# Pattern Shape Pictures

Practices
Noticing and using mathematical structure
Representing and connecting
Topics
Subtraction
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2
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4
Use an App
Pattern Shapes
3MVX-15G1

How many points is each pattern shape worth?

Ava made pictures using three pattern shapes. Each pattern shape is worth a different number of points.  She found the total number of points for each picture.

• How many points is each shape worth?
• Can you make a picture worth 20 points?
• Can you make a picture worth 10 points?
How could you get started?
• What do you notice about the three pictures that Ava made? What do all three pictures have in common?
• What could the green triangle and blue rhombus be worth in the first picture? Do these values work with the other pictures?
• Choose another shape in the app. Decide how many points it is worth. Make a picture that includes your shape. Tell a friend the total number of points your picture is worth and ask them to figure out how many points the new shape is worth.
• Suppose the values of the pictures changed to 48, 50, and 52. How would the values of the shapes change?
• Suppose the values of the pictures changed to 17, 19 ½, and 21. How would the values of the shapes change?
• Choose your own numbers for the values of the shapes. Tell a friend the values of the pictures and ask if they can find the values of each shape.

In this task, students use algebraic reasoning and number sense as they engage with composite pictures to find the values of each shape. They will also use the relationship between addition and subtraction as they systematically test different values for the shapes. Working with combinations of values to produce the known picture values reinforces basic facts and mental computation.

There are many ways students may approach finding the values of each shape. They might notice:

• Half of the hexagon in the first picture is made with a triangle and a blue rhombus. So the combined values of the two shapes must be half of 14. Using this relationship, they may choose pairs of values for the triangle and blue rhombus and make adjustments if their first choice does not work.
• Pictures have shapes in common. For example, pictures 2 and 3 have three shapes in common—1 blue rhombus, 1 black rhombus, and 1 triangle. The only difference is that picture 2 has another triangle and picture 3 has another blue rhombus. Since the third picture is worth 1 more point than the second picture, students can reason that the value of the blue rhombus must be 1 more than the value of the triangle. They can then test pairs of numbers that have a difference of 1 for the values of the triangle and blue rhombus.

As students compare the shapes in the pictures, they may notice that there is at least 1 triangle and 1 blue rhombus in each picture. Recognizing that this combination is half of 14, they can find the values of part of each of the pictures. Here’s the beginning of a solution that uses this visual approach.

To extend students’ thinking about number relationships, ask How could you use the relationship between addition and subtraction to help you find the point value of the black rhombus? Based on the first picture, why does it make sense that the blue rhombus and triangle together are worth half of 14? To extend students' algebraic reasoning, ask questions like We know that the second picture is worth 1 more than the first picture. What does this mean for the values of the blue and black rhombi in the second picture? What similar relationship can you find by comparing the second and third pictures? Or the first and third pictures?