strategies in math, but it is highly effective.

You can watch a break down of the 3 reads strategy in my YouTube video below.

First Read: What is the problem about?

Teacher reads the problem stem out loud only.

The most important concept for your students to understand is that with the first read, they will NOT be able to see the math problem. The teacher will read the question orally while your students listen carefully. They will begin to think about what the problem is about.

I always provide an opportunity after the first read for my students to turn and talk with their shoulder partner about what they just heard.

Overall, the main objective of this step is for students to get a general, brief overview of the word problem. Do not worry if your students don’t pick up on the small details yet. We will work on that in the next step.

3 Reads Strategy Anchor Chart

During the first read, I like to guide students with this 3 reads strategy anchor chart. This will help you visually represent the various pieces of information you’re looking for during the reads and help guide your students’ thinking.

Second Read: What are the quantities (numbers) & units?

Teacher displays the math problem and tells students to focus on the numbers within the problem..

Throughout the second read of the 3 reads strategy, students will now be given the first opportunity to see what the math problem looks like. You can share it on your whiteboard or under your document camera. Sometimes I would print the problem on a piece of paper and have it face down on my students’ desks. During this step they would be allowed to flip the paper over and look at the problem.

In the course of the second read, I also like to have the whole class participate in a choral read of the math problem. Since this is the first time they are actually seeing the problem, it can be helpful to read it aloud together. One of the key factors for this step is to inform students that there are always quantities or numbers that need to be counted within mathematical problems.

3 Reads Strategy Interactive Notebook Activity

Students can use this interactive notebook activity to break down the 3 reads problem solving strategy in math. You can have students cover the word problem during the first read or have them glue it to the top of their page during the 2nd read.

They can lift the flap and record their thoughts about what the problem is about. They will use this activity to guide their thinking through all of the reads. This goes along perfectly with the whole class 3 reads strategy anchor chart.

3 Reads Strategy to Problem Solve interactive notebook activity

Third Read: What are the possible math questions you could ask about this problem?

Teacher chooses one student to read the question one last time..

As this student is reading the question for the final time, it’s crucial for your class to think about any math questions they can come up with about the problem. I always like to use this step to ask my students, “Can we come up with multiple questions that we can ask about this problem?”

You will notice that you may have to do a little extra digging to really get students thinking during this step. Sometimes probing a few extra questions is necessary so that your class can think more critically and still stay on track. List all the possible questions that your students come up with and decide which one you want to tackle together.

problem solving strategies in math 3 reads strategy template

How To Teach Word Problems

You can have this FREE 3 Reads Strategy problem solving template sent straight to your inbox to use with your own students. All you need to do is fill out the form below. I recommend using your home email address because school spam filters like to block these emails sometimes.

After you’ve done the 3 reads strategy for problem solving a few times as a whole group, give students the opportunity to try it on their own with a partner. Working on problem solving strategies in math is one of the best ways to set students up for success!

If you’re looking for resources where you can incorporate this strategy, be sure to check out my problem solving units ! You can find examples of word problems comparing part-part whole-whole unknown as well as comparing difference unknowns !

You can also check out my problem solving strategies in math posters to help your students break down what to look for when solving word problems.

Shop This Post

Check out these problem solving resources.

Problem Solving Strategy Posters

2nd Grade Word Problems for the Year – Comparing Numbers, Part Part Whole, etc.

A Year’s Worth of Problem Solving Journal Prompts Bundle

Looking for more tips for teaching problem solving strategies in math?

Check out these blog posts:

The Power of Numberless Word Problems
Why I Stopped Teaching Keywords in Word Problems

Feel free to pin the image below to save this post for later!

3 Reads Strategy to Problem Solve template

You might also like...

3 Fun Ways to Use Addition and Subtraction With Regrouping Worksheets (That Aren’t Boring!)

Teaching 3 Digit Subtraction Strategies | 4 Effective Ways to Teach Regrouping

Teaching 3 Digit Addition Strategies in 2nd Grade | 4 Effective Ways to Tackle This Tricky Concept

Say Goodbye to Timed Tests

Math should be fun, not stressful. Ditch the timed math fact tests and replace them with math games that will help your students learn and retain information more effectively.

Site search

Three-read protocol.

Watch a video of a Three-Read Protocol

Pre-K/K sample statements in 3 read protocol

Pre-K and K

The Three-Read Protocol is one way to do a close read of a complex math word problem or task. This strategy includes reading a math scenario three times with a different goal each time. The first read is to understand the context. The second read is to understand the mathematics. The third read is to elicit inquiry questions based on the scenario. Why would I use this strategy? The Three-Read Protocol is designed to engage students in sense-making of language-rich math problems or tasks. It deepens student understanding by surfacing linguistic as well as mathematical clues. It focuses attention on the importance of understanding problems rather than rapidly trying to solve them. It allows for the use of authentic, instead of overly simplified, text. This strategy also allows for natural differentiation within a class of diverse learners. When do I use this strategy? This strategy can be used for math tasks that include complex language structures or language that lends itself to a variety of interpretations. While this is a particularly useful strategy for English Language Learners, all students can benefit from the deeper understanding of word problem structures and open-ended questioning. How do I use this strategy? The Three-Read Protocol uses the “problem stem” of a word problem. This is essentially the word problem without the question at the end. The purpose of presenting the problem stem alone is to have students focus on the contextual and mathematical information before dealing with any question that is involved. This gives students the freedom to create their own questions for a given scenario, which is an excellent skill to develop both in math and in reading in general. It is important that the teacher choose the problem carefully and anticipate potential linguistic and mathematical roadblocks the students may encounter. 1. First Read: Teacher reads the problem stem orally. The teacher may have visuals to accompany the oral read of the problem stem. Students listen to the story with the goal of turning to a partner and sharing what they remember of it. Memorizing it is not necessary. Students may act out the problem if that helps them grasp the context. Key Question: What is this situation about? After the Turn-and-Talk, the teacher asks students to volunteer information they remember from the story. Teachers and students ask clarifying questions about the vocabulary as needed.

Read more about the Three-Read Protocol. 2. Second Read: Class does choral read or partner read of the problem stem. The teacher projects the problem stem so the whole class can see it. The teacher leads the class either in a choral read of the problem or has partners read the problem orally to each other. Choral read is preferable because it allows all students to participate without excessive pressure, but a partner read can work fine if that is a better fit to the classroom culture or age of students. The teacher explains that math stories usually have information about quantities (numbers) and the units that are being counted. Key Question: What are the quantities in the situation? An example is 25 cats, where “25” is the quantity and “cat” is the unit. Sometimes the quantities are implied. For example, “some cats” implies a quantity but we do not know what it is. There can also be implied units. An example is “I have one at home.” The implied unit in this case depends on the context of the story. Bottom line: The discussion of quantities and units can be important for focusing student attention, but how deeply the teacher delves into the explicit and implicit information depends on the math and language objectives. 3. Third Read: Partner or choral read the problem stem orally one more time. The teacher asks students to do one more read of the “story” and asks them to think, “What is missing to make this a good math problem?” Students volunteer their answers to that question. Responses will likely vary because many students assume there is a question without actually reading one. Without correcting student responses, the teacher probes until the class decides that a question is missing. The teacher asks, “Is there only one question that we can ask of this story?” Students responses may vary, but there are usually many different questions that can be asked of almost any scenario. Key Question: What mathematical questions can we ask about the situation? The teacher asks partners to determine at least two questions that can be asked using the problem stem. Students share their questions. The teacher writes a couple of the questions and clarifies language as appropriate. After each question, the teacher asks the class, “Can this question be answered with the information from this story?” and the class discusses why or why not. 4. Students work in collaborative groups on the problem. Students work in groups to solve a question based on the problem stem. The teacher may assign a specific question for all groups to answer, or groups may choose a question from the list asked by the class. If groups are asked to choose their own questions, it is important that the teacher circulate and clarify expectations for the work. This can be an opportunity to differentiate the math work because the range of possible questions to a problem stem is broad.

