This lesson is a continuation of Day 4: How Many Apples?. Today, students will solve more problems that require them to find equivalent conversions between grams and kilograms. By applying the concept of measurement to an everyday object, apples, this helps students also develop the practice of modeling with mathematics (Math Practice 4).
I left the drawing of the Apple Farmer on the board to help remind students of problems we began solving yesterday. I reviewed the goal: Fourth graders, today, we are going to continue working on the same goal as yesterday: I can solve word problems involving grams and kilograms. To review the number of grams per apple, I drew a whole apple, half an apple, and a fourth of an apple on the board: Splitting Apples. I asked: How many grams are in one apple? "200!" I labeled the whole apple 200 grams and continued: How many grams are in half an apple then? "100!" I labeled the half apple 100 grams. Knowing that finding the number of grams in 1/4 an apple would be more challenging, I asked students to Turn & Talk: How many grams would be in one fourth of an apple? I listened closely to students and supported students who needed further support by drawing a picture. After a couple minutes, we came back together: How many grams are in 1/4 an apple? One student said, "50 grams." Another student explained, "Because 1/2 an apple is 100 grams so 1/4 an apple is 50 grams."
Before providing students with independent work time, I reviewed the expectations and drew attention to students who were paying attention to the mathematical practices: Please make sure you solve each problem using two strategies. Can anyone remind me why we try to use two strategies when solving problems? A student responded, "Just in case we make a mistake. Another student said, "So we can check our work." I celebrated a couple of students from the day before: I noticed David attended to precision (Math Practice 6) when she labeled her number line. I also noticed that Sarah persevered (Math Practice 1), even when she was a little confused. Today, I also want you to all focus on correctly labeling your answers. Yesterday I noticed a few students wrote 5 grams instead of 5 apples. In addition, pay attention when you get two different answers. Yesterday, I noticed a student got two different answers. One of the answers was correct. The other answer was incorrect. This is an amazing opportunity to ask "Does this make sense?" I have such great respect for students who can go back, find a mistake, and revise their work (Math Practice 1).
I explained, "Yesterday, most of you solved the first two apple problems. Today, the problems are going to get a little harder, step by step. Students became excited for the challenge! I then passed out the third problem from Apple Problem Solving and said: Here's your next challenge! When you finish, you can go onto the next problem. You can either work together or with others. Students went right to work, pasting down problem #3 on the top of a new page in their journals. Then, they drew a line at half a page and created two boxes to show two strategies they used to solve problem #3. Here's a Student Example.
During this time, I met with almost all students. Many times, I would ask a student a probing question, move on to the next student, and then come back to see if the first student figured out the answer to the question. The goal was to give students time to grapple with the problems and develop the skills to solve complex tasks (which supports Math Practice 1, Make sense of problems and persevere solving them).
Here's a perfect example of the importance of patience, encouragement, and time. I conferenced with this student several times before she arrived at the answer.
Here, First Conference, the student thought, "If 2 apples = 400 grams, then 2 1/2 = 450 grams." I also loved watching her go back to previous problems to help her make sense of the problem.
Next, Second Conference, the student was on the right track, but got confused with the in & out table. After she remembered that 1,000 grams = 5 apples, I wanted to see if she could go back and correct the incorrect numbers in her table (4, 6, 8).
During the Third Student Conference, the student understood the number of grams in a kilogram, but struggled with finding that 2 1/2 kilograms = 12 1/2 apples. This concept is the same as 2 1/2 x 2.5 = 12 1/2. We have yet to get to our fractions unit in fourth grade so this is especially a challenge for these students. However, by connecting concepts to an everyday object and scenario (making a 2.5 kilogram bag of apples), I noticed students were able to solve these complex problems when given time. Before walking off, I asked the student to represent her thinking using a number line.
Here, Fourth Conference, the student confused 1/2 kilogram with 1/4 kilogram. I drew a picture of five apples (1 kilogram) and asked her to show me 1/2 of a kilogram. She split the kilogram in 1/2 and said "2 and a half." Then she changes 1/4 back to 1/2 and says, "Oh, I was right." At this point, she need more time to process the concepts.
During the Fifth Conference, you'll hear me review what we've already gone over one more time. When she gets the incorrect answer, I simply asked her to count it up again. This time, she got it!!! Not only did this student get the correct answer, but she demonstrated how to make sense of a problem and persevere (Math Practice 1).
Here's another student who grappled with this problem, but finally Got It!. She excitedly came up to me and said, "I figured it out!"
These students work together to find How much is 0.25 kilogram?. I drew a model on the board and said: Could we say that this rectangle is one kilogram? What would half a kilogram be? What decimal is equal to 1/2 kilogram? The girls took it away from there. I could tell that they understood the concepts, but just needed a model to solidify their thinking before completing word problem #5.
For an exit slip today, I asked students to answer the following questions on a half sheet of paper. I wanted to see which students could solve the following problems outside of a word problem.
1. How many grams is equal to 1 1/2 kilogram?
2. How many grams is equal to 2.5 kilograms?
I collected the exit slips and graded them. Most importantly, I identified students who needed more support with these concepts. Many students demonstrated an understanding of fractional conversions: Proficient Example. I noticed others, Nearing Proficient, got at least one of the problems correct. I think this student misread the first problem. However, this shows how important attending to precision (Math Practice 6) truly is! Finally, there were some kids who didn't quite connect ideas today: Novice Example. This student was able to grasp: 1/2 a kilogram = 500 grams. However, it became challenging for him to transfer this understanding to 1 whole and a half = 1,500 grams.