UNIT 4

SMALLER NUMBERS CAN BE FOUND HIDING IN BIGGER NUMBERS

Contents

Unit description and duration

Student prior learning

Lesson overview and resources

Lesson 1: Counting on

Daily number sense: Benchmark numbers – 10 minutes

Forwards and backwards counting – 15 minutes

Counting on – 15 minutes

Consolidation and meaningful practice: Real counting on – 20 minutes

Lesson 2: Rekenrek numbers

Daily number sense: Number line – 20 minutes

Rekenreks – 30 minutes

Discuss and connect the mathematics – 10 minutes

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Consolidation and meaningful practice: Bar model – 20 minutes

Lesson 3: Domino numbers

Daily number sense: Rekenrek doubles – 15 minutes

Part whole dominoes – 30 minutes

Consolidation and meaningful practice: Real counting on – 15 minutes

Lesson 4: Bar model

Daily number sense: Odd and even numbers – 10 minutes

Building towers – 30 mi

Lesson 5: Domino triangles

Daily number sense: Dot Talk 1 – 15 minutes

Domino triangles – 35 minutes

Discuss and connect the mathematics – 10 minutes

Lesson 6: Doubles

Daily number sense: Dot Talk 2 – 10 minutes

Doubles – 40 minutes

Discuss and connect the mathematics – 10 minutes

Lesson 7: Near doubles

Daily number sense: Rekenrek number talk – 10 minutes

Near doubles – 40 minutes

Discuss and connect the mathematics: Anchor Chart – 10 minutes

Lesson 8: Power dot pro

Daily number sense: Counting strategies – 15 minutes

Power dot pro – 30 minutes

Consolidation and meaningful practice – 15 minutes

Contents

Resource 1: Numbers cards 1

Resource 2: Numbers cards 2

Resource 3: Recording sheet

Resource 4: Rekenrek problems

Resource 5: Bar model

Resource 6: Dot talk 1

Resource 7: Domino triangles 1

Resource 8: Domino triangles 2

Resource 9: Dot talk 2

Resource 10: Doubles memory

Resource 11: Rekenrek number talk

Resource 12: Near doubles

Resource 13: Recording table

Resource 14: Numbers cards 3

Resource 15: Graphic organiser

Syllabus outcomes and content

References

Unit description and duration

This two-week unit provides opportunities to further develop student knowledge, understanding, and skills of combinations of numbers that add up to a given number. Students are provided opportunities to:

• count on from the largest number to find the total of 2 numbers
• recognise and recall different combinations of 2 numbers that add up to a given number
• identify patterns to find all combinations for a given number
• use the bar model to represent parts of a number
• identify and combine numbers which make doubles facts
• describe combinations for numbers using words such as ‘more than’ and ‘less than’, and ‘double’
• find smaller numbers inside larger numbers
• identify near doubles by doubling the smaller number and adding one more or doubling the larger number and subtracting one less
• recognise and record numbers in different representations including words, numerals, and known structures such as dice, rekenreks, and ten-frames.

Student prior learning

Before engaging in these teaching and learning activities, students would benefit from prior experience with:

• identifying combinations of 2 numbers that add up to numbers less than 10
• representing numbers 0-20 with drawings, numerals, symbols, and words
• counting with one-to-one correspondence, recognising that the last number name represents the total number in the collection
• using a ten-frame to represent smaller parts of a number
• exposure to representing numbers on a rekenrek.

Lesson overview and resources

The table below outlines the sequence and approximate timing of lessons; syllabus focus areas and content groups; and resources.

Lesson 1: Counting on

Core concept: Count-by-one strategies help to solve addition problems.

The table below contains suggested learning intentions and success criteria. These are best co-constructed with students.

Daily number sense: Benchmark numbers – 10 minutes

1. Build student understanding of the relationship between numbers by locating the position of them on a number line.
2. Students draw a blank number line with the benchmark numbers 5 and 10 at either end. Ask students to plot the number 7.
3. Students share where they plotted the number 7 and justify its placement.
4. Students then work out how many more from 5 to 7 and how many less from 10 to 7. Choose students to share their responses and how they worked it out.
5. Students draw another blank number line with the benchmark numbers, 15 and 20 at either end. Ask students to plot the number 17.
6. Students share where they plotted the number 17 and justify its placement.
7. Students then work out how many more from 15 to 17 and how many less from 20 to 17. Choose students to share their responses and if they can draw any links between the 2 number lines (see Figure 1).

