Episode 2

Michael Shayer; Mundher Adhami

Lesson 8 Ladders and slides

Episode 2





Reasoning
Resources: Worksheet 2 on OHT and copies for pupils





Whole class preparation


Understanding the diagram. The left to right distance from start.

Focus on steepness of slides, and how the height of upright supports changes.The column of heights is similar to the lists of numbers in Episode 1.
Introduce Worksheet 2 on OHT as a diagram of three slides which need to be supported by a system of scaffolding. One slide is for seniors, one for juniors and one for infants. Ask about the difference between the slides and why they are different. Encourage ideas about different heights and steepness of slides. Explain that the diagram is not complete: it needs upright rods at 6, 18 and 30 along the ground. Draw in the upright rod at distance 6 to and how the height of demonstrate and fill in the table for that upright: Infant 3, Junior 5, Senior 6. Key question: What relationships can we find between these vertical rods?



Pair and group work 


Using the intuitively recognised relationship to go beyond the diagram.

Finding multiplicative relationships in a spatial or visual context.

Using the context to explore the relationships between sets of numbers derived from the diagram.
Pupils draw the uprights and fill in the table. They look for a relationship between the heights of the upright rods to each slide and the distance along the ground. They are also asked to extrapolate to the rod at 42 which is not on the diagram.

Key question:What relationships are there?



Whole class sharing/discussion


Exploring ratio in a visual context. The complex topic of ratio is approached lightly.
Introduce Worksheet 2 on OHT as a diagram of three slides which need to be supported by a system of scaffolding. One slide is for seniors, one for juniors and one for infants. Ask about the difference between the slides and why they are different. Encourage ideas about different heights and steepness of slides. Explain that the diagram is not complete: it needs upright rods at 6, 18 and 30 along the ground. Draw in the upright rod at distance 6 to demonstrate and fill in the table for that upright: Infant 3, Junior 5, Senior 6.

Key question: What relationships can we find between these vertical rods?



Pair and group work

Pupils draw the uprights and fill in the table. They look for a relationship between the heights of the upright rods to each slide and the distance along the ground. They are also asked to extrapolate to the rod at 42 which is not on the diagram.

Key question: What relationships are there?


Applying what has been learnt to a real-life practical context.

Making connections between relationships between lists of numbers and relationships between distances in a diagram. The table format provides a practical link between the contexts.
Whole class sharing/discussion.

Some ideas may be: Compared with the distance along the bottom the infant slide goes up by 1/2 and the senior side the same up as along. For the junior slide: divide by 6, then multiply by 5: x 5/6 for every 6 along the bottom, there are 5 in the upright.

Write (distance along)/height as a ratio for each slide. Infant slide: distance along/height = 2. We call this the ratio of distance along to height. The ratio is the same for each upright rod in this slide. Similarly, junior slide ratio is 6/5. Senior slide ratio is 1.

Show that the correct responses and ratios are equivalent. To consolidate, discuss what would happen at distance 42 if everything was extended. The relationships between the heights of the vertical rods of the different slides can be multiplied by 5... and equivalents. Junior to infant: divide by 5, then multiply by 3... Junior to senior: divide by 5, then multiply by 6... Senior to junior: divide by 6, then multiply by 5...




Whole class reflection

Ask the class if there is a difference between multiplication and ratio. The aim is to highlight the near identity, but that multiplication is an operation while a ratio often keeps the two numbers separate, showing how they relate to each other.



EPISODE 3 (EXTENSION)

Using ratio for finding heights of buildings (10-20 mins)
Resources: Worksheet 3 on OHT and copies for pupils



Whole class preparation

Show the OHT of Worksheet 3. How does this relate to what we've just been doing with the diagram of the slides? How can you use the distances you can measure to find the height of the building which you can't measure directly?

Pupils work together to try to solve the problem. Their ideas can then be shared with the rest of the class.





End of Lesson Reflection

What was similar in the two/three activities we have done today Both/all used multiplication or division for patterns between numbers, use of table. What was different? First was just with numbers, second and third had diagrams and were visual. How did working on the first exercise (lists) help you with the second (and third)?

