Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’...
Transcript of Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’...
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Strand 3: Number
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• Half a cup of tea• Quarter of an hour• Three quarters full
Fractions are Easy!
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00000 =2 4 of ?3 5
Point to Ponder!
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Chief Examiners’ Reports
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LC HL - 2000
"Candidates answering showed weaknesses in the following specific areas: handling fractions"
Chief Examiners’ Reports
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LC HL - 2000
"Candidates answering showed weaknesses in the following specific areas: handling fractions"
Chief Examiners’ Reports
LC HL - 2005
“Incorrect cancelling in algebraic fractions.”
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Chief Examiners’ Reports
LC HL - 2005
“Incorrect cancelling in algebraic fractions.”
![Page 8: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/8.jpg)
Chief Examiners’ Reports
LC HL - 2005
“Incorrect cancelling in algebraic fractions.”LC FL - 2005
“Fractions again caused problems for many.” “..it would appear that fractions are not well understood.”
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Chief Examiners’ Reports
LC FL - 2005
“Fractions again caused problems for many.” “..it would appear that fractions are not well understood.”
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Chief Examiners’ Reports
LC FL - 2005
“Fractions again caused problems for many.” “..it would appear that fractions are not well understood.”
JC OL - 2006
“Common errors were: mishandling either fraction or square root; adding before multiplying.”
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Chief Examiners’ Reports
JC OL - 2006
“Common errors were: mishandling either fraction or square root; adding before multiplying.”
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Chief Examiners’ Reports
JC OL - 2006
“Common errors were: mishandling either fraction or square root; adding before multiplying.”
JC HL - 2006
“Areas of weakness in candidate performance:simplifying algebraic fractions.”
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Chief Examiners’ Reports
JC HL - 2006
“Areas of weakness in candidate performance:simplifying algebraic fractions.”
![Page 14: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/14.jpg)
Spoken Symbols
ManipulativeModels
Real Life Situations
Written Symbols
Pictures
Lesh’s Translation Model
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Spoken Symbols
ManipulativeModels
Real Life Situations
Written Symbols
Pictures
Lesh’s Translation Model
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Spoken Symbols
ManipulativeModels
Real Life Situations
Written Symbols
Pictures
Lesh’s Translation Model
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Spoken Symbols
ManipulativeModels
Real Life Situations
Written Symbols
Pictures
Lesh’s Translation Model
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Spoken Symbols
ManipulativeModels
Real Life Situations
Written Symbols
Pictures
Lesh’s Translation Model
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Spoken Symbols
ManipulativeModels
Real Life Situations
Written Symbols
Pictures
Lesh’s Translation Model
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Spoken Symbols
ManipulativeModels
Real Life Situations
Written Symbols
Pictures
Lesh’s Translation Model
![Page 21: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/21.jpg)
Spoken Symbols
ManipulativeModels
Real Life Situations
Written Symbols
Pictures
Lesh’s Translation Model
![Page 22: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/22.jpg)
Spoken Symbols
ManipulativeModels
Real Life Situations
Written Symbols
Pictures
Lesh’s Translation Model
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Cara has 4 pizzas for her party.3She decides that a serving will be of a pizza.5
2 2Will she get 6 or 6 servings from the 4 pizzas?3 5
Unifix Cubes
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Cara has 4 pizzas for her party.3She decides that a serving will be of a pizza.5
2 2Will she get 6 or 6 servings from the 4 pizzas?3 5
Unifix Cubes
![Page 25: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/25.jpg)
Spoken Symbols
ManipulativeModels
Real Life Situations
Written Symbols
Pictures
Lesh’s Translation Model
![Page 27: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/27.jpg)
Fraction Strips & Fraction Circles
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Spoken Symbols
ManipulativeModels
Real Life Situations
Written Symbols
Pictures
Lesh’s Translation Model
![Page 29: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/29.jpg)
Prior knowledge
Overview of Fractions
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Diagnostic Test
Prior knowledge
Overview of Fractions
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00000
3Which of these rectangles has shaded in? 4
Is it more than one rectangle?