Read more about the Three-Read Protocol in the Math Teaching Toolkit .

The original source materials for the Three Read Protocol is the Routine for Reasoning Fostering Mathematical Practices in All Students , where you will find additional resources.

This page was last updated on February 29, 2024

Our Mission

3 Simple Strategies to Improve Students’ Problem-Solving Skills

These strategies are designed to make sure students have a good understanding of problems before attempting to solve them.

Research provides a striking revelation about problem solvers. The best problem solvers approach problems much differently than novices. For instance, one meta-study showed that when experts evaluate graphs , they tend to spend less time on tasks and answer choices and more time on evaluating the axes’ labels and the relationships of variables within the graphs. In other words, they spend more time up front making sense of the data before moving to addressing the task.

While slower in solving problems, experts use this additional up-front time to more efficiently and effectively solve the problem. In one study, researchers found that experts were much better at “information extraction” or pulling the information they needed to solve the problem later in the problem than novices. This was due to the fact that they started a problem-solving process by evaluating specific assumptions within problems, asking predictive questions, and then comparing and contrasting their predictions with results. For example, expert problem solvers look at the problem context and ask a number of questions:

What do we know about the context of the problem?
What assumptions are underlying the problem? What’s the story here?
What qualitative and quantitative information is pertinent?
What might the problem context be telling us? What questions arise from the information we are reading or reviewing?
What are important trends and patterns?

As such, expert problem solvers don’t jump to the presented problem or rush to solutions. They invest the time necessary to make sense of the problem.

Now, think about your own students: Do they immediately jump to the question, or do they take time to understand the problem context? Do they identify the relevant variables, look for patterns, and then focus on the specific tasks?

If your students are struggling to develop the habit of sense-making in a problem- solving context, this is a perfect time to incorporate a few short and sharp strategies to support them.

3 Ways to Improve Student Problem-Solving

1. Slow reveal graphs: The brilliant strategy crafted by K–8 math specialist Jenna Laib and her colleagues provides teachers with an opportunity to gradually display complex graphical information and build students’ questioning, sense-making, and evaluating predictions.

For instance, in one third-grade class, students are given a bar graph without any labels or identifying information except for bars emerging from a horizontal line on the bottom of the slide. Over time, students learn about the categories on the x -axis (types of animals) and the quantities specified on the y -axis (number of baby teeth).

The graphs and the topics range in complexity from studying the standard deviation of temperatures in Antarctica to the use of scatterplots to compare working hours across OECD (Organization for Economic Cooperation and Development) countries. The website offers a number of graphs on Google Slides and suggests questions that teachers may ask students. Furthermore, this site allows teachers to search by type of graph (e.g., scatterplot) or topic (e.g., social justice).

2. Three reads: The three-reads strategy tasks students with evaluating a word problem in three different ways . First, students encounter a problem without having access to the question—for instance, “There are 20 kangaroos on the grassland. Three hop away.” Students are expected to discuss the context of the problem without emphasizing the quantities. For instance, a student may say, “We know that there are a total amount of kangaroos, and the total shrinks because some kangaroos hop away.”

Next, students discuss the important quantities and what questions may be generated. Finally, students receive and address the actual problem. Here they can both evaluate how close their predicted questions were from the actual questions and solve the actual problem.

To get started, consider using the numberless word problems on educator Brian Bushart’s site . For those teaching high school, consider using your own textbook word problems for this activity. Simply create three slides to present to students that include context (e.g., on the first slide state, “A salesman sold twice as much pears in the afternoon as in the morning”). The second slide would include quantities (e.g., “He sold 360 kilograms of pears”), and the third slide would include the actual question (e.g., “How many kilograms did he sell in the morning and how many in the afternoon?”). One additional suggestion for teams to consider is to have students solve the questions they generated before revealing the actual question.

3. Three-Act Tasks: Originally created by Dan Meyer, three-act tasks follow the three acts of a story . The first act is typically called the “setup,” followed by the “confrontation” and then the “resolution.”

This storyline process can be used in mathematics in which students encounter a contextual problem (e.g., a pool is being filled with soda). Here students work to identify the important aspects of the problem. During the second act, students build knowledge and skill to solve the problem (e.g., they learn how to calculate the volume of particular spaces). Finally, students solve the problem and evaluate their answers (e.g., how close were their calculations to the actual specifications of the pool and the amount of liquid that filled it).

Often, teachers add a fourth act (i.e., “the sequel”), in which students encounter a similar problem but in a different context (e.g., they have to estimate the volume of a lava lamp). There are also a number of elementary examples that have been developed by math teachers including GFletchy , which offers pre-kindergarten to middle school activities including counting squares , peas in a pod , and shark bait .

Students need to learn how to slow down and think through a problem context. The aforementioned strategies are quick ways teachers can begin to support students in developing the habits needed to effectively and efficiently tackle complex problem-solving.

Just one more step to access this resource!

Get your free sample task today.

Ready to explore Exemplars rich performance tasks? Sign up for your free sample now.

The Three Reads Protocol: Understanding vs. Solving

Written By: Nakoa Wiley, Professional Learning Specialist and Curriculum Designer, GA

With a rapid transition into virtual teaching and learning, teachers are facing challenges with finding instructional resources that employ the same rigor and critical thinking received from students in the traditional classroom setting. Since it seems that virtual teaching is the “new norm,” many believe it is in best practice that teachers focus on the mastery of previously taught standards vs. introducing new standards. With that being said, implementing performance tasks into virtual teaching and learning will continue to: (1) elicit critical thinking and authentic learning from students, (2) allow for more personalized teaching for each student (responsive teaching), and (3) allow for learning activities or assessments to be scaffolded for mastery. Performance tasks can be used as a part of daily instruction or for assessing skills taught.

The Three Reads Protocol is a stand-alone strategy that deepens a student’s understanding of the structure of rich tasks. It encompasses open-ended questioning which embodies the framework for all performance tasks. Unlike commonplace procedures such as CUBES or SOLVE, this instructional routine is evidence-based and sets the precedence for our students to “think like mathematicians”.

The Three Reads Protocol

The Three Reads Protocol is one way to do a close read of a complex math task. This strategy includes reading a math scenario three times with a different goal each time. The first read is to understand the context. The second read is to understand the mathematics. The third read is to elicit inquiry questions based on the scenario. The Three Read Protocol is designed to engage students in sense-making of language-rich math problems or tasks. It deepens student understanding by surfacing linguistic as well as mathematical clues. It focuses attention on the importance of understanding problems rather than rapidly trying to solve them. It allows for the use of authentic, instead of overly simplified, text. This strategy also allows for natural differentiation within a class of diverse learners. (SFUSD Mathematics Department, June 2015, sfusdmath.org)

What it Looks Like in the Virtual Learning Platform

Student behaviors can be modified for primary grade levels and based on student need. The Three Reads Protocol supports standards-based instruction and helps scaffold teaching and learning in any educational platform. This strategy will also help students develop constructive academic conversations around mathematical content.

