Patterns, rules and graphs

This lesson can be presented with the accompanying teacher’s slides and teaching notes, which provide further suggestions for teaching the lesson. Use the teaching slides and accompanying teaching notes to complement the lesson.

• To begin, inform students they will be learning by playing, an effective teaching strategy for this level, but deserves learning intentions.
• Use slide 2 to inform students of the learning intention and what that might look like by the end of the lesson (success criteria). Assure students that they will be exposed to new terminology but not to be perturbed. In the teaching notes of the teacher's slides, definitions and further advice is given.
• Students are served a teaser about the board game ‘Battleships’ (slide 3). You may have to give a brief description of the game. Ask students if they can think of reasons why they will be playing a board game in class and what it has to do with algebra.
• Move to slide 4, skills check to ensure all students have been exposed to the Cartesian plane. Students are asked to draw a Cartesian plane using a pencil and ruler, and to plot various plots. Refer to the prerequisite knowledge to support or challenge the variability of students in your class.
• Slide 5 provides the link to the GeoGebra lesson, Battleship – Cartesian coordinates. Provide the link to students and allow them an extended period to play battleship against the computer. The game will reinforce their ability to plot coordinate pairs on the Cartesian plane. It may be worthwhile demonstrating a few turns on the screen. Further advice and differentiation are given in the teaching notes.

Note: further explanation, suggestions and ideas for iterations and differentiation appear in the teaching notes of the teacher's slides.

• Slide 6 introduces the students to the idea of combining mathematics, design, and space and location. The task is to design a straight-line dot-to-dot drawing that will be ultimately mapped out on the Cartesian plane. It is recommended that students sketch their drawings freehand and are allowed to let their imaginations run free and not limited to straight-line drawings only.
• Provide grid paper to every student to provide further practice in graphing skills for students, or have students use dynamic graphic software, such as GeoGebra, Wolfram or Desmos, another important skill for students to develop. Students must label each of their coordinates correctly. If time allows, have students list their coordinates and trade with another student. Have their partner draw their drawing according to the list provided to see if the original design can be re-created by someone else. Finished designs could be displayed in the classroom and used as a gallery walk, depending on available time. Further notes for differentiation are given in the teaching notes.

Differentiation (extend): Students can use a four-quadrant plane if they are comfortable with negative numbers or use one quadrant if they would prefer to. Students may wish to draw complicated designs but mention that unfinished work could be finished at home if they would like to.

• Instruct students to copy the table shown in slide 7 into their notebooks and fill the table out by considering the expanding pattern displayed. Ask students to describe the pattern they can see in the figures in the table. Highlight the interesting differences students may use with their language to describe the figures in the table.
• Slide 8 further connects students to algebraic thinking. The pattern can be described using terms and integers. By showing the students the progression of the pattern and its linear growth (outward), this linear growth can be shown using algebraic expressions to form a linear equation.
• Slide 9 turns the tables. Students are to find five patterns that could be described as increasing or decreasing by a linear or constant amount. Try visualpatterns.org. Instruct students to draw up a table of values for each pattern as shown previously, filling it in, and then descriptively writing the rule. Then students use algebra notation to describe the relationship they have discovered in numbers and symbols. Alert students to the fact that part of this activity will be used as an assessment.
• Slide 10 is the next step. Instruct students to draw up a further table of ordered pairs, rules (equations), initial values and the size of increase of each pattern. Students are to plot 3 ordered pairs from each of their linear rules, connect them with a line and label the line with the rule. Provide support for students who are require more scaffolding at this stage. You may wish to draw an example on the board.

Note: Teachers do not need to sign up to the mini-workshop on the visual patterns website to use this content.

• Ask students to reflect on what they have learnt. Have them write down the big ideas in the lesson. Encourage students to describe these big ideas using the language of maths, and make connections to visual pattern recognition, and the concept of representing pattern as mathematical sentences or equations and the connection to linear relationships.
• Discuss this reflection and reiterate the main concepts as a class.
• Challenge students to go out into the world and take notice of patterns and relationships that are linear. Explain that some of them may notice relationships that are not explicitly visual patterns they can see, but abstract patterns that are implied showing an increasing or decreasing linear relationships. Ask students to report back for the following class.
• Note the students that connected with the content and those that found it challenging. Think of other ways you could support those students to understand the important concept behind modelling linear relationships using algebra. Ask what you now know about your students and how you can provide feedback. Consider how you might iterate this for the following lesson, Graphs: formulas and variables.