There are lots of times you are surprised as a teacher. Today I did some NAPLAN revision of decimal numbers with my year 9 class. It really surprised me how difficult students find the concept of decimal numbers.
Here's something to try with your child.
Draw a number line and place 4.5 at one end and 4.6 at the other.
Place a marker in the middle and ask your child what number would go there.
The answer is 4.55 and many students may get this right, but many would not be 100% sure.
Split the number line again so that it is now in four equal sections. Ask your student to label the new sections.
You may get a wide variety of answers and weird looks.
The answer is 4.5, 4.525, 4.55, 4.575 and 4.6
If your child cannot do this they are not alone. Try again using whole numbers and break it into ten equal sections. Try asking for points between intervals.
Errors like these indicate an issue with both division and place value. It can easily be remedied with some place value exercises (to check if they understand that 4.6 is bigger than 4.59), some estimation exercises (to check if their answers are feasible/reasonable), determining how to find the width of set intervals (using division), learning how to add on intervals and how to find midpoints of intervals.
Monday, February 7, 2011
Friday, February 4, 2011
Fractions and year 10
We're reviewing fractions and my academic 10's sheepishly owned up to not being confident at fractions. The issue was traced back to poor tables (without it students get hopelessly stuck with LCD methods).
PARENTS NOTE: TEACH YOUR CHILDREN TABLES.
I'm shouting because it's seemingly not PC to rote learn anything. It is hard to get this message heard. People are too busy to do the little things. Curriculum is too full to teach tables in lower school (nonsense), parents are working multiple jobs and don't have time (you can't afford to not find the time), students are too lazy (they have always been too lazy, this hasn't changed), students have little discipline. We are setting students up to fail if we don't take minimum effort to assist them learn key content.
Anyhow, the second element of students not knowing fractions is a lack of actual teaching of what fractions are and how they work. After 60 mins of learning time they could add subtract and multiply fractions and there were a lot of happier students in the room. Here's the method I used.
I started by drawing two objects, one in halves, one with two quarters (colouring in the selected parts) and described fractions as a way of describing the proportion of an object selected. Both objects were the same size and were split into equal parts. I wrote 1/2 and 2/4 (vertically) next to the objects and discussed numerators were the parts selected and denominators were the number of equal parts in each object
I then asked students what would happen if I added the two objects. Students responded that I would have a whole of an object. This was good as it indicated that they had some understanding of a fraction. We discussed how we would expect 2/2 and 4/4 for a whole.
I then added the numerators and denominators and students could see that this was wrong (3/6). I drew what 3/6 would look like.
I then split the 1/2 into quarters and relabelled the 1/2 object 2/4. We talked about equivalent fractions and lowest common multiples at some length.
I then added the numerator and denominators again. This time we had 4/6. I drew this. It was still wrong. Students pointed out not to add the denominators. We noted that adding denominators made no sense as the denominator described the number of parts. Good! We now had 4/4.
We then talked about multiplication. They were happy to accept that to multiply fractions, multiply the numerators and multiply the denominators.
Now we discussed the effect of multiplying by one, how 2/2, 3/3, 4/4 was really one; and used this fact and multiplication to construct equivalent fractions. I pointed out that without tables it was difficult to find lowest common multiples or factors (for denominators) and that simplifying large fractions was a poor alternative for knowing multiples and factors. We then looked back at the cross multiplication method that many had been taught and how that aligned with what we were doing.
Students completed 60 questions of increasing difficulty. All completed working and checked their own answers. Note that there was no "fractions" specific method (such as cross multiplication and lowest common denominator) used here. It simply flowed from their own mathematical understandings.
Finally we discussed that order was important with subtraction. Division was left for another lesson. Formal notes were then given. 60 mins. Happy faces. Job done. Tick.
I'm not saying that this would work with students that have no understanding of fractions. I am saying that proper consolidation of teaching done in upper primary and lower secondary is not difficult with average students and this topic.
The trick will be to consolidate this in algebra, indices and trigonometry topics so that key concepts are not lost in future.