Note 1: * Denotes changes to protocol to align to virtual teaching and learning.

Note 2: Google offers a variety of collaboration tools that may be used by students such as the chat and comment feature in Docs, Gmail, and Hangouts. Platforms such as Zoom, Skype, and FaceTime may also be used to promote communication as well as the chat feature in a school's LMS.

Tackling Wordy Problems: How the Three Reads Math Language Routine Supports Access for All Learners

by Joe Herbert

“These problems are great, but they’re just so wordy. My students can’t handle all that reading!”

Does this feel familiar? If so, you’re not alone! Many students and teachers struggle with the language demands of IM 6–12 Math when they first start their implementation journey.

Students may not be used to the level of reading and writing in math that this curriculum requires, and teachers may be unsure of how to support students with these literacy demands, particularly if their students are more comfortable in a language other than English.

Luckily, the curriculum authors have integrated eight Mathematical Language Routines (MLRs)* into the curriculum to support teachers in amplifying, assessing, and developing students’ language.

While these routines are specifically tagged as supporting Access for English Learners, they are actually beneficial to all learners.

One of the core tenets of language support in this curriculum is that we amplify rather than simplify language. I’m going to be honest: when I first heard the phrase “ amplify rather than simplify ,” I was confused by what it meant. But by digging into one of the Mathematical Language Routines— MLR6: Three Reads**—I finally understood the philosophy, and why it is so powerful in making sure that all students have access to mathematics, while keeping the focus on authentic sense-making.

In this blog post, I’m going to walk through the mechanics of implementing MLR6: Three Reads using an example from the curriculum. I’ll also share some anecdotes and tips from my experience facilitating this routine—both in a GenEd classroom, and in a sheltered class dedicated to newcomers who are just beginning to learn and use English.

While I’m going to illustrate this routine using a high school example, this same routine structure can be applied to middle school and even elementary school contexts.

The example we’ll look at in this post comes from Geometry Unit 5 Lesson 17 . I think it’s always helpful to know where students are in their learning trajectory, especially when looking at an example that comes so late in a unit. Before this lesson, students have thought deeply about volume, and know how to find the volume of many solids. Early in this lesson, they’re introduced to the concept of density as an application of volume.

With that in mind, here is the activity:

The best average density for the species of fish that will go in the tank is 16 fish per 100 gallons of water. This provides enough room for the fish to swim while making sure that there are plenty of fish for people to see.

The aquarium has 275 fish available to put in the tank. Is this the right number of fish for the tank? If not, how many fish should be added or removed? Explain your reasoning.

If your head is spinning from all of the information you’re trying to keep track of in this problem, you’re not alone! Every teacher can probably think of at least one student (and maybe a whole class of students!) who would get intimidated by this problem and not know where to start.

That’s where MLR6: Three Reads comes into play. Rather than simplifying the problem and removing some of the academic rigor, Three Reads provides a structured format for teachers and students to partner in amplifying the language in the problem, so that students are able to understand it and engage in authentic sense-making.

Let’s dig into how the routine does this. As the name suggests, in the Three Reads routine, we engage in three structured readings of the same problem context, with a slightly different and more focused lens each time.

Read 1: Understand the Story Context

In the first read, the goal is just to understand the context of the problem.

To help students focus solely on the story context, we remove the question. Thus, in our example, we would just share the following with students:

The aquarium has 275 fish available to put in the tank.

When doing this first read, I often read out loud to the class. Alternatively, you can have a student read out loud. If you teach English Learners, this is the time to bring in visual aids, such as pictures from your local aquarium.

The discussion that follows Read 1 has a very clear goal: make sure students understand the story context of the problem, and address any non-mathematical vocabulary words that might get in students’ way.

After the shared reading, we ask students “What is this situation about?” or “What is the story of this problem?”

Then, scribe their ideas on the whiteboard or some other shared space.

Students sometimes start jumping to talking about the numbers that are in the problem, but I specifically ask them to tell me what the story of the problem is without referencing any numbers at all. By the end of this discussion, I want to make sure students know what a fish tank is, and that they recognize the unique shape of the particular fish tank in this problem. I might even draw a little stick figure in the inner cylinder to really drive the point home.

Read 2: Identify Quantities

The second time we read the problem together, we again omit the question so that students can focus exclusively on making sense of the context. The mechanics for how I implement Read 2 sometimes differ depending on the audience.

With newcomers who are just learning how to read (and speak!) in English, I like to do a choral read aloud. This gives students a low-stakes opportunity to practice reading in English, because everyone is reading at the same time so no one individual voice can be easily heard.

In my experience, in a GenEd classroom, high schoolers prefer to read the problem together in small groups, rather than choral reading.

Now for the important part: the questions we ask and discussion we have after Read 2.

Here is where we ask students, “What can be counted or measured?”, without focusing on any specific numbers. In our Geometry example, this might include the volume of the tank, the density of fish, the dimensions of the tank, and the number of fish available.

When working with newcomer English learners, I have had some trouble explaining what I mean by the question “What can be counted or measured?”. In that case, I modify the routine slightly to ask students what quantities they see in the problem, and I specify that a quantity is a number and its unit***. Thus, rather than just identifying the number 3, for example, students will identify that the distance between the inner and outer cylinders is 3 feet.

Regardless of the specific prompt you give students, the main goal for Read 2 is for students to begin to think more deeply about the various quantities in the problem, how they’re related to each other, and how they relate to the overall story context. As in Read 1, we scribe students’ ideas on the whiteboard to help make our collective thinking visible.

Read 3: Reveal the Questions and Plan Solution Strategies

Read 3 is where we read the context one last time, and then reveal the actual questions students will investigate.

The mechanics for Read 3 can be whatever works best for your students: choral reading, reading in small groups, or one volunteer reading out loud.

After the shared reading, the discussion prompt for Read 3 is, “What are some ways we might solve this?” The goal is for students to do some high-level planning and strategizing before they jump into the nitty gritty of computation.

In our example, we might have students who talk about finding the total volume of the big cylinder, and then subtracting the volume of the smaller cylinder. Alternatively, they might discuss finding the area of the base of the fish tank and multiplying it by the height to find the volume.

By discussing these ideas and strategies in partner pairs or as a whole class, we set kids up for success when we release them to independent or group work.

Amplify, Don’t Simplify

I hope this breakdown of MLR6: Three Reads makes clear how we can use Mathematical Language Routines to amplify rather than simplify language. By engaging in this routine, we’re not changing the wording of the problem to make it simpler. Rather, we’re using the three structured reads and discussion prompts to amplify the language in the problem and help students make sense of the problem for themselves.

The first time I tried Three Reads in the classroom, I had several students who had arrived in the US earlier that week. They were just starting to learn English, and did not feel very comfortable with it yet. However, after engaging in the routine, they made sense of the problems authentically, and offered meaningful contributions to their groups. (I of course encouraged students to discuss the problems with peers in whatever language they felt most comfortable with.)

