TGHS Maths
Level 1 Mathematics: Teachers' page
This is a full NCEA Mathematics course. Any student who is capable of achieving the 4 Internals (= 12 credits) is encouraged to follow this course, and will therefore be entered for the MCAT. We see this as an inclusive policy.
The resources for this course at the moment rely too heavily on the text book; teachers should make a positive effort to share non-text resources.
Before teaching, teachers should be familiar both with the assessment and especially with the assessment notes (in google docs). These are password protected - request access if necessary.
Trigonometry
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Resources |
Time Scale (approx.) |
Pythagoras |
GM: Ex. 20.01 p266-267 |
2 lessons |
Rearranging formulas |
GM: Ex. 7.06 p. 89 GM: Ex. 7.07 p. 91 |
2 lessons |
Simple Trig (finding a “non-denominator” side) SOHCAHTOA |
10T: L7-8 P5 p32 (section F) |
2 lessons |
Finding a “denominator side” |
10T: L7-8 P5 p33 |
1 lesson |
Finding angles |
GM: Ex. 21.02 p279-280 (more on 10T: L7-8 P5 p34) |
2 lessons |
Mixed “side” problems |
GM: Ex. 21.01 p277-278 |
1 lesson |
Similar Triangles |
GM: Chapter 19 |
3 lessons |
Measuring heights: by estimation by similar triangles by trig |
Use a number of sites/buildings around the school grounds |
4 lessons |
Pythagoras and Trig in 3D |
GM: Chapter 23 (more on 10T: L9-10 P2 p3-6, including some useful models) |
2 lessons |
Applications |
GM: Chapter 24 |
3 lessons |
Revision or Excellence Extension* |
Revision 10T: L7-8 P5 p35-37 Extension worksheet: No Right Angles |
2 lessons |
Practice Assessment |
2 lessons + 1 lesson feedback |
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Real Assessment |
2 lessons |
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Total: |
30 lessons (about 6 weeks) |
* It is important a) that students decide which option to take b) all students are encouraged to take a risk.
Much of this is covered in Year 10, so should be revision (See Year 10 Teacher Handbook).
Also note that the teaching of trigonometry allows an opportunity to teach “rearranging formulas” in a relevant context. Please avoid methods that seek to remove this opportunity!
Resource References:
GM = Gamma Mathematics.
Bivariate Data
The data should be processed with Excel or Google Sheets for this standard.
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Resources |
Time Scale (approx.) |
The easiest way to teach this standard is to run through a series of example activities.
Note that a) The standard demands that students are familiar with the PPDAC cycle and b) The question is given.
An investigation with the PPDAC cycle: |
Is there a connection between: Height and armspan? Jumping from a dominant and non-dominant leg? Throwing a tennis ball with a dominant and non-dominant arm?
Poster: PPDAC cycle |
The first example will take about 8 lessons, and the next two about 3 each = 14 lessons |
Problem: A simple statement of the question. But you need to decide the independent/dependent variables. |
Worksheet: Dependent/independent variables |
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Plan: A description of what you are going to do. Must include “managing variations”. |
Teacher notes: Managing Variation Student Notes: Achieved checklist
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Data: A simple table of data, including the units in the column heading. Might include some averaging. An excellence response might include a discussion of averaging (which average?) |
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Analysis: The scatterplot. It must have the dependent variable on the y-axis. Describing the connection, first without the trendline, then with the trendline. |
Worksheet: Cafe Sales Spreadsheet: Fake Cafe Sales (allows the teacher to create more worksheets similar to cafe sales with random data). Teacher notes: analyzing the data Student notes: analysis writing frame |
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Conclusion: - Restate connection/strength (Achieved) - Was the connection expected? (Excellence) - Comment on unusual points (Excellence) - Writing an algebraic model (Merit) - Explaining the gradient and y-intercept (Merit/Excellence) - Making predictions (Merit) - Reliability of predictions (Merit) - Unforeseen problems (Excellence) - Limits of the model (Excellence) - Critique of linear model (Excellence) - Discusses causality issues (Excellence) |
Teacher notes: an excellence Conclusion exemplar Student notes: a Conclusion exemplar Student notes: a Conclusion checklist |
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Practice Assessment |
3 lessons + 1 lesson feedback |
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Real Assessment |
4 lessons |
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Total: |
22 lessons (about 5 weeks) |
Hopping Example, which includes an example of a mark schedule.
Helicopter Seeds A simulation of the flight of sycamre seeds
PPDAC framework for real sycamore seeds
PPDAC framework for Leaf Investigation
PPDAC framework for Rocks Investigation
10T= The Ten Ticks worksheet packs.