Figure 1 – Benchmark numbers

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Note: Highlight to students that if it is 2 more from 5 to 7, it is also 2 less from 7 to 5. Students should start to see the pattern between numbers which end in 7.

8. Challenge students to complete the same activity and plot the number 37.

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Forwards and backwards counting – 15 minutes

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Stand students in a circle. Choose a target number between 5-10.

9. Stand students in a circle. Choose a target number between 5-10.

10. Students take turns to count forwards to the target number, sitting down as they say a number. Once the target has been reached, the student who said the target number then stands back up and the count goes back down. Each student stands up again as they say a number.

11. Continue to choose different target numbers and different students to start from within the circle.

Note: Have students start their count at numbers other than zero.

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Counting on – 15 minutes

12. Display Resource 1: Number cards 1. Ask students how they would work out the total. Provide thinking time and then have students turn and talk to discuss their strategy.

13. Select students to share and model their strategy.

14. If a student models counting on from the largest number, focus on this strategy and the explicit steps they undertook. If a student does not model this, explicitly demonstrate counting on.

Counting on: Counts on from the larger number to find the total of 2 numbers.

Note: The first advanced count-by-one strategies students use for addition and subtraction are counting on and counting back.

15. Display Resource 2: Number cards 2. Ask students to use counting on to solve the problem. Students may use fingers, counters, or an individual whiteboard to keep track when counting on.

16. Choose a student to model how they solved the problem, highlighting the counting on strategies.

Resource 1: Numbers cards 1

Resource 2: Numbers cards 2

Consolidation and meaningful practice: Real counting on – 20 minutes

This activity has been adapted from Real Counting On by Van de Walle et al. (2019)

17. Have a deck of cards (0-9), Resource 3: Recording sheet, and counters. Sitting in a circle, choose different students to play against.

18. Turn over a card and place the indicated number of counters in the red circle and place the card above the circle. The student turns over a card and places the indicated number of counters in the blue circle and place the card above.

19. Together, determine which number is the largest number. Students then count on from the larger number and record the total (see Figure 2).

Figure 2 – Game play

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2O Once total has been determined, clear the board, and turn over new cards. Play again with a different student.

22. Once students are confident with the activity, provide pairs of students with a deck of cards (0-9), Resource 3: Recording sheet, and counters. Students play with their partner.

Note: Using a reusable sleeve with Resource 3: Recording sheet will allow for continual use. An ace can represent zero.

22. While students are playing, ask:

• How are you finding the total?
• How do you know your total is correct?
• Which number are you counting on from?

The table below details assessment opportunities and differentiation ideas.

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Resource 3: Recording sheet

LESSON 2:

REKENREK NUMBERS

CORE CONCEPT:

PATTERNS HELP TO IDENTIFY NUMBER COMBINATIONS.

• Lesson 2: Rekenrek numbers
• How do you know your total is correct?
• Which number are you counting on from?
• The table below details assessment opportunities and differentiation ideas.

• The table below contains suggested learning intentions and success criteria. These are best co-constructed with students.

Daily number sense: Number line – 20 minutes

1. Build student understanding of numeral identification and order by correctly sequencing numbers.
2. Write 9, 11, 15, 17, and 20 on the board in random order. Using their individual whiteboards, ask students to order these numbers from smallest to largest on a blank number line.
3. Choose students to share where they placed their numbers and justify the position of their placement.

Note: It is important to look at the placement of numbers on the number line. Check if students have considered the missing numbers or placed all the numbers together.

4. Ask students to identify and add the missing numbers.

5, Repeat the above steps for different collections of numbers.

The table below details assessment opportunities and differentiation ideas.

Rekenreks – 30 minutes

This activity has been adapted from Introducing rekenreks (11:53) from Thinking Mathematically.

Watch Introducing rekenreks (11:53) prior to teaching this lesson.

6. Display a 20-Bead rekenrek and ask students what they notice.

The table below outlines prompts to generate conversation about the topic, along with anticipated responses from students.