This was when I truly understood the incredible power of this routine to help create a world where all learners know, use, and enjoy mathematics.

I hope that this leaves you eager to try MLR6: Three Reads in your classroom, regardless of the student population you teach.

If you’re feeling excited but still a little intimidated, I’d like to draw your attention to the Illustrative Mathematics teacher notes. I think it’s helpful to see how much support the curriculum authors have given teachers in making IM 6–12 Math accessible to all.

This is the suggestion to support Access for English Learners for the fish tank activity:

Notice how this suggestion goes beyond just saying, “do Three Reads,” and instead highlights the important ideas that you want to draw out of the conversation after each of the three reads.

Does MLR6: Three Reads take some extra instructional time? Yes, of course. However, if your students are able to dive straightaway into authentic sense-making after engaging in the routine, then it is time well spent.

So the next time you get overwhelmed thinking about how your students are going to struggle with a wordy problem, take a look at the suggestions for access that are built into the curriculum. Whether or not you teach English learners, you might find a suggestion for a Mathematical Language Routine that provides the perfect support to help you amplify rather than simplify language for your students!

*Zwiers, J., Dieckmann, J., Rutherford-Quach, S., Daro, V., Skarin, R., Weiss, S., & Malamut, J. (2017). Principles for the Design of Mathematics Curricula: Promoting Language and Content Development

**Kelemanik, G, Lucenta, A & Creighton, S.J. (2016). Routines for reasoning: Fostering the mathematical practices in all students. Portsmouth, NH: Heinemann.

*** This modification for Read 2 comes from the Three Reads protocol distributed by San Francisco Unified School District

to the IM Blog and Newsletter

Popular Tags

Feb 18, 2022

How to Help Students Understand Word Problems Using the 3 Read Protocol

The 3 Read Strategy is a word problem strategy I use in my classroom to help my students really understand and analyze word problems. I love it because you can use this strategy to teach so many math concepts such as addition, subtraction, and comparing numbers.

What is the 3 Read strategy and why should you use it? The 3 read protocol is a close read of a word problem. It gives students a chance to analyze what is happening in a word problem without feeling the immediate need to find the answer. This strategy is extremely helpful for English language learners because of all the visuals and the focus on the actions and vocabulary in word problems. It is beneficial for ALL students in helping them solve math problems. The 3 read protocol is engaging for students and requires them to think critically and ask questions. It can also be easily differentiated for all students.

How do you use the 3 Read strategy? As the name implies, this strategy requires the the teacher and students to read the word story 3 times. Each read has a different goal or focus question. The procedure starts with the story and 3 empty boxes. See example below: Read #1 –The teacher reads the story aloud. The goal of this read is for the students to think about the question: What is the story about?

Students turn and talk with a partner after each read. For the first read, I provide them the sentence frames: The story is about ________________. The characters are _______________. The setting is ___________________. Students share what the story is about, and the teacher can write a short summary of the problem in the first box and/or place the photo and word cards in the first box. The word cards and pictures should include the characters, the setting, and any objects or living things mentioned in the story. New vocabulary or verbs can be taught during this time as well. This is very helpful for ELLs in understanding a story. After the 1st read, the chart paper should look something like this:

Read #2 –The teacher reads the story a second time (the students can read along too). For this read, the focus question is: What are the numbers in the story? After the 2nd read, the students turn and talk. I provide the sentence frame: The numbers are _____ and ______. The students share the numbers, and the teacher posts the numbers in the 2nd box along with visuals representing the numbers. You can quickly draw the quantities or tape pictures of the quantities next to the number cards. After the 2nd read, complete the “what are the numbers?” column of the chart.

Read #3 -The teacher and students read the story for a third time. The 3rd read, the focus question is: What do you wonder about the story? After the 3rd read, the students turn and talk. I provide the sentence frames: How many ___________? How many more/less does ________ have than ________? Who ___________? What _______________? Creating a question is very challenging for students, especially in kindergarten. -It will take about 4-5 times using the 3 read protocol before students will start to create their own questions. -The first 1-3 times, students will probably give you facts about the story, and you can help them change those into questions. -Write down all questions students give you in the 3rd box. In the beginning, students will ask questions that cannot be answered by the story, such as what color are the houses? Write all questions on the chart, but point out that the story cannot help us answer that question. Student questions will become more focused and complex as they learn the procedure. After questions are created, write them down in the third column of the chart paper.

You can pick 1 or more question(s) to focus on and answer after the 3rd step. This can be a question that a student created, or a questions you created prior to the lesson. At the beginning of kindergarten, I have the students start by solving a problem in pairs using manipulates. Students will come up with the answer to the question with their partners, and then different pairs will share their thinking. After a couple months of solving problems this way, I have my kindergarteners use whiteboards (and manipulatives if they want them) to show their thinking. Students solve the problem on their own and then share their different strategies on the document camera. Here are some examples of a problem we did in December.

Here are two examples of students answering the question, "How many gingerbread men did they decorate in all?" All students showed their work on whiteboards, and then I picked about 2-3 different strategies to have students share their work with their classmates on the document camera. This is an important way for students to realize that there are many different ways to solve problems and different strategies. It is also a good opportunity for students to practice explaining their thinking.

After students feel comfortable showing their work on whiteboards, I introduce them to my 3 read graphic organizer. They can use the strategies on the organizers available and practice showing their thinking in many different ways.

Students can use this to follow along as the 3 reads procedure is completed together as a class. Students can talk to a partner after each read, and then write down what they spoke about in each column.

Students in kindergarten can use this to show their work and solve the word problem. This will challenge them to show their work in many ways (pictures, number line, ten frame, etc.). In kindergarten, I usually will write or type the question I want the students to solve at the top before I make the copies. Otherwise, students can take up to 10 minutes copying the question on their own (which is not the purpose of this activity).

Here is a graphic organizer for 1st graders to use with a number line to 20 and two ten frames.

Creating questions is how you can differentiate the word problems to meet a variety of student needs. My 3 Read resources can be for kindergarteners or 1st graders, depending on which questions you and your students create. In the beginning of kindergarten, just getting students to ask any question will be an objective all on its own. You can help students ask such as: How many gingerbread men did Samuel decorate on Saturday? How many stockings did Julio have? When students are able to ask and answer those types of questions, you can progress to more difficult questions such as: How many gingerbread men did they have altogether? How many gingerbread men would they have if Sam ate 3 of them? First grade might progress from addition and subtraction questions to comparing questions such as: How many more candy canes did Jane have than Samantha? I hope you enjoyed reading about how to use 3 reads in your classroom! Please email me at [email protected] if you have any questions or would like more details about anything. If you are interested in my 3 read resources, please visit my Teachers Pay Teachers site at Inspiring Little Learners. I have 3 read resources for 10 months out of the year or a bundle of the 10 months together. 3 Read Resources in my store

3.1 Use a Problem-Solving Strategy

Learning objectives.

By the end of this section, you will be able to:

Approach word problems with a positive attitude
Use a problem-solving strategy for word problems
Solve number problems

Be Prepared 3.1

Before you get started, take this readiness quiz.

Translate “6 less than twice x ” into an algebraic expression. If you missed this problem, review Example 1.26 .

Be Prepared 3.2

Solve: 2 3 x = 24 . 2 3 x = 24 . If you missed this problem, review Example 2.16 .

