Apple tree multiplication: A sample differentiated single task lesson
An apple farmer plants his trees in straight lines. He likes his lines to have equal numbers of trees. He has 72 new trees to plant in two fields. One field is smaller than the other.
How could he plant out his two fields?
Assumed knowledge: partitioning, arrays
Learning Intention: To understand “groups of” in an organised manner (arrays). To link visual and written representations of times tables.
What the lesson might look like:
- Have someone read the question. Ask for questions/clarifications. Allow those who can make a start to go and start.
- Provide one extra step of clarification for others then expect everyone to make a start.
- Rove and answer questions as necessary. Provide enabling or extending prompts as necessary.
- Select some students to put their solution on the board for all to see (student writes their solution on whiteboard, photograph solution for projector)
- Pause the class working and allow the students you have chosen to share their solutions to explain their working.
- Highlight arrays and patterns in student models.
- Relate arrays and patterns to times tables algorithms. Record times tables.
- Allow students more time to work on the problem.
- Rove and assist as needed. Redirect any misconceptions.
- Expect students to share their solution with at least one other person.
- Stop the class for final discussions of solutions and strategies. Connect solutions to times tables.
- Reflect on what we’ve learnt about times tables. Students to record own learnings.
Possible discussion topics:
- What is an array? How does the apple field show an array? How do arrays show times tables?
- What are good numbers of trees to have in each different field? Why are they good numbers?
- How can you record your working? Show it in more than one way (diagram and number sentence)?
- Find a pattern in your solution. How does the pattern relate to times tables?
- Vary your pattern to show a different times table. Show a different times table in each field.
- How does your solution compare to someone else’s. What is the same, what is different?
- Explain why you think the farmer should plant his fields using your design.
- Relate the solutions to division as well as multiplication.
Enabling question 1: how many trees will you plant in each field, remember one is bigger than the other. (This questions gives a starting point to those who don’t have one)
Scaffold 1: draw a picture – how many trees are there, how are they arranged, how many more are needed to make 72, how could they be arranged?
Extension 1: what if the farmer had 108 trees (takes the number into 3 digits and out of regular times tables for year 3)
Extension 2: You decide how many trees the farmer will plant in three fields. Show how the trees will be planted using a different pattern in each field.
Going further: start making patterns with the number of apples on each tree. Create word problems about apple trees and numbers of apples.
Why 72? It’s large enough, but familiar enough, for year 3s to manipulate. It has lots of factors. It partitions into numbers that have lots of factors. Scaffold 1 – the picture has 24 trees, which leaves 48 trees (double 24) still to be planted.
Download a pdf of the assessment rubric here.