While we’re working away on ACARA, here’s another post to keep readers occupied. Below are released “benchmark” test items from TIMSS 2019. (Further details about the benchmarking can be found in the full report (pp 35-59, 172-198).
OK, hands up who thought there was ever gonna be a second NotCH?
We’re not really a puzzle sort of guy, and base ten puzzles in particular tend to bore us. So, this is unlikely to be a regular thing. Still, the following question came up in some non-puzzle reading (upon which we plan to post very soon), and it struck us as interesting, for a couple reasons. And, a request to you smart loudmouths who comment frequently:
Please don’t give the game away until non-regular commenters have had time to think and/or comment.
Start by writing out a few terms of the standard doubling sequence:
More or less by accident, this post is the beginning of a new series: Not Crap Here.
A couple of people have suggested that we could occasionally include Dr. Jekyll material on this blog. You know, helpful stuff. It’s a decent idea, if our current thoughts weren’t so influenced by misanthropic disgust and murderous rage. Still, we’ve received two specific requests for the same old Jekyll material,* and which entailed some digging. Having finally dug, we’ve decided to post the material here, for whomever is interested. Whether or not there will ever be a NotCH 2 is anybody’s guess.
The previous A-Crash consisted of everything we could find in the Daft Curriculum on the algebraic treatment of polynomials and polynomial equations. This companion A-Crash consists of everything we could find in the Year 10 draft on the same material.
CONTENT (Year 10)
expand and factorise expressions and apply exponent laws involving products, quotients and powers of variables. Apply to solve equations algebraically
reviewing and connecting exponent laws of numerical expressions with positive and negative integer exponents to exponent laws involving variables
using the distributive law and the exponent laws to expand and factorise algebraic expressions
explaining the relationship between factorisation and expansion
applying knowledge of exponent laws to algebraic terms, and simplifying algebraic expressions using both positive and negative integral exponents to solve equations algebraically
CONTENT (Year 10 Optional Content)
numerical/tabular, graphical and algebraic representations of quadratic functions and their transformations in order to reason about the solutions of
connecting the expanded and transformed representations
deriving and using the quadratic formula and discriminant to identify the roots of a quadratic function
identifying what can be known about the graph of a quadratic function by considering the coefficients and the discriminant to assist sketching by hand
solving equations and interpreting solutions graphically
recognising that irrational roots of quadratic equations of a single real variable occur in conjugate pairs
Well, the plan to post each day lasted exactly one day.* We have an excuse,** but we won’t make excuses. We’ll try to do better.
This A-Crash consists of two Content-Elaboration combos for Year 9 Algebra.
expand and factorise algebraic expressions including simple quadratic expressions
recognising the application of the distributive law to algebraic expressions
using manipulatives such as algebra tiles or an area model to expand or factorise algebraic expressions with readily identifiable binomial factors, for example,or
recognising the relationship between expansion and factorisation and identifying algebraic factors in algebraic expressions including the use of digital tools to systematically explore factorisation fromwhere one oforis fixed and the other coefficient is systematically varied
exploring the connection between exponent form and expanded form for positive integer exponents using all of the exponent laws with constants and variables
applying the exponent laws to positive constants and variables using positive integer exponents
investigating factorising non-monic trinomials using algebra tiles or strategies such as the area model or pattern recognition
graph simple non-linear relations using graphing software where appropriate and solve linear and quadratic equations involving a single variable graphically, numerically and algebraically using inverse operations and digital tools as appropriate
graphing quadratic and other non-linear functions using digital tools and comparing what is the same and what is different between these different functions and their respective graphs
using graphs to determine the solutions to linear and quadratic equations
representing and solving linear and quadratic equations algebraically using a sequence of inverse operations and comparing these to graphical solutions
graphing percentages of illumination of moon phases in relationship with Aboriginal and Torres Strait Islander Peoples’ understandings that describe the different phases of the moon
This is another quick one, but it keeps the bullets flying while we prepare a more substantial post for tomorrow(ish). It can be considered a companion to the previous ACARA Crash a Content-Elaborations for Year 8 Number.
recognise and investigate irrational numbers in applied contexts including certain square roots and π
recognising that the real number system includes irrational numbers which can be approximately located on the real number line, for example, the value of π lies somewhere between 3.141 and 3.142 such that 3.141 < π < 3.142
using digital tools to explore contexts or situations that use irrational numbers such as finding length of hypotenuse in right angle triangle with sides of 1 m or 2 m and 1 m or given area of a square find the length of side where the result is irrational or the ratio between paper sizes A0, A1, A2, A3, A4
investigate the Golden ratio as applied to art, flowers (seeds) and architecture
We’re still desperately trying to find the time to properly go through the Daft Curriculum, and we hope to have some longer posts in the coming week. Until then we’ll try to keep things ticking over, sniping a little each day.
This is a short one, and can be thought of as a WitCH or a PoSWW. It is a Content-Elaboration for Year 6 Algebra. We won’t comment now, except to note that we cannot see how any competent and attentive mathematician (or grammaticist) would sign off on this. The consideration of possible corollaries is left for the reader.
recognise and distinguish between patterns growing additively and multiplicatively and connect patterns in one context to a pattern of the same form in another context
investigating patterns on-Country/Place and describing their sequence using a rule to continue the sequence such as Fibonacci patterns in shells and in flowers.
The following is just a dumb exercise, and so is probably more of a PoSWW. It seems so lemmingly stupid, however, that it comes around full cycle to be a WitCH. It is an exercise from Maths Quest Mathematical Methods 11. The exercise appears in a pre-calculus, CAS-permitted chapter, Cubic Polynomials. The suggested answers are (a) , and (b) 81/32 km.