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CGI problems

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A revolutionary, online math program comprising games, animated books and downloadable materials.

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Solving Word Problems without Relying on Key Words

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Solving Word Problems without Relying on Key Words

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http://teachingtoinspire.com/2016/02/solving-word-problems.html

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Teaching to Inspire with Jennifer Findley

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One of the toughest things about teaching fifth grade is definitely the word problems. By fifth grade, a lot of students have become so dependent on using key words that they no longer even read for meaning when they're solving word problems. However, as fifth grade teachers know, key words only take the students so far.

I have read many articles and blog posts that are adamant against teaching key words, but many of them do not offer an alternative. About three years ago, I created a strategy for teaching my students to solve word problems that does not rely on key words, and I want to share it with you today.

I really want my students to understand what the problem is asking them to do. Keeping this in mind, I teach word problems in terms of what the situation of the word problem is versus what key word is in the word problem. Before I taught this strategy, many of my students read word problems in order to find the key words. They did not read to understand what the problem was really telling them or asking them. To combat this, I teach them to think of word problems more as situations. When a student looks at a word problem from a situation standpoint, they are reading for meaning and really understanding what operation is required to solve the problem.

To get students to stop relying on key words and think of situations instead, I do an introductory lesson involving four word problems (shown above). Each of the word problems use the word total. However, the word problems each require a different operation. When discussing the word problems, we always have a big discussion about how each of the word problems uses total but they are not all adding or even multiplying. This really gets the students to understand that key words alone cannot always be relied upon.

During this lesson, I stress the importance of really understanding the situation that the word problem is describing to figure out which operation to use. A link to download the printable of the four word problems I use will be available at the end of this post.

After teaching the lesson involving the four word problems, I move right into discussing different situations and how those situations can be translated through a word problem. As a class, we discuss different situations and determine which operation would be used to solve the word problems that involve that situation. Together, we create an anchor chart of different situations under the operation that would be used to solve the situation. We add to this anchor chart as the year progresses and the students are exposed to more word problems with varying situations (For example: taking part of a part when multiplying fractions).

I use the printable chart above to help me generate a list of situations to discuss with students. Many of the fraction situations I don't introduce until later on in the year once we start fractions.

Speaking of fractions, when we really start digging into multiplication and division of fractions, I always have to revisit the idea of using situations to help solve word problems versus key words. The fifth grade level fraction word problems are really tricky and many of my students revert back to key words because they are overwhelmed. At this point, I typically give students the printable version of the chart for them to refer to on a daily basis as they solve word problems.

Moving away from key words and having students think about operations in terms of situations instead has made a huge difference in the way my students think about and solve word problems. When they are solving a tricky word problem, I always remind them to revisit the situation chart and see which situation matches the word problem. This gets them away from relying solely on key words and builds their confidence with word problems. It also helps them be more successful when solving multi-part word problems.

To read more about how I teach multi-part word problems, click here. You can download the word problems and the printable chart by clicking on the image below.

What do you think of this strategy? Do you think this is something your students would benefit from being introduced to? Let me know in the comments.

P.S. If you are in need of word problems, click here to see the sets I have in my TeachersPayTeachers store.

You might also enjoy these posts:

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Solving Word Problems Without Relying on Key Words - Teaching to Inspire with Jennifer Findley

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One of the toughest things about teaching fifth grade is definitely the word problems. By fifth grade, a lot of students have become so dependent on using key words that they no longer even read for meaning when they're solving word problems. However, as fifth grade teachers know, key words only take the students so far.

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https://bstockus.wordpress.com/2016/02/21/purposeful-numberless-word-problems/

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Teaching to the Beat of a Different Drummer

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UPDATE - You can find all of my numberless word problem sets on this page.] This year I read Sherry Parrish's Number Talks from cover to cover as I prepared to deliver introductory PD sessions to K-2 and 3-5 teachers in November. She outlines five key components of number talks; you can read about them...

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Open Middle - Challenging math problems worth solving

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Open Middle offers interactive math problems designed to foster critical thinking. Explore unique challenges that boost problem-solving skills and creativity.

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Math story problems can be classified into various problem-types using the Cognitively Guided Instruction approach. Read on to learn more about the different types of math story problems. After a brief description of each problem type, there are several math story problems provided for example.

