Workshop presented at National Numeracy Facilitators Conference February 2008

Teresa Maguire, Jonathan Fisher, Alex Neill February 2008

Workshop presented at

National Numeracy Facilitators Conference

February 2008

Teresa Maguire, Jonathan Fisher and Alex Neill

Unpacking Student Responses for

Teacher Information


OutlineOutline

Supporting teachers with the ARBs (10 min)

Resources with fractions (20 min)

Resources with algebra (20 min)

Other information on resources (25 min)

Discussion (15 min)


Supporting teachers with the ARBsSupporting teachers with the ARBs

How the ARBs can be used to support teachers?

Concept maps

Animation / CD

Next steps booklet

Support material

Teacher information pages


Concept mapsConcept maps

Provide information about the key mathematical ideas involved

Link to relevant ARB resources

Suggest some ideas on the teaching and assessing of that area of mathematics

Are “Living” documents


Concept mapsConcept maps


Animation CD/exeAnimation CD/exe


Next steps bookletNext steps booklet


Support materialsSupport materials


Teacher information pagesTeacher information pages

Task administration

Answers/responses

Calibration easy (60-79.9%)

Diagnostic and formative information

(common wrong answers and misconceptions)

Strategies

Next steps

Links to other resources/information and to concept maps


Eating fractions of pie, pizza and cakeEating fractions of pie, pizza and cake ((NM1251)NM1251)

The questions

Student responses and misconceptions

Strategies

Suggested Next steps

Other resources

Discussion



167 students

Year 6

Nationwide

Range of deciles



Petra ate two-fifths (2/5) of a pizza and Sarah ate one-fifth (1/5). Show how to work out how much pizza they ate altogether.

3/5

(64%, and 66% showed working)



Lima and Paul each had the same sized cake. Lima ate four-fifths (4/5) of his cake and Paul ate three-fifths (3/5) of his cake. Show how to work out how much cake they ate altogether.

7/5 or 1 2/5

(38%, and 52% showed working)



Bill ate one-fifth (1/5) of a whole apple pie. Show how to work out how much pie was left.

4/5 (64%, and 70% showed working)



Andrew started with one and a half pizzas (1 1/2) and ate three-quarters (3/4) of a whole pizza. Show how to work out how much pizza is left.

3/4 (46% - 53% showed working)


Eating fractions of pie, pizza and cakeEating fractions of pie, pizza and cake (Misconceptions)(Misconceptions)

Numbering the Pieces only

3 7 4 3Whole number (top and bottom)

3/10 7/10Other whole number relationship/system

13 17 or 4 8 or 4/10 8/10 or 4/0 8/0Varying referent whole

1/2 or 3/6And the “Size of the pieces”


Eating fractions of pie, pizza and cakeEating fractions of pie, pizza and cake(Next steps)(Next steps)

Partitioning

Part-whole relationships

Referent whole

Whole class discussion

Explanation and justification

Diagrams


Eating fractions of pie, pizza and cakeEating fractions of pie, pizza and cake(Other resources)(Other resources)

Link to other ARB resource (keywords)

Fractional thinking concept map

NEMP

Book 7: Teaching Fractions, Decimals and Percentages, 2006

Figure it out


Balance pans & SolvingBalance pans & Solving simple equations simple equationsAL 7111 & AL7124AL 7111 & AL7124

=

www.nzcer.org.nz/arb


Strategies with ZeroStrategies with Zero

a) i) 25 + 16 – 16 =

ii) 28 + 36 – 36 + 52 – 52 =

iii) 62 + 74 – 62 =

iv) 78 – 44 + 44 =

v) 67 + 23 + 55 – 23 – 67=

b) Explain what you did

Student strategies

Usage and success rates using the additive identity concept

In our sample, 37% of students included at least something in part b) to indicate that they were employing the concept of the additive identity. These accounted for 43% of the strategies students described.

- 13% of students had an explanation including reference to the concept of the additive identity and no calculations were used. These students got an average of 94% of their answers in part a) correct, and maintained their success throughout the five parts of the question.

- 19% applied the additive identity rather than a calculation but were unable to clearly explain what they had done. 85% of these students' answers in part a) were correct, but they were slightly less successful with the harder questions.

- 5% had an explanation that included reference to the additive identity but they also did some calculations. Only 67% of these students' answers in part a) were correct, but they were less successful with the harder questions.