A. B. C. D.
Answer:
Diagnostic Test
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00000
What fraction of the fraction circle is marked ‘f’?
a
bc
d
e
f
Diagnostic Test
Answer:
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00000
(a) (b) (c) (d)
(a) (b) (c) (d)
The shaded part of this diagram could represent the numbers:1 5 15 2 12 8 4
Identify the unit in each case by drawing:
Diagnostic Test
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Concepts
Operations
Connections
Prior knowledge
Overview of Fractions
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Diagnostic Test
Concepts
Operations
Connections
Prior knowledge
Overview of Fractions
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Partitioning
Diagnostic Test
Concepts
Operations
Connections
Prior knowledge
Overview of Fractions
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Ordering
Partitioning
Diagnostic Test
Concepts
Operations
Connections
Prior knowledge
Overview of Fractions
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Equivalence
Ordering
Partitioning
Diagnostic Test
Concepts
Operations
Connections
Prior knowledge
Overview of Fractions
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Add Subtract
Equivalence
Ordering
Partitioning
Diagnostic Test
Concepts
Operations
Connections
Prior knowledge
Overview of Fractions
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Multiply & Divide
Add Subtract
Equivalence
Ordering
Partitioning
Diagnostic Test
Concepts
Operations
Connections
Prior knowledge
Overview of Fractions
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Decimals
Multiply & Divide
Add Subtract
Equivalence
Ordering
Partitioning
Diagnostic Test
Concepts
Operations
Connections
Prior knowledge
Overview of Fractions
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Percentages
Decimals
Multiply & Divide
Add Subtract
Equivalence
Ordering
Partitioning
Diagnostic Test
Concepts
Operations
Connections
Prior knowledge
Overview of Fractions
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Ratio
Percentages
Decimals
Multiply & Divide
Add Subtract
Equivalence
Ordering
Partitioning
Diagnostic Test
Concepts
Operations
Connections
Prior knowledge
Overview of Fractions
![Page 44: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/44.jpg)
Ratio
Percentages
Decimals
Multiply & Divide
Add Subtract
Equivalence
Ordering
Partitioning
Diagnostic Test
Concepts
Operations
Connections
Prior knowledge
Overview of Fractions
AlgebraStrand 4
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• Partitioning
• Ordering
• Equivalence
Reasons for rules ~Checking tools
Concepts
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00000
3Is always the same?4
Concepts
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00000
3Is always the same?4
Only if the unit is the same.
Concepts
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00000
3Is always the same?4
Only if the unit is the same.
Concepts
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00000
3Is always the same?4
Only if the unit is the same.
Concepts
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3 5Which is bigger: or ?7 7
2 2Which is bigger: or ?3 5
5 3Which is bi
5 1 becau
gger:
se there are
or ?9 7
more 7 7
1 1 2 23 5 3 5
5 1 3 1 5 3 and 9 2 7 2 9 7
> ⇒ >
> < ⇒ >
Concepts
![Page 51: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/51.jpg)
3 5Which is bigger: or ?7 7
2 2Which is bigger: or ?3 5
5 3Which is bi
5 1 becau
gger:
se there are
or ?9 7
more 7 7
1 1 2 23 5 3 5
5 1 3 1 5 3 and 9 2 7 2 9 7
> ⇒ >
> < ⇒ >
Concepts
![Page 52: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/52.jpg)
3 5Which is bigger: or ?7 7
2 2Which is bigger: or ?3 5
5 3Which is bi
5 1 becau
gger:
se there are
or ?9 7
more 7 7
1 1 2 23 5 3 5
5 1 3 1 5 3 and 9 2 7 2 9 7
> ⇒ >
> < ⇒ >
Concepts
![Page 53: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/53.jpg)
3 5Which is bigger: or ?7 7
2 2Which is bigger: or ?3 5
5 3Which is bi
5 1 becau
gger:
se there are
or ?9 7
more 7 7
1 1 2 23 5 3 5
5 1 3 1 5 3 and 9 2 7 2 9 7
> ⇒ >
> < ⇒ >
Concepts
![Page 54: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/54.jpg)
3 5Which is bigger: or ?7 7
2 2Which is bigger: or ?3 5
5 3Which is bi
5 1 becau
gger:
se there are
or ?9 7
more 7 7
1 1 2 23 5 3 5
5 1 3 1 5 3 and 9 2 7 2 9 7
> ⇒ >
> < ⇒ >
Concepts
![Page 55: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/55.jpg)
3 5Which is bigger: or ?7 7
2 2Which is bigger: or ?3 5
5 3Which is bi
5 1 becau
gger:
se there are
or ?9 7
more 7 7
1 1 2 23 5 3 5
5 1 3 1 5 3 and 9 2 7 2 9 7
> ⇒ >
> < ⇒ >
Concepts
![Page 56: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/56.jpg)
Concepts
3 2Which is bigger: or ?5 3
=
=
> ⇒ >
3 6 1If we double , we get 15 5 52 4 1If we double , we get 13 3 3
1 1 2 3 .3 5 3 5
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Whole Number Multiplication
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Whole Number Multiplication
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Liz wants to give each of her 3 friends 4 bars of chocolate. How would you work out how many bars she needs?
