Hunting for Patterns and Rules

1. Functions and Relations

Suggested Learning Intentions

  • To understand that functions describe the relationships between numbers
  • To understand that a relation is a set of inputs and outputs
  • To understand that a function is a relation with one output for each input

Sample Success Criteria

  • I can explain that the relationship between one number and another number is described by the functions performed on them
  • I can articulate that a relation is a set of inputs and outputs
  • I can determine the output of a number given a known input
  • I can demonstrate the effect of a function being performed using a variety of manipulatives
  • Function Machine slide show: pptx (NB: this is an animated MS PowerPoint - use presentation mode for correct functionality)
  • Function Machine Repairs - 4 levels of worksheet or 4 levels of laminated function machine: docx PDF
  • White board pens or counters

Mystery number

Begin by asking the students, ‘What is the relationship between 7 and 21?’

Some expected answers might be:

  • 21 is 14 more than 7
  • 7 is 14 less than 21
  • 21 divided by 7 is 3

Summarise by saying, ‘We can say that the relationship between these two numbers is described by the function/operation performed on them – addition, subtraction, division etc. Do you think this is true for the relationship between all numbers?’

Share the following task with students:

Pick a number between 1 and 10 and double it. Add 9. Now subtract 3. Now divide your answer by 2. Finally subtract the number you originally thought of.

When they have completed these series of calculations, ask them to share their results. Ask them, ‘What number did you begin with? What number did you end up with?’

Once a few results have been given and recorded on the board, ask students to discuss the relationship between the original number they chose and the result, and the functions performed on the original numbers with a partner. What do they notice?

The broken function machine

In this activity students are introduced to function machines. Function machines are a visual representation of the relationship between numbers. They display the function that is performed between the two numbers. Students are encouraged to work in pairs.

Say to students:

In mathematics we can use a 'function machine' to show how maths operations have been applied to a number and the order that they have happened. The four functions (addition, subtraction, multiplication, and division) take input values (numbers) and convert them into output values (other numbers).

Show students the opening slide of the Function Machine slideshow provided in the Materials and texts section. Ensure that you are running the Powerpoint slides in presentation mode, so that animations are active. You will need to click the space bar to change slides. Set the context for students by saying:

The local Function Machine factory has encountered a problem. It appears some of their machines are starting to malfunction. We have been asked to help them identify which machines are broken and which machines are working correctly. Work with a partner and let me know when you see a Function Machine that is broken. Here goes ….

Run the Powerpoint slides in presentation mode. Machines A and B (Slides 2-5) are functioning correctly. Machines C-F (Slides 6-21) represent malfunctioning machines. Pause between each set of slides (two per machine) to give students time to determine if that function machine is broken or not. Ask students to share whether the machine is functioning correctly, and how they determined this. Malfunctioning machines are identified with an 'Alert' screen, followed by an additional example which outlines the type of malfunction. 

Show students the Technical Inspection Report that summarises which machines were malfunctioning.  Once this is complete, continue with the scenario:

Today you have been employed as Function Machine Technicians and you will be helping to repair some machines which have malfunctioned. Some of you may have more experience as Function Machine Technicians. Think about how comfortable you were identifying the malfunctioning machines. Based on how difficult you found that task, you can decide which level of Function Machine Technician you are. There are four levels of Function Machine Technicians. 

Ask students to reflect on how difficult they found each level of the task, and to use this to guide their selection of an appropriate function machine repair activity sheet (worksheets are available in Materials and texts). Students who found identifying malfunctions difficult should choose Level 1. Students who want the highest level of challenge should choose Level 4.

When students have completed the activity sheet bring the students together to reflect on the task. Ask the students to describe the relationship between the input and output numbers. What functions were performed to obtain the output numbers?

Enable students by providing them with counters and a cup to act as the machine. Encourage students to model each machine and check their answers using counters.

Extend students by encouraging them to select function worksheet 3 or 4. Each machine has one malfunction. Students must correctly identify the malfunction of the machine and record each equation. If time permits, encourage students to create their own machine malfunctions for their partners. Challenge students to write multi-step malfunction problems. 

When presenting the function machine activity sheets to students, consider cutting out each machine and laminating them to form a deck of cards. Once students have correctly identified all the Function Machines which have malfunctioned ask them to record the equations in their books.

Areas for further exploration

Explore more function machines using the following website Function Machines. Students have the option of selecting from several different function machines. This supports all students to further consolidate their understanding and participate in the learning objective.

To consolidate students’ understanding that the relationship between input and output numbers is represented by the function performed on them, have students complete this in a table. Provide students with the following scenario as a worked example:

For each orange that is placed in a juicing machine, 100mls of juice comes out. We can record this relationship in a table. We can record the IN values and the OUT values.

Ask students to help you add the answers to the bottom row.

IN (number of oranges)

1 2 3 4 5 6
OUT (juice in ml) 100

Provide students with a range of tables. For each table, have them use the given rule to find the missing OUT numbers.

Here are some examples to get you started:

  • Add 4 to each IN number.
  • Subtract 4 from each IN number.
  • Multiply each IN by 3.
  • Divide each IN number by 2.
  • Multiply each IN number by 2 and then add 3.

For example:

Add 4 to each IN number in the table.

IN 1 2 3 4 5 6

Enable students by simplifying the rules provided. For example:

  • Add 2 to each IN number.
  • Subtract 1 from each IN number.

Extend students by increasing the complexity of the rules provided and tables provided. See examples below:

Change the IN values

IN 4 5 8 11 12 14

Provide students with a partially filled in table and ask them to fill in the remaining boxes and then to write out what they think the rule is.

IN 4 8 11 12 14
OUT  12 24

Ask the students to fill out these tables individually. These tables can be collected and used to assess students' level of understanding.

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