Mathematics Developmental Continuum
Guess-Check-Improve Strategy: 2.5
Supporting materials
- Related Progression Points
- Developmental Overview of Working Mathematically (PDF - 31Kb)
- Developmental Overview of Structure (PDF - 35Kb)
Indicator of Progress
Success depends on students being confident in using the guess-check-improve strategy to find answers for appropriate problems.
Prior to achieving this level, students are unable to use the improve aspect of the strategy to refine initial guesses to work towards answers. They can make a guess and then check whether the guess correctly answers the problem, but if it doesn’t then the initial guess is not used to inform the next guess.
Illustration 1: Guess-Check-Improve is not just Trial and Error
Students may be familiar with trial and error or guess and check , but they may not appreciate how the improve step works. The improve phase takes the results from earlier guesses to inform the next guess. The crucial step here is recognizing that each result from previous guesses can provide information for improving the next guess.
As teachers observe students using this strategy, they can talk to students to see that they are not randomly guessing, but are gathering information for improving subsequent guesses.
Illustration 2: Working systematically is important
The importance of working systematically is reinforced when using the guess-check-improve strategy. A systematic approach enables students to work in an efficient way, rather than choose numbers randomly with the hope that they will eventually hit upon the answer to a problem. In a numerical problem, systematic working can be related to choice of numbers and the way that results are recorded. They will learn to use tables to organise results and identify useful numbers to use as the next guess.
As students use this strategy, teachers can observe whether they record their work systematically.
Teaching Strategies
Teaching a strategy for problem solving is a long term endeavour, revisited with mathematics from different dimensions. Students need to be given experiences in solving problems for themselves, and key points about the strategy can be drawn out from the experience. There is also a place for students to practise strategies, such as guess-check-improve, which apply to a wide range of problems. The key points to emphasise are listed in Activity 3.
For these activities, allow students to use calculators for any challenging calculations, to enable a focus on the strategy rather than spending considerable time performing multiplication.
Activity 1: Mystery number is an initial activity to highlight the importance of the improve step when using guess-check-improve as a strategy.
Activity 2: Ducks and horses aims to get students to recognise the usefulness of a systematic approach and the importance of careful organisation of results.
Activity 3: Additional problems provides some other suggestions.
In this activity students use the guess-check-improve strategy to find unknown numbers. The importance of the improve phase of the guess-check-improve strategy is highlighted so that students can recognise that this strategy will provide answers more quickly than random guessing. It is important to use problems where students cannot immediately see the answer otherwise there is no need to use the strategy of guess-check-improve.
Choose a problem beyond the range of students’ mental calculation skills, such as the following. There are many different possibilities (see notes at the end of this activity, and also Activity 3).
Ask a student to guess the mystery number in the following problem quickly.
"If you multiply 14 by this number you get 252. What is the number?" [Ans: 18]
Get all students to check if the number offered is the correct answer. Use calculators, so that the focus is on the strategy, rather than being distracted by by-hand calculation. Discuss whether the result is too high or too low and what this means about the answer to the problem. For example, if the number selected was 11, then the students will have used their calculators to find that 14 × 11 = 154, so 11 is too small. This information can be used to improve the next guess. They now know that the number needed is greater than 11. Focus on the use of the words guess, check and improve.
Now ask another student for a second guess: it should be a number greater than 11. If they suggest 25 (for example), they multiply 14 by 25, get 350, and note it is too big. Discuss what this means for the mystery number. The students should now recognise that the number they want to find is between 11 and 25.
Ask for another number, now between 11 and 25, gradually funneling onto the answer of 18. Focus on the word improve so that students can see that they are using their results at each stage to refine their guesses.
Provide students with some additional problems with whole number answers, such as:
"What number multiplied by 15 gives 405?" [Ans: 27]
"If you multiply 8 by this number you get 256. What is the number?" [Ans: 32]
"I have 288 eggs in boxes which each hold a dozen eggs. How many boxes?" [Ans: 24]
Discussion should focus on how students used their checking to improve their guesses. Focus on the cycle, guess-check-improve so that students recognise the importance of using results to inform them to work towards the required numbers.
