During my second year of teaching, in the early 90s, our state testing began to include a greater focus on problem solving and writing in math. Over the next couple of years, we used standard sentence starters to help students practice explaining their problem solving process, so they would be well-prepared for the test. These were starters like:
“In this problem, I need to….”
“From the problem, I know….”
“I already know…”
“To solve the problem, I will…”
“I know my answer is correct because…”
Using these starters, students ended up with several paragraphs (some short, some long) to explain how they approached and solved the problem and how they knew they were correct. Sometimes this took quite a long time, but it was helpful, because it made many students slow down and think a bit more about what they were doing. They took a little more time to analyze the problem (rather than picking out the numbers and guessing at an operation!). I was teaching 5th grade at this time, in elementary school, and we had a full hour for math every day. So, fitting in problem solving practice a few times a week was easy, after students understood the process. I really liked spending the time on these problems, because they often led to discussion of other concepts, and they reinforced concepts already learned.
I used problems from a publication that focused on various strategies, like Guess and Check, Work Backwards, Draw a Picture, Use Logical Reasoning, Create a Table, Look for a Pattern, Make an Organized List. I LOVED these…
I really did (do)! And the students I taught during those years became very good problem solvers.
When I moved to 6th grade in middle school, I tried to keep teaching these strategies, but our math periods are only 44 minutes. I tried to use the problem solving as warm-ups some days, but it would often take 30 minutes or more, especially if we got into a good discussion, leaving little time for a lesson. I found that spending too many class periods using the problem solving ended up putting me too far behind in the curriculum (though I'd argue that my students became better thinkers:-), so I had to make some alterations. I liked the format of the sentence starters, but the biggest time-consumer was the writing part. We had to decrease it. Instead of writing so much, we started to:
* highlight/underline the question in the problem
* shorten up the writing to bullet points
* highlight/underline the important information in the problem
Problem Solving Steps
Now, when I teach these problem solving strategies, our steps are:
Find Out, Choose a Strategy, Solve, and Check Your Answer.
When they Find Out, students identify what they need to know to solve the problem. They underline the question the problem is asking them to answer and highlight the important information in the problem. They shouldn’t attempt to highlight anything until they’ve identified what question they are answering – only then can they decide what is important to that question. In this step, they also identify their own background knowledge about the concepts in that particular problem.
Choose a Strategy
This step requires students to think about what strategy will work well with the question they’ve been asked. Sometimes this is tough, so I give them some suggestions for when to use these particular strategies:
Make an Organized List: when there are many possible answers/combinations; or when making a list may help identify a pattern.
Guess and Check: when you can make an educated guess and then use an incorrect guess to help you decide if the next guess should be higher or lower. This is often used when you’re looking for 2 unknown numbers that meet certain requirements.
Work Backwards: when you have the answer to a problem or situation, but the “starting” number is missing.
Make a Table: when data needs to be organized; with ratios (ratio tables).
Draw a Picture or Diagram: when using the coordinate plane; with directional questions; with shape-related questions (area, perimeter, surface area, volume); or when it’s just hard picture in your mind.
Find a Pattern: when numbers in a problem continue to increase, decrease or both.
Write an Equation: when the missing number(s) can be expressed in terms of the same variable; when the information can be used in a known formula (like area, perimeter, surface area, volume, percent).
Use Logical Reasoning: when a “yes” for one answer means “no” for another; the process of elimination can be used.
Students use their chosen strategy to find the solution.
Check Your Answer
I've found that many students think "check your answer" means to make sure they have an answer (especially when taking a test), so practice several strategies for checking:
* Reread the question; make sure your solution answers the question.
* Redo the problem and see if you get the same answer.
* Check with a different method, if possible.
* If you used an equation, substitute your answer into the equation.
* Ask - does your answer make sense/is it reasonable?