7. Provide pairs with a 20-bead rekenrek.

Explain that beads are moved across to represent quantities.

Note: Find out How to make a rekenrek (5:29) or access a digital rekenrek.

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8. Challenge students to represent 4 on their rekenrek.

• Look for different representations from students and choose them to explain how they represented 4. Record the different representations and draw attention to the pattern of the numbers (see Figure 3).

Figure 3 – Rekenrek pattern

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9. Choose different numbers for students to represent and share with the class. Continue to record and look for patterns.

10. Display and read Resource 4: Rekenrek problems. Students work in pairs to find as many solutions as possible using their rekenrek for both problems. Students record their thinking (see Figure 4).

Resource 4: Rekenrek problems

Discuss and connect the mathematics – 10 minutes

11. Summarise the lesson together, drawing out some key mathematical ideas. Ask students:

• How did the rekenrek help to solve the problem?
• How do you know you have all the solutions?
• How do patterns help to solve problems?
• How did you work like a mathematician today?

Note: When discussing the question about whether students have represented all the solutions, focus on the structure of the pattern. The pattern and order of combinations might not come naturally. Support students to understand where to start and end.

The table below details assessment opportunities and differentiation ideas.

Lesson 3: Domino numbers

Core concept:

Patterns help to identify number combinations.

• The table below contains suggested learning intentions and success criteria. These are best co-constructed with students.

Daily number sense: Rekenrek doubles – 15 minutes

This activity has been adapted from Doubling from Beadstring Mathematics by Swan (2020).

1. Build student understanding of doubles by representing numbers on a rekenrek.

2. Provide each student with a rekenrek or a digital rekenrek and ask them to represent double 4.

3. Students share their representation and explain how they see it. Record student responses (see Figure 5).

Figure 5 – Student working

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Provide other numbers under 10 for students to represent as doubles.

Figure 5 – Student working

4. Provide other numbers under 10 for students to represent as doubles.

Part whole dominoes – 30 minutes

This activity has been adapted from Representing part whole with dominoes (2022).

5. Provide pairs with a large collection of dominoes, more than 20. Students work together to organise their collection to find dominoes that form combinations for a given number.

6. When dominoes have been organised, students select one number and write all the combinations (see Figure 6).

Figure 6 – Domino patterns

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Images sourced from Canva and used in accordance with the Canva Pro Content Licence.

Note: In Lesson 2, students were exposed to the concept of a pattern to represent all combinations. Students may naturally organise their thinking using a systematic pattern, however, do not prompt students to do this at this stage of the lesson.

7. Students display their work and go on a gallery walk to look at how others have structured their combinations. Ask students how they can be confident that they have all the combinations.

Note: Through class discussion, students consider the idea of the pattern and reflect on their work to see if they have applied this. Being able to flexibly partition numbers is critical for building number sense. Combining numbers in set patterns helps with the recall of mental calculations.

8. Students reflect on their working and make changes to reflect the pattern. Students share their working with the class.

The table below details assessment opportunities and differentiation ideas.

Consolidation and meaningful practice: Real counting on – 15 minutes

This activity has been adapted from Real Counting On by Van de Walle et al. (2019)

9. Provide students with number cards (0-9), Resource 3: Recording sheet, and counters and revise the rules from Lesson 1.

10. Students play with their partner, counting on from the largest number. Challenge students to draw on their knowledge of doubles and combinations of numbers to also assist them when solving problems.

Note: Provide students with 2 decks of number cards, 0-9 and 10-20. This will provide opportunities for students to count on and solve addition problems involving one- and two-digit numbers.

Lesson 4: Bar model

Core concept: Concrete materials help to represent the smaller numbers which make up larger numbers.

• The table below contains suggested learning intentions and success criteria. These are best co-constructed with students.

Daily number sense: Odd and even numbers – 10 minutes

This activity has been adapted from Open-Ended Maths Activities by Sullivan et al. (2017).

1. Build student understanding of numbers by identifying if a number is odd or even.
2. Tell students you are thinking of a number between 20 and 30. It is even. Students record any possible responses on their individual whiteboards.
3. Students share their responses.