Solve: 3 x + 8 = 14 . 3 x + 8 = 14 . If you missed this problem, review Example 2.27 .

Approach Word Problems with a Positive Attitude

“If you think you can… or think you can’t… you’re right.”—Henry Ford

The world is full of word problems! Will my income qualify me to rent that apartment? How much punch do I need to make for the party? What size diamond can I afford to buy my girlfriend? Should I fly or drive to my family reunion?

How much money do I need to fill the car with gas? How much tip should I leave at a restaurant? How many socks should I pack for vacation? What size turkey do I need to buy for Thanksgiving dinner, and then what time do I need to put it in the oven? If my sister and I buy our mother a present, how much does each of us pay?

Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student below?

When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.

Start with a fresh slate and begin to think positive thoughts. If we take control and believe we can be successful, we will be able to master word problems! Read the positive thoughts in Figure 3.3 and say them out loud.

Think of something, outside of school, that you can do now but couldn’t do 3 years ago. Is it driving a car? Snowboarding? Cooking a gourmet meal? Speaking a new language? Your past experiences with word problems happened when you were younger—now you’re older and ready to succeed!

Use a Problem-Solving Strategy for Word Problems

We have reviewed translating English phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. We have also translated English sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. We restated the situation in one sentence, assigned a variable, and then wrote an equation to solve the problem. This method works as long as the situation is familiar and the math is not too complicated.

Now, we’ll expand our strategy so we can use it to successfully solve any word problem. We’ll list the strategy here, and then we’ll use it to solve some problems. We summarize below an effective strategy for problem solving.

Use a Problem-Solving Strategy to Solve Word Problems.

Step 1. Read the problem. Make sure all the words and ideas are understood.
Step 2. Identify what we are looking for.
Step 3. Name what we are looking for. Choose a variable to represent that quantity.
Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation.
Step 5. Solve the equation using good algebra techniques.
Step 6. Check the answer in the problem and make sure it makes sense.
Step 7. Answer the question with a complete sentence.

Example 3.1

Pilar bought a purse on sale for $18, which is one-half of the original price. What was the original price of the purse?

Step 1. Read the problem. Read the problem two or more times if necessary. Look up any unfamiliar words in a dictionary or on the internet.

In this problem, is it clear what is being discussed? Is every word familiar?

Step 2. Identify what you are looking for. Did you ever go into your bedroom to get something and then forget what you were looking for? It’s hard to find something if you are not sure what it is! Read the problem again and look for words that tell you what you are looking for!

In this problem, the words “what was the original price of the purse” tell us what we need to find.

Step 3. Name what we are looking for. Choose a variable to represent that quantity. We can use any letter for the variable, but choose one that makes it easy to remember what it represents.

Let p = p = the original price of the purse.

Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Translate the English sentence into an algebraic equation.

Reread the problem carefully to see how the given information is related. Often, there is one sentence that gives this information, or it may help to write one sentence with all the important information. Look for clue words to help translate the sentence into algebra. Translate the sentence into an equation.

Step 5. Solve the equation using good algebraic techniques. Even if you know the solution right away, using good algebraic techniques here will better prepare you to solve problems that do not have obvious answers.

Step 6. Check the answer in the problem to make sure it makes sense. We solved the equation and found that p = 36 , p = 36 , which means “the original price” was $36.

Does $36 make sense in the problem? Yes, because 18 is one-half of 36, and the purse was on sale at half the original price.

Step 7. Answer the question with a complete sentence. The problem asked “What was the original price of the purse?”

The answer to the question is: “The original price of the purse was $36.”

If this were a homework exercise, our work might look like this:

Pilar bought a purse on sale for $18, which is one-half the original price. What was the original price of the purse?

Joaquin bought a bookcase on sale for $120, which was two-thirds of the original price. What was the original price of the bookcase?

Two-fifths of the songs in Mariel’s playlist are country. If there are 16 country songs, what is the total number of songs in the playlist?

Let’s try this approach with another example.

Example 3.2

Ginny and her classmates formed a study group. The number of girls in the study group was three more than twice the number of boys. There were 11 girls in the study group. How many boys were in the study group?

Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was 3 more than twice the number of notebooks. He bought 7 textbooks. How many notebooks did he buy?

Gerry worked Sudoku puzzles and crossword puzzles this week. The number of Sudoku puzzles he completed is eight more than twice the number of crossword puzzles. He completed 22 Sudoku puzzles. How many crossword puzzles did he do?

Solve Number Problems

Now that we have a problem solving strategy, we will use it on several different types of word problems. The first type we will work on is “number problems.” Number problems give some clues about one or more numbers. We use these clues to write an equation. Number problems don’t usually arise on an everyday basis, but they provide a good introduction to practicing the problem solving strategy outlined above.

Example 3.3

The difference of a number and six is 13. Find the number.

The difference of a number and eight is 17. Find the number.

The difference of a number and eleven is −7 . −7 . Find the number.

Example 3.4

The sum of twice a number and seven is 15. Find the number.

Did you notice that we left out some of the steps as we solved this equation? If you’re not yet ready to leave out these steps, write down as many as you need.

The sum of four times a number and two is 14. Find the number.

The sum of three times a number and seven is 25. Find the number.

Some number word problems ask us to find two or more numbers. It may be tempting to name them all with different variables, but so far we have only solved equations with one variable. In order to avoid using more than one variable, we will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.

Example 3.5

One number is five more than another. The sum of the numbers is 21. Find the numbers.

One number is six more than another. The sum of the numbers is twenty-four. Find the numbers.

Try It 3.10

The sum of two numbers is fifty-eight. One number is four more than the other. Find the numbers.

Example 3.6

The sum of two numbers is negative fourteen. One number is four less than the other. Find the numbers.

Try It 3.11

The sum of two numbers is negative twenty-three. One number is seven less than the other. Find the numbers.

Try It 3.12

The sum of two numbers is −18 . −18 . One number is 40 more than the other. Find the numbers.

Example 3.7

One number is ten more than twice another. Their sum is one. Find the numbers.

Try It 3.13

One number is eight more than twice another. Their sum is negative four. Find the numbers.

Try It 3.14

One number is three more than three times another. Their sum is −5 . −5 . Find the numbers.

Some number problems involve consecutive integers. Consecutive integers are integers that immediately follow each other. Examples of consecutive integers are:

Notice that each number is one more than the number preceding it. So if we define the first integer as n , the next consecutive integer is n + 1 . n + 1 . The one after that is one more than n + 1 , n + 1 , so it is n + 1 + 1 , n + 1 + 1 , which is n + 2 . n + 2 .

Example 3.8

The sum of two consecutive integers is 47. Find the numbers.

Try It 3.15

The sum of two consecutive integers is 95 . 95 . Find the numbers.

Try It 3.16

The sum of two consecutive integers is −31 . −31 . Find the numbers.

Example 3.9

Find three consecutive integers whose sum is −42 . −42 .

Try It 3.17

Find three consecutive integers whose sum is −96 . −96 .

Try It 3.18

Find three consecutive integers whose sum is −36 . −36 .

Now that we have worked with consecutive integers, we will expand our work to include consecutive even integers and consecutive odd integers. Consecutive even integers are even integers that immediately follow one another. Examples of consecutive even integers are:

Notice each integer is 2 more than the number preceding it. If we call the first one n , then the next one is n + 2 . n + 2 . The next one would be n + 2 + 2 n + 2 + 2 or n + 4 . n + 4 .