Teaching the Strategies
I teach problem solving strategies as a unit, teaching and practicing each one, and then incorporate the strategies and our 4-step process as students approach problems throughout the year. They keep reference sheets in their binders, so they can quickly refer to the steps and strategies. Some strategies are used more frequently than others (Draw a Picture, Write an Equation, Make a Table), but it's important to know that others are possible. During the unit, I like to show them the same problem, solved with different strategies. For example, I often find that a 'Guess and Check' problem can be solved algebraically, so we’ll do the guessing and checking together first, and then we’ll talk about an algebraic equation - some students can follow the line of thinking well, and will try it on their own the next time; for others, the examples are exposure, and they’ll need to see several more examples before they give it a try.
This year, I'm trying something new - I created a set of Doodle Notes to use during our unit. The first page is a summary of the steps and possible strategies. Then I created a separate page for each strategy, with a problem to work through, as well as an independent practice page for each. I also created a blank template, so I can create homework for students throughout the year, using the same format. I'm hoping that using the Doodle Notes format will make the strategies a little more fun, interesting, and easy to remember.
To check out the Doodle Notes, click either of the images!
Free Number Puzzle!
Turning decimals into a fun, Sudoku-like puzzle is a great way to help students work their way through the different decimal operations.
I love solving Sudoku puzzles using whole numbers, and I encourage students to work on them as well. But when I replace the whole number with decimal problems, I’m able to create a logic puzzle that also gives students a new way to practice skills that can be a challenge to master.
Sudoku puzzles are fun and interesting. They require students to problem solve, in order to ensure that every row, column and group of squares only has one of each number. Some students develop strategies when approaching a puzzle; some learn to use guess and check quite often:-)
When working with decimals in a Sudoku puzzle, students need to consider their target numbers, as well as how to complete the necessary operation. This makes decimal practice a bit more interesting and engaging than working on one problem after another on a worksheet.
As with whole number Sudoku, I start decimal Sudoku with a few squares filled in, so students have a starting point. (If students have never tried Sudoku puzzles before, I recommend starting with a regular Sudoku puzzle, to teach students how they work.)
So far, I’ve only used a 4 X 4 grid, which makes figuring out the target numbers fairly easy. All of the squares have a decimal problem with a missing addend in them (in this example). Within each 2 x 2 section, there is one completed problem, with the target number of 1, 2, 3, or 4 already filled in. The object is for students to figure out the target number for each square and then find the missing decimal number in each individual square. Every row and column must each contain 1, 2, 3 and 4 as the answers to the decimal problems (these are the target numbers). Every 2 x 2 section must also contain 1, 2, 3 and 4.
For example, on the answer key shown here, you can see the target numbers of 1-4 in each row, column and section, and you can see the completed decimal problems. Everything in black (target numbers and decimal numbers) is given. Everything in green is what the students must find (target numbers and missing decimal addends):
I've only used the addition problems (which actually require them to subtract:-), but I plan to try the other operations as well and create larger puzzles (6 x 6 and maybe 9 x 9).
To make students really think things through, I may mix up the operations! For instance, in a 4 X 4 puzzle, the first column could be two multiplication problems and two addition problems. This should keep students paying close attention.
These could be great to complete on mini dry erase boards - easy to erase any guesses that don't work out!
This is a post I wrote back in 2013 (now revised), on my other blog, so the observation I refer to was quite a while ago now...how time flies!
I was observed by one of my assistant principals today (a Friday). After 20 years, I don't get super-worried when I'm going to be observed, but I still feel a little anxious. Today, I decided to have the students complete a problem solving activity and then start a "Footloose" activity, even though they wouldn't finish....Footloose normally takes about 40 minutes, so I figured they could do about half and then finish on Monday. (I do this fairly often, to give students flexibility in their work time - they can take as long as needed to complete problem solving, but if they get done quickly, they can move on). Things went so well during the observation...AP commented that there was so much going on in the room, and that the kids were so engaged! I was happy:)
During the class, students worked on group problem solving, (which they have done previously, with other math skills). These particular problems involved comparing and ordering fractions. Our procedure was as follows:
1) Each group received a different sheet with a problem "situation" and 3-4 questions about that situation. (I have five different sheets so that we can do the problem solving several different days with the same concepts, if needed and if time allows).