Note: Look at how students record numbers, haphazardly or using a systematic pattern.

1. Repeat for other number ranges on and off the decade and for odd numbers.

Building towers – 30 minutes

This activity has been adapted from Building towers (7:22) from Thinking Mathematically.

• On 4 sticky notes, have the numbers 5, 3, 11, and 7 written on a set for the class and on another set to model playing against the class.

Note: Watch Building towers (7:22) https://sites.google.com/education.nsw.gov.au/get-mathematical-stage-1/contexts-for-practise/building-towers

for an example of how to play the game.

6. Take turns with students rolling the die and taking the corresponding number of interlocking cubes. Consider where to place the interlocking cubes to build a tower. The aim is for every number on the sticky note to have a tower made up of the same number of interlocking cubes. Model thinking aloud with students about how to separate numbers to create the given number and how many more are needed to get to the target number.

7. When towers have been completed, reflect on the combinations of interlocking cubes used to build the tower and record.

8. Once students are confident with the understanding of the game, put students with a partner. Students divide their individual whiteboard into quarters and write the same 4 numbers as their partner. Guide students to choose 4 numbers under 15.

9. Provide students with a collection of interlocking cubes and a die. Students take turns to play the game.

10. Students reflect and record the combination of numbers in each tower (see Figure 7).

11. Regroup as a class. Students should keep their whiteboard and one of their towers.

• Ask students to turn their tower on the side and create a bar model representation (see Figure 8).

Figure 8 – Student bar model

Note: Part-whole bar model involves one whole divided into 2 or more parts using bars to represent part or whole numbers

12. Students share examples of their bar model. Ask students what the bar model might remind them of. Students Think-Pair-Share.

Note: Through discussion, draw the link between the bar model and a number line.

The table below details assessment opportunities and differentiation ideas.

Consolidation and meaningful practice: Bar model – 20 minutes

• Display the first task card from Resource 5: Bar model and ask students to represent 9 using interlocking cubes and their whiteboard (see ). Students share their representation of 9 and discuss how they have represented it using the bar model.
• Display other tasks cards from Resource 5: Bar model and create other examples. Students complete these using interlocking cubes to represent the bar model and then draw their working.

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See next slide for Resource 5

Resource 5: Bar model

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Lesson 5: Domino Triangles

Core concept: a quantity can be described by talking about its smaller parts.

• The table below contains suggested learning intentions and success criteria. These are best co-constructed with students.

Daily number sense: Dot Talk 1 – 15 minutes

This activity has been adapted from Seeing spots by Boaler et al. (2021).

• Build student understanding of subitising, composing and decomposing numbers by engaging in a dot talk.
• Display Resource 6: Dot talk 1 for a few seconds. Ask students how many dots they saw. Students turn and talk to a partner about how they saw the dots. See next slide
• Students share their thinking by showing or describing the different clusters they saw. Colour code the different groups they saw to help others see their thinking (see ).

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Figure 9 – Recording dot talk

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Note: Students may use the symmetry of the dots, use their knowledge of dice patterns, decompose in rows, or other strategies. These connections support students thinking visually about number and making the connection between the physical objects and the number used to represent them.

Domino triangles – 35 minutes

This activity has been adapted from Domino Squares by Swan (2001).

• Display Resource 7: Domino triangles 1. Explain to students that the 3 dominoes must add up to the number in the middle, for example, 8.
• Allow students thinking time to work out possible solutions using their individual whiteboards. Students share how they have represented the number 8. If needed, model some additional solutions (see ).

Figure – Representing 8

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Resource 7: Domino triangles 1

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Images sourced from Canva and used in accordance with the Canva Pro Content Licence.

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6. Remind students that all numbers are composed of smaller numbers and smaller numbers make up larger numbers. Encourage students to use the language of ‘more than’ or ‘less than’. For example, 10 is 2 more than 8 or 9 is one less than 10.

7. Discuss that, when a ten-frame is full, the total is always 10. Fill the digital ten-frame

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https://sites.google.com/education.nsw.gov.au/math-manipulative/ten-frames-1-20

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with red counters and change one of the counters to yellow. Ask the students what the total is, how many red and how many yellow. Continue this a few times, discussing and recording the total number of counters, the number of red counters and the number of yellow counters (see ).