The remaining 63% of students described computational methods, other methods, or gave no explanation in part b). These accounted for 57% of the strategies students described. These students had lower success rates that varied from 0% to 68%, and their success rates generally dropped off on the harder questions.

For more details on the success rates of student strategies, click on the link Appendix of student strategies for AL7123


Strategies with ZeroStrategies with ZeroDescription of strategy in part b) Number of

studentsPercent of strategies1

Percent of students2

Success rate3

Additive identity strategies 1.Explanation includes reference to the additive identity and no calculations used.2.Applies the additive identity rather than a calculation but unable to clearly explain what they had done.3.Explanation includes reference to the additive identity but some calculations are also included.

24

35

9

15%

22%

6%

13%

19%

5%

94%

85%

67%

Computation strategy1.Calculates left to right2.Uses vertical algorithm3."Used my fingers"4.Other calculation5.Incorrect calculation (e.g., ignores some numbers or operators)

4782

234

29%5%1%

14%2%

25%4%1%

12%2%

49%68%50%32%0%

Other explanations 10 6% 5% 54%

TOTAL strategies used 162 100% 86% 62%

No explanation 26 14% 27%

TOTAL 188 100% 100% 57%Based on a representative sample of 188 students


Car maintenance & Post a parcelCar maintenance & Post a parcelAL6156 & AL 7129AL6156 & AL 7129

Exemplars of student responses- Graphs

Multiple representations- Graphs, tables, and equations

Other goodies


Car maintenance Car maintenance (AL6156 – Level 4)(AL6156 – Level 4)

b) Draw a line graph

Incorrect line graphs:

Another type of graph:

Step graph


Diagnostic and formative informationDiagnostic and formative informationCommon error Likely misconception

a) i) ii)

110, 150, 190, 230, 270, 310145, 205, 265, 325, 385, 445

Confuses intercept (cost for a first look) and slope (hourly cost).

Adds on $40 or $60 for each hour of servicing.

a) i) ii)

140, 210, 280, 350, 420, 490170, 265, 350,435, 520, 605

Ignores the intercept (40 or 60) and assumes the total charge for 1 hour equals the slope.

b) Incorrect use of line graphs: joins origin to end point; bend the line to pass through the origin; starting at 1 hour.

Ignores or misinterprets the role of the intercepts ($40 or $70), preferring an intercept of $0.

b) Uses an other type of graph: Histogram Bar graph (often a series of vertical lines) Scatterplot Relationship graph

c) Makes no comparison between the two garages.e.g., "It's cheaper to use the garage at the start because the longer you're

in the garage, the more expensive it is."

c) Honore's, because it has a lower hourly cost.Example: "Honore's Garage is cheapest because for every hour it's only

$25."

Ignores the effect of the lower intercept (set up cost of $40) for Honore's garage.

c) Kakariki's because it is cheapest at the beginning of the table. Ignores the effect of the lower slope (hourly rate of $25) for Honore's garage.

c) Identifies Honare's or Kakariki's as cheapest with no justification.

Based on a representative sample of 201 students.


Car maintenance – Car maintenance – Student ResponsesStudent Responses

joins origin to end point

bend the line starting at 1 hour


Scatterplot

Relationship graphHistogram

Bar graph


Car MaintenanceCar MaintenanceStep graph

This graph could be seen as correct if only complete hours are charged


Post a parcelPost a parcel(AL7129 – Level 5)(AL7129 – Level 5)

Multiple representations

Graphs

Tables

Equations

Two courier companies

a) This graph shows the price Peru.

Show how to use this graph to find the weight of a parcel that both companies would charge the same

.


Post a parcel-Student strategiesPost a parcel-Student strategies

.Graphical interpretation [part a)]Indicating the x- and the y- coordinates of the intersection of the lines (5%).Indicating the x-coordinate only of the intersection of the lines (35%).Indicating the intersection of the lines only (26%). Using graphical interpolation to get a more accurate answer (19%). This was often used in conjunction with one of the other three strategies. About 80% of students who showed interpolation obtained a correct answer, while only about 50% of those whose working did not show interpolation got a correct answer. Many of the latter were satisfied with 2.5 as their answer, even though the break-even point was clearly somewhat less than that.

Table interpretation [part b)]Interpolation of the table between 1 and 2 kg (4%).Using the differences between 1 and 2 kg for the two companies (9%).Averaging the costs at 1 and 2 kg (1%).