Addition Problems

Addition problems involve a direct or implied action in which a quantity is increased by a particular amount. They are sometimes referred to as "Joining" problems in CGI literature. There are three types of basic addition problems:

1. Result Unknown 15+32+___

2. Change Unknown 15+___=47 (a slightly more difficult problem)

3. Start Unknown ___+32=47 (the most difficult of the basic addition problems)

Below are some sample story problems for each type. These problems can be changed to include familiar names, actions or situations with your students. Be creative! The numbers in the problems can also be altered, depending on the child's mathematical abilities. Feel free to try different combinations or even several numbers to make the problem more challenging.
Result Unknown:

You have 4 books to read. You get 3 more books from the library. How many books do you have now?

Robin has 98 toy cars. Her parents gave her 45 more toy cars for her birthday. How many toy cars will she have then?

There are 320 boys in a club. 29 girls join the club. How many children are in the club altogether?

Jane gave 10 pieces of candy to Sam. Fred gave Sam 6 pieces of gum. Mary gave Sam 12 pieces of gum. How many pieces of gum does Sam have now?
Change Unknown

Kristin had 17 apples. How many more apples will she need to have 39 apples all together?

Eric has 72 golf balls. He finds some more in the basement. Now Eric has 183 golf balls. How many golf balls did Eric find in the basement?

Columbus saw 25 whales in the ocean. How many more whales will he need to see to make it 3 dozen whales all together?

There were 450 kids in the school. Some more kids came to the school. Now there are 920 kids at the school. How many more kids came to the school?
Start Unknown

Clifford has some bones. Emily gave him 23 more bones. Now Clifford has 46 bones. How many bones did Clifford have to start with?

Mrs. Wistrom saw some ducks. Then she saw 156 more ducks. All together, Mrs. Wistrom saw 234 ducks. How many ducks did she see to begin with?

Mom had baked some cookies. Then she baked 4 dozen more cookies. All together, mom has baked 58 cookies. How many cookies did mom bake to begin with?

Jeff had some leaves in a bag. Then he found 16 more leaves, and put those in the bag. Now Jeff has 76 leaves. How many leaves did Jeff find to begin with?

Subtraction Problems

Subtraction problems involve action in which the initial quantity is decreased over time. They are sometimes referred to as "Separating" problems in CGI literature. There are three types of basic subtraction problems:

1. Result Unknown 12-6+___

2. Change Unknown 12-___=6 (slightly more difficult problem type)

3. Start Unknown ___-6=6 (the most difficult of the basic subtraction problems)

Below are some Cognitively Guided Instruction sample story problems for each type. These problems can be changed to include familiar names, actions or situations with your students. Be creative! The numbers in the problems can also be altered, depending on the child's mathematical abilities. Feel free to try different combinations or even several numbers to make the math story problems more challenging.
Result Unknown

Jessica had 14 toy cars. She lost 11 of the cars. How many cars does she have left?

Melissa had 78 pumpkin seeds. She gave 23 of them to her brother. How many pumpkin seeds does Melissa have left?

There were 156 ornaments on the tree. 120 of the ornaments fell off the tree. How many ornaments are left on the tree?

Dan has 386 papers in his desk. He threw away 17 of them. How many papers does Dan have left in his desk?
Change Unknown

Mikey had 16 leaves. Some of her leaves blew away. Now she has 7 leaves left. How many of her leaves blew away?

Sara had 99 goldfish. She gave some away. Now she has 32 goldfish left. How many goldfish did Sara give away?

320 children wore their Halloween costumes to school. Some of the children took their costumes off at recess. There were 210 children left wearing costumes. How many children took their costumes off at recess?

There were 42 cookies on a plate. Dad ate some of the cookies. Now there are 29 cookies left on the plate. How many cookies did Dad eat?
Start Unknown

Ashley had some guppies. She gave 28 guppies to Timmy. Now Ashley has 12 guppies left. How many guppies did Ashley have to start with?

Ryan had some trick-or-treat candy. He gave 123 pieces of candy to his sister. Now Ryan has 100 pieces of candy left. How many pieces of candy did Ryan have to begin with?

Mrs. Wistrom had some pencils. She lost 24 of them. Now she has 6 left. How many pencils did Mrs. Wistrom have to begin with?

A clown had some balloons. 56 of the balloons floated away. Now the clown has 45 balloons left. How many balloons did the clown have to start with?

Multiplication Problems

Grouping and Partitioning problems involve three distinct quantities, as illustrated by the following examples of math story problems:
Megan has 5 bags of cookies. There are 3 cookies in each bag. Altogether, Megan has 15 cookies.
The three quantities in the problems are: the number of bags, the number of cookies in each bag, and the total number of cookies. In a problem, any one of the three quantities can be an unknown. When the total number of cookies Megan has all together is unknown, the problem is a multiplication problem.