Figure 11 – Digital ten frame

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8. Provide students with dominoes and Resource 8: Domino triangles 2. These may be cut into 3 separate strips of single dominoes, double dominoes, and triple dominoes. See next slide

9. Students use their understanding of combinations of numbers that add up to 10 and record different solutions (see ).

Images sourced from Canva and used in accordance with the Canva Pro Content Licence.

Resource 8: Domino triangles 2

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The table below details assessment opportunities and differentiation ideas.

Discuss and connect the mathematics – 10 minutes

• 0. Students share and explain their working. Ask students:
• What did you notice?
• Did you check your solution? How?
• How many possible solutions did you find?
• Could there be any other solutions? How do you know?

Lesson 6: Doubles

Core concept: Doubles facts are an efficient way to combine quantities.

The table below contains suggested learning intentions and success criteria. These are best co-constructed with students.

The table below contains suggested learning intentions and success criteria. These are best co-constructed with students.

Learning intentions

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Students are learning that:

• doubles facts are an efficient way to combine quantities
• communication is important when explaining mathematical thinking.

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Success criteria

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• Students can:
• find smaller numbers inside larger numbers
• identify and combine numbers which make doubles facts
• describe combinations of numbers using the word ‘double’.

Daily number sense: Dot Talk 2 – 10 minutes

This activity has been adapted from Dot talks from Boaler et al. (2021).

1. Build student understanding of subitising, composing, and decomposing numbers by engaging in a dot talk.
2. Display Resource 9: Dot talk 2 for a few seconds. Ask students how many dots they saw. Students turn and talk to a partner about how they saw the dots.
3. Students share their thinking by showing or describing the different clusters they saw. Colour code the different groups they saw to help others see their thinking.

Resource 9: Dot talk 2

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Doubles – 40 minutes

This activity has been adapted from Doubles concentration (2002).

4. Using 2 digital ten-frames, demonstrate doubles facts from 0-9 (see ).

Figure 13 – Doubles ten frames

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• For each of the doubles facts, ask students:
• What do you see?
• How can you use what you know about the ten-frame structure to help you determine the total?
• Can you explain your strategy?
• Is there another way to find the total?
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• Encourage students to visualise moving the counters between the 2 ten-frames to firstly make 10 before adding any additional counters in the second ten-frame (see Figure 14).
• Demonstrate this to students using different coloured counters to show how the different amounts have been partitioned and joined.

Figure 14 – Counting on

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6. In small groups, students are provided with Resource 10: Doubles memory. (See next slide)

Students shuffle the cards and lay them out in an array, face down.

Resource 10: Doubles memory

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Discuss and connect the mathematics – 10 minutes

1. As a class, students show and explain their working for doubles facts. Ask students:
1. What strategy did you use to find the total?
2. How did bridging to 10 help you find the total?
3. Are there other strategies you could use to find the total?
4. How did you work like a mathematician today?

The table below details assessment opportunities and differentiation ideas.

Lesson 7: Near doubles

Core concept: Near doubles are an efficient way to combine quantities.

Daily number sense: Rekenrek number talk – 10 minutes

1. Build student understanding of number combinations by looking at how the beads are represented on a rekenrek.
2. Display Resource 11: Rekenrek number talk and students use individual whiteboards to record their thinking.
3. As students share their thinking, ask:
1. What do you notice in the image?
2. What strategy did you use to work out the total?
3. Can you see a pattern?
4. Is your solution different to others? Why/why not?

Resource 11: Rekenrek number talk

Near doubles – 40 minutes

Adapted from Finding known facts (2002).

4. Revise doubles facts from Lesson 6 by displaying a digital rekenrek with different doubles facts, for example, 7 beads on the top line and 7 beads on the bottom line.

5. Display Resource 12: Near doubles. Students turn and talk to determine the total using their knowledge of doubles. Choose students to share their strategy with the class. Highlight the use of near doubles as a strategy with the class.

Note: Near doubles facts can be built upon the idea of one more or one less. For example, 5 and 4, think double 5 is 10 and one less is 9 or for 7 and 8, think double 7 is 14 and one more is 15 (Siemon et al. 2021).