Using equations [part c)]Algebraic formulation of the problem (9%).Trial and improvement methods (15%). The included a range of methods including moving sequentially towards the solution in single integers, to jumping several numbers to speed up getting to the answer.


Graphical interpretatonGraphical interpretaton

Indicating the intersection of the lines only

Indicating the x-coordinate only

Indicating the x- and the y- coordinates

Graphical interpolation


Table interpretationTable interpretation

Interpolation of the table

Using the differences between 1 and 2 kg

70 - 60 = 10100 - 90 = 10

Averaging

(60 + 100) / 2 = 80(70 + 90) / 2 = 80

CompanyWeight of parcel (in kilograms)

0 1 2 3 4 5

FastAir $20 $60 $100 $140 $180 $220

SafeWay $50 $70 $90 $110 $130 $150


Using equationsUsing equations -- Algebraic formulationAlgebraic formulation FastAir: Cost = 7x + 14

SafeWay: Cost = 4x + 35

Formulation only 7x + 14 = 4x + 35

Formulation and attempt to solve 7x + 14 =

4x + 35 7x = 4x +

49 3x = 49 x = 16.33

Formulation and a successful solution 7x + 14 = 4x + 35 3x = 21 x = 7


Using equationsUsing equations –– Trial and improvementTrial and improvement Initial guess only (sometimes

correct)

Initial guess then other guesses

Starting from 1 and iterating in steps of 1

7 x 1 + 14 = 21 7 x 2 + 14 = 28

4 x 1 + 35 = 39 4 x 2 + 35 = 43

Initial guess then iterating with a step size of 1

Jumping towards the solution.


Future areasFuture areas

Statistics

Geometry and Measurement


EnlargementEnlargement(GM5118)(GM5118)

Terminology of enlargement

Scale factor??


Next steps GM5118

Uses an additive model Students need to look carefully at the language clues in the question. The phrase "how many times bigger … compared with" in parts b)ii) and c)ii) needs to be interpreted as a multiplicative question. If students have misinterpreted the question this way, see if they can perform the question once this has been clarified.

Unfamiliar with the term "scale factor" or of the concept of enlargement Work with students to understand that enlargement increases each dimension linearly by an amount known as the scale factor. Get the students working with enlargement on grid paper. GM5013 describes the process without using the term scale factor. Click on Level 3 and 4, enlargement AND scale factor for further resources that assess this. Also click on the Figure it out resource Enlargement Explosion (Geometry, L4+, Book 2, page 12).

Unfamiliar with the effects of enlargement on area or volumeStudents firstly need to know that "scale factor" is a linear measure. They then need to explore the relationship between scale factor and area. The increase in area goes up by the square of the scale factor. This is because area is a square measure (m2, cm2 etc). Click on scale factors AND area OR invariant properties for further resources on this relationship. Resources GM5113, GM5038, and GM5054 are particularly useful for teaching this principle. Also click on the Figure it out resource Growing Changes (Geometry, L3, page 24).The increase in volume goes up by the cube of the scale factor. This is because area is a cubic measure (m3, cm3 etc). Click on scale factors AND volume for further resources on this relationship. Resources GM5113, and GM5038

are particularly useful for teaching this principle.


Perimeter Perimeter (MS2178)(MS2178)

Composite shapes


Numerical value Fractional form Date and Origin

3 3/1 2000 BC Biblical

3.125 25/8 2000 BC Babylonian

3.16049 (16/9)2 2000 BC Egyptian

3.14285 22/7 250 BC Archimedes

3.14167 377/120 150 AD Ptolemy

3.14159292 355/113 480 AD Chung Chi

The first four diagnostics above or other incorrect answers indicate that the student does not know how to calculate the circumference of a circle. These students need to have experiences in how to calculate the circumference. ARB resource MS2107 and Figure it out resource Circle Links (Measurement, L4, Book 1, page 3) each give an activity that gets students physically measuring the circumferences and diameters of different circles and exploring the relationship between them.

The classical definition of π is geometrical, and it is the ratio between the circumference and the diameter of any circle.

π = circumference ÷ diameter (of any circle)

So a circle with a diameter of 1 unit has a circumference of π units.

Students could also be asked to explore π (pi) on the internet. The following are some historical fractional approximations to pi:

Next steps MS2178


Assessment Resource BanksAssessment Resource Banks

www.arb.nzcer.org.nz

Username: arb

Password: guide

Documents

Workshop presented at National Numeracy Facilitators Conference February 2008