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 acids and bases. SI units of measure are used to provide concrete reinforcement for the knowledge and skills used in rates and ratios and the application of the gradient of a linear relationship to solve problems.

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 bases.

Students will solve problems using proportions described as a ratio or as a decimal value and to solve problems involving rates. Students will develop and apply their understanding of the multiplicative relationship between different rates and measures, scaling up or down as is appropriate to the situation given.

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.

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 introductory activity 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):

Given the acid HCl (0.1 molL^{-1}) and the base NAOH (0.1 molL^{-1}), form each of the following solutions in a test tube. Place the test tubes with these solutions in the order specified. When this process is complete, add one drop of Universal Indicator to each test tube to measure the pH of the solution formed. Record all your results in a table. Graph the pH of the solution against the proportion of NaOH used to form that solution.

Make up 5 mL solutions in test tubes, formed in ratio of HCl:NaOH as follows:

- 3:1
- 3:2
- 1:1
- 2:3
- 1:3

In this activity, the teacher will be able to locate their students on the Multiplicative Thinking learning progression by observing how students manage ratios (scaling up to form a multiple of 5 parts) and how they find and express a proportion. They will be able to locate their students on the Patterns and Relationships learning progression using the approach students take to recording data on a table and the appropriateness and accuracy of their graphing.

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. The equipment that should be used by each group of students is:

- a rack of 5 test tubes
- droppers or pipettes
- universal indicator
- a universal indicator colour chart giving the whole number pH scale 1-14
- HCl (0.1 molL
^{-1}) and NaOH (0.1 molL^{-1}) solutions, in beakers or bottles.

If the equipment is available, this could be carried out as a titration, with the acid and base each in a burette. The results will not be linear, rather they will show a titration curve. The universal indicator results should be close to the values expected from calculation (1, 2, 7, 12, 13 respectively).

Mathematical discussions that should follow this activity involve:

- How was the problem of measuring a 5 mL solution based on a ratio of 4 (3:1) or 2 (1:1) parts resolved?
- What was the shape of the graph of proportion of NaOH solution vs pH? (If linear, the students will have ‘curved’ their axes, so this issue will need to be addressed). Ask the students: “If we only made three solutions, the 3:2, 1:1, and 2:3, what type of relationship would we think we had”, as an opening to discussion of when we can be certain of a relationship being linear and talk about some non-linear relationships having a linear section.
- As an extension - given that pH stands for ‘power of the concentration H
^{+}, this could link into discussion around standard form and the shape of a graph of an exponential relationship.

### Session two

Focussing on using rates and ratios, reasoning with linear proportions to solve a problem.

**Focus activity**

Plan how to make up solutions of NaOH of concentrations of 0.25 molL^{-1}, 0.2 molL^{-1} and 0.1 molL^{-1}, given a solution of NaOH at 0.5 molL^{-1} and any amount of distilled water.

Discussion arising from activity:

- How do we understand the measurement molL
^{-1}as a concentration? - How does this compare with gL
^{-1}? - Which measurement would be simpler to use mathematically?
- Why might chemists prefer to use this than gL
^{-1}?

**Building ideas**

- How many mol of NaOH are needed to make a solution, in 1L of distilled water, at a concentration of 0.1 molL
^{-1}? - How many mol of NaCl (table salt) are needed to make a solution, in 1L of distilled water, at a concentration of 0.1 molL
^{-1}? - How many mol of HCl are needed to make a solution, in 1L of distilled water, at a concentration of 0.1 molL
^{-1}?

**Reinforcing ideas**

- How many mol of NaCl (table salt) are needed to make a solution, in 0.5L of distilled water, at a concentration of 0.1 molL
^{-1}? - How many mol of NaCl (table salt) are needed to make a solution, in 0.5L of distilled water, at a concentration of 0.2 molL
^{-1}? - How many mol of NaCl (table salt) are needed to make a solution, in 0.25L of distilled water, at a concentration of 0.1 molL
^{-1}? - How many mol of NaCl (table salt) are needed to make a solution, in 0.25L of distilled water, at a concentration of 0.2 molL
^{-1}?

**Extending ideas**

- How many
**grams**of NaCl (table salt) are needed to make a solution, in 1L of distilled water, at a concentration of 0.1**molL**?^{-1} - Convert the concentrations of NaCl at 0.25 molL
^{-1}, 0.2 molL^{-1}and 0.1 molL^{-1}, into gL^{-1}. - Find a rule to generalise the conversion of any concentration of NaCl given in molL
^{-1}to gL^{-1}. Would this conversion work for any substance, or just NaCl?