Whole Number Multiplication
![Page 60: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/60.jpg)
Liz wants to give each of her 3 friends 4 bars of chocolate. How would you work out how many bars she needs?
Whole Number Multiplication
![Page 61: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/61.jpg)
Liz wants to give each of her 3 friends 4 bars of chocolate. How would you work out how many bars she needs?
Whole Number Multiplication
![Page 62: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/62.jpg)
Liz wants to give each of her 3 friends 4 bars of chocolate. How would you work out how many bars she needs?
Whole Number Multiplication
![Page 63: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/63.jpg)
Liz wants to give each of her 3 friends 4 bars of chocolate. How would you work out how many bars she needs?
• 3 multiplied by 4 or 3 times 4
Whole Number Multiplication
![Page 64: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/64.jpg)
Liz wants to give each of her 3 friends 4 bars of chocolate. How would you work out how many bars she needs?
• 3 multiplied by 4 or 3 times 4
• 4 + 4 + 4 (Repeated Addition)
Whole Number Multiplication
![Page 65: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/65.jpg)
Liz wants to give each of her 3 friends 4 bars of chocolate. How would you work out how many bars she needs?
• 3 multiplied by 4 or 3 times 4
• 4 + 4 + 4 (Repeated Addition)
• 3 groups of 4
Whole Number Multiplication
![Page 66: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/66.jpg)
Multiplying a Whole Number by a Fraction
Barry is having 4 of his friends over to his house for pizza. 2He is going to give them of a pizza each.3
How is this like the last problem? Draw a picture to model this situation.
(a)(b)
(c) 2If you have “4 groups of ", how many ‘one thirds’ do you have?3
![Page 67: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/67.jpg)
Multiplying a Whole Number by a Fraction
Barry is having 4 of his friends over to his house for pizza. 2He is going to give them of a pizza each.3
How is this like the last problem? Draw a picture to model this situation.
(a)(b)
(c) 2If you have “4 groups of ", how many ‘one thirds’ do you have?3
![Page 68: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/68.jpg)
Multiplying a Whole Number by a Fraction
Barry is having 4 of his friends over to his house for pizza. 2He is going to give them of a pizza each.3
How is this like the last problem? Draw a picture to model this situation.
(a)(b)
(c) 2If you have “4 groups of ", how many ‘one thirds’ do you have?3
![Page 69: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/69.jpg)
Multiplying a Whole Number by a Fraction
Barry is having 4 of his friends over to his house for pizza. 2He is going to give them of a pizza each.3
How is this like the last problem? Draw a picture to model this situation.
(a)(b)
(c) 2If you have “4 groups of ", how many ‘one thirds’ do you have?3
![Page 70: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/70.jpg)
Multiplying a Whole Number by a Fraction
Barry is having 4 of his friends over to his house for pizza. 2He is going to give them of a pizza each.3
How is this like the last problem? Draw a picture to model this situation.
(a)(b)
(c) 2If you have “4 groups of ", how many ‘one thirds’ do you have?3
![Page 71: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/71.jpg)
Picture/Words
Multiplication sentence
...groups of...
Repeated addition
Poster/ White Boards
![Page 72: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/72.jpg)
00000If I multiply what incorrect answer do you think I might get?
Pair Work
×243
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00000
Pair Work
4 2 8 1 3 3How come 4 2 24 ?1 3 3
× = + =
and
![Page 74: Strand 3: Number · 2018. 9. 3. · simplifying . algebraic fractions.” Chief Examiners’ Reports. ... Equivalence. Ordering . Partitioning . Diagnostic Test. Concepts. Operations.](https://reader036.fdocuments.us/reader036/viewer/2022071218/604eecabeaa7cb65e846e79e/html5/thumbnails/74.jpg)
00000
Fraction x Fraction
2Cara has of her rectangular birthday cake left 5
over from her party. 3She ate of the leftover cake.4
How much of the full cake was this?