Mathematical modelling: Year 1: Planning tool

Expected level of development

Australian Curriculum Mathematics V9: AC9M1N05, AC9M1N06

Numeracy Progression: Counting processes: P6, Additive strategies: P6, Multiplicative strategies: P2, Understanding money: P3

At this level, students use mathematical modelling to solve practical problems involving additive situations including simple money transactions. They also use mathematical modelling to solve practical problems involving equal sharing and grouping. They represent the situations with diagrams, physical and virtual materials, and use calculation strategies to solve the problem.

Provide novel contexts, such as story problems and challenging tasks, to apply and consolidate addition and subtraction. Story problems involving additive situations can emerge as a result of recent classroom experiences, discussions or stories read. Where possible, explore stories involving additive situations in different places and cultures.

Create opportunities for role-playing additive situations, such as keeping track of the number of people in a bus at each stop, as people get on and off. Include scenarios involving money transactions using whole dollar amounts. For instance, set up a shop and engage students in buying and selling groceries.

Explore various ways to model problems including using physical and virtual materials and a think board diagram. Be explicit about the value of mathematical modelling for confidently representing and communicating thinking and ideas.

Use similar approaches to explore and model multiplicative situations – those involving sharing and grouping. Use modelling materials and draw a picture when working to solve grouping problems like ‘How many wheels are there on four bicycles?’ as well as sharing problems like ‘How many gumnuts will we each have if we share them equally between the two of us?’ Include a focus on equal sharing in situations involving money. For instance, ensure a collection of coins is shared equally and fairly between four people.

Once students are confidently solving initial problems, encourage them to use the ‘What if …?’ stem as a way of extending themselves. For instance, after solving the bicycle problem above, they could create a more challenging question, such as ‘What if there were eight bicycles?’ or ‘What if there were four tricycles (instead of bicycles)?’ This can be a powerful way to facilitate self-management and self-differentiation.

There is an opportunity to connect existing knowledge and skills in addition and subtraction (AC9M1N04) and multiplication and division (AC9M1N03).

Teaching and learning summary:

Explore and model additive and multiplicative situations that are meaningful to students using materials, stories, role-play, games, challenging tasks and regular classroom talks.
Build habits of reasoning by inviting students to think about solving a problem in more than one way.
Ensure a playful culture of learning where mistakes and time for sense-making are seen as valuable to learning.

Students:

model, describe and solve a variety of addition and subtraction problems
model simple money problems involving whole dollar amounts with addition and subtraction
model problems that involve repeated grouping
model money problems involving equal sharing using whole dollar amounts.

Some students may:

not yet grasp that assigned money values don’t necessarily relate to size or colour. To help address this, hold regular conversations that build on existing knowledge of fairness. For instance, ask: Would it be fair for me to give you this $1 coin in exchange for that $2 coin? Why/why not?
have trouble interpreting a problem due to the literacy demands involved. To help address this, use problems that are accompanied by visual representations. Or use photos involving number, for example, a scene with countable animals, and provide key language such as ‘whole group’, ‘altogether’ and ‘how many …’ to invite students to generate their own story problems. This is one way of building understanding and language over time.
have trouble interpreting exactly what a problem is asking. To address this, provide regular opportunities to ask questions that give less information than is required to solve the problem. For instance, Felicity bought a carton of milk. How much change did she get from $5? Questions like this with incomplete information provoke discussions that involve thinking deeply about the situation. Over time, these deep-thinking opportunities about contexts will support general ability to solve everyday problems.

The Learning from home activities are designed to be used flexibly by teachers, parents and carers, as well as the students themselves. They can be used in a number of ways including to consolidate and extend learning done at school or for home schooling.

Learning intention

We are learning to solve everyday problems that involve making calculations.

Why are we learning about this?

We add, subtract, make groups and share in everyday life.

What to do:

When engaged in creative play, notice situations involving addition, subtraction, grouping and sharing. Model curiosity and pose questions such as:

How many books have we here altogether? What if we shared them out equally between the two of us? I wonder if it would be an equal share? I wonder how many we’d each get?
There are 4 wheels on the truck and another 4 wheels on the car. I wonder how many wheels altogether? How can we know for sure?
How many gumnuts have you there? What if I took these 4 away, how many would be left?
Here we have a collection of 12 pebbles. Is it possible to share these equally between the two of us? I wonder how many we’d each get. What if there was three of us? What if there were four of us?

Read storybooks and discuss situations involving addition, subtraction, grouping and sharing. Model curiosity and pose questions such as:

Dog is joined by his friends. How many animals are there now?
Oh, the goat just ate another apple! There were 5 apples. How many are left now?

Link making calculations to everyday situations. Examples might include the following.

Provide opportunities to play with, sort and add up money.
When shopping, or when playing shops, talk about the total number of items in the basket. For example, 'there are 4 apples, 5 bananas and 2 oranges. How many pieces of fruit altogether?'
When drawing pictures, notice and describe additive aspects such as the number of legs on two dogs (4 and 4) and the number of legs on the person (2) makes a total of 10 legs.
Make money calculations using whole dollars. For example, 'the milk costs $3 and the bread costs $3. So altogether, how much do we need to pay? If we pay with a $10 note, how much change will we get?'

Think about and discuss different ways of solving addition and subtraction. For example, to solve 7 + 6 you might decide to:

count up from 7, 6 times (... 8, 9, 10, 11, 12, 13)
break the 6 up into 3 and 3 to make a ‘friendly 10’, then 7 + 3 = 10, then 10 + 3 = 13
start with a known doubles fact of 6 + 6 = 12, then add 1 more to get 13.

Know that mistakes are a normal part of building mathematical understanding. Be curious about mistakes and see them as opportunities to learn. Take plenty of time to solve problems.

Success criteria

I can:

think about what’s happening in a situation and explain how I might solve it
use my knowledge of number and making calculations to work through everyday situations
consider whether an answer seems correct by thinking carefully about the situation
repeat the process of solving a problem if it doesn’t seem right.

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