In this unit students investigate the different number pairs that numbers can be broken into, using the context of frogs in ponds. They list all possible combinations for a given number, working with numbers up to 9.

- Give many names for the same number, using the strategies of drawing a picture, acting it out or using equipment.
- Use the mental image of a given number to work out a missing number in a number pair.
- Separate a set of up to 9 objects into two or more parts.

This unit is all about how numbers are made up of other, smaller numbers, an essential concept underlying addition and subtraction. The unit helps develop two ideas:

- there is a finite number of number pairs for a given number (for example 5 can be thought of as 0 and 5, 1 and 4, 2 and 3 and no other pairs can be found)
- numbers are uniquely paired (if 2 is one of the parts of 5, the other part must be 3).

Students need to investigate these relationships many times. Once students believe that 2 and 3 is always 5 they see a real reason to remember it.

Students working on this unit will be using the strategy of Count All, or counting from one, to solve simple addition and subtraction problems. Students at this stage have a counting unit of one and given a joining or separating problem they represent all objects in both sets, then count all the objects to find an answer. Objects may be represented by materials, or later, in their mind as an image.

From this stage of counting all, students will move to Counting On, a stage where they realise that a number can represent a completed count that can be built on.

- Copymaster of the problems.
- Frogs: up to nine for each group of students. These can be plastic models or photocopied and laminated using the frogs copymaster.
- Equipment to use as a pond to hide the frogs: pieces of blue fabric or paper would be appropriate, alternatively ice-cream containers could be used.
- Paper for students to record their solutions.
- Materials for making a wall chart or big book in the final session. Alternatives include paint, crayon and dye, glue, paper, scissors etc.

#### Getting Started

- Introduce the problem:
5 frogs live in a pond.

If 2 of the frogs are sitting on the rock, how many are hiding in the pond?

How many different ways are there for the frogs to be, in and out of the water?Solution:

If there are 2 frogs on the rock, there must be 3 hiding in the pond as 2 + 3 = 5.

There are 6 ways for the frogs to be in and out of the water: none on rock and 5 in pond, 1 on rock and 4 in pond, 2 on rock and three in pond… etc.

- Brainstorm for ways to solve the first part of the problem. Strategies of drawing a picture, using equipment or acting it out could be raised.
- Encourage the students to tell you how they know the number of frogs hiding in the pond. Allow the students to describe their ideas and encourage explanations.
*How did you know how many frogs were hiding?*

Tell us about your thinking.

Could there be any other number of frogs hiding if 2 are on the rock?

How do you know?

- Have the students plan ways to record their solution. Possibilities include drawing a picture, a diagram, some writing or a flip-fold page.
- Read the second part of the problem and let the students try to solve this, in pairs or on their own. (The frogs need to be treated as identical or there are multiple solutions for each number pairing.) Let the students experiment with the pairings of the digits. The following questions may help support their problem solving:
*How do you know how many frogs are on the rock?*

Does there always have to be a frog on the rock? Or hiding in the pond?

How are you keeping track of the ways that you find?

- To conclude the session, have several students share their findings, including the method of recording, with the class. Ensure several different methods of recording are presented and discuss the different ways students used to think about the hiding frogs.

#### Exploring

Over the next two to three days, revisit the problem with the frogs in the pond, varying the number of frogs living in the pond and sitting on the rock. Explain that because the pond is such a nice place to live, more frogs keep moving in.

Three appropriate number combinations to use would be:

6 frogs live in the pond, begin with 3 on the rock.

8 frogs live in the pond, begin with 2 on the rock

9 frogs live in the pond, begin with 4 on the rock.

These problems are provided on the problem copymaster.

Each day follow a similar lesson structure to the introductory session, with students becoming more independent in their search for solutions as the week progresses. Conclude each session by having students share their solutions and compare their different ways of working.

#### Sharing

As a conclusion to the weeks work, have the class work together to make a wall chart illustrating the different combinations of frogs in and out of the water, when 7 frogs are living in the pond (8 possible combinations):

- Introduce the problem – have a large copy for all to use, appropriate for display at the end of the session.

#### Problem:

Seven frogs live in a pond. They like to sit on the rock in the middle of the pond or hide in the water.

How many different ways are there for the frogs to be, in and out of the water?

- Have students work in pairs or individually to come up with a solution.
- Pose the following questions as the students work on the solution:
*How many frogs are there altogether?*

How many are on the rock? How many are hiding?

How do you know?

How could you find out?

How are you keeping track of the ways that you find?*Tell me about your thinking.* - As a class discuss the different combinations possible and list these together.
- Split the students into groups, with each group responsible for illustrating one of the number combinations. Illustrations could use a range of media (paint, crayon and dye etc) or the frog copymaster could be provided for students to use.
- Have the students share their work.
- Display the students' illustrations alongside the problem and revisit the work as appropriate. Alternatively, the illustrations could be made into a big book, using the problem as a cover.

Dear Family and Whānau,

At school this week we are completing a maths unit on Frogs and Ponds. This unit is all about how numbers are made up of other, smaller numbers, an essential concept underlying addition and subtraction. The unit helps develop two ideas:

- there is a finite number of number pairs for a given number (for example 5 can be thought of as 0 and 5, 1 and 4, 2 and 3 and no other pairs can be found)
- numbers are uniquely paired (if 2 is one of the parts of 5, the other part must be 3)

Children need to investigate these relationships many times. Once children believe that 2 and 3 is always 5 they see a real reason to remember it.

At home this week I would like your child to solve the Inside and Outside the house problem. Encourgae your child to record the numbers and draw pictures to show people inside and outside.

_____ people live in my house.

If there are 2 people inside my house then _____ people are outside.

If there are _____ inside my house then _____ people are outside.

If there are _____ inside my house then _____ people are outside.

If there are _____ inside my house then _____ people are outside.