Measurement
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Resources |
Time Scale (approx.) |
Units and Estimation Estimation Conversion |
GM: Exercise 10.01 p. 134
Note students should be discouraged from converting units for areas where-ever possible. The units of length should be converted FIRST and then the area calculated. Q2 in Exercise 10.02 can be used as a discussion point. |
2 lessons |
Rearranging formulas |
GM: p. 89 Ex. 7.06 GM: p. 206 - 208 Ex. 16.04 |
2 lessons |
Perimeter/Circumference Perimeter Circumference |
GM: Exercise 11.01 p. 140 GM: Exercise 11.02 p. 143 |
2 lessons |
Area
Common quadrilateralas and triangles Circle Compound Areas Surface area Applications |
All formulas are provided in the assessment. Teachers should note the form used in the formula sheet provided.
GM: Exercise 12.01 p.146-148 GM: Exercise 12.02 p. 151 GM: Exercise 12.03 p. 152-153 GM: Exercise 12.04 p. 156 GM: Exercise 12.5 p. 157-160
Students should have experienced examples with mixed units. They should be familiar with Surface Area as a measure of material used (eg in the construction of a box) or as a painted surface. |
5 lessons |
Volume Prisms
Pyramids, cones and spheres Applications Capacity |
GM: Exercise 13.01 p. 162-163 Worksheet: Irregular Volumes. Answers. GM: Exercise 13.02 p. 165-167 GM: Exercise 13.03 p. 167-169 GM: Exercise 14.01 p. 173-175
Whilst pyramids, cones etc are not explicitly assessed, they commonly occur in algebra assessments. This exercise will provide some timely revision! |
4 lessons |
Rates |
GM: Exercise 15.01 p. 182-183 |
1 lesson |
Limits of Accuracy |
GM: Exercise 16.03 p. 203-205 |
2 lessons |
Practice Assessment |
Document: Practice |
2 lessons + 1 lesson feedback |
Real Assessment |
Document: Real |
2 lessons |
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Total: |
23 lessons (about 5 weeks) |
Algebra I
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Resources |
Time Scale (approx.) |
Indices |
GM: p. 75 Ex. 6.02 GM: p. 126 -127 Ex. 11.01 – 11.02 |
3 lessons |
Forming and solving linear equations |
It is enormously important that students have lots of practice in forming, as well as solving, equations. Simultaneous equations (by substitution) also fit here, but were covered by the Linear Algebra Standard. Some revision - particularly of 'word problems' might be in order.
GM: p. 78-79 Ex. 6.03 An introduction to forming expressions (ie not equations yet)
GM: p. 85-87 Ex. 7.01 - 7.04 GM: p. 91-93 Ex. 7.07 – 7.09
GM: p. 198 - 204 Ex. 16.01 - 16.03 |
5 lessons |
Inequations |
Increasingly popular with MCAT writers. Again, forming the inequality is important.
GM: p. 95 Ex. 7.11 GM: p. 210-211 Ex. 16.05 |
3 lessons |
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Total: |
11 lessons (about 3 weeks) |
Linear Graphs
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Resources |
Time Scale (approx.) |
Linear graphs from equations (Equation to table to graph)
y=mx+c form |
(Note: for this standard, ax+by=c is NOT required)
10T: L6P1 p27-28 GF: p. 162 Ex. 13.04 onwards (students should be encouraged to answer the questions by PLOTTING. Rearranging can be used as an extension). |
2 lessons |
y=mx+c (graph to equation) |
Worksheet: Linear equations from graphs |
2 lessons |
Gradient and y-intercept (and interpretation) |
Worksheet: Linear Graphs Intro. Worksheet: Example |
3 lessons |
Simultaneous Equations
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GF: Ex. 10.01 to 10.03 p. 119-123 GF: Ex. 10.04 to 10.05 p. 124-125 |
5 lessons |
Applications |
Worksheet: Four example questions (at achieved/merit) Worksheet: Taxis Worksheet: Coffee Machine |
5 lessons |
Revision |
Worksheet: Exam Questions (from the old NCEA External) |
2 lessons |
Practice Assessment |
2 lessons + 1 lesson feedback |
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Real Assessment |
2 lessons |
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Total: |
24 lessons (about 5 weeks) |
Much of this is covered in Year 10, so should be revision (See Year 10 Teacher Handbook).
Also note that the teaching of linear graphs allows the teaching of simultaneous equations. Although only simple substitution methods are required here, there is an opportunity to teach the other techniques required by the MCAT. It is also appropriate to teach some of the material required for AS91028, Investigate relationships between tables, equations and graphs (including continuous/discontinuous graphs).