Below are some examples of math story problems for each type. These problems can be changed to include familiar names, actions or situations with your students. Be creative! The numbers in the problems can also be altered, depending on the child's mathematical abilities. Feel free to try different combinations or even several numbers to make the problem more challenging.

Mrs. Wistrom has 4 boxes of candy. There are 10 pieces of candy in each box. How many pieces of candy does she have now?

Tommy has 4 packages of Pokemon cards. There are 7 cards in each package. How many cards does Tommy have altogether?

Our classroom has 6 jars. There are 8 butterflies in each jar. How many butter flies are in our room altogether?

Farmer Ted has 10 hens. There are 6 eggs under each hen. How many eggs are there altogether?

Division Problems

As we learned above, Grouping and Partitioning problems involve three distinct quantities. This is illustrated in the example:
Megan has 5 bags of cookies. There are 3 cookies in each bag. Altogether, Megan has 15 cookies.
To refresh, the three quantities in the problems are: the number of bags, the number of cookies in each bag, and the total number of cookies. In a problem, any one of the three quantities can be unknown. In a Measurement Division problem, the total number of objects and the number of objects in each group is given. What is unknown, is the number of groups (in our example, this would be the number of bags of cookies.)

Here are some examples of Measurement Division problems:

Meagan has 15 cookies. She puts 3 cookies in each bag. How many bags can she fill?

Karen has 18 Gummy Worms. She puts 3 in each Halloween treat bag. How many bags will she fill?

Room 124 is going on a hayride. There are 27 kids in the class. 9 kids can fit on a wagon. How many wagons must they take?

The hens laid 20 eggs. There are 4 eggs under each hen. How many nests are there in the hen-house?

In a Partitive Division problem, the total number of objects and the number of groups is given. The number of objects in each group (in our case, the number of cookies per bag) is unknown.

Here are some examples of Partitive Division problems:

Megan has 15 cookies. She puts the cookies into 5 bags, with the same number of cookies in each bag. How many cookies are in each bag?

Karen has 16 caterpillars. She puts the caterpillars into 4 jars, with the same number of caterpillars in each jar. How many caterpillars are in each jar?

Mom has 12 marshmallows. She puts the marshmallows on 4 sticks, to roast. How many marshmallows are on each stick?

Mrs. Bell has 20 stickers. She gives them to 10 children, so that they each have the same amount. How many stickers did each child get?

For additional examples of math story problems using Cognitively Guided Instruction, read more in this series.

CGI Math In Your Classroom

Learn more about the Cognitively Guided Instruction approach and how it can be used in your classroom to effectively further student learning. This series will explore the principles behind CGI math, identify related concepts, and offer a multitude of problems you can use in your own classroom.

1. CGI - An Approach to Teaching Mathematics

2. Teaching Different Types of Math Story Problems

3. Using Cognitively Guided Instruction for Teaching and Assessment

4. Math Skills Assessment for Division Problems Using Cognitively Guided Instruction

5. Elementary Math - Assessment of Strategies for Solving Math Problems

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Cognitively Guided Instruction for Math Story Problems Using CGI Math

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Math story problems can be classified into various problem-types using the Cognitively Guided Instruction approach. Read on to learn more about the different types of math story problems. After a brief description of each problem type, there are several math story problems provided for example.

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A list of online resources for math word problems and problem solving

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Find here an annotated list of problem solving websites and books, and a list of math contests. There are many fine resources for word problems on the net! have personally checked & reviewed each website, to make sure it is truly useful.

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Word Problems with Addition and Subtraction are Easy, Right?

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Word Problems with Addition and Subtraction are Easy, Right?

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https://aerecord.wordpress.com/2016/10/06/word-problems-with-addition-and-subtraction-are-easy-right/

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Word problems with addition and subtraction are easy, right?

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Over the course of my 13 years as a grade 5 teacher and now in my third year as a Math Coach, I have witnessed the same situation over and over....a student is presented with a word problem, their brain freezes and they say, "I don't get it."

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https://drive.google.com/file/d/0B8DtIUhMGc9qbktSNElWOF9oRGs/view

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Teaching Word Problems with Situations Poster.pdf

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http://www.bemidji.k12.mn.us/wp-content/uploads/2013/11/Word-Problem-ID.pdf

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https://bstockus.wordpress.com/numberless-word-problems/

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Addition and Subtraction Structures-Part 2

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Addition and Subtraction Structures-Part 2

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http://www.mathcoachscorner.com/2012/05/addition-subtraction-structures-part-2/

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Yesterday, I posted about an alternative approach to using key words to identify addition and subtraction in word problems based on the work of John Van de Walle. This is a continuation of that post, so it you didn't read it, you might want to catch up before reading on!
Another Case Against Key Words
So, yesterday I outlined join and separate problems. Today, I want to finish up with part-part-whole and compare problems. You will notice strong similarities between join and part-part-whole problems. To be honest, I think we probably don't often distinguish between the two.

