# MAKING 10

The Making 10 activity is designed to have a low floor and a high ceiling and may be used across a number grades.

• We first did this activity in grades 4/5 classrooms.
• Then in grades 6-8 classrooms.
• And in a grade 9 classrooms.
• We also did part of this activity in grades 1-3 classrooms with concrete materials, as shown in the concept map below.

1. Try this:

• Roll a number cube to get a number 1-6
• Place the number rolled in the first blank of __ + __ = 10
• Calculate and record the missing number for the second blank
• For example, if you rolled a 3, your number sentence would become 3 + 7 = 10
• Repeat this until you exhaust all possibilities
• 3 + 7 = 10
• __ + __ = 10
• __ + __ = 10
• Record the pairs of numbers in a table, and also as ordered pairs
• Plot the ordered pairs on a grid, as shown below for (3, 7): 3 across and 7 up

2. View the video below, which summarizes and extends the activity above.

• What do you notice about the graph?
• Offer an explanation.

3. Run the RANDOM and the CONSTRAINED code blocks at https://scratch.mit.edu/projects/408092534/editor

• What is the difference in the graphs produced by the 2 code blocks?
• What does the RANDOM code do?
• How does it do it?
• What does the CONSTRAINED code do?
• How does it do it?

4. Change x + y = 100 in the CONSTRAINED code as shown below:

• x – y = 100
• x + y < 100
• x + 2y > 100

5. Make your own edits and run the code.

• What did you learn?
• What else do you want to know?

6. Try the simulation at https://imaginethis.ca/mathncode/sims-randpairs.html

• Click on the number cube to run the code
• Edit the + and the 10 in x + y = 10 and notice how the graph changes.

7. Try the game at https://imaginethis.ca/mathncode/games-randpairs.html

• Select a level.
• Edit the code to try to hit the points with water balloons.

# INEQUALITIES

###### x > 3 in 1 Dimension

When introducing the topic of inequalities in grades 5 and 6, use masking tape to form a number line on the classroom floor.

Where on the number line are numbers greater than 3?

###### x > 3 in 2 Dimensions

Then create a second number line on the floor, perpendicular to the first one.

How many coordinates do we need to identify locations on the floor?

Where on the floor is the first coordinate greater than 3?

###### x > 3 in 3 Dimensions

Use string to create a third number line, starting where the first two number lines meet, then extending vertically to the ceiling.

How many coordinates do we need to identify locations in the classroom?

Where in the classroom is the first coordinate greater than 3?

###### Low floor & high ceiling

This activity offers a low floor (minimal prerequisite knowledge), by anchoring student understanding to the physical environment around them.

This activity also offers a high ceiling, by engaging students with multiple meanings and representations of x > 3, in 1 dimension, 2 dimensions, and 3 dimensions.

The different meanings of x > 3 engage student interest and imagination, and motivate them to attend deeply and to understand them.

#### G. Give students code that worKS

Try the code in the video at https://scratch.mit.edu/projects/665007043/editor

#### H. Coding offers cognitive simplicities

For instance, code that does something mathematical, like the code that plots inequalities, does it dynamically.

We can easily change one of the parameters, click on the code, and immediately see the effect. It would be very time-consuming to do this on paper.

This allows students to engage playfully and creatively with mathematical concepts and relationships and to develop conceptual understanding.

Andy diSessa (2000, 2018) defines the phenomenon where coding simplifies and automates a process needed for mathematical investigation, as a cognitive simplicity.

Cognitive simplicities bring mathematics to life in new ways.

They change what and how mathematics is done, and who can do it and when.

As one teacher noted: It’s really neat because it extends their thinking, in a natural way.

#### I. PUZZLE 1

• Run the code. The output is shown below.
• What does the code do? How does it do it?

9. Alter the code to get each of the following.

#### J. PUZZLE 2

• Run the code. The output is shown below.
• What does the code do? How does it do it?

11. Alter the code as shown below.

• Predict how the output will change.
• Run the code to test your prediction.
• Explain the result.

A.

B.

12. Alter the code to get the output shown below.

13. Alter the code as shown below. [Notice that AND changed to OR]

• Predict how the output will change.
• Run the code to test your prediction.
• Explain the result.

#### K. PUZZLE #3

• The code and the output as shown below. Run the code.
• What does the code do? How does it do it?

15. Alter the code to get each of the following.

#### L. PUZZLE #4

• The code and the output as shown below. Run the code.
• What does the code do? How does it do it?

17. Alter the code to get each of the following.

Complete the following.

A. Alter the code as shown below.

• Predict how the output will change.
• Run the code to test your prediction.
• Explain the result.

B. Alter the code as shown below.

• Predict how the output will change.
• Run the code to test your prediction.
• Explain the result.

C. Alter the code as shown below.

• Predict how the output will change.
• Run the code to test your prediction.
• Explain the result.

18. Review the activities above.

• What surprised most you mathematically? Why?
• What new ideas, concepts or relationships did you better understand? Explain.

16. Share your favourite inequality experience with family and friends.

• How will you share your experience so others may also:
• Experience mathematical surprise?
• Better understand a math idea, concept or relationship?