Help! (sum of three numbers)

[TwoDownTwoToGo]

I have tutored two of my children the DIY tutoring method. Both are now in GS's. About a month before the exams, I taught my kids (respectively) how to employ algebra in questions with 'unknown' quantities, just like the question that started this thread.

Once they grasped the concept ie lets call the three numbers a, b and c, they were able to understand the question as though it was represented in a diagram, rather than a confusing 'wordy' question!

It may or may not be necessary to use algebra in some cases. However, both my children resorted to using the 'algebra' method in every 'unknown' quantity type question thereafter.

Noticable results thereafter:
1) Increased confidence in answering wordy questions.
2) Their percentage marks in past papers shot up
3) Both were offered OOC GS places

I am now working with DC3 and have already started the 3 apples and 4 bananas is the same as 3a+4b method.

Hope this helps prospective DIY 11+ parents.

[Guest55]

3 apples and 4 bananas is the same as 3a+4b method.

Sorry, as a mathematician this is not an advised approach - what does a^2 mean?!

Algebra is generalised arithmetic not fruit salad.

Children who automatically use algebra often over complicate a question; it is preferable to use algebra only when it is needed.

Extract from paper by Helen Chick

"One of the best known of the misconceptions is the “letter as object” misconception, described by Küchemann (1981), in which the letter, rather than clearly being a place-holder for a number, is regarded as being an object (the abbreviated word misconception is such an example). The term “fruit salad algebra” is sometimes used for this misconception, infamously presented in examples such as “a is apples and b is bananas, and so 3a + 2b is like 3 apples and 2 bananas, and since you can’t add apples and bananas we just write it as 3a + 2b.” One difficulty here is that 3a in algebra does not represent 3 apples, but three times an unknown number. The second difficulty here concerns the mathematical idea of closure: in saying we cannot add apples and bananas we contradict the fact that 3a + 2b is the sum."

[mystery]

Yes, I could never understand why some teachers at my senior school talked about apples and bananas when the letter represented a number. They were graduate mathematicians but some had better explanations than others.

[Proud_Dad]

A typical question would more likely talk about working out the cost of apples and bananas.

e.g. 3 apples and 4 bananas cost £1.20

In which case it is perfectly reasonable to write 3a + 4b = 120,

where a is the cost of one apple and b is the cost of 1 banana.

[Guest55]

Indeed - and here the variable represents an amount of money ie a value not an 'apple' or a 'banana'.

I encourage students to define variables so this is reinforced.

[equilibro]

The term “fruit salad algebra” is sometimes used for this misconception

Who knows, it could even be the source of the humorous "Time flies like an arrow but fruit flies like a banana"

In truth, although this quote is often erroneously attributed to the late, great, Groucho Marx, it apparently originates in the work of an early researcher in artificial intelligence, Anthony Oettinger, who was trying to get a computer to manipulate the English language. In the magazine Scientific American in 1966 (I was too busy watching us win the World Cup that year to have read it myself!) it seems that Oettinger described a computer program that takes sentences as input and attempts to determine which words are acting as verbs, nouns, adjectives, and adverbs, etc.

Here is what he said:

"A grammar that pretends to describe English at all accurately must yield a structure for "Time flies like an arrow" in which "time" is the subject of the word "flies" and "like an arrow" is an adverbial phrase modifying the verb. "Time" can also serve attributively, however, as in "time bomb", and "flies" of course can serve as a noun. Together with "like" interpreted as a verb, this yields a structure that becomes obvious only if one thinks of a kind of flies called "time flies", which happen to like an arrow, perhaps as a meal. Moreover, "time" as an imperative verb with "flies" as a noun also yields a structure that makes sense as an order to someone to take out his stopwatch and time flies with great dispatch, or like an arrow."

Someone subsequently juxtaposed the two sentences in question to form the quoted phrase, and it is possible that Marx said the words but there is no direct evidence of this.

Reference: http://quoteinvestigator.com/2010/05/04 ... ies-arrow/" onclick="window.open(this.href);return false;

[mystery]

I'm sure you could use this to develop a level 7 spag question.

Plus I think you are a nutty as a fruitbat.

[TwoDownTwoToGo]

I only used apples and bananas as an introduction to the concept that unknowns can be represented as a letter. I did not go beyond that with regards an algebra lesson. Only this morning, my son said to me, I think i'm going to enjoy algebra. I can't wait till you start teachimg me.

I totaly agree with proud_dad, 'In which case it is perfectly reasonable to write 3a + 4b = 120,' where a and b represent numerical values. That would have been leeson number two!!

I too am a secondary mathematics teacher. Many times over I have seen students gaze at a page of algebra as though it was a 'foreign language!' I take the approach of breaking it down gently using the fruit bowl only to expel this fear. Once that is accomplished, they are more willing to listen to the concept of no longer using fruit but accepting, "Hey why don't we now imagine that 'a' is the cost of the apple!!!

But i'd like to add one more point... I have tutored many students who have fallen behind in maths. In every single case, I have had to tread very carefully, intricate layer by layer, until my students have finally gained some confidence in their own ability and eventually gained a love of the subject.

In the same way that G55 says 'Children who automatically use algebra often over complicate a question;', I think my introduction to algebra has also been over analysed.

Apples and oranges has worked for me. My students have ALL gone on to love maths.

[Proud_Dad]

Algebra is a very easy concept and once DCs master it they will be able to apply to various types of questions. They need to know that there are more than one way of solving problems. They have to understand and find what works for them.

I agree. I don't think there should be any "rules" about when to use algebra and when not to. Different methods work best for different people, particularly with worded problems.

There are some people who like thinking in words and real life problems and find algebra too abstract and complicated.

However there are other people who are good at maths but struggle with worded maths problems. If they are able to translate these worded problems into a simple algebraic equation it suddenly makes more sense and they can then solve it. Algebra can often be used to simplify a problem not complicate it.

[moved]

I think with all things in maths it is fine to teach children concepts as long as they are taught well.

I wrote some questions recently that were trialled with year 6 children across 15 schools. Teacher feedback was part of the the trial - more than one teacher told me this question:
32 - 12 x 2 = 8
was poor as I hadn't included the 'necessary' brackets making the question ambiguous and that I should therefore should have allowed 40 as an acceptable answer. The question was deliberately written to test the order of operations from the new national curriculum.

Fortunately, the children seem to have this concept reasonably securely.