Russ.
PARENTS NOTE: TEACH YOUR CHILDREN TABLES.
I'm shouting because it's seemingly not PC to rote learn anything. It is hard to get this message heard. People are too busy to do the little things. Curriculum is too full to teach tables in lower school (nonsense), parents are working multiple jobs and don't have time (you can't afford to not find the time), students are too lazy (they have always been too lazy, this hasn't changed), students have little discipline. We are setting students up to fail if we don't take minimum effort to assist them learn key content.
Anyhow, the second element of students not knowing fractions is a lack of actual teaching of what fractions are and how they work. After 60 mins of learning time they could add subtract and multiply fractions and there were a lot of happier students in the room. Here's the method I used.
I started by drawing two objects, one in halves, one with two quarters (colouring in the selected parts) and described fractions as a way of describing the proportion of an object selected. Both objects were the same size and were split into equal parts. I wrote 1/2 and 2/4 (vertically) next to the objects and discussed numerators were the parts selected and denominators were the number of equal parts in each object
I then asked students what would happen if I added the two objects. Students responded that I would have a whole of an object. This was good as it indicated that they had some understanding of a fraction. We discussed how we would expect 2/2 and 4/4 for a whole.
I then added the numerators and denominators and students could see that this was wrong (3/6). I drew what 3/6 would look like.
I then split the 1/2 into quarters and relabelled the 1/2 object 2/4. We talked about equivalent fractions and lowest common multiples at some length.
I then added the numerator and denominators again. This time we had 4/6. I drew this. It was still wrong. Students pointed out not to add the denominators. We noted that adding denominators made no sense as the denominator described the number of parts. Good! We now had 4/4.
We then talked about multiplication. They were happy to accept that to multiply fractions, multiply the numerators and multiply the denominators.
Now we discussed the effect of multiplying by one, how 2/2, 3/3, 4/4 was really one; and used this fact and multiplication to construct equivalent fractions. I pointed out that without tables it was difficult to find lowest common multiples or factors (for denominators) and that simplifying large fractions was a poor alternative for knowing multiples and factors. We then looked back at the cross multiplication method that many had been taught and how that aligned with what we were doing.
Students completed 60 questions of increasing difficulty. All completed working and checked their own answers. Note that there was no "fractions" specific method (such as cross multiplication and lowest common denominator) used here. It simply flowed from their own mathematical understandings.
Finally we discussed that order was important with subtraction. Division was left for another lesson. Formal notes were then given. 60 mins. Happy faces. Job done. Tick.
I'm not saying that this would work with students that have no understanding of fractions. I am saying that proper consolidation of teaching done in upper primary and lower secondary is not difficult with average students and this topic.
The trick will be to consolidate this in algebra, indices and trigonometry topics so that key concepts are not lost in future.
Russ.
Labels:
fractions
Attacking a subject
I always tell my students to attack a subject and it worries me when I get a class of passive students - especially in stage three courses.
Students that are attacking a course:
a) come in bright eyed and bushy tailed
b) are on time
c) have all of their resources (books, calculators, pens ...) ready on day 1
d) attend regularly
e) have pre-read the chapters
f) get stuck into their coursework and are not afraid to have a go
g) natter about their current question with other students
Students that wait to be prompted and expect to be spoonfed, wait for you to find that they are stuck and look like deers in headlights make me concerned. Students that seek personal information from the teacher, natter during instruction, dawdle in late, are disrupting the whole class with nonsense annoy me. They make me think "Is this student in the right place?". This is after all senior school, the pointy end of education.
My 9's, 10's and 2C course are going gangbusters. They demand notes on everything. They attempt questions that I haven't asked them to do as well as the ones I have. They are working on revision books. They are playing with their calculators. Good for them.
My 1B's and 3A courses are another story. Where's the ego? Where's the work ethic? Where is the focus? Hopefullly they're more awake next lesson.
