Replace the Shapes
What numbers could make this equation true?
Each of these symbols represents a number. Only one of the numbers is a whole number. What could the three numbers be?
Hint: A whole number is a number such as 0, 1, 2, 3....
- How might you represent the relationships in this equation?
- Think about how you could start. What happens if you explore values for the heart and circle? What happens if you instead choose a value for the square?
- Which symbol do you think could make sense as the only whole number?
- Is your answer the only solution? How do you know?
- If you keep your whole number the same, how might you change the other two numbers?
- What if none of the numbers are whole numbers? Then what could the three numbers be?
- Try creating your own number puzzle and sharing it with a friend. Consider using division or subtraction in your equation.
In this task, students need to determine one or more sets of values for three symbols in an equation, only one of which is a whole number. Students decompose the decimal number 2.36 into a product (of the heart and the circle) and an addend (the square). Students will draw upon their understanding of operations, inverse operations, and decimals to make sense of this equation and come to a possible solution.
Since there are many ways to decompose 2.36, there are many unique combinations of values for the heart, circle, and square. Students may:
- Build a model for the total (2.36), remove a value for the square, and work backward by building equal groups or an array for the remaining amount.
- Build equal groups or an array with a value less than the total and determine how much more is needed to make 2.36.
- Separate a model for 2.36 into two convenient numbers (e.g., 2 and 0.36) and find equal groups or build an array with a product of either 2 or 0.36.
Students may choose to use an app to explore this problem and model the equation.
- In the Number Pieces app, students might use the 10 x 10 piece to represent 1 whole, and then use the pieces to build 2.36. From there, they could determine how to express 2.36 as a product of two numbers and an additional addend, as shown here.
- The Number Pieces app might also be used to model multiplication of decimals on a decimal grid. Here are several ways to multiply two decimals to get a product of 0.36, with both the heart and the circle as decimal numbers.
- In the Number Line app, they might choose to use the “jump” feature to model either the multiplication, the addition, or both. Here’s one visual model, representing the operations without a specific solution.
- In the Money Pieces app, they might choose to make equal groups of coins to start. Then, they would determine the missing addend to make $2.36.
To extend their thinking about the relationships among operations, consider how students approached their solution. If they start with a value for the square, ask: How does the relationship between addition and subtraction tell you something about the product? If students start with the product of the heart and the circle, ask: How does the relationship between addition and subtraction tell you something about the value of the square? No matter the solution process, ask: How do your models of multiplication and addition look different? Pay attention to how students model and describe multiplication and addition, as this will provide insights into their understanding of these operations.