Ula lole
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Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about estimation.
Task administration:
This task can be completed with pencil and paper or online.
This assessment activity is designed to be carried out with a group of students or a whole class. A suggested lesson sequence:
- Give the students the problem of estimating 34 × 46 to identify whether they use the front-end method in multiplication.
- Discuss students' methods. Use the language of estimation in the Computational estimation concept map for each different method.
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Introduce the front-end strategy in multiplication, i.e., use just the first digit of each number. You may wish to write this down as you discuss it.
Example: Estimate 34 × 46
The front-end of 34 is 30 and the front-end of 46 is 40
The answer is at least 30 × 40 = 1 200 (or 3 × 4 = 12, so it's about 1 200) - Students complete the sentence with the word "front-end", and then answer parts b) to d) to assess if they can apply it. Emphasise they are to use the front-end method only (i.e., without trying to adjust the estimate to make it more accurate).
For more information about discussions see Classroom discourse (Mathematics)
Level:
4
Curriculum info:
Key Competencies:
Keywords:
Description of task:
Students estimate a multiplication problem, share their methods, discuss the front-end method, and then do more problems using front-end.
Curriculum Links:
This resource can help to identify students' ability to apply multiplicative strategies flexibly to whole numbers when estimating multiplication problems.
Key competencies
This resource involves recording the strategies students use to estimate in multiplication problems. This relates to the Key Competency: Using language, symbols and text.
For more information see https://nzcurriculum.tki.org.nz/Key-competencies.
Learning Progression Frameworks
This resource can provide evidence of learning associated with within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.Answers/responses:
|
a) |
Makes a reasonable estimate close to 1,200 and gives a reasonable explanation of their method of estimating it. The front-end method would be: |
|
b) |
160 or 200 |
|
c) |
600 |
|
d) |
20,000 or 25,800 |
NOTE:
- Other strategies are possible. If they don't use the front-end methods, conference them to see if they can, even if they prefer other strategies.
- Students may use a combination of estimation and computation (i.e., computations that they cannot easily do in their heads and may need to write down).
- Do not accept the exact answers unless they have first made an estimate using front-end and then adjusted it using compensation. It is possible to estimate and then compensate to get an exact answer, especially in b) (e.g., 23 × 8 20 × 8 = 160. This is 3 × 8 too small so it's 184! This shows estimation thinking as long as it is all done mentally).
Diagnostic and formative information:
| Common error | Likely calculation | Likely misconception | |
|
a) b) c) d) |
1,564 184 962 33,282 |
34 × 46 23 × 8 37 × 26 258 × 129 |
Performs exact calculation rather than estimation. |
|
c) d) |
1,200× 30,000× |
40 × 30 300 × 100 |
Uses rounding rather than front-end. |
|
c) d) |
60 or 6,000 3 000 or 300,000 |
Result out by an order of magnitude (10 times too big or too small). Does not process the number of 0's in the estimate. |
Information on front-end estimation
- Front-end always underestimates the true results because both the numbers being multiplied are smaller than the numbers in the original product.
- Final-end uses just the first digit of each number.
- Post-compensation (i.e., updating the estimate to a more accurate one) is easier to do with the front-end method than with rounding because it always involves adding on extra amounts to compensate.
Prior knowledge needed
- A firm base of multiplicative basic facts.
- Good place value concepts.
- Understanding that estimation is a quick mental way of getting an answer close to the exact one.
- Ability to evaluate the order of magnitude of products (number of trailing 0's).
For similar ARB resources about front-end estimation refer to:
- Estimating sums of money (front-end with addition);
- Estimating team scores (front-end with compensation, to give closer estimates by making adjustments using the second most significant figures).
For more information about computational estimation see the Computational estimation concept map.
Links with Numeracy Project.
Using front-end estimation with multiplication involves advanced additive/early multiplicative part-whole (Stage 6) knowledge and strategies (see The Number Framework, Book 1 (2004, p. 10 & 14). Wellington: Ministry of Education).