Consecutive odd integers are odd integers that immediately follow one another. Consider the consecutive odd integers 77, 79, and 81.

Does it seem strange to add 2 (an even number) to get from one odd integer to the next? Do you get an odd number or an even number when we add 2 to 3? to 11? to 47?

Whether the problem asks for consecutive even numbers or odd numbers, you don’t have to do anything different. The pattern is still the same—to get from one odd or one even integer to the next, add 2.

Example 3.10

Find three consecutive even integers whose sum is 84.

Try It 3.19

Find three consecutive even integers whose sum is 102.

Try It 3.20

Find three consecutive even integers whose sum is −24 . −24 .

Example 3.11

A married couple together earns $110,000 a year. The wife earns $16,000 less than twice what her husband earns. What does the husband earn?

Try It 3.21

According to the National Automobile Dealers Association, the average cost of a car in 2014 was $28,500. This was $1,500 less than 6 times the cost in 1975. What was the average cost of a car in 1975?

Try It 3.22

U.S. Census data shows that the median price of new home in the United States in November 2014 was $280,900. This was $10,700 more than 14 times the price in November 1964. What was the median price of a new home in November 1964?

Section 3.1 Exercises

Practice makes perfect.

Use the Approach Word Problems with a Positive Attitude

In the following exercises, prepare the lists described.

List five positive thoughts you can say to yourself that will help you approach word problems with a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.

List five negative thoughts that you have said to yourself in the past that will hinder your progress on word problems. You may want to write each one on a small piece of paper and rip it up to symbolically destroy the negative thoughts.

In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question.

Two-thirds of the children in the fourth-grade class are girls. If there are 20 girls, what is the total number of children in the class?

Three-fifths of the members of the school choir are women. If there are 24 women, what is the total number of choir members?

Zachary has 25 country music CDs, which is one-fifth of his CD collection. How many CDs does Zachary have?

One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?

There are 16 girls in a school club. The number of girls is four more than twice the number of boys. Find the number of boys.

There are 18 Cub Scouts in Pack 645. The number of scouts is three more than five times the number of adult leaders. Find the number of adult leaders.

Huong is organizing paperback and hardback books for her club’s used book sale. The number of paperbacks is 12 less than three times the number of hardbacks. Huong had 162 paperbacks. How many hardback books were there?

Jeff is lining up children’s and adult bicycles at the bike shop where he works. The number of children’s bicycles is nine less than three times the number of adult bicycles. There are 42 adult bicycles. How many children’s bicycles are there?

Philip pays $1,620 in rent every month. This amount is $120 more than twice what his brother Paul pays for rent. How much does Paul pay for rent?

Marc just bought an SUV for $54,000. This is $7,400 less than twice what his wife paid for her car last year. How much did his wife pay for her car?

Laurie has $46,000 invested in stocks and bonds. The amount invested in stocks is $8,000 less than three times the amount invested in bonds. How much does Laurie have invested in bonds?

Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than three times the amount she earned from her job at the college. How much did she earn from her job at the college?

In the following exercises, solve each number word problem.

The sum of a number and eight is 12. Find the number.

The sum of a number and nine is 17. Find the number.

The difference of a number and 12 is three. Find the number.

The difference of a number and eight is four. Find the number.

The sum of three times a number and eight is 23. Find the number.

The sum of twice a number and six is 14. Find the number.

The difference of twice a number and seven is 17. Find the number.

The difference of four times a number and seven is 21. Find the number.

Three times the sum of a number and nine is 12. Find the number.

Six times the sum of a number and eight is 30. Find the number.

One number is six more than the other. Their sum is 42. Find the numbers.

One number is five more than the other. Their sum is 33. Find the numbers.

The sum of two numbers is 20. One number is four less than the other. Find the numbers.

The sum of two numbers is 27. One number is seven less than the other. Find the numbers.

The sum of two numbers is −45 . −45 . One number is nine more than the other. Find the numbers.

The sum of two numbers is −61 . −61 . One number is 35 more than the other. Find the numbers.

The sum of two numbers is −316 . −316 . One number is 94 less than the other. Find the numbers.

The sum of two numbers is −284 . −284 . One number is 62 less than the other. Find the numbers.

One number is 14 less than another. If their sum is increased by seven, the result is 85. Find the numbers.

One number is 11 less than another. If their sum is increased by eight, the result is 71. Find the numbers.

One number is five more than another. If their sum is increased by nine, the result is 60. Find the numbers.

One number is eight more than another. If their sum is increased by 17, the result is 95. Find the numbers.

One number is one more than twice another. Their sum is −5 . −5 . Find the numbers.

One number is six more than five times another. Their sum is six. Find the numbers.

The sum of two numbers is 14. One number is two less than three times the other. Find the numbers.

The sum of two numbers is zero. One number is nine less than twice the other. Find the numbers.

The sum of two consecutive integers is 77. Find the integers.

The sum of two consecutive integers is 89. Find the integers.

The sum of two consecutive integers is −23 . −23 . Find the integers.

The sum of two consecutive integers is −37 . −37 . Find the integers.

The sum of three consecutive integers is 78. Find the integers.

The sum of three consecutive integers is 60. Find the integers.

Find three consecutive integers whose sum is −3 . −3 .

Find three consecutive even integers whose sum is 258.

Find three consecutive even integers whose sum is 222.

Find three consecutive odd integers whose sum is 171.

Find three consecutive odd integers whose sum is 291.

Find three consecutive even integers whose sum is −36 . −36 .

Find three consecutive even integers whose sum is −84 . −84 .

Find three consecutive odd integers whose sum is −213 . −213 .

Find three consecutive odd integers whose sum is −267 . −267 .

Everyday Math

Sale Price Patty paid $35 for a purse on sale for $10 off the original price. What was the original price of the purse?

Sale Price Travis bought a pair of boots on sale for $25 off the original price. He paid $60 for the boots. What was the original price of the boots?

Buying in Bulk Minh spent $6.25 on five sticker books to give his nephews. Find the cost of each sticker book.

Buying in Bulk Alicia bought a package of eight peaches for $3.20. Find the cost of each peach.

Price before Sales Tax Tom paid $1,166.40 for a new refrigerator, including $86.40 tax. What was the price of the refrigerator?

Price before Sales Tax Kenji paid $2,279 for a new living room set, including $129 tax. What was the price of the living room set?

Writing Exercises

What has been your past experience solving word problems?

When you start to solve a word problem, how do you decide what to let the variable represent?

What are consecutive odd integers? Name three consecutive odd integers between 50 and 60.

What are consecutive even integers? Name three consecutive even integers between −50 −50 and −40 . −40 .

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!

…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential—every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.

As an Amazon Associate we earn from qualifying purchases.

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/elementary-algebra-2e/pages/1-introduction

Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis
Publisher/website: OpenStax
Book title: Elementary Algebra 2e
Publication date: Apr 22, 2020
Location: Houston, Texas
Book URL: https://openstax.org/books/elementary-algebra-2e/pages/1-introduction
Section URL: https://openstax.org/books/elementary-algebra-2e/pages/3-1-use-a-problem-solving-strategy

© Jan 23, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

school Campus Bookshelves
menu_book Bookshelves
perm_media Learning Objects
login Login
how_to_reg Request Instructor Account
hub Instructor Commons
Download Page (PDF)
Download Full Book (PDF)
Periodic Table
Physics Constants
Scientific Calculator
Reference & Cite
Tools expand_more
Readability

selected template will load here

This action is not available.

