Today, I'm doing an evaluating expressions review.
When students enter the classroom, the entry work is posted on the board. The morning message asks students to:
Choose the numerical expression that matches the phrase on the board, I provide multiple choices to help the students think about the small nuances of the phrase, be prepared to explain your thinking. If you have extra time, please make up an algebraic expression and then write 3 possible choices to match it.
Seven less than twice the amount of d dollars in a piggy bank.
7 - 2d
(7 - 2) x d
2d - 7
The students and I gather on the carpet in a circle. Each child is given a small handful of place value "ones". I ask the students to place "one" in front of their work space. Then add one more. I continue with this until they have ten in their work space. Then I ask the students to remove two. I continue with the pattern of adding and removing and eventually arrive back at ten once again.
What operations were we practicing then we were adding and removing these blocks? (Addition and subtraction.)
I noticed when we were doing this that is was hard to keep your row of blocks neat. Before be move on, lets make a bundle (or group) to represent ten, rather than 10 ones. How can we make a group (or bundle) of ten using the place value cubes? I'm looking for students to suggest trading 10 ones for 1 ten. At this point I give each student one "group of ten".
Now, you each have 1 group of 10. How did we form this group? I'm headed toward a discussion about equal groups and the operations of multiplication and division.
A tens rod represents 1 group of 10. Students understand that multiplication represents a number of groups. For example, 4 x 5 means 4 groups of 5. This understanding has been established through prior lessons. To reinforce this, before trading 10 ones for a rod (group of 10), I write 1 x 10 on the whiteboard and ask students what this number sentence states. We agree that it means 1 group of 10.
Ask the students to divide the group in half. This will spark discussion because some students will think it is impossible, since the 10 ones are now in one rod. Others might say it is impossible without regrouping. My objective here is for students to discover that the connections made when operations form groups are stronger than when we add and subtract.
Before now, did you know that different operations have different strengths? Let's make a list of the operations based on their strength. Lead the discussion, with the models, so students agree that addition and subtraction are of equal strength (because they are opposite operations), multiplication and division represent groups, these connections are stronger than addition and subtraction, but multiplication and division are of equal strength. When solving problems with more than one operation, we have to remember to solve the strongest connections first.
During the Launch, I solve a few problems with the students' participation. One student shares, "I don't know why you are solving the problem that you don't know, before you are solving the problem that you do know".
Together, we generate an example: 13 x 7 + 3. This opens a great conversation, comparing strategies we have encouraged students to use when there is one operation such as "make ten" or "look for a basic fact", with what we have learned when solving problems with more than one operation. Students' questions are so important, because they help make connections for their peers. Providing opportunities for students to ask questions is invaluable to growing conceptual understanding.
Students return to their seats for guided practice. To help students practice what we discovered in the launch, I start with problems with only the 4 operations.
We will solve about 4 problems of this type together:
1) 5 + 4 x 2
2) 15 - 6 / 2 (use the division symbol and the / interchangeably when writing on the board)
3) 24 - 2 x 6 + 5
4) 3 + 7 x 6 / 3 - 4
As I model each problem with the students, I demonstrate the need for organizing to solve one operation at at time. Also, I emphasize the need to reevaluate the connections (operations) after each piece is solved.
I remind the students that we know about other mathematical connections that we haven't talked about yet. I lead them to remember exponents and also parenthesis. We discuss the strength of these connections.
Then I share that mathematicians long ago made the same discoveries that we did today, they developed an order of operations that must be used to evaluate expressions with more than one operation, but I have intentionally not taught the students PEMDAS. This is because I don't want them to focus on this mnemonic.
Let's try a problem with these other connections...
In this model, I struggle with disorganization to intentionally demonstrate the importance of being precise (MP6). After struggling and becoming "frustrated", I introduce the organization tool "PIZZA" that some former students created. This requires students to rewrite each "new expression" under the last every time a piece has been solved. Because each equation gets smaller and smaller it begins to form a triangle, the answer will be the solution to the last expression. When an expression is entirely solved, the student can outline the triangular shape and add some crust. It gets students excited about simplifying expressions and also helps with organization.
Now, you will have a chance to try solving some expressions with more than one operation. Remember, unlike with reading, you do not always move from left to right. You must look for the strongest connection first.
Draw a picture or cartoon to show explain the various strength different operations have on numbers.
Each day I use the ticket out to gauge the students understanding of the lesson. Today, I don't ask students to simplify an expression because I want something more open ended that provides me a sense of how they feel about simplifying expressions or what stuck with them the most.
I have some examples of student work here. In each example, I describe how that ticket out helps me understand each student better.