This unit uses linear and area spatial models to represent and solve linear and simple quadratic equations.
The concept of a variable can be difficult for students to grasp. A variable is a quantity that can take up different values, however, in algebra variables are represented as letters. We often refer to a letter symbol as a variable, when we mean an unknown specific value, a generalised unknown (lots of possible values), or a variable that changes in relation to other variables in a situation.
Before beginning this unit, students have probably solved linear equations. They should recognise the equals sign as an indicator of balance or sameness of quantity, and should understand that equality is conserved if the same operation is applied to both sides of the equation. They might also recognise the importance of inverse operations, i.e. subtraction undoes addition, and division undoes multiplication. Students should also be accustomed to arrays being used as a representation of whole number multiplication.
Research indicates that students often ‘invent’ inappropriate rules for operations on numbers and letters, e.g. 5x + 4x = 20x and 3x – (x + 4) = 2x + 4. Algebraic conventions are not as flexible as those in arithmetic. Therefore physical and diagrammatic representations should be used to support students in checking that their operations make sense.
This unit should follow delivery of linear algebra and linear graphing at Level 5. Students learn to expand and factor quadratic expressions. Quadratic functions have many practical applications, particularly in physics such as speed of a falling body, braking distance of vehicles, and Einstein’s famous formula connecting mass and energy. In this unit the teaching of skills occurs within support of an area model that provides opportunities for discussion and dual development of procedures and concepts. Students who are quick to grasp these concepts will benefit from extension tasks such as those found in the Rich Learning Activities.
The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:
The contexts for this unit involve collections of discrete objects (e.g. people, scoops of mochaccino mix). These contexts might be supplemented by, or adapted to, better reflect students' interests, cultural backgrounds, or to make connections to learning from other curriculum areas. Examples might include ratios in cooking, ratios of positions in sports teams or genders in a class, or in the dimensions of human faces. Consider how you can utilise these ratio and rate problems as a way to make connections between mathematics and your students' 'real-world' contexts.
Te reo Māori kupu such as ōwehenga (ratio), hautanga (fraction), whakarea (multiplication, multiply), and pāpātanga (rate) could be introduced in this unit and used throughout other mathematical learning.
It is expected that students will understand why the area of a rectangle is found by multiplying side lengths, and that the product counts the number of units of area that fill the space. At Level 5 students are also expected to have well-developed calculation strategies with whole numbers. Understanding, and applying, the distributive property of multiplication is particularly important.
Key points are:
l | w |
1 | 210 |
2 | 105 |
3 | 70 |
5 | 42 |
6 | 35 |
7 | 30 |
… | … |
You may choose to plot the points using a graphing tool to see what pattern emerges.
Be open to students creating non-integral values in the ordered pairs, e.g. (1.5, 140), (28, 7.5). It is interesting to zoom in on the point defined by the ordered pair to check that it lies on the graph of lw = 210. Note that in this situation both l and w are variables. This is worth pointing out to students.
In this session students connect expressions with area representations. The expressions involve the distributive property of multiplication at different levels of complexity, and the use of specific unknowns.
24 x 16 (20 + 4)(10 + 6) (20 x 10) + (20 x 6) + (4 x 10) + (4 x 6)
35 x 23 (30 + 5)(20 + 3) (30 x 20) + (30 x 3) + (5 x 20) + (5 x 3)
(k + 7) (k + 9) (k x k) + (k x 9) + (7 x k) + (7 x 9) k2 + (9 + 7)k + 63
In this session specific unknowns are introduced using array diagrams. Students are tasked with finding the value of those unknowns.
The problems are as follows. Students are expected to use their own strategies so the algebraic solutions are for your information.
8r = 840 → r = 105
15(m+4) = 300 → m + 4 = 20 → m = 16
w(w+4) = 437 → w2 + 4w- 437 = 0 → (w + 23)(w - 19) = 0 → w = 19
h(h – 6) = 832 → h2 – 6h – 832 = 0 → (h + 26)(h – 32) = 0 → h = 32
(k + 6)(k + 5) = 240 → k2 + 11k -210 = 0 → (k + 21)(k – 10) = 0 → k = 10
(v + 7) (v – 3) = 459 → k2 + 4k – 480 = 0 → (k + 24)(k – 20) = 0 → k = 20
In this session the students use an array model to expand quadratic equations that are in factor form.
Important points to address are:
This session is devoted to factorising quadratic expressions.
Solutions are as follows:
x2 + 8x + 15 = (x + 5)(x + 3) x2 + 6x + 5 = (x + 3)(x + 2)
x2 + 10x + 21 = (x + 3)(x + 7) x2 + 9x + 20 = (x + 5)(x + 4)
x2 + 5x - 14 = (x + 7)(x - 2) x2 - 5x - 24 = (x - 8)(x + 3)
x2 - 25 = (x - 5)(x + 5) x2 + 6x - 16 = (x + 8)(x - 2)
x2 - 10x + 25 = (x - 5)(x - 5) x2 - 13x + 22 = (x - 11)(x - 2)
x2 - 12x + 36 = (x - 6)(x - 6) x2 - 20x + 64 = (x - 16)(x - 4)
Hello Parents and Caregivers
This week our class is learning about algebra. We will be using an area model to help us to expand and factorise quadratic expressions. Those skills are very important if we want to succeed in future mathematics, particularly calculus.
Please take the opportunity to discuss with your student what they are learning. They may be willing to demonstrate how they solve some of the problems from class.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/array-quadratics at 8:41pm on the 26th February 2024