Whole unknown
NOTE: This is the simplest form of part-part-whole problem and is the type of problem typically found in primary classrooms.
There are 4 ducks and 3 egrets in the pond. How many birds are in the pond altogether?
Think about the difference between this problem and the similar join problem. With the join problem, there were ducks, more ducks came, and we counted the total ducks. In a part-part-whole problem, we had ducks, we had egrets, and we were counting birds. Not a significant difference, but it might present a challenge for some kiddos, so it's good to be aware of it.
There are 7 birds in the pond. Four of the birds are ducks and the rest are egrets. How many egrets are in the pond?
Compare Problems
Ah, compare problems. We all know how difficult these can be for kiddos. How many more will always be easier than how many fewer, but we also need to use the word difference.

Difference unknown
NOTE: This is the simplest form of compare problem and is the type of problem typically found in primary classrooms.

There are 4 ducks and 3 egrets in the pond. How many more ducks are in the pond
than egrets?
There are 4 ducks and 3 egrets in the pond. How many fewer egrets are in the pond There are 4 ducks and 3 egrets in the ponds. What is the difference in the number of ducks and egrets in the pond? There is 1 more duck than egrets in the pond. There are 3 egrets in the pond. How many ducks are in the pond? There is 1 fewer egret than ducks in the pond. There are 3 egrets in the pond. How many ducks are in the pond? There is 1 more duck than egrets in the pond. There are 4 ducks in the pond. How many egrets are in the pond? There is 1 fewer egret than ducks in the pond. There are 4 ducks in the pond. How many egrets are in the pond? than ducks?
Phew! That's a lot to digest! Now that you have all the types of problems, design some of your own scenarios and rewrite each scenario using all the types of structures.

One word of caution. Remember that when using these problem types in the classroom, be mindful of the CRA sequence of instruction. In the problems I've included here, you would want students to have counters and story mats to act out these problems. In other words, the learning should be very concrete as they learn the various structures.

You might also like these posts:

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Addition & Subtraction Structures, Part 2 - Math Coach's Corner

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Yesterday, I posted about an alternative approach to using key words to identify addition and subtraction in word problems based on the work of John Van de Walle. This is a continuation of that post, so it you didn't read it, you might want to catch up before reading on!

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Modeling Addition and Subtraction Structures

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Modeling Addition and Subtraction Structures

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http://www.mathcoachscorner.com/2015/05/modeling-addition-and-subtraction-structures/

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I am finishing out the school year with my first and second graders by trying to give them a solid foundation with the different structures for addition and subtraction problems. I've been using a marvelous free resource called Thinking Blocks. They offer a variety of free iPad apps, but I've been using the web version on my interactive white board because I wanted to bring in some concrete learning.

The Thinking Blocks program allows you to choose the type of problem to work with, and we began with part/whole problems with two parts. I made part/whole mats for the kiddos to use (on the back side is a part/whole mat with three parts), because I wanted them to be able to act out the models. My first graders used cubes while my second graders used base-10 blocks to find their solutions.

The problems cover all the structures the students need to be familiar with. All too often, we only expose them to join problems where the parts are known and they are asked to find the sum. But they also need to be able to solve missing part problems. Thinking Blocks mixes up the problem types, which I really like.

For each problem, we went through a series of steps that I hope will become a habit for the students as they work problems independently. First, we read the problem straight through. When we finished reading, I asked them to tell me what the story was about with one word (pizzas, shoes, pages, etc.). This helps to focus them on what we're counting in the story. Next, I asked them to summarize the story. This was really tough! I was looking for them to just retell the story without numbers or details. For example: Maria had some marbles. She gave some to her brother and she kept some.

Next, the Thinking Blocks program asks students to place the labels in the model. I love that this is the first step. Too often, students focus only on the numbers and lose the meaning of the problem. We re-read each sentence and discussed if it was a part or the whole and a student placed the labels in the model. After the labels are placed, the program offers a check feature.