### Session three

Focussing on using rates and ratios, reasoning with linear proportions to solve a problem.

**Focus activity**

Encourage students to set their working out in a table.

For a solution of NaOH and HCl to be neutral (pH = 7) there needs to be an equal amount of substance (number of moles) of each. How much is NaOH is needed in order to neutralise 1 mL of HCl of concentration 0.1 molL^{-1} if the concentration of NaOH is:

- 0.1 molL
^{-1}? - 0.2 molL
^{-1}? - 0.25 molL
^{-1}?

Discussion arising from activity:

- Did students solve this problem by considering the concentrations of each reactant and what scaling is needed for the NaOH to have the same amount of substance (number of mol) as the HCl?
- How did they apply proportional reasoning to solve this problem?
- What were the advantages of setting their working out in a table?

**Building ideas**

Calculate the number of moles of a substance present in:

- 1 L of substance at a concentration of 0.1 molL
^{-1}. - 0.2 L of substance at a concentration of 0.1 molL
^{-1}. - 10 mL of substance at a concentration of 0.1 molL
^{-1}.

**Reinforcing ideas**

Calculate the number of moles of a substance present in:

- 2 mL of substance at a concentration of 0.1 molL
^{-1}. - 2 mL of substance at a concentration of 0.2 molL
^{-1}. - 10 mL of substance at a concentration of 0.2 molL
^{-1}.

Calculate the volume of a solution if it has:

- a concentration of 0.1 molL
^{-1}with 1 mol of substance in the solution. - a concentration of 0.1 molL
^{-1}with 0.1 mol of substance in the solution. - a concentration of 0.2 molL
^{-1}with 0.1 mol of substance in the solution.

**Extending ideas**

- Calculate the number of moles of a substance present in:
- 2 mL of substance at a concentration of 0.1 molL
^{-1}. - 5 mL of substance at a concentration of 0.2 molL
^{-1}. - 10 mL of substance at a concentration of 0.05 molL
^{-1}.

- 2 mL of substance at a concentration of 0.1 molL
- Give a general rule for finding the number of moles of a substance present in a given volume of a known concentration.
- Calculate the volume of a solution if it has:
- a concentration of 0.1 molL
^{-1}with 0.1 mol of substance in the solution. - a concentration of 0.1 molL
^{-1}with 0.2 mol of substance in the solution. - a concentration of 0.2 molL
^{-1}with 0.25 mol of substance in the solution.

- a concentration of 0.1 molL
- Could you have used the rule that you found in task 2 to answer task 3?

### Session four

Focusing on relating a linear pattern shown in a table and to the graph of that same data.

**Focus activity**

Look at this graph of volume of HCl added (vertical axis) vs volume of 0.1 molL^{-1} NaOH (horizontal axis), to form a neutral solution (all volumes are in mL).

Describe the shape and trend of this graph. What is the concentration of the HCl?

Discussion arising from activity:

- What is the shape of the relationship shown in the graph?
- What is the gradient of the graph? What is the unit of measure of the gradient?

**Building ideas**

Locate the position on the graph where 10 mL of HCl was added to neutralise the NaOH. We are going to work out the concentration of the HCl from this point.

What is the volume of NaOH that was neutralised by 10 mL of HCl?

The concentration of NaOH is 0.1 molL^{-1}. How many moles of NaOH is present in the volume at that point on the graph?

How many moles of HCl are present at that point on the graph?

How many moles of HCl are present in 10 mL of the HCl solution?

How many moles of HCl are present in 1 L of the HCl solution?

What is the concentration of the HCl solution?

**Reinforcing ideas**

How would the graph change if:

- the concentration of HCl is doubled?
- the concentration of HCl is halved?
- the concentration of NaOH is doubled?
- the concentration of NaOH is halved?

**Extending ideas**

Give a general rule for the graph of volume HCl vs volume NaOH shown in the starter activity.

How would the **general rule** change if:

- the concentration of HCl is doubled?
- the concentration of HCl is halved?
- the concentration of NaOH is doubled?
- the concentration of NaOH is halved?

### Session five

Focusing on relating a linear pattern shown in a table and to the graph of that same data. Within the science learning area, the focus is on understanding pH as the power of H^{+} ions.

**Focus activity**

Note that there are two H+ ions in H_{2}SO_{4}, but only one OH^{-} ion in NaOH.

