Problems to Pose to your Students at the Beginning of the School Year
Some of you are beginning your first CGI journey with a group of students. Others are beginning a new CGI journey with a new group of students. If you have been using CGI for a while and haven’t changed grade levels, you probably already have some favorite problems to use at the start of the year. If you are new to CGI or have changed grade levels or schools, here are some problems that might be good ones to kick off your CGI journey this year.
We recommend that you pose these problems without providing instruction on how to solve them. Students who are used to having teachers show them how to solve problems may struggle to get started. There is a note at the end of the post about some ways that you can support these students without showing or leading them through a solution strategy.
Grade levels are provided as a suggestion. Please adjust the numbers or choose problems from different grade levels to meet your students’ needs. We recommend that you don’t make the numbers much smaller than those provided. When the numbers are too small, children can have trouble explaining their strategies.
Hopefully, these problems will serve as a launching point for your CGI journey. Whether you are new to CGI, or an experienced CGI teacher, it is exciting to see the strategies of a new group of students. These strategies provide a valuable window into children’s thinking and serve as tool for informing our instruction.
Kindergarten and First Grade
Peggy had ___ grapes. She ate ___ of them. How many grapes does Peggy have left?
Possible numbers: 5, 2 10, 2 15, 6
Suggested materials: Individual counters; paper and pencil if you wish
Rationale for using this problem: Many kindergarteners and some first graders are still learning to count. Separate Result Unknown problems are good problems to use with students who can count a collection but struggle to make a collection. It is no more difficult to directly model a Separate Result Unknown problem than it is to directly model a Join Result Unknown problem.
If you see students who can’t get started and you think they are struggling to make a set to represent the starting amount, you can reread the problem and ask, “Can you show me Peggy’s 5 (or 10) grapes?” If they struggle to make a set, you can place a set of 5 (or 10) counters in front of them and say, “Let’s pretend that these are Peggy’s grapes. How many grapes does she have?” Reread the story and say, “Could you use those blocks to show what would happen if Peggy ate 2 of those grapes?” Young children can often make a set of 2 even if they weren’t able to make a set of 5 or 10. Not all children who struggle to make sets will be successful with these prompts, but these prompts might help them understand other students’ strategies for solving this problem. Be careful not to provide this extra help until you are sure that the child needs help making sets. Allow students who miscount but feel that they have the right number of grapes in front of them to continue solving the problem. At this time of the year, it’s more important to develop students’ beliefs that they can generate their own solutions to math problems than it is for them to always get the right answer. Furthermore, getting the wrong answer may motivate some students to be more attentive to their counting in the future.
Second and Third Grade
Tyrone has ____ boxes of books with 10 books in each box. How many books does Tyrone have?
Possible first number: 4 8 13 23
Suggested materials: Base ten blocks; paper and pencil
Provide the base ten materials without giving specific directions on how to use them. You don’t need to point out that there are 10 ones in the ten stick. You may need to provide some directions about the materials related to classroom management such as, “So that everyone can hear me read the problem, please don’t touch the blocks until I am done reading the problem,” or “If the blocks drop on the floor, please pick them up right away so that we don’t lose the blocks.” If your base ten blocks came with mats to organize the materials, don’t distribute the mats at this time.
Rationale for starting with this problem: Base ten number concepts are crucial for many of the second and third grade math standards. Many second and third graders don’t fully understand how our base ten number system is structured around the idea that 1 ten is the same as 10 ones. A student’s strategy for solving this problem will give you a window into that student’s understanding of base ten that can then inform your instruction.
This problem is also a good starting problem, because it’s a multiplication problem. Few students will have been taught a standard procedure for solving this type of problem. It will be easier for you and your students to start your CGI journey without having to make decisions about how to respond to students who use strategies that they may not understand.
If this problem proves useful, you could also pose a problem such as:
Mr. Lee has ____ pencils. It takes 10 pencils to fill a pencil box. How many pencil boxes could Mr. Lee fill with his pencils?
Possible first number: 32 91 124 341
Although adults sometimes see these problems with a remainder as more challenging, children are typically not bothered by the extra pencils.
Fourth and Fifth Grades
The base ten problem for second and third grades would make a good starting problem for most fourth and fifth grade classes. Base ten continues to be important in the upper elementary school, and many fourth and fifth graders aren’t completely solid in their understanding of our base ten system through 1,000.
Another good problem to start with would be:
4 people want to share _____ cookies so that everyone get the same amount of cookie. How many cookies should each person get?
Possible second number: 5 11
Suggested materials: Paper and pencil; You could provide colored pencils or crayons. (Colors can help some children keep track of who is getting which portion of the cookies.) Do not provide fraction manipulatives, because these are already divided. Students need to develop the idea that fractions are the results of dividing wholes.
Rationale for starting with this problem: This problem is a good starting problem, because students most likely won’t have been taught a standard procedure for solving this type of problem. It will be easier for you and your students to start your CGI journey without having to respond to students using strategies that they may not understand. There is a chance that some students will use a long division algorithm to solve this problem. If they use this algorithm, you could say, “If you had those cookies in front of you, would you really cut them into hundredths?” or “Is there another way that these people could share these cookies?” If your students are ready, you might even share the long division algorithm as a strategy for solving this problem to launch a discussion of choosing strategies that are appropriate to the problem. (If you don’t want a student to feel that their long division strategy was being criticized, you could share the strategy and say something like, “My teacher friend at another school saw some kids solve this problem like this.”) The standard long division algorithm is clearly not a good strategy for this problem.
Some students may provide an answer in picture form only. Consider accepting the picture as a valid way to show your answer. As the year progresses, students will need to learn to represent their answers using fraction notation, but for now, it might be more important to honor their thinking than to insist on mathematical notation for representing the answer. Some students may be able to tell you the answer in words (e.g., each person gets one and one-fourth) you could choose to write these answers in text for now (e.g., 1 and 1 fourth) rather than insist that students write their answers as 1¼.
If your students don’t get started because they are waiting for you to tell them what to do…
If your students are used to having teachers show them how to solve math problems, they may struggle to get started. Pose the problem at first without providing any support to give you a sense of which students are willing to jump into problem solving. For students who aren’t getting started, you might reread the problem and ask questions about the problem such as, “How many boxes does Tyrone have?” “What do we know about these boxes?” “How many cookies are there?” and “How many people are sharing them?” If they still struggle to get started, you could link the story to their experiences with further questions like, “Have you ever put things into boxes to carry them?” or “Have you ever shared cookies with other people? What are some of the things that you have done?” If needed, you could follow with questions such as, “If you were Tyrone and wanted to know how many books you had, what would you do?” “Could you draw a picture to help you figure out how many books you have?” and “Could you draw a picture to show how you would share cookies, if you were one of the people in the story?”
Do your best to resist the temptation to lead students who struggle through a strategy for solving the problem. You can say something like, “I think you are going to be able to solve this problem next month. Let’s see if listening to your classmates will help.” When students see their classmates generate strategies for solving the problem, they will learn that children can generate strategies for solving the problem which eventually will lead them to believe that they can generate strategies for solving problems. Right now, it’s more important for them to learn that mathematical ideas can come from them than it is for them to solve this one problem correctly.
Chapter 5 in Children’s Mathematics: Cognitively Guided Instruction has additional suggestions for getting started with CGI. If you participated in a CGI Math Teacher Learning Center session this summer, the plans your made for using problems with your students contain additional suggestions for getting started.
Authored by Linda Levi, Director, CGI Math Teacher Learning Center