Apparently Farmer Frank is forgetful! He can’t remember the dimensions of his pumpkin patch! He knows the area because he just spread 15 square meters of fertilizer in one section and 10 square meters in the other. Help him figure out the dimensions. (He knows the sides are all longer than 1 meter).
To begin, I highlight the left side of the rectangle. I ask, "What could the measure of this side be?" I label it with their suggestion no matter if it is correct or not. Then, I will ask, "What would the length of just that section have to be in order to make an area of 15 m^2?" (I often cover the other section at this point to help my students focus.) Then, I show the other section and I ask, "What would the length of this section have to be in order to make an area of 10m^2?"
I follow this sequence of questions because if my students have made a mistake in their reasoning, then they will give a different value for the right and left sides of the rectangle (for example h=5, l=3 and h=2, l=3). This mistake happens often enough that it is worth building it into my plan. When it occurs, I wait for a student to see this mistake. I want them to process all of the information in the problem. Often, the dimensions they gave do give the correct areas, but they are not possible dimensions given the constraint that the patch is a rectangle.
Next, I ask my students what the dimensions could be if the two sections have areas of 15 & 3.
As my students think about this I draw the area model on the board and I label the areas. When we go over this one I cover the smaller section, highlight the dimensions of the larger section, and ask my students to suggest some what are some possible dimensions. I will list all the possibilities for both sides as they give me the factors.
I then cover the first section and uncover the second and follow the same process. Once we have considered both sections separately, I reveal the whole figure and I highlight just the left side of my diagram. I ask, "Which of the dimensions are not possible for this side of the patch?"
Someone will eventually point out that since both left and right sides are the same they can only be the factors they have in common. (If no one does, I will pick an impossible one. I will ask if this left side is 5 what would the right side have to be? And is 5 one of the options for this side? (no) Why not? (it’s not a factor of 3). So 5 is not possible for this dimension, let’s cross it out.)
Once we have narrowed down to the only possible ways I go through each dimension of the area model and have students determine what its value must be if 5 is the height. When we’ve answered the question I ask what the dimensions of the larger rectangle and I highlight the height and the entire length across the top. Then I ask what multiplication we would do to find the area of this large rectangle.
I plan to do two more of these problems as a class. Each time I will ask students to come up and cross off impossible values and explain their reasoning (MP3, MP6). Then, we test out each possible length vs. height combination to determine the remaining dimensions. Finally, I will ask the questions from above about the dimensions of the larger rectangle and the multiplication that represents its area. One example asks what the dimensions could be if the two sections have areas of 18 and 6x and the second example is in three sections 10c, 20x, and 5.
For vocabulary development I use a strategy I have modified from the vocabulary guru, Kate Kinsela. The strategy is a Direct Teaching Method for introducing a single term or family of terms. I give students the target vocabulary in context and then I have my students create their own sentences using the term by practicing in pairs.
The term I chose for this lesson is common. I chose the term partly because it has multiple meanings and is used both within mathematical context and in their daily life as well. These types of words are sometimes particularly problematic for kids, because the term already has an assigned meaning for them and the meaning, though similar in math, may be different. In their daily life constant is an adjective, but in math we are using it as a noun. In addition, students are most likely not used to using it to distinguish between things that vary and things that remain unchanged, which is the point of this lesson.
When we do this type of vocabulary development in class I project the term, its definition, etc., and the sentences on the screen vocabulary common. I read each one using a method that Kate Kinsela refers to as "shared" reading in which when I stop reading midsentence students read the one next word. Once we get through all the sample sentences I give students 2 silent minutes to either write their own sentence using the sentence frame provided. If within those 2 minutes they don't come up with one of their own they write down one of mine. Then they turn to a partner and take turns sharing their sentence or the one of mine they chose. Then we take a few minutes sharing out their sentences or their partners sentence. Afterwards I like to collect the sentences they wrote and post a few on the wall on sentence strips.