6.Explain that the strategy is to double the smaller number and add one, or to double the larger number and take away one. Model using the example 6 and 7 (see ).

Note: When displaying Resource 12: Near doubles, draw students’ attention to the number 6 playing card and how it can also look like a 9. Remind students to read the number in the top left corner, count the number of hearts, and demonstrate rotating the card to show how the 9 turns into a 6.

7. Using playing cards, select examples of near doubles and model the strategy for different combinations.

8. In pairs, provide students with a packet of playing cards, removing the picture cards, and Resource 13: Recording table.

9. Students take turns to deal 4 cards to each other. Students turn over their cards and look for any known facts they can see. For example, combinations to 10, doubles and near doubles.

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Resource 12: Near doubles

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• 0. For every known fact a student finds, they explain their thinking to their partner who records the information on Resource 13: Recording table. ( See next slide)

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The partner checks and confirms their working on Resource 13: Recording table and gives a counter for each correct fact they found (see ).

Figure 17 – Recording table

Resource 13: Recording table

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11. Students take turns and complete 5 rounds each. The student with the largest number of counters at the end is the winner.

The table below details assessment opportunities and differentiation ideas.

Discuss and connect the mathematics: Anchor Chart – 10 minutes

• Revise doubles and near doubles. As a class create an anchor chart to be displayed with known doubles and near doubles facts (see Figure 18).

Figure 18 – Anchor chart

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Images sourced from Canva and used in accordance with the Canva Pro Content Licence.

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Lesson 8:

Power dot pro

Core concept: There are many ways to combine quantities to find a total.

The table below contains suggested learning intentions and success criteria. These are best co-constructed with students.

Daily number sense: Counting strategies – 15 minutes

1. Build student understanding of smaller numbers make up larger numbers by providing opportunities for students to consolidate and explain their thinking through Resource 15: Graphic organiser.
2. Revise with students the strategies they have learned over the previous lessons including:
1. counting on
2. number bonds to 10
3. doubles and near doubles.
1. Display Resource 14: Numbers cards 3 and provide students with Resource 15: Graphic organiser. Students use drawings, numerals, or words to show how they have solved the problem using known strategies.
2. As a class discuss solutions and share working.

Resource 15: Graphic organiser

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Resource 14: Numbers cards 3

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The table below details assessment opportunities and differentiation ideas.

Power dot pro – 30 minutes

Adapted from Power dot pro by Thinking Mathematically. This game can be played with Tiny Dot Starter Kit or dominoes. Watch the Power dot pro (6:47) video to learn how to play.

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https://sites.google.com/education.nsw.gov.au/get-mathematical-early-stage-1/targeted-teaching/tiny-polka-dots

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5. Model Power dot pro to the whole class by playing against a small group. Share the tiny dot cards evenly amongst players.

6. To start with, turn over 2 cards. As students become more confident with the game, they can increase the number of cards they turn over.

7. All players turn over the assigned number of cards from their deck. Each player combines the total of their cards and the player with the largest total wins the round. The winner of the round places their used cards at the bottom of their pile. The other players put their cards in a discard pile. If there is a tie, each player turns over another card and adds it to their previous total.

8. The game is over when a player runs out of cards.

When students are confident with the understanding of the rules, divide them into small groups with a collection of tiny dot cards.

Note: Print cards on cardboard so that they can be reused in future maths activities.

10. During the game, ask students:

• How did you know what number was represented?
• How did you see the representation?
• Which colour of cards/structures are the easiest to combine for you? Why?
• What do you think is the highest total you could get in this round? Why?

The table below details assessment opportunities and differentiation ideas.

Consolidation and meaningful practice – 15 minutes

As a whole class select 3 cards from the deck, do not show students the cards.

11. Ask students, ‘If we were to combine the quantities represented on these 3 cards’:

What is the smallest possible total?

What is the largest possible total?

What do you think the total might be?

12. Reveal one card and ask students:

• Has your prediction changed?
• What might the total be now?
• Would you revise your thinking? Why/why not?

13. Reveal another card and ask the same questions before revealing the total. Students reflect on their prediction and the final total.

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Resource 14: Numbers cards 3

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