This unit integrates student learning in the mathematics and science learning areas. Students will be developing their use of rates and ratios, exploring linear relationships using tables and graphs, in the context of chemical reactions involving common household acids and bases.

Students develop their skills and knowledge on the mathematics learning progressions; multiplicative thinking, and patterns and relationships, in the context of science, understanding the material world; investigating the chemical properties of acids and base

Students will model a real life situation in a scientific investigation. They will measure and record data to find the relationship between two variables. Students will recognise a pattern, from a table and/or a graph and describe it as being linear (or not). They will plot a linear relationship on an appropriate set of axes and describe the key features of the graph. They will be encouraged to generalise the pattern, forming a rule from which predictions can be made and tested. Students will solve problems using the relationship shown on a graph and/or described by an algebraic equation.

Note that the context of neutralising acids and bases does not yield the exact linear relationships hinted at in this unit, but does so approximately when only strong acids and bases are used. It can be argued that students will gain a better understanding of chemical processes overall, if they can first look for clear patterns in such constrained settings.

The practical work outlined in the practical activities requires solutions and equipment that should be available in secondary school laboratories. In the writing of this unit, it is assumed that the teachers using this activity will have access to the expertise of a science teacher and the necessary resources for this practical work.

### Structure

This cross-curricular, context based unit has been built within a framework that has been developed with input from teachers across the curriculum to deliver the mathematics learning area, while meeting the demands of differentiated student-centred learning. The unit has been designed around a six session focus on an aspect of mathematics that is relevant to the integrating curriculum area concerned. For successful delivery of mathematics across the curriculum, the context should be meaningful for the students. With student interest engaged, the mathematical challenges often seem more approachable than when presented in isolation.

The first session is an introductory activity that is aimed to spark the imagination of students, to introduce the need for a particular idea or technique in mathematics that would enable them to explore deeper into that context. It is expected that rich discussion may be had around the context and around the nature of the mathematics involved.

The following five sessions are each based around a model of student-centred differentiated learning.

- There is a starting problem to allow students to settle into the session and to focus on the mathematics within the chosen context. These starting problems might take students around ten minutes to attempt and/or to solve, in groups, pairs or individually.
- It is then expected that the teacher will gather the students together to review the problem and to discuss ideas, issues and mathematical techniques that they noticed during the process. It may be helpful to summarise key outcomes of the discussion at this point.
- The remaining group of activities are designed for differentiating on the basis of individual learning needs. Some students may have managed the focus activity easily and be ready to attempt the reinforcing ideas or even the extending ideas activity straight away. These could be attempted individually or in groups or pairs, depending on students’ readiness for the activity concerned. The students remaining with the teacher could begin to work through the building ideas activity together, peeling off to complete this activity and/or to attempt the reinforcing ideas activity when they feel they have ‘got it’.
- It is expected that once all the students have peeled off into independent or group work of the appropriate selection of building, reinforcing and extending activities, the teacher is freed up to check back with the ‘early peelers’ and to circulate as needed.

### Introductory session

(This activity is intended to motivate students towards the context/integrated learning area and to inform teachers of students' location on the learning progressions):

Bee Sting Problem:

One summer’s day Charlie’s little sister Lilly comes running into the house, crying. She has been stung by a bee! There is no consoling Lilly. Charlie remembers something he heard once about how he can add some household ingredient to take the pain away...but what was it? And why? And how much?

What can you find out about bee stings?

- Are they acidic or basic?
- Would an acid or a base need to be applied to ‘neutralise’ the bee sting?
- Are there any products that could neutralise a bee sting that may be commonly found at home?

### Session two

Learning about household acids, bases, and neutral solutions

#### Focus activity

We can measure how acidic or basic a solution is using an acid-base indicator. One indicator can be made from the water that was used to cook red cabbage. (Strain the juice after cooking, and leave this coloured water to cool for an hour.) Acids and bases will each turn this cabbage juice indicator a different colour.

- Test water and solutions of vinegar, baking soda to find what colour the juice goes when a solution is neutral, acidic or basic.

#### Building ideas

- White vinegar is acidic. Put a finger height of white vinegar in a clean test tube and note the colour of this solution. Add a few drops of cabbage juice indicator to the vinegar. Note the colour change.
- What colour is the cabbage juice indicator when it is in an acidic solution?
- Baking soda is basic. Dissolve a spatula full of baking soda in water, in a clean test tube and note the colour of this solution. Add a few drops of cabbage juice indicator to the baking soda solution. Note the colour change.
- What colour is the cabbage juice indicator when it is in an basic solution?
- Water is neutral. Put a finger height of water in a clean test. Add a few drops of cabbage juice indicator to the water. Note the colour change.
- What colour is the cabbage juice indicator when it is in a neutral solution?

