In this unit students represent a quadratic relationship using spatial patterns, tables, graphs and equations. In doing this, they recognise the key features of a parabola, find unknowns from the graph of a parabola, and find unknowns from simple quadratic equations. Procedural fluency and conceptual understanding are developed simultaneously.
The concept of a variable is a difficult one for students to grasp. A variable is a quantity that can take up different values. In algebra variables are represented as letters. We often refer to a letter symbol as a variable, when sometimes we mean an unknown specific value, sometimes a generalised unknown (lots of possible values), and sometimes a variable that changes in relation to other variables in a situation.
Before beginning this unit, students probably have constructed linear graphs with integer values and will have solved simple linear equations. Graphs provide an accessible visual representation of the relationship between two variables. Connecting graphs and equations from the same context supports students to understand of how multiple representations connect and are useful for solving problems.
For a structured situation, research indicates that finding a rule for the general term can be a difficult for many students. Students often first notice the iterative (recursive) rule that gives the value of a term of a sequence from the previous term. However, they are sometimes unable to find the general relationship between the independent and dependent variables, or apply the inverse relationship.
This unit should follow delivery of linear algebra and linear graphing at Level 5. The teaching of skills occurs within contexts that provide opportunities for discussion and development of procedures and concepts. As each learning outcome is explored, there may be a need for consolidation through more traditional exercises, such as those found in text books, worksheets or online practice activities. 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 context for this unit can be adapted to suit the interests, experiences, and cultural makeup of your students. The unit is framed around the context of Mary's garden and that plants it contains. Following this discussion, you could work with the students and whānau to link this learning to meaningful contexts from their lives. Possible contexts could include patterns of students sitting in different arrangements at assembly, patterns of students sleeping at the marae, arrangements of different types of produce for the school garden, and so on.
Te reo Māori kupu such as tauira (pattern), ture (formula, rule), whārite pūrua (quadratic equation), and unahi (parabola) could be introduced in this unit and used throughout other mathematical learning.
This unit of work is built on the pattern of square numbers which are the foundation of quadratic equations. These activities are intended to progress students from writing a quadratic rule from a visual pattern, towards forming a quadratic rule from a description and using equations to solve problems.
For each session a possible sequence for students to follow is:
Each session is initially presented as a problem or series of problems for students to solve. Instructions and questions relating to the ideas above, that teachers may give students, are italicised. As stated earlier, it is not suggested that all the learning outcomes should be taught during each session. Therefore, you do not need to provide students with all of the instructions initially. If the first session is to be used for diagnostic assessment, then most of the instructions can be worked through, as a class, to find what the students are capable of. In subsequent activities, one or more of the instructions may be focused on, but other instructions may be given, either as enabling or extending prompts.
Adapt the context reflected in this unit to suit the interests and backgrounds of your students. You might introduce this problem as written, and then work with students to re-contextualise it. Mary plants this sequence of cabbage plants because she likes patterns. The first group contains one plant, the next contains four, the next contains nine plants, etc.
Group number (g) | Number of cabbages (c) |
1 | 1 |
2 | 4 |
3 | 9 |
4 | 16 |
5 | 25 |
Further Activities (extensions)
Use these to extend students who have grasped the foundational concepts. You might choose to work with other students, developing their foundational knowledge, during this time.
This might lead more knowledgeable students to consider how to find the sum of n square numbers. That involves some complicated algebra about series. This is usually dealt with at Level 7.
Expect students to note that a continuous line has replaced discrete points and that the parabola has two ‘sides’ as the domain of the online graph extends into negative real numbers. Point out that the online graph is a result of imagination rather that practical application to Mary’s problem. Use the online graph to read out solutions to problems like:
If g = 13 what is c? If c = 100 what is g? (Note this has two solutions +10 and -10)
Emphasise these important points as they arise in discussion:
The first pattern is c = g2 – 1. The pesky rabbit has eaten one cabbage from each group. The result is that each value for c in the table is one less than the corresponding value in the table for c = g2.