Now it's time to add numbers to the model. Again, we re-read each sentence to make sure we got the numbers in the correct place on the model. A question mark represents the missing number. Once the numbers are placed, the program offers a check feature and also re-sizes the blocks. That allows for some great discussions about what the size of the block tells us about the missing number.

After we had the completed model in Thinking Blocks, I had the students show the problem on their part/whole mat and use the manipulatives to find the missing number (sometimes the whole and sometimes a part).

After working with 2- and 3-part part/whole problems, we tackled comparison problems. This time, the kiddos used two colors of cubes to build trains to match the model. We used blue cubes for the difference and yellow cubes for the two quantities being compared. Again, sometimes the difference is known and one of the quantities being compared is missing, and sometimes the difference is missing.

We are working to the point when we will mix all the problem types. We've had such great discussions, and I'm excited to see their mathematical understanding grow each day!

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Modeling Addition and Subtraction Structures - Math Coach's Corner

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I am finishing out the school year with my first and second graders by trying to give them a solid foundation with the different structures for addition and subtraction problems. I've been using a marvelous free resource called Thinking Blocks. They offer a variety of free iPad apps, but I've been using the web version on ...

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http://www.mathcoachscorner.com/2013/09/modeling-addition-structures/

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The chart shown above shows all the different structures for addition and subtraction problems (adapted from Van de Walle). Unfortunately, we don't often get past the first row of structures, which are the least challenging. Students need to experience all the different structures on a regular basis.

Bar model drawing is a great tool for helping kiddos understand the different problem structures. Let's look at the three Part-Part-Whole structures. The most straight-forward structure is Whole Unknown, or Missing Sum. In this structure, we know both parts and we are looking for the total. Notice the labeling in the model. This very important component of model drawing makes the numbers in the problem real for students-through repeated practice with model drawing, students come to understand that the numbers represent something in the story. Teaching students to label their models also addresses CCSS Standards for Mathematical Practice 6, Attend to Precision (CCSSI 2010).

There are 14 boys and 10 girls in Mrs. White's class. How many students are in Mrs. White's class?

Moving down the Part-Part-Whole column, we see Part Unknown. In this structure, we know the total and one of the parts, but we are missing the other part. This is also commonly called a Missing Addend problem. We would often solve this problem using subtraction, but a counting up strategy works well also. Once the model is drawn, elicit multiple strategies for solving the problem!

Mrs. White has 24 students in her class. Fourteen students are boys and the rest are girls. How many girls are in Mrs. White's class?

As we move to the bottom of the chart, it gets really interesting! Both Parts Unknown. This is a GREAT problem structure for composing and decomposing larger numbers. Think of the wonderful mathematical conversations you could have with this structure!

Mrs. White has 24 students in her class. Some are boys and some are girls. How many boys and girls could be in Mrs. White's class?

One other note, while these example all have two addends, you can also use more than two addends to provide more of a challenge. Think: There are 425 students in the cafeteria. One hundred twenty-seven students are 5th graders, 146 are 4th graders, and the rest are 3rd graders. How many 3rd graders are in the cafeteria? The model drawing technique makes these more difficult problems accessible to students of many ability levels.

You might also like these posts:

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Modeling Addition Structures - Math Coach's Corner

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The chart shown above shows all the different structures for addition and subtraction problems (adapted from Van de Walle). Unfortunately, we don't often get past the first row of structures, which are the least challenging. Students need to experience all the different structures on a regular basis.

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https://docs.google.com/presentation/d/1uTtrTIEf52g-1qqlkLbBNEfxdN3W55W8R-WFXPKhBdM/edit#slide=id.p4

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Prezi-Word Problems and Underlying Structures

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Prezi-Word Problems and Underlying Structures

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https://prezi.com/sja3wwqo5tdi/math-word-problems-and-underlying-structures/

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Math Word Problems and Underlying Structures

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for our students to truly be successful. But we have understand these ideas so deeply.... In classrooms and on the tests: How does this relate to the content standards? Solving word problems is present at every grade level Usually in operations and algebraic thinking In Title and Sped Groups: How does this relate to the practice guides for RTI in math?

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Solving Word Problems Using Story Structure

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Solving Word Problems Using Story Structure

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http://www.cpalms.org/Public/PreviewResourceLesson/Preview/37197

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Solving Word Problems Using Story Structure

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https://assets.pearsonschool.com/asset_mgr/current/201034/Randy%20Charles%20Monograph.pdf

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CGI and Understanding the Structure of a Word Problem

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CGI and Understanding the Structure of a Word Problem

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http://www.uwosh.edu/coehs/cmagproject/ethnomath/legend/documents/teacher_guide.pdf