2.3: Use a Problem Solving Strategy

Last updated
Save as PDF
Page ID 114096

Learning Objectives

By the end of this section, you will be able to:

Use a problem solving strategy for word problems
Solve number word problems
Solve percent applications
Solve simple interest applications

Be Prepared 2.4

Before you get started, take this readiness quiz.

Translate “six less than twice x ” into an algebraic expression. If you missed this problem, review Example 1.8.

Be Prepared 2.5

Convert 4.5% to a decimal. If you missed this problem, review Example 1.40.

Be Prepared 2.6

Convert 0.6 to a percent. If you missed this problem, review Example 1.40.

Have you ever had any negative experiences in the past with word problems? When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. Realize that your negative experiences with word problems are in your past. To move forward you need to calm your fears and change your negative feelings.

Start with a fresh slate and begin to think positive thoughts. Repeating some of the following statements may be helpful to turn your thoughts positive. Thinking positive thoughts is a first step towards success.

I think I can! I think I can!

While word problems were hard in the past, I think I can try them now.

I am better prepared now—I think I will begin to understand word problems.

I am able to solve equations because I practiced many problems and I got help when I needed it—I can try that with word problems.

It may take time, but I can begin to solve word problems.

You are now well prepared and you are ready to succeed. If you take control and believe you can be successful, you will be able to master word problems.

Use a Problem Solving Strategy for Word Problems

Now that we can solve equations, we are ready to apply our new skills to word problems. We will develop a strategy we can use to solve any word problem successfully.

Example 2.14

Normal yearly snowfall at the local ski resort is 12 inches more than twice the amount it received last season. The normal yearly snowfall is 62 inches. What was the snowfall last season at the ski resort?

Try It 2.27

Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was three more than twice the number of notebooks. He bought seven textbooks. How many notebooks did he buy?

Try It 2.28

We summarize an effective strategy for problem solving.

Use a Problem Solving Strategy for word problems.

Step 1. Read the problem. Make sure all the words and ideas are understood.
Step 2. Identify what you are looking for.
Step 3. Name what you are looking for. Choose a variable to represent that quantity.
Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
Step 5. Solve the equation using proper algebra techniques.
Step 6. Check the answer in the problem to make sure it makes sense.
Step 7. Answer the question with a complete sentence.

Solve Number Word Problems

We will now apply the problem solving strategy to “number word problems.” Number word problems give some clues about one or more numbers and we use these clues to write an equation. Number word problems provide good practice for using the Problem Solving Strategy.

Example 2.15

The sum of seven times a number and eight is thirty-six. Find the number.

Did you notice that we left out some of the steps as we solved this equation? If you’re not yet ready to leave out these steps, write down as many as you need.

Try It 2.29

The sum of four times a number and two is fourteen. Find the number.

Try It 2.30

The sum of three times a number and seven is twenty-five. Find the number.

Some number word problems ask us to find two or more numbers. It may be tempting to name them all with different variables, but so far, we have only solved equations with one variable. In order to avoid using more than one variable, we will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.

Example 2.16

The sum of two numbers is negative fifteen. One number is nine less than the other. Find the numbers.

Try It 2.31

The sum of two numbers is negative twenty-three. One number is seven less than the other. Find the numbers.

Try It 2.32

The sum of two numbers is negative eighteen. One number is forty more than the other. Find the numbers.

Some number problems involve consecutive integers . Consecutive integers are integers that immediately follow each other. Examples of consecutive integers are:

1 , 2 , 3 , 4 −10 , −9 , −8 , −7 150 , 151 , 152 , 153 1 , 2 , 3 , 4 −10 , −9 , −8 , −7 150 , 151 , 152 , 153

Notice that each number is one more than the number preceding it. Therefore, if we define the first integer as n , the next consecutive integer is n + 1 . n + 1 . The one after that is one more than n + 1 , n + 1 , so it is n + 1 + 1 , n + 1 + 1 , which is n + 2 . n + 2 .

n 1 st integer n + 1 2 nd consecutive integer n + 2 3 rd consecutive integer etc. n 1 st integer n + 1 2 nd consecutive integer n + 2 3 rd consecutive integer etc.

We will use this notation to represent consecutive integers in the next example.

Example 2.17

Find three consecutive integers whose sum is −54 . −54 .

Try It 2.33

Find three consecutive integers whose sum is −96 . −96 .

Try It 2.34

Find three consecutive integers whose sum is −36 . −36 .

Now that we have worked with consecutive integers, we will expand our work to include consecutive even integers and consecutive odd integers . Consecutive even integers are even integers that immediately follow one another. Examples of consecutive even integers are:

24, 26, 28 24, 26, 28

−12 , −10 , −8 −12 , −10 , −8

Notice each integer is two more than the number preceding it. If we call the first one n , then the next one is n + 2 . n + 2 . The one after that would be n + 2 + 2 n + 2 + 2 or n + 4 . n + 4 .

n 1 st even integer n + 2 2 nd consecutive even integer n + 4 3 rd consecutive even integer etc. n 1 st even integer n + 2 2 nd consecutive even integer n + 4 3 rd consecutive even integer etc.

Consecutive odd integers are odd integers that immediately follow one another. Consider the consecutive odd integers 63, 65, and 67.

63, 65, 67 63, 65, 67

n , n + 2 , n + 4 n , n + 2 , n + 4

n 1 st odd integer n + 2 2 nd consecutive odd integer n + 4 3 rd consecutive odd integer etc. n 1 st odd integer n + 2 2 nd consecutive odd integer n + 4 3 rd consecutive odd integer etc.

Does it seem strange to have to add two (an even number) to get the next odd number? Do we get an odd number or an even number when we add 2 to 3? to 11? to 47?

Whether the problem asks for consecutive even numbers or odd numbers, you do not have to do anything different. The pattern is still the same—to get to the next odd or the next even integer, add two.

Example 2.18

Find three consecutive even integers whose sum is 120 120 .

Try It 2.35

Find three consecutive even integers whose sum is 102.

Try It 2.36

Find three consecutive even integers whose sum is −24 . −24 .

When a number problem is in a real life context, we still use the same strategies that we used for the previous examples.

Example 2.19

A married couple together earns $110,000 a year. The wife earns $16,000 less than twice what her husband earns. What does the husband earn?

Try It 2.37

According to the National Automobile Dealers Association, the average cost of a car in 2014 was $28,400. This was $1,600 less than six times the cost in 1975. What was the average cost of a car in 1975?

Try It 2.38

US Census data shows that the median price of new home in the U.S. in November 2014 was $280,900. This was $10,700 more than 14 times the price in November 1964. What was the median price of a new home in November 1964?

Solve Percent Applications

There are several methods to solve percent equations. In algebra, it is easiest if we just translate English sentences into algebraic equations and then solve the equations. Be sure to change the given percent to a decimal before you use it in the equation.

Example 2.20

Translate and solve:

Try It 2.39

Try It 2.40

Translate and solve: ⓐ What number is 55% of 60? ⓑ 8.5% of what amount is $3.06? ⓐ 126 is what percent of 72?