Episode 2

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Reasoning	Resources: Worksheet 2 on OHT and copies for pupils
	Whole class preparation
Understanding the diagram. The left to right distance from start. Focus on steepness of slides, and how the height of upright supports changes.The column of heights is similar to the lists of numbers in Episode 1.	Introduce Worksheet 2 on OHT as a diagram of three slides which need to be supported by a system of scaffolding. One slide is for seniors, one for juniors and one for infants. Ask about the difference between the slides and why they are different. Encourage ideas about different heights and steepness of slides. Explain that the diagram is not complete: it needs upright rods at 6, 18 and 30 along the ground. Draw in the upright rod at distance 6 to and how the height of demonstrate and fill in the table for that upright: Infant 3, Junior 5, Senior 6. Key question: What relationships can we find between these vertical rods?
	Pair and group work
Using the intuitively recognised relationship to go beyond the diagram. Finding multiplicative relationships in a spatial or visual context. Using the context to explore the relationships between sets of numbers derived from the diagram.	Pupils draw the uprights and fill in the table. They look for a relationship between the heights of the upright rods to each slide and the distance along the ground. They are also asked to extrapolate to the rod at 42 which is not on the diagram. Key question:What relationships are there?
	Whole class sharing/discussion
Exploring ratio in a visual context. The complex topic of ratio is approached lightly.	Introduce Worksheet 2 on OHT as a diagram of three slides which need to be supported by a system of scaffolding. One slide is for seniors, one for juniors and one for infants. Ask about the difference between the slides and why they are different. Encourage ideas about different heights and steepness of slides. Explain that the diagram is not complete: it needs upright rods at 6, 18 and 30 along the ground. Draw in the upright rod at distance 6 to demonstrate and fill in the table for that upright: Infant 3, Junior 5, Senior 6. Key question: What relationships can we find between these vertical rods?
	Pair and group work Pupils draw the uprights and fill in the table. They look for a relationship between the heights of the upright rods to each slide and the distance along the ground. They are also asked to extrapolate to the rod at 42 which is not on the diagram. Key question: What relationships are there?
Applying what has been learnt to a real-life practical context. Making connections between relationships between lists of numbers and relationships between distances in a diagram. The table format provides a practical link between the contexts.	Whole class sharing/discussion. Some ideas may be: Compared with the distance along the bottom the infant slide goes up by 1/2 and the senior side the same up as along. For the junior slide: divide by 6, then multiply by 5: x 5/6 for every 6 along the bottom, there are 5 in the upright. Write (distance along)/height as a ratio for each slide. Infant slide: distance along/height = 2. We call this the ratio of distance along to height. The ratio is the same for each upright rod in this slide. Similarly, junior slide ratio is 6/5. Senior slide ratio is 1. Show that the correct responses and ratios are equivalent. To consolidate, discuss what would happen at distance 42 if everything was extended. The relationships between the heights of the vertical rods of the different slides can be multiplied by 5... and equivalents. Junior to infant: divide by 5, then multiply by 3... Junior to senior: divide by 5, then multiply by 6... Senior to junior: divide by 6, then multiply by 5...
	Whole class reflection Ask the class if there is a difference between multiplication and ratio. The aim is to highlight the near identity, but that multiplication is an operation while a ratio often keeps the two numbers separate, showing how they relate to each other.
	EPISODE 3 (EXTENSION) Using ratio for finding heights of buildings (10-20 mins) Resources: Worksheet 3 on OHT and copies for pupils
	Whole class preparation Show the OHT of Worksheet 3. How does this relate to what we've just been doing with the diagram of the slides? How can you use the distances you can measure to find the height of the building which you can't measure directly? Pupils work together to try to solve the problem. Their ideas can then be shared with the rest of the class.
	End of Lesson Reflection What was similar in the two/three activities we have done today Both/all used multiplication or division for patterns between numbers, use of table. What was different? First was just with numbers, second and third had diagrams and were visual. How did working on the first exercise (lists) help you with the second (and third)?