Michael Drake MMP Day 2008 Teaching algebra for meaning.
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Transcript of Michael Drake MMP Day 2008 Teaching algebra for meaning.
![Page 1: Michael Drake MMP Day 2008 Teaching algebra for meaning.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649e835503460f94b84e18/html5/thumbnails/1.jpg)
Michael Drake
MMP Day 2008
Teaching algebra for meaning
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Algebra
• What is algebra?
Think…
Discuss in pairs…
• If you ask parents (or Y11 students), what would they say?
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How do we use letters in mathematics?
• Have a chat about it…
• Tell me what you have come up with…
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How do we use letters in mathematics?
1) A letter can be used to name something
• In the formula for the area of a rectangle, the base of the rectangle is often named b
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The development of formulae
Area = base × heightA = b × hA = bh
3 cm
5 cm A = 5 × 3
A = 15cm2
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How do we use letters in mathematics?
2) A letter can be used as a variable
• This happens in patterning
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Developing patterning
• Write the next three terms of the sequence 3, 6, 9, 12, …
1 3
2 6
3 9
4 12
Down rule
Across rule
Write a sentence explaining how to get from one term to the next. Start with “To get the new term …”
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• Write a sentence for the across rule. Start it with “the number of matches is found by taking the number of triangles and…”
1 3
2 6
3 9
4 12
Number of triangles
Number of matches
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How many terms?• What is the next term of this
sequence?1, 2, _
• What is the next term of this sequence?
1, 2, 4, _
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Differentiating…• Write the next four terms of the
sequence 10 507, 10 627, 10 747, 10 867,
• Write a sentence that explains how to get from one term in the sequence to the next
• Can we write this more briefly, and still know what we mean?
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How do we use letters in mathematics?
3) A letter can be used to stand for a specific unknown number that needs to be foundIn a triangle, x is often used for the angle students need to find
• This happens when we solve equations
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When do we need to use letters to solve problems?
• Jemima has some sweets. She eats 5 and has 1 left. How many did she start out with?
•Rewi has 47 super 14 rugby cards. 19 of them are repeats that he can swap. How many different cards does he have?
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• In numeracy schools, student may have seen an algebraic form of recording problems since Year 1…
6 + = 10
…and may have even learned to record problems in this format
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How do we use letters in mathematics?
4) A letter can be used as a variable that can take a variety of possible values
(This also happens with equations)
+ = 10
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How do we use letters in mathematics?
5) A letter is really a number
c = speed of light
e
πOr if you are a student, in substitution a = 1…
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How do we use letters in mathematics?
6) In the equation y = 2x + 1, x can be any number, a whole number, a fraction, integer or decimal
7) In y = mx +c, m and c act as parameters
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Lessons
• Letters mean different things in different situations
• Algebra is not something that should be taught in a single block…
•Algebra often makes sense when it arises from a context…
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The algebra domains
Patterns and
relationships
Formula developmen
t
EquationsNumber property
generalisation
Introduction to the conventions of algebra
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•Time to do some mathematics…
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• Think of a numberStarters
• Add 3 to it
• Double it
• Subtract two
• Halve it
• Subtract the number you started with• Add 6
• Multiply the number by 3
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Think of a number…• How do we write down that we are
thinking of a number – without giving away the number we are thinking of?
•What would one more than the number look like?
•What would 3 less than the number look like?
•What would double the number look like?
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What is?
Starters
3 3
52
6 6
7 7
102
2 4
5 7
6 4
8 6
9 11
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Generally generalising
• 6 + 6 + 6 + 6 + 6 + 6 + 6 = 7 6
• How can we show that this always works – no matter the number?
• Give me another example
• Explain why the sentence is true
Thinking time: discuss in pairs
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• Did we always have to use 7 of them?
• What other generalisations could we have made?
• How can we show this always works – regardless of the number of things we are multiplying?
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The development of algebraic thinking
Using and exploring number
properties
Generalising number properties
Generalising algebra properties
First order abstraction
Second order abstraction
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• What are your answers now?
What is algebra?
• Generalised number
A way of describing trends and patterns we find in sets of similar problems
• The structure of the number system
A way of looking behind (underneath) a set of number problems to see what is really going on
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Write me some…
• Write me a story problem which has the answer x = 4. Try using a context that tells us something about your culture or your family
Sharing time…
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• So how does this algebra work?
•Why does it help us to work out what is going on?
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Teaching volume
• How do we each teach volume?
• How might these resources be useful?
• How can we use learning about solids as a context for algebra development?
•What prior knowledge is needed for this approach to be useful?
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What is 4 4?
• What about 16 16?
• 250 250?• Does this always work?
• Can you write this as a rule in words?
• Explain why this works using a drawing or some equipment from the box
• Write a sentence with symbols to show your rule