2) Each group read their situation and each of the questions together.
3) Each student spent 5-7 minutes, thinking/working individually to solve the questions, writing their work on their own recording sheet.
4) When students completed their individual thinking time, they compared their ideas (and answers if they had them), discussed any differences in thought, and worked to agree on final answers.
5) The final answers (with work) were written onto a group answer sheet to hand in.
When we did this type of group problem solving the first time (with decimal problems), we spent about 5 days on the problem solving, with each group working on a different problem sheet each day. The students really like the problem solving, partly because they are able to talk out their answers with each other. It's great to hear their communication about math and how they are able to point out the steps a group member needs to complete or the concepts that they may have missed.
Today, it was great to hear them say "Oh, we're doing this again. I like this!" My AP commented that he listened to hear what they were talking about, to see if they were focused, and he could hear one student explain to another how the work that they had done was different from another student.
The problem solving took about 15 minutes, and then as each group finished their problem, they moved on to Comparing and Ordering Fractions Footloose. This is a great game for keeping students engaged, but moving! Students start out with one card and a sheet of paper with 30 blank "blocks" in which to write answers to the questions on the cards. Each card has a number on it, and students record the answer to each card in the same number block as the number on the card. After answering the question on the card they start with, students put the card on the chalk ledge and pick up another card with another question to answer. Students continue answering and returning cards until they have answered all 30 questions. Students work so quietly when they are doing this activity! My AP said it was like "night and day" when they switched from the problem solving to Footloose - they were talking about the p.s., but as soon as they started the Footloose, it was sooo quiet.....and I didn't have to say anything for it to be this way - it just happened.
As I mentioned, I don't really get worried when an observation comes around, but it was great to hear the positive feedback for these activities that I create for my students!
I used this week's problem in class today (6th grade), for early finishers. Because we haven't gotten too "into" a particular topic, I made the problem a mix of operations - mostly division and multiplication, but I saw students using addition as well.
I really enjoy talking with my students about what they are thinking when they try to solve problems, for a few reasons - because 1) they think about problems in a different way than I do; 2) it makes me rethink the wording of the questions I ask (which makes me improve); and 3) I learn that there will be several ideas to share with class.
I noticed a few different things when the students were solving the different parts of this week's problem:
For part A, I multiplied 85 times 3 to get the total number of cookies and then divided by 24 (when I wrote the problem, I wanted the students to have to interpret the quotient, so I approached it with a desire to use division). And most students did the same thing (except for the few that multiplied 24 x 3 - that gave me some good info: -), but one student was just sitting and thinking, so I asked him what he was thinking. He started to say he divided 24 by 3 and then paused - I almost interrupted his thinking to redirect him to my way, but I successfully restrained myself, and asked why. He said he was thinking about how many baggies could be filled with one batch, and since the numbers worked nicely, he could definitely say that one batch would fill 8 baggies. I really liked his thinking process, because it hadn't occurred to me to do it that way. Now, if the numbers hadn't worked out evenly, it might not have been the best approach, but we can expand our class discussion to explore that. After deciding he could fill 8 baggies per batch, he added on sets of 8 until he reached the correct number of batches.
As some students worked on part C (below), I started to think that I should adjust the wording of the problem. When I wrote the problem, I thought it would be clear that the number of cookies for part C was the same as part A, but some students thought of the part C as using 85 baggies of 2 cookies (same number of baggies), instead of using the same number of cookies. As more students worked on it though, other students seemed to understand that the number of cookies should be the same as the original number they were working with, so I haven't changed it yet. If you use the problem, please let me know what you think.
Again, a few students approached this part in a different way than I did - they said that in both cases, the cookies cost 25 cents each. Using this reasoning, some students said the cost was the same, while others did not - again, a great opportunity for discussion, both in small groups and as a whole class.
To see and/or use the entire problem and answer key, click on the link below the picture.
Click to download this freebie!
To access all of the Problem of the Week problems (previous and future), click here!
Have a great week!
Hi, I'm Ellie! I've been in education for 25 years, teaching all subject areas at both the elementary and middle school levels.