The second pattern is c = 2g2. Mary has allowed for her rabbit problem by planting double the number of cabbages in each group. The result is that each value for c in the table is two times the corresponding value in the table for c = g2.
The third pattern is c = (g + 2)2 + 1. Mary is expecting a rabbit invasion! The result is that to find each value for c in the table find the corresponding value two groups ‘ahead’ in the table for c = g2 and add one.
The effect on each graph is as follows:
Translation down by 1 Increase in scaling by factor 2 Translation across of -2
Further Activities
Use these to extend students who have grasped the foundational concepts. You might choose to work with other students, developing their foundational knowledge, during this time.
Ask students to explore transformations to the equation y = x2 and the effect of those transformations on the graphs. Let them use a graphing tool to do so.
What happens to the graph if you add or subtract a constant to x2? Try these examples:
y = x2 + 3 y = x2 – 5 y = x2 + 1.5
What happens to the graph if you multiply x2 by a co-efficient? Try these examples:
y = 3x2 y = 1.5x2 y = -1x2 y = -3x2
What happens if you add or subtract a number from x before you square it? Try these examples:
y = (x + 1)2 y = (x - 1)2 y = (x + 4)2 y = (x - 4)2
What happens if you combine these transformations? Try these examples:
y = (x - 2)2 + 3 y = (x + 1)2 – 5 y = 2(x - 3)2 - 4
In this session the spatial patterns of lettuces drive the inquiry into relationships between the variables, group number and number of lettuces. The priority is for students to structure the spatial patterns to create rules and equations for the nth term.
You may need to explain the last instruction. That is, if given any value for the group number, how would you work out the total number of lettuces?
The numeric expressions are useful for developing general terms. Organising the expressions vertically can help students to relate the group number to numbers in the expressions.
Group Number | Pattern One | Pattern Two | Pattern Three |
1 | 1 x 2 or 12 +1 | 1 x 2 - 2 | 1 x 2 + 1 |
2 | 2 x 3 or 22 +2 | 2 x 3 - 2 | 2 x 3 + 2 |
3 | 3 x 4 or 32 +3 | 3 x 4 - 2 | 3 x 4 + 3 |
|
|
|
|
n | n(n+1) or n2 + n | n(n+1) – 2 | n(n + 1) + n |
For each pattern ask:
Different ways of structuring the patterns lead to different general rules. This provides rich opportunities for manipulating the expressions. For example, consider these three ways to structure Pattern Three.
n(n + 1) + n (n + 1)2 – 1 n2 + 2n
Further Exploration
Students proficient at manipulating expressions may enjoy proving algebraically that the expressions for general terms are equivalent. For example:
(n + 1)2 – 1 = n2 + 2n + 1 – 1
= n2 + 2n
Record the general rules from the previous pattern activities.
Pattern One | Pattern Two | Pattern Three |
n(n+1) | n(n+1) – 2 | n(n + 1) + n |
n2 + n | n2 + n - 2 | (n + 1)2 – 1 |
|
| n2 + 2n |
The next phase of the lesson requires students to use abductive reasoning. Abduction involves seeing structure in one member of a pattern and applying that structure to other members. Therefore, it is quite different to the inductive reasoning that students have been expected to apply so far. In the attached PowerPoint one or two members of each pattern are shown.
Encourage students to look for ‘chunks’ in the figure and relate the chunks to general rules they may have discovered so far. For example, Pattern two might be chunked any of these ways:
As students work watch for:
You might use equivalent expressions to practise rearranging expressions. For example:
Graphs and equations can be used to solve problems with specific unknowns. For example, A general rule for Pattern Three is:
(From the surrounding square less the nth triangular number)
How many cabbages are in the 15th pattern?
In equation form this is:
In graphic form this is:
Dear families and whānau,
Recently, we have been learning about how to represent a quadratic relationship using spatial patterns, tables, graphs and equations. We have learnt to recognise the key features of a parabola, find unknowns from the graph of a parabola, and from simple quadratic equations. Ask your child to tell you about their learning.
Printed from https://nzmaths.co.nz/resource/mary-s-garden at 2:06am on the 21st April 2024