Negative gradients should be taught, but do not appear on the assessment. Solving simultaneous equations is NOT required for Achieved (strictly, neither is it required for Merit/Excellence, but it helps!).
Resource References:
GM = Gamma Mathematics.
10T= The Ten Ticks worksheet packs.
Algebra II
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Resources |
Time Scale (approx.) |
Expanding brackets |
GM: p. 97-101 Ex. 8.01-8.05 |
2 lessons |
Factorizing |
Factorization of non-quadratic expressions appears to have become popular with MCAT writers.
To be at Level 6, a quadratic ax2 + bx + c must have a not 1.
GM: p. 102 - 103 Ex. 8.06 Non-quadratics GM: p. 104 - 107 Ex. 8.08 – 8.11 Quadratics |
4 lessons |
Forming and solving quadratic equations |
GM: p. 110 Ex. 9.01 Solve GM: p. 111 Ex. 9.02 Factorize and solve GM: p. 112 Ex. 9.03 Rearrange, factorize and solve GM: p. 113 Ex. 9.04 Form and solve GM: p. 212-214 and p. 216 Ex. 16.06 |
4 lessons |
Simple exponential equations |
This is a non-calculator paper, so these will always be simple guess-and-check.
GM: p. 115 -117 Ex. 9.05 – 9.06 |
2 lessons |
Algebraic fractions |
GM: p. 128 - 129 Ex. 11.04 Cancelling factored expressions GM: p. 130 - 131 Ex. 11.06 – 11.07 Adding fractions |
3 lessons |
15 lessons (about 3 weeks) |
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| MID YEAR EXAMINATIONS | ||
Revision These resources can be used either at the end of the unit, in tutorials, or in the run-up to the real MCAT. Will need some work on Solving Simultaneous Equations by Elimintaon (by substitution covered in Linear Algebra).
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These resources need backing up with some more application type questions. Some consecutive number type problems (eg n, n+1, n+2.... etc) would be good. I have also not put in any work on sequences as yet.
Students should be getting a good grounding in equations (including applications) and expanding brackets during year 10.
Resource References:
GM = Gamma Mathematics.
10T= The Ten Ticks worksheet packs.
Google docs. Inequalities: word problems.
Tables, Equations and Graphs
The external assessments for this standard have tended to be quite "idiosyncratic". For this reason, teaching materials are often hard to come by. Emphasis should be given to revising from the past papers available, especially since this is the last standard taught in the year. If teachers are running short of time, then these papers themselves often provide the best teaching materials.
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Resources |
Time Scale (approx.) |
Straight line y=mx+c (revision) |
Worksheet: Linear equations from graphs |
1 lesson |
Linear sequences: Relation between table, graph and times table. |
Worksheet: Growing Patterns
In most sequences, the points should be plotted discretely. The external questions have been somewhat inconsistent on this issue, but students should be consistent in their answers. |
2 lessons |
Piecewise defind and step-functions |
There is very little on this in the textbooks or workbooks available.
Worksheet: Parcels NCEA External 91028 2011 Question 1 NCEA External 91028 2012 Question 2 NuLake EAS 1.3 p. 35 Question 130
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2 lessons |
Quadratic graphs: transformations |
If possible, spend some time on this. It introduces some important ideas required by Level 2.
GM: p.182-185 |
3 lessons |
Quadratic graphs: Intercept (factorized form) |
GM: p.187-189 |
1 lesson |
Quadratic graphs: Equation of quadratic from Graph (including finding coefficient of x2). |
In reality this does not seem to be a common question in recent Externals.
GM: p.229-231 There are also many examples (all at Excellence) in the pre-2011 external papers: 90148 Sketch and interpret graphs. |
1 lessons |
Quadratic Sequences |
This topic now seems to be assessed less often, which is a good thing in my opinion. It is hard for students at Level 1 to see why the difference method works. However, in case it ever comes up:
GM p.145-146
Ten Ticks Level 7/8 Pack 1 Pages 40-44. Pages 43 and 44 have some good "mini-investigations" that may be worthwhile. |
2 lessons |
Exponential: Patterns, graphs (and their) transformations |
GM p. 151 Q7 and 9 |
2 lessons |
Summary: Linear, quadratic and exponential |
Worksheet: Linear, quadratic and exponential sequences. |
2 lessons |
Revision |
NCEA Externals 91028 2015-2011 (work backwards)
Worksheet: Tilings |
as many lessons as possible! |
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Total: |
17 lessons (about 4 weeks without revision) |