H_{2}SO_{4} neutralises NaOH in approximately the ratio of 1:2 H_{2}SO_{4}:NaOH molecules.

Look at this graph of volume of H_{2}SO_{4} added (vertical axis) vs volume of 0.1 molL^{-1} NaOH (horizontal axis), to form a neutral solution (all volumes are in mL).

Describe the shape and trend of this graph. What is the concentration of the H_{2}SO_{4}?

Discussion arising from activity

What is the shape of the relationship shown in the graph?

What is the gradient of the graph? What is the unit of measure of the gradient?

**Building ideas**

Locate the position on the graph where 10 mL of H_{2}SO_{4} was added to neutralise the NaOH. We are going to work out the concentration of the H_{2}SO_{4}from this point.

- What is the volume of NaOH that was neutralised by 10 mL of H
_{2}SO_{4}? - The concentration of NaOH is 0.1 molL
^{-1}. How many moles of NaOH is present in the volume at that point on the graph? - How many moles of H
_{2}SO_{4}are present at that point on the graph? - How many moles of H
_{2}SO_{4}are present in 10 mL of the H_{2}SO_{4}solution? - How many moles of H
_{2}SO_{4}are present in 1 L of the H_{2}SO_{4}solution? - What is the concentration of the H
_{2}SO_{4}solution?

**Reinforcing ideas**

How would the graph change if:

- the concentration of H
_{2}SO_{4}is doubled? - the concentration of H
_{2}SO_{4}is halved? - the concentration of NaOH is doubled?
- the concentration of NaOH is halved?

**Extending ideas**

Give a general rule for the graph of volume H_{2}SO_{4} vs volume NaOH shown in the starter activity.

How would the **general rule** change if:

- the concentration of H
_{2}SO_{4}is doubled? - the concentration of H
_{2}SO_{4}is halved? - the concentration of NaOH is doubled?
- the concentration of NaOH is halved?

### Session six

Focussing on using rates and ratios, reasoning with linear proportions to solve a problem.

Nb For the purposes of simplification of the idea of neutralisation, only strong acids and strong bases have been used in these examples. eg H_{2}SO_{4} neutralises NaOH in approximately the ratio of 1:2 H_{2}SO_{4}:NaOH molecules.

**Focus activity**

Given 2 mL of each of the following acid solutions;

- 0.1 molL
^{-1}HCl - 0.2 molL
^{-1}HCl - 0.1 molL
^{-1}H_{2}SO_{4} - 0.2 molL
^{-1}H_{2}SO_{4}

what volume of 0.1 molL^{-1} NaOH would be needed to neutralise the acid?

Discussion arising from activity:

How did you solve these problems?

Were you using proportional reasoning?

What pattern or relationship did you notice when solving these problems?

**Building ideas**

- What volume of 0.1 molL
^{-1}NaOH would be needed to neutralise 1 mL of 0.1 molL^{-1}HCl? - What volume of 0.1 molL
^{-1}NaOH would be needed to neutralise 2 mL of 0.1 molL^{-1}HCl? - What volume of 0.2 molL
^{-1}NaOH would be needed to neutralise 1 mL of 0.1 molL^{-1}HCl? - What volume of 0.2 molL
^{-1}NaOH would be needed to neutralise 2 mL of 0.1 molL^{-1}HCl?

**Reinforcing ideas**

Given 5 mL of each of the following acid solutions;

- 0.1 molL
^{-1}HCl - 0.2 molL
^{-1}HCl - 0.1 molL
^{-1}H_{2}SO_{4} - 0.2 molL
^{-1}H_{2}SO_{4}

what volume of 0.2 molL^{-1} NaOH would be needed to neutralise the acid?

**Extending ideas**

Given 5 mL of each of the following acid solutions;

- 0.2 molL
^{-1}HCl - 0.05 molL
^{-1}HCl - 0.25 molL
^{-1}H_{2}SO_{4} - 0.01 molL
^{-1}H_{2}SO_{4}

What volume of 0.1 molL^{-1} Ca(OH)_{2} would be needed to neutralise the acid?

How would your answers change if Al(OH)_{3} was used instead of Ca(OH)_{2}?

nb – For the purpose of simplification of the idea of neutralisation, assume that:

HCl neutralises Ca(OH)_{2} in **approximately **the ratio of 2:1 HCl:Ca(OH)_{2}, and

HCl neutralises Al(OH)_{3} in **approximately **the ratio of 3:1 HCl:Al(OH)_{3}.