The do now is a problem from illustrative mathematics that involves finding equivalent expressions. Students will be looking at the expressions and simplifying them to find ones that are the same. If they find an expression that does not have an equal expression, they are to create one on their own (SMP 8).
I really liked this problem for two reasons. First, it’s a review of combining like terms from a previous lesson and second, it will help keep the notion of equivalent expressions in their mind for today’s lesson.
For this problem, students will find out the a, b, and d are all equivalent expression. They will need to use both the commutative property and distributive property to find this out.
For c and e, the students will need to write an equivalent expression because they do not have a match.
c) 2x + 4, students can decompose this expression and re-write it using the distributive property 2(x + 2)
e.) x + 4, students may have trouble with this because there are no common factors. Ask students if there is another way to represent 4? Examples: x + 2² or x + 3 + 1.
Tools: DO NOW example
There are 5 problems that I will be using for this activity. Some problems came from illustrative math and some are my own. According the CCSS, students should be able to evaluate an expression using real world examples and be able to understand that in a formula, without parenthesis, the orders of operations still apply. Students will be working independently to get started (SMP 1), after I will have them partner up to talk strategies and solutions (SMP3). As a follow up to each question, I will have random students come to the board to explain their strategies.
I captured this in the following problems.
Problem 1. Perimeter of a Rectangle
The students are asked to write an expression that adds the length and width and doubles the sum. I’m expecting answers to look like this 2l +2w or 2(l + w). (SMP 2) Then they are asked to find the perimeter given a length and a width. Students can substitute in their value for the variable to find the answer. (SMP 6).Students need to know that perimeter means to add up all the sides. As students are working out this problem, look to see if there are different expressions written, shown above. If there are, have these students come to the board to show the different expressions being used still result in the same answer. Ask students why this happens? If you don’t see different expression being used, then you can pose the question on the board for them.
Problem 2: Finding the perimeter of a rectangle
In this problem, students have to decide which expressions show how to find the perimeter of a rectangle. I’m anticipating that students may have some trouble with this. For the students that are struggling, encourage students to plug in the same value for the l and w for each expression to see which ones match. Then the students have to explain what each person might have been thinking. Allow any reasonable responses.
Problem 3: Finding the distance
This one is going to be a zinger! Let students try to work on their own. Offer some assistance by reminding students to circle key words in the problem. For example: to and from school, 1 rainy day/ week, and over 4 weeks. Have the students write the expression for 1 week first: (d +d)+(d+d) +(d+d)+(d +d) +(d+d) = 10d. The next part of the problem says that there is always 1 rain day per week. So the problem becomes 10d – 2d, for one week. They need to find the distance for 4 weeks. One expressions could look like 4(10d – 2d+ or (40d – 8d). The other expression would be to figure that each week (minus a rain day) would look like 8d and then multiply it by 4 weeks = 4(8d). Both expressions are equivlanet.
Volume of a cube
The problem gives them the formula V = l x w x h and asks the students to write an equivalent expression. Students will need to know that a cube has all sides the same. If all sides are the same then you would be multiplying by 3 of the same number which would result in the expression V=s³.
Then through substitution, the students will find the volume of a cube given a side length. They can use both formulas to prove their answer is correct. (SMP 3 and 6)
Surface Area of a cube
Given the side lengths of different cubes, the students will be substituting in for the variable. I’ve chosen to use fractions and decimals for the substitution to help students recall this mathematical process. Remind students they can use the standard algorithm for decimals and a visual for the fractions. (SMP 4)
Tools: Problem Solving Questions
The students will be completing a thinkpairshare for each of these questions.
In question 1, I want the students to realize that the variable can stand for any number. If there are two of the same variables in a expression, they would also represent the same quantity. (SMP 2)
In question 2, I want the students to say that the expression represents 6 times the side squared. The variable stands for the side length and the expression finds the surface area of a cube.
In question 3, I’m looking for students to know that they will need to apply the orders of operations to find a solution. So they would substitute 2 in for s. Then they would square 2 and finally multiply by 6.
Tools: Closure Questions