To start students thinking about the importance of working systematically and organising results it would be helpful to show how guess-check-improve can be tabulated. For example, if trying to find a number that can be multiplied by 35 to give 840 the following table allows students to see how writing down the results in an organised way can help to find the answer.
| Guess | Check (Target is 840) | Improve: Guess is ... |
| 20 | 35 × 20 = 700 | Too small |
| 30 | 35 × 30 = 1050 | Too big |
| 25 | 35 × 25 = 875 | Too big, but close |
| 24 | 35 × 24 = 840 | Correct |
NOTES:
- Using division: The problem above can solved immediately by using division. This activity has been suggested because it provides a simple context and at this level students are unlikely to identify the division strategy. However, if a student does suggest using division to solve the problem, show all students that this is correct and the best way, but then challenge the student to find the answer to the problem without using division.
- Variations: All the examples above involve multiplication, but addition or subtraction examples may be more appropriate for many students. The examples here also all involve only one operation, but this too can be varied (e.g. I thought of a number, added 12, then added my number again, and the answer was 50. What was my number? Ans: 19)
This activity is useful for showing how systematic recording of results can assist in solving problems using guess-check-improve where students have to do more than multiply one number.
Start by posing the problem for students.
"A farmer has some ducks and some horses. Altogether the ducks and horses have 40 legs and 14 heads. How many ducks and horses are there on the farm?" [Ans: 6 horses and 8 ducks]
 |
Altogether these ducks and horses have 18 legs and 6 heads. |
Begin with the first step of problem solving: understanding the problem. Students need to recognise that ducks have two legs and one head and horses have four legs and one head. Some students might also say that if there are four legs then this could be for one horse or two ducks. Some students might want to guess the answer at this early stage (e.g. 3 horses and 3 ducks, as in the picture). It is always a good problem solving strategy to try an example, so work out the number of legs and heads for this first case. In this way, students will fully understand the problem requirements.
Now ask students to attempt the problem and then discuss some of the strategies used.
Many students will guess both the numbers of horses and ducks. For example, they might draw 8 horses and 10 ducks and count the legs (52) and heads (18), and then make another guess of both numbers.
A better strategy is to guess the number of horses and calculate the number of ducks (or vice versa) to get the right number of heads, and then check whether the number of legs is correct. For example, if a student guesses that there are 8 horses then there must be 6 ducks to get 14 heads. They can draw these animals, or just calculate that there are too many legs. Making a drawing or diagram is a strategy that will be used by children right through to secondary school, so it is a useful strategy to learn. Encourage students to record their results in a table. Discuss how students can improve their guess now, as they know that the number of horses must be less than 8.
| Number of horses | Number of ducks | Check 14 heads | Number of horse legs | Number of duck legs | Total number of legs | Comment |
| 8 | 6 | 8 + 6 = 14 √ | 32 | 12 | 44 | Too many legs so need fewer horses. |
A final table might look like this:
| Number of horses | Number of ducks | Check 14 heads | Number of horse legs | Number of duck legs | Total number of legs | Comment |
| 8 | 6 | 14 √ | 32 | 12 | 44 | Too many legs so need fewer horses. |
| 5 | 9 | 14 √ | 20 | 18 | 38 | Not enough legs, so need more horses |
| 6 | 8 | 14 √ | 24 | 16 | 40 | Right! |
At each stage discuss how the table is being used to organise the data to systematically work towards the answer. Discuss how it is possible to improve the guess as more information is available to narrow down the possibilities. Finally students need to answer the problem and say that there are 8 ducks and 6 horses. Always encourage students to check that they have answered the actual problem.
Another systematic way is to start with one horse, then two, then three and so on. This is certainly systematic, but not often an efficient method for solving the problem. It is good to encourage students to think about what a reasonable guess might be for the first number, and then work from there.
Activity 3: Additional problems
Some additional problems to practise use of working systematically when using guess-check-improve are given below. Many variations can easily be constructed.
- I thought of a number, added 12, then added my original number again, and the answer was 50. What was my number? Ans: 19
- Some galahs were sitting in a tree. Half of them flew away, and then another 5 flew away. Then 10 more arrived, and there were 16 galahs in the tree. How many were there to start with? (Ans: 22)
- A number multiplied by 42 gives 798. What is the number? (Ans: 19)
- On a farm there are ducks and horses. Altogether the ducks and horses have 108 legs and 36 heads. How many ducks are there on the farm? (Ans: 18 horses & 18 ducks)
- A farmer has some ducks and some horses. Altogether the ducks and horses have 106 legs and 39 heads. How many ducks and horses are there on the farm? (Ans: 14 horses & 25 ducks)
- Some birds and spiders are in a shed. Altogether they have 64 legs and 17 heads. How many spiders are there? (Ans: 5 spiders & 12 birds)
|
There are some important principles to help students develop when using guess-check-improve:
|
|---|