This lesson follows one in which students looked for patterns in division (day 3). Several claims came out of that examination that are partially correct. This lesson helps students make connections between the numbers and gives them the opportunity to critique and revise claims. This is key to creating viable arguments. Articulating their own thinking & questioning the thinking of others is not something many students have not had much practice in during math classes in their past. Students are asked to test claims by coming up with their own problems to try and to decide if the results of their tests provide evidence for or against our claims. Working together helps ELL students formulate responses. Being able to point to their work as evidence is also helpful.
Students begin the warm up when they enter. I have three fractions that I ask students to set up as division and complete (1/12, 3/12, 5/12). As they finish up I start a discussion with the students who have finished. I remind them that we looked for some patterns in division in a previous lesson (day 3). As we go through the observations I record the responses on a poster to show what happens with the decimal when we divide by 2, 4, 8, 3, 6, 9, and 5. I expect them to have noticed that dividing by 2, 4, and 8 resulted in decimals that ended with a 5 in the tenths, hundredths, and thousandths place respectively. When dividing by multiples of 3 the decimals all repeated and dividing by 5 terminated in the tenths place. They also may bring up that dividing by 7 and 11 resulted in repeating decimals with some pretty cool patterns.
We start by going over the three warm up problems and I ask them which decimal answer is not like the others. This is a good way to get them to focus on making observations. This shift can be hard for students who are used to following algorithms and not looking for connections. They tell me number two does not repeat. I ask why we might have expected dividing by 12 to result in a repeating decimal. Asking this question helps students express their ideas about math, also something they are unaccustomed to) I expect students to say that since we observed that dividing by 3, 6, & 9 resulted in repeating decimals we thought all multiples of three might". "If problem 2 does not support our claim that dividing by multiples of 3 repeat it looks like our claim might not be entirely accurate and we need to revise it". I ask students to discuss with their math family groups what is similar about the two problems that do support our claim and different from the one that doesn't to figure out why. Asking students to find similarities and differences will produce more observations than asking them simply what they notice. They are less likely to stop after finding one. I expect students to notice that the fraction in number 2 can be simplified, that 3 is a factor of both numbers.
Now I tell them all of our claims are now in question and we had better test them.
One claim is that "dividing by 8 will result in a decimal that ends with a 5 in the thousandths place". I have given them three examples to try. As I circulate I ask groups which ones support our claim and which ones don't. Then I ask them to test out some on their own that they think might provide additional evidence against our claim. After a few minutes I ask students what different ways they found the decimal could end when dividing by 8 (tenths, hundredths, thousandths, ones) and I write those on the board. I ask them to give me some evidence (division problems) for each category and I record that on the board as a fraction. Then I ask them to decide in their group how we should revise our claim. I tell them to write a new claim based on the new evidence.
We do a similar exploration of dividing by 3 in which they are asked again to revise the claim based on the evidence.
As I circulate I ask prompting questions like "what did you notice about the division that did support our claim?" This leads to statements like "when dividing an odd number by 8 the decimal will terminate with a 5 in the thousandths place". I also ask what they noticed about the examples that did not support our claim which leads to statements like "when dividing an even number by 8 (the fraction can be simplified) it will still end with a 5, but not in the thousandths place". They may say that it will end in the tenths or hundredths and maybe even discuss that it depends on whether the denominator simplifies to a 2 or 4. They will probably also mention that when dividing multiples of 8 by 8 it will result in a whole number with no decimal remainder. Sentence starters are helpful for any students who are not accustomed to writing in math, but especially for the ELL students. Division pattern sentence frame starters.docx I tell them they may need 3 or 4 separate sentences for each claim and they can use a sentence frame more than once. I only give them the two sample sentences. Otherwise they would just fill in the blanks of however many I give them and call it done. I want them to decide when their explanation is complete.
As a whole group we go over the revisions together and write new claims. It takes me longer to record what they say, so while I am writing I ask students to show each other the evidence that supports the statement I am writing and then look for more evidence against. When they think we have a complete claim I ask if we have accounted for all the possible results when dividing by 8 or 3.