Students that are attacking a course:
a) come in bright eyed and bushy tailed
b) are on time
c) have all of their resources (books, calculators, pens ...) ready on day 1
d) attend regularly
e) have pre-read the chapters
f) get stuck into their coursework and are not afraid to have a go
g) natter about their current question with other students
Students that wait to be prompted and expect to be spoonfed, wait for you to find that they are stuck and look like deers in headlights make me concerned. Students that seek personal information from the teacher, natter during instruction, dawdle in late, are disrupting the whole class with nonsense annoy me. They make me think "Is this student in the right place?". This is after all senior school, the pointy end of education.
My 9's, 10's and 2C course are going gangbusters. They demand notes on everything. They attempt questions that I haven't asked them to do as well as the ones I have. They are working on revision books. They are playing with their calculators. Good for them.
My 1B's and 3A courses are another story. Where's the ego? Where's the work ethic? Where is the focus? Hopefullly they're more awake next lesson.
Labels:
engaging students
Wednesday, February 2, 2011
Multiplying and dividing by powers of 10
I had my academic year nine class for the first time today and had a lot of fun. I had been warned about a few students, but they were arms deep in the trenches having a good go.
I took an experimental approach today with the class examining how to multiply and divide powers of 10. The idea was to create algorithms in student terms for solving simple equations.
We started with simple examples using whole numbers
5 x 10 = ...
"When multiplying by 10, 100, 1000... count the zeroes and put them on the end of the number you are multiplying."
25 ÷ 10 = ...
This time students considered the position of the decimal point:
"Count the zeroes in the number after the division symbol [divisor] and move the decimal point right of the other number [the dividend] that many digits"
We then looked at the case
2.5 x 10 =
and discovered our first algorithm didn't work as by our first algorithm 2.5 x 10 = 2.50
This lead us to a similar algorithm as for our division case:
"Count the zeroes in the multiplier and move the decimal point left of the other number [the factor] that many digits"
Using the whole number cases gave students an additional method of multiplying powers of ten than the messy loops moving decimal places method. The idea of this lesson was not to deny them mathematical language - but to give them an opportunity to explore a mathematical concept before formal language was introduced. It was a lot of fun for me and engaged them during the lesson.
We then looked at a few cases where the multiplier and divisor were not powers of 10. This exposed that students had difficulty with long division and long multiplication and were over dependent on calculators - which has the possibility of causing issues in non-calculator sections in upper school. We'll now go own to examining factor trees and ease into indices.
We also looked at 250 ÷ 10 where we converted the expression to a fraction and cancelled the zeroes - although we didn't consider why this works and will need to revisit it later.
I took an experimental approach today with the class examining how to multiply and divide powers of 10. The idea was to create algorithms in student terms for solving simple equations.
We started with simple examples using whole numbers
5 x 10 = ...
"When multiplying by 10, 100, 1000... count the zeroes and put them on the end of the number you are multiplying."
25 ÷ 10 = ...
This time students considered the position of the decimal point:
"Count the zeroes in the number after the division symbol [divisor] and move the decimal point right of the other number [the dividend] that many digits"
We then looked at the case
2.5 x 10 =
and discovered our first algorithm didn't work as by our first algorithm 2.5 x 10 = 2.50
This lead us to a similar algorithm as for our division case:
"Count the zeroes in the multiplier and move the decimal point left of the other number [the factor] that many digits"
Using the whole number cases gave students an additional method of multiplying powers of ten than the messy loops moving decimal places method. The idea of this lesson was not to deny them mathematical language - but to give them an opportunity to explore a mathematical concept before formal language was introduced. It was a lot of fun for me and engaged them during the lesson.
We then looked at a few cases where the multiplier and divisor were not powers of 10. This exposed that students had difficulty with long division and long multiplication and were over dependent on calculators - which has the possibility of causing issues in non-calculator sections in upper school. We'll now go own to examining factor trees and ease into indices.
We also looked at 250 ÷ 10 where we converted the expression to a fraction and cancelled the zeroes - although we didn't consider why this works and will need to revisit it later.
Labels:
teaching methods
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