Now that we have a problem solving strategy to refer to, and have practiced solving basic percent equations, we are ready to solve percent applications. Be sure to ask yourself if your final answer makes sense—since many of the applications we will solve involve everyday situations, you can rely on your own experience.

Example 2.21

The label on Audrey’s yogurt said that one serving provided 12 grams of protein, which is 24% of the recommended daily amount. What is the total recommended daily amount of protein?

Try It 2.41

One serving of wheat square cereal has 7 grams of fiber, which is 28% of the recommended daily amount. What is the total recommended daily amount of fiber?

Try It 2.42

One serving of rice cereal has 190 mg of sodium, which is 8% of the recommended daily amount. What is the total recommended daily amount of sodium?

Remember to put the answer in the form requested. In the next example we are looking for the percent.

Example 2.22

Veronica is planning to make muffins from a mix. The package says each muffin will be 240 calories and 60 calories will be from fat. What percent of the total calories is from fat?

Try It 2.43

Mitzi received some gourmet brownies as a gift. The wrapper said each brownie was 480 calories, and had 240 calories of fat. What percent of the total calories in each brownie comes from fat? Round the answer to the nearest whole percent.

Try It 2.44

The mix Ricardo plans to use to make brownies says that each brownie will be 190 calories, and 76 calories are from fat. What percent of the total calories are from fat? Round the answer to the nearest whole percent.

It is often important in many fields—business, sciences, pop culture—to talk about how much an amount has increased or decreased over a certain period of time. This increase or decrease is generally expressed as a percent and called the percent change .

To find the percent change, first we find the amount of change, by finding the difference of the new amount and the original amount. Then we find what percent the amount of change is of the original amount.

Find percent change.

Step 1. Find the amount of change. change = new amount − original amount change = new amount − original amount
Step 2. Find what percent the amount of change is of the original amount. change is what percent of the original amount? change is what percent of the original amount?

Example 2.23

Recently, the California governor proposed raising community college fees from $36 a unit to $46 a unit. Find the percent change. (Round to the nearest tenth of a percent.)

Try It 2.45

Find the percent change. (Round to the nearest tenth of a percent.) In 2011, the IRS increased the deductible mileage cost to 55.5 cents from 51 cents.

Try It 2.46

Find the percent change. (Round to the nearest tenth of a percent.) In 1995, the standard bus fare in Chicago was $1.50. In 2008, the standard bus fare was 2.25.

Applications of discount and mark-up are very common in retail settings.

When you buy an item on sale, the original price has been discounted by some dollar amount. The discount rate , usually given as a percent, is used to determine the amount of the discount. To determine the amount of discount , we multiply the discount rate by the original price.

The price a retailer pays for an item is called the original cost . The retailer then adds a mark-up to the original cost to get the list price , the price he sells the item for. The mark-up is usually calculated as a percent of the original cost. To determine the amount of mark-up, multiply the mark-up rate by the original cost.

amount of discount = discount rate · original price sale price = original amount – discount price amount of discount = discount rate · original price sale price = original amount – discount price

The sale price should always be less than the original price.

amount of mark-up = mark-up rate · original price list price = original cost + mark-up amount of mark-up = mark-up rate · original price list price = original cost + mark-up

The list price should always be more than the original cost.

Example 2.24

Liam’s art gallery bought a painting at an original cost of $750. Liam marked the price up 40%. Find ⓐ the amount of mark-up and ⓑ the list price of the painting.

ⓐ

Try It 2.47

Find ⓐ the amount of mark-up and ⓑ the list price: Jim’s music store bought a guitar at original cost $1,200. Jim marked the price up 50%.

Try It 2.48

Find ⓐ the amount of mark-up and ⓑ the list price: The Auto Resale Store bought Pablo’s Toyota for $8,500. They marked the price up 35%.

Solve Simple Interest Applications

Interest is a part of our daily lives. From the interest earned on our savings to the interest we pay on a car loan or credit card debt, we all have some experience with interest in our lives.

The amount of money you initially deposit into a bank is called the principal , P , and the bank pays you interest , I. When you take out a loan, you pay interest on the amount you borrow, also called the principal.

In either case, the interest is computed as a certain percent of the principal, called the rate of interest , r . The rate of interest is usually expressed as a percent per year, and is calculated by using the decimal equivalent of the percent. The variable t , (for time) represents the number of years the money is saved or borrowed.

Interest is calculated as simple interest or compound interest. Here we will use simple interest.

Simple Interest

If an amount of money, P , called the principal, is invested or borrowed for a period of t years at an annual interest rate r , the amount of interest, I , earned or paid is

I = interest I = P r t where P = principal r = rate t = time I = interest I = P r t where P = principal r = rate t = time

Interest earned or paid according to this formula is called simple interest .

The formula we use to calculate interest is I = P r t . I = P r t . To use the formula we substitute in the values for variables that are given, and then solve for the unknown variable. It may be helpful to organize the information in a chart.

Example 2.25

Areli invested a principal of $950 in her bank account that earned simple interest at an interest rate of 3%. How much interest did she earn in five years?

I = ? P = $ 950 r = 3 % t = 5 years I = ? P = $ 950 r = 3 % t = 5 years

Try It 2.49

Nathaly deposited $12,500 in her bank account where it will earn 4% simple interest. How much interest will Nathaly earn in five years?

Try It 2.50

Susana invested a principal of $36,000 in her bank account that earned simple interest at an interest rate of 6.5 % . 6.5 % . How much interest did she earn in three years?

There may be times when we know the amount of interest earned on a given principal over a certain length of time, but we do not know the rate.

Example 2.26

Hang borrowed $7,500 from her parents to pay her tuition. In five years, she paid them $1,500 interest in addition to the $7,500 she borrowed. What was the rate of simple interest?

I = $ 1500 P = $ 7500 r = ? t = 5 years I = $ 1500 P = $ 7500 r = ? t = 5 years

Try It 2.51

Jim lent his sister $5,000 to help her buy a house. In three years, she paid him the $5,000, plus $900 interest. What was the rate of simple interest?

Try It 2.52

Loren lent his brother $3,000 to help him buy a car. In four years, his brother paid him back the $3,000 plus $660 in interest. What was the rate of simple interest?

In the next example, we are asked to find the principal—the amount borrowed.

Example 2.27

Sean’s new car loan statement said he would pay $4,866,25 in interest from a simple interest rate of 8.5% over five years. How much did he borrow to buy his new car?

I = 4,866.25 P = ? r = 8.5 % t = 5 years I = 4,866.25 P = ? r = 8.5 % t = 5 years

Try It 2.53

Eduardo noticed that his new car loan papers stated that with a 7.5% simple interest rate, he would pay $6,596.25 in interest over five years. How much did he borrow to pay for his car?

Try It 2.54

In five years, Gloria’s bank account earned $2,400 interest at 5% simple interest. How much had she deposited in the account?

Access this online resource for additional instruction and practice with using a problem solving strategy.

Begining Arithmetic Problems

Section 2.2 Exercises

Practice makes perfect.

In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question.

There are 16 girls in a school club. The number of girls is four more than twice the number of boys. Find the number of boys.

There are 18 Cub Scouts in Troop 645. The number of scouts is three more than five times the number of adult leaders. Find the number of adult leaders.