#### Reinforcing ideas

Make up solutions of each of the following and test with cabbage juice indicator to decide whether they are acids, bases or neutral:

- Lemon juice
- Dishwashing liquid
- Soda water
- Fizzy tablet
- Hand soap
- Laundry detergent
- Shampoo
- Apple juice

#### Extending ideas

From your research you have learned that bee stings are acidic.

- What could you do to neutralise the acidity of Lilly’s bee sting?
- Suggest an emergency procedure for neutralising a bee sting using common household products?
- Use cabbage juice indicator to test whether these products are acids or bases.
- How could you be sure that the sting site was neutralised and no longer acidic, nor made basic?

The activity itself has a science focus, with mathematical skill and knowledge needed to measure, record and process appropriately. Although it need not be carried out in a science laboratory, this activity should be carried out in an area that has washable surfaces and a sink for cleaning up spills.

### Session three

Learning about pH indicators

Charlie has learned that bee stings are acidic. He knows that applying the appropriate amount of a base will neutralise the acid of the sting. To be sure of how much of a basic solution is needed to neutralise the sting, Charlie needs to know more about measuring acidity. Acids, bases and neutral solutions are all measured on the pH scale.

#### Focus activity

The concentration of hydrogen ions (H^{+}) in a solution is measured on a scale called pH. An acid solution has a pH of < 7, a basic solution has a pH of > 7. We have found that cabbage juice indicator goes pink-red-crimson-purple in acid solutions, purple in neutral solutions and blue-green-yellow in basic solutions.

The graph below shows the how the concentration of hydrogen ions (H+) in a solution relates to the pH of that solution. Mark the regions of the graph in which a solution would sit if it:

a) Turned cabbage juice indicator yellow

b) Turned cabbage juice indicator pink

c) Turned cabbage juice indicator purple

d) Turned cabbage juice indicator green

The pH scale goes from 1-14, representing the concentration of hydrogen ions (H+) in a solution. A solution which has a pH of 1 has 10x more hydrogen ions (H+) than the same volume of a solution of pH of 2. A solution which has a pH of 1 has 100x more hydrogen ions (H+) than the same volume of a solution of pH of 3.

#### Building ideas

- Which solution, a solution of pH = 2 and and solution of pH = 3 has a higher concentration of hydrogen ions?
- Which solution, a solution of pH = 5 and and solution of pH = 3 has a higher concentration of hydrogen ions?
- Which solution, a solution of pH = 7 and and solution of pH = 3 has a higher concentration of hydrogen ions?

#### Reinforcing ideas

- How does the concentration of hydrogen ions in a solution of pH = 3 compare with the concentration of hydrogen ions in a solution of pH = 5?
- How does the concentration of hydrogen ions in a solution of pH = 3 compare with the concentration of hydrogen ions in a solution of pH = 7?
- How does the concentration of hydrogen ions in a solution of pH = 3 compare with the concentration of hydrogen ions in a solution of pH = 10?

#### Extending ideas

- How does the concentration of hydrogen ions in an acidic solution of pH = 1 compare with the concentration of hydrogen ions a neutral solution?
- How does the concentration of hydrogen ions in an acidic solution of pH = 1 compare with the concentration of hydrogen ions a basic solution of pH = 8?
- Describe the relationship between the concentration of hydrogen ions in an acidic solution of pH = 1 and the concentration of hydrogen ions a solution of pH = n (where n may be 1, 2, 3, …14)?

This activity aims to develop students’ understanding of a science concept through interpretation of a mathematical relationship displayed on a graph. The relationship itself is exponential, but is applicable for students working at level 4 of mathematics in the NZC because the numbers are used are whole numbers from 1-14 and the powers of ten.

### Session four

Making a pH indicator scale

Charlie wants to treat his sister Lilly’s bee sting. He knows that applying the appropriate amount of a base will neutralise the acid of the sting. Charlie has been looking at how we measure acidity. Now he needs to work out just how much of a basic solution to is needed to neutralise Lilly’s sting.

#### Preparatory activity

Working in small groups, gather together; 6 test tubes on a rack, 3 droppers.

Also get 50mL of each of the solutions; cabbage juice indicator, vinegar (acetic acid) and dissolved baking soda. Use one dropper for each of these solutions and be careful not to mix the droppers up.

- Add 10 drops of vinegar (acetic acid) to one test tube. Add 5 drops of cabbage juice indicator. This is your acid ‘control’. Store this test tube on the far left of your test tube holder. This is the colour of your acid.
- Add 10 drops of neutral cabbage juice to a test tube. Store this test tube on the far right of the test tube holder. This is the colour of your neutral.

