Garden Design

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Objective

SWBAT use the area model to write two equivalent expressions using the distributive property.

Big Idea

Students will understand the form behind the distributive property and transition from the physical model to the mathematical model.

Warmup

15 minutes

As students enter they begin the warm-up (projected on the screen) which shows a rectangular garden plot divided into four sections (warmup garden plot). My students are asked to identify all of the possible areas we might have planted with tomatoes if we planted two of the four sections with tomatoes.

While students are working on the warm-up I circulate and spot check three answers from their homework. I have asked them to highlight these three problems for me (Problems 2, 5, and 7).

  • In problem 2 I am looking to see if they have the terms in the correct order. 
  • I am checking number 5 to see if they noticed the subtraction. 
  • I am checking problem 7 for two common mistakes, combining the constant and the variable term and forgetting to distribute to the second term.

If they have made any mistakes on these homework problems they have a chance to correct them now after I have given them some helpful feedback.

Going over the warm up I begin by asking, "What total area(s) could be planted with tomatoes?" Possible answers are: 55 sq.ft., 22 sq.ft., 28 sq.ft., 49 sq.ft., 43 sq.ft., and 34 sq.ft. I am not as interested in the last two because they are on the diagonal and don't demonstrate the Distributive Property, but these are valid answers and we want to be complete.

For each response I ask students to describe what those sections of the garden plot look like. This pushes students to think about the dimensions. As they describe I model it on the board and ask for the dimensions of each side. I label the unique dimension and ask if I need to label the shared dimension on the diagram or if labeling it on one side is sufficient (this helps prepare for our next geometry unit).

Next I ask students what calculations they did and I model these: 5(7+4) or 5(7)+5(4). The most important questions to ask now are:

  • Why does it make sense that both of these expressions can correctly represent the same thing?
  • Where do we see each expression represented in the visual model?
  • How does the expression represent the garden?

These questions help them make sense (MP1) of the distributive property as well as recognize it's mathematical structure (MP7). They are also mathematically modeling (MP4) real world situations.

As we continue discussion each of the expressions I will make sure to have at least one student come to the front and provide an explanation. The more we have students practice explaining, even if it is the same explanation someone else gave, the better they will get at articulating mathematical ideas and presenting and argument (MP3).

Exploration

20 minutes

Next, I plan to draw a diagram that is very similar to the diagram from the warmup. As a group we will choose dimensions for each section. In this case, however, we will label one of the sections with a variable. I typically shade each section differently so they are easy to refer to visually without dimensions.

Once we have this diagram prepared, I will draw a single rectangular section with one variable dimension. I'll ask, "How can we represent the area of this rectangle with an expresssion?" If students suggest that we can't calculate the area without knowing the value of the variable, I will ask why? This question is usually a stumper, which gets everyone's attention. Once I have it, I will remind the class that we do know how to calculate an area for any value of x.  So, I'll ask if anyone can share an expression that proves this?

Next I draw two sections together and ask them to write two different expressions to show how we might calculate the total area, which I write on the board as they tell me or I ask them to write it up. When we have both expressions on the board, I ask what we call it when two different expressions can be used to find the same answer. If they don't say equivalent expressions,  I will ask if they both have the same value. I will write an equal sign between them and ask, "Are they equal?" Once we arrive at the idea of equivalence for the first pair of sections in our diagram, I will repeat this process with the other pairs to reinforce the idea of writing equivalent algebraic expressions.

The final task pushes students a little further in the direction of using variables to generalize relationships. I erase the numeric dimensions on the diagram and replace them all with variables. I go through the same process as above but this time I ask for "two different but equivalent expressions" to model the use of the term. And for the next few days I will encourage them to do this every chance I get.

Problem solving

19 minutes

I ask them to describe the terms 16 and then 5x. They have had problems like this before, but I remind them they can use words to describe them or they can use math to describe them.  We started on a problem (homework consecutive sums) in an earlier lesson (Number System Assessment) and they have worked on it at home. These types of problems are designed to get students to collaborate with others, listen to and critique their ideas, and to persevere in solving problems in the long term. This type of problem that is spread over several days helps them take a break when they get stuck or frustrated and not give up. They are not used to this type of work and the breaks really help them reenter the problem after being stuck.

I write down all of their descriptions. They may describe 16 as an even number, a constant number, a number that could be the answer to an addition, multiplication, division, or subtraction problem, a number that is consecutive to 15 and 17, a double digit number, a square number, a multiple of 8, etc. They may give math problems like 8+8, 8x2, 4x4, 15+1, 2x3+10, etc. What
I’m interested in is the prime factors, because it will help them see the pattern in the “consecutive sums” problem we will be working on, so I would underline the 8x2 and 4x4 and say that these are both ways of multiplying two numbers together to make 16, is there a way to multiply 3 numbers to make 16. If they are stuck I would draw __x__x__ under the 8x2 and maybe even branch out the 8 if need be to get them to say 4x2x2. Then I ask if there is a way to
multiply 4 numbers together to get 2x2x2x2. They may suggest now 2^4, but if not I would ask if there is another way to write that. If they don’t see it I would underline the 4x4 and ask if there is another way to write that and then come back to the 2x2x2x2.

They may describe 5x as a variable term, a term that has an unknown value, they may say is could represent 5 monkeys, etc. They may say it is 5 times x or x+x+x+x+x or 4x + x, etc. This one I just want them to look a little closer at a variable term.

Here we pull out our earlier work on the “consecutive sums” brainstretcher problem. Students have worksheets and we have a whole class chart that has two sides, one for numbers we can make by adding positive consecutive whole numbers and one for the numbers we can't make. I ask students what numbers they have made so far and ask how they did it so the class can critique whether the method fits the problem's criteria. Once we have a good list I have them look for all the numbers in a certain range (1 - 20 for example) and ask which numbers are missing. I tell them I will write them on the "can't make" and then I task them to try to make them so we can cross them off and move them to the "can make" side. Students share when they make one and as we check the math I remind them we can use the number properties we discovered in our number talks (let’s talk addition, let’s break it down, ) and ask them to identify the property we used by referring to our properties posters. If their method works we cross the number off the “can’t make” side and move it over to the other.  Then I have them look at the next “range” of numbers and see if there are any gaps on the “can make side”. I write these on the “can’t make” side and challenge them to try to make them. I tell them if they can make any of the numbers on the list I will give them the homework to start early.