Thought for the day: Prompts that lead students to compare and contrast expressions or equations can be powerful learning experiences!
Concepts: representations of addition and subtraction, properties of addition and subtraction, fact-families, properties of numbers and operations, equality and equivalent names for numbers; rules
I notice two expressions that have the same numbers.
I notice that both expressions have adding and subtracting.
I notice that both expressions start with 10.
I notice that the third number is 7 in both of them.
I notice that the 2 and 4 trade places.
I wonder if both expressions have the same answer.
I notice that both expressions equal 11.
I wonder if there are other ways to change the order of the numbers and keep it equal to 11.
I wonder how many different ways there are to make it equal 11 by just changing the order.
I wonder if I could draw pictures to show why the answers are the same.
I wonder if I could make up stories for these two expressions.
I wonder if my two stories could be about the same thing.
I wonder if I have to start with 10.
I notice that the 2 and the 4 are both subtracted.
I wonder if I could start with the 2 or the 4 and keep the answer equal to 11.
I wonder how many different answers I can make by adding and subtracting these 4 numbers.
I wonder what would happen if I subtracted three of the numbers instead of 2.
As they notice and wonder, students may create and explore their own expressions like the ones in the prompt. Many of their creations may flow out of things that they have wondered about. For example, they may
Create stories for the expressions. Compare and contrast stories with those of other students.
Create as many equations as possible having an answer of 11 by adding and subtracted these four numbers.
Choose four new numbers and use them to create many expressions having the same answer. Compare and contrast with the original four numbers. (Some students may be interested in trying larger numbers.)
Create longer strings of addition and subtraction. Explore what happens after changing the order of their numbers.
Reflecting and Extending
I notice that there are many ways to add and subtract the same four numbers and get the same answer.
I notice that it matters that I always subtract the same numbers, even when put the numbers in a different order.
I notice that I do not get the same answer when I start with one of the numbers that was subtracted.
I wonder if I can find patterns to help me predict which expressions will always give the same answer.
I wonder what happens if I use five numbers.
I wonder what happens if I use parentheses to tell me which operations to do first.
I wonder how using the parentheses would change the stories that I wrote.
Both expressions have a value of 11. In fact, there are many ways to add or subtract the same four numbers to get 11! How many can your students find?
10 + 7 – 2 – 4 10 – 2 + 7 – 4 10 – 4 + 7 – 2 7 + 10 – 2 – 4 7 – 2 + 10 – 4 7 – 4 + 10 – 2
10 + 7 – 4 – 2 10 – 2 – 4 + 7 10 – 4 – 2 + 7 7 + 10 – 4 – 2 7 – 2 – 4 + 10 7 – 4 – 2 + 10
Note: Trying to think of a way to organize the expressions may be helpful, though it may also be very challenging! Consider having your students write each new expression on a 3 x 5 card as they discover it. Then they can try to arrange the cards in a way that helps them notice expressions they may have missed.
Ask students to compare and contrast their expressions in order to help them realize that there are patterns or rules that result in equivalent expressions (expressions that give the same answer). In this case:
You must always start with or add the 10 and the 7. You must always subtract the 2 and the 4.
A few students may suspect (correctly) that it is possible to start with the 2 or the 4 if you make it negative! For example:
–2 + 10 + 7 – 4 or – 4 + 7 – 2 + 10
This opens up many more possible expressions! It is best not to teach rules for adding and subtracting negative numbers to young children. However, if they bring up the idea, they can explore it to their heart’s content! They can try to understand what expressions like these might mean, and they may even discover some rules for themselves!
Asking students to create stories for expressions with multiple parts like this is a wonderful way to help them understand why certain expressions give same answer and others do not. Suggest that they create an idea for a single story and then change to fit each new expression. Then ask them to compare and contrast their stories and explain why they all have the same result (11).
The work that students do on this project will prepare them for understanding the equivalence of certain algebraic expressions such as
a + b – c – d = a – c + b – d = b – d – c + a, etc.
By changing the numbers and/or the lengths of their strings, there is no limit to how much students can explore!