- Which test tube represents a bee sting? What colour is the indicator in this solution?
- What are we using as a base to neutralise the bee sting?
- Which test tube represents a neutralised bee sting? What colour is the indicator in this solution?

With the four remaining test tubes, add the following.

- 5 drops of vinegar, 5 drops of indicator
- 10 drops of vinegar, 5 drops of indicator
- 15 drops of vinegar, 5 drops of indicator
- 20 drops of vinegar, 5 drops of indicator

#### Building ideas

To each test tube, carefully add one drop at a time of baking soda until the solution just turns purple neutralised (purple). Count and record the number of drops of baking soda that could be added to the test tube until it was purple

#### Reinforcing ideas

To each test tube, carefully add one drop at a time of baking soda until the solution is neutralised (purple). Record the number of drops of baking soda solution that could be added to the test tube until it was purple. One more drop would make it go blue and be one drop too many. If your solution turned purple with x drops of indicator, record the number needed to neutralise as x – 1.

#### Extending ideas

To each of the four test tubes, add just enough baking soda solution to neutralise the solution in that tube. Record the results so that you will be able to graph volume of vinegar against volume of baking soda solution needed to produce a neutral solution.

This activity has an emphasis on practical technique for science, with the need for careful measurement and recording being a major outcome. Encourage students to take accurate measurements and to record their results in a table. If time and resources allow, they should repeat the procedure several times so that they may use the average of their individual results for each test tube.

### Session five

Graphing the results of the practical investigation

In session four, students gathered data from a practical investigation, measuring the quantity of a given basic solution needed to neutralise an acidic solution. This investigation was modelling how much of a sting treatment (base) would be needed to to neutralise different amounts of a bee sting (acid).

#### Focus activity

Looking at the results from the practical work in session four:

- Do you expect every group to have similar or different results?
- If we graph these results, how many data points would there be for one group’s results?
- What will the labels of the axes of this graph be?
- What shape do you think these data points would form when they are graphed?

#### Building ideas

- On squared paper, rule a set of axes
- Label the horizontal axis number of drops of vinegar (the independent variable)
- Label the vertical axis with number of drops of baking soda solution (the dependent variable)
- Choose a suitable scale for each of your axes
- Plot the data you recorded from the investigation.

#### Reinforcing ideas

- On squared paper, rule a set of axes
- Choose a suitable scale for each of your axes. The independent variable will be on the horizontal axis.
- Plot the data you recorded from the investigation.

#### Extending ideas

- Graph the data you recorded from the investigation.
- What type of relationship is shown on this graph?
- Think about the range of the data. If this graph is extended to show all the possible results, should this be done with more dots (discrete data) or would it be appropriate to fit a line (continuous data)?

### Session six

Interpreting and using the graph of result of the practical investigation

In session five, students graphed their data from a practical investigation, measuring the quantity of a given basic solution needed to neutralise an acidic solution. This investigation was modelling how much of a treatment (base) would be needed to to neutralise different amounts of a bee sting (acid). Their graphs should show a linear relationship between the amount of sting (acid) and the amount of treatment (base) needed to neutralise the sting.

#### Focus activity

- What type of relationship is shown in your graph?
- Describe the relationship you have discovered by graphing your results.
- Using
**a**for the sting (acid, vinegar) and**b**for the treatment (base, baking soda solution), write an**equation**to show the relationship between**b**and**a**.

#### Building ideas

After taking Lilly to the doctor, Charlie has discovered that the unusual bee that stung Lilly actually injected 30 drops of venom when it stung Lilly! This is a rare bee indeed!

Use your graph to tell Charlie how much baking soda solution he needs to add to perfectly neutralise the bee sting and make Lilly’s pain go away.

#### Reinforcing ideas

After taking Lilly to the doctor, Charlie has discovered that the unusual bee that stung Lilly actually injected 37 drops of venom when it stung Lilly! This is a rare bee indeed! Can you figure out how to help Charlie? How much baking soda solution does he need to add to perfectly neutralise the bee sting and make Lilly’s pain go away?

#### Extending ideas

After taking Lilly to the doctor, Charlie has discovered that the unusual bee that stung Lilly actually injected n drops of venom when it stung Lilly! This is a rare bee indeed! Can you figure out how to help Charlie? How much baking soda solution does he need to add to perfectly neutralise the bee sting and make Lilly’s pain go away?

This session focusses on the generalisation and application of a linear relationship. Because the relationship has been derived from a practical context, students should be well placed to discuss the domain and range of the relationship. Looking at the shape of the graph, they should recognise a linear trend. Their data may not fit exactly along the trend-line, which will give the opportunity to discuss experimental uncertainty and the accuracy of their measurements. Discussion could compare the relative merits of using their graph (interpolation, extrapolation) with the use of the algebraic rule of the trend-line to solve a problem.