Today's Number Talk
For a detailed description of the Number Talk procedure, please refer to the Number Talk Explanation. For this Number Talk, I am encouraging students to represent their thinking using an array model on student white boards.
Task 1: 6 x 2
For the first task, 6 x 2, I modeled how to use the array model. Although I have previously introduced this model, I wanted to review it before continuing with our Number Talk so that students would be more successful at modeling their thinking. I began by explaining: When I solve 6 x 2, I visualize a 6x2 array. See how I could draw lines to separate the rectangle into two rows of six?
Task 3: 6 x 4
We moved on to 6 x 4. During the next task, I modeled one array we could draw: When I solve 6 x 4, I decompose 4 in my head. Then, I visualize a 6 x 2 (just like our previous task) and another 6 x 2: 2(2x6). Who can tell me what you are visualizing?! As each of the following students shared, I modeled students' thinking on the board. One student said, "I decomposed the 6 into a 3 and a 3. I see one 3 x 4 which equals 12. And I added on another 3 x 4 which equals 12. Altogether, 3x4 + 3x4 equals 24." Another student said, "I decomposed the 4 into two 2s. Then I visualized a 6x2 + 6x2, which equals 24. Then, a student said, "I decomposed the 4 into a 3 and a 1. Then, I added a 6x3 + 6x1. I just love how one student drew an array to model 6x4 = 4(2x3)! How amazing! Do all these strategies equal 24?
Task 3: 6 x 8
During the next task, we discussed 4 x 8. This time, I didn't model any students' thinking. Instead, I asked students to show multiple strategies on their white boards. Students excitedly came up to me and asked if they could share their strategies on the front white board. Soon, about half our class was showing their work! Here, a student develops a deeper understanding of modeling with arrays: Change 4 Rectangles into 1 Rectangle. Another student showed: 6x8=4(3x4). My favorite teaching opportunity was when I worked together with a student to show an array model for Doubling & Halving.
To begin this lesson, I introduced the goal: I can make and test conjectures! I reminded students: A conjecture is a judgement based on incomplete evidence. For example, I noticed that David has a green water bottle. Based upon this evidence, I might form the conjecture: David's favorite color is green when it really could be any color. Have I collected enough evidence to know that David's favorite color is green? David giggled and said, "It's not my favorite color!" I continued: So you see, a conjecture is a judgement. In math, it is important to collect more evidence to either prove our conjectures to be true or to disprove our conjectures. When David told me that green is not his favorite color, I was able to gather more evidence that disproved my original conjecture. This is what you get to do today in math! So let's jump in and begin making conjectures! Note: This entire lesson is inspired by Math Practice 3: Construct viable arguments and critique the reasoning of others. Proficient math students can "make conjectures" and "explore the truth of their conjectures."
I began a list of student conjectures on the Conjectures Poster on the board. (You'll see some corrections in the picture. These corrections will be made later on in the lesson!) I asked students: Based upon your investigations yesterday, what can you tell me about ounces, cups, pints, quarts, or gallons? Without looking at any of their notes, students were able to come up with an amazing list of conjectures! Each time students shared a conjecture, I wrote the first letter of their name (for Grace, I wrote "G") to encourage student ownership, and then wrote the student's conjectures.
Even though the Conjectures Poster became full quickly, many students were still eager to share! They really enjoyed making judgements and sharing their ideas! We just had to move on though! I then asked: Who sees a conjecture that is true? Every hand went up! Who sees a conjecture that is incorrect? Again, every hand went up!
I explained: Today, you will get the opportunity to either prove or disapprove these conjectures using a capacity number line!
I assigned partners I knew they would work well with. I always take into consideration math skills, behavior, leadership skills, and communication skills when assigning partners. I passed out the following materials to students: a large laminated paper, a meter stick, dry-erase markers, and Pictures of a Gallon, Quart, Pint, and Cups. I asked students to draw a line on their large laminated sheet to represent their number line and advised students to make a mark every two inches to represent one cup. Then, I asked students to cut out the Pictures of a Gallon, Quart, Pint, and Cups and place them accordingly on the number line: Capacity Number Line. As students finished, I asked them to continue adding multiple units, such as 2 pints, 3 pints, 4 pints, etc: Adding on to Number Line.
During the creation of the number line, many students had to draw upon information they learned from investigating with capacity during yesterday's lesson. Others chose to remeasure units using a balance scale or refer to the gallon set containers to check their thinking. Here, I try to guide a student to discover his Misconception. Instead of telling him the correct answer, I asked him to show me his thinking: Correction of Misconception. I believe the most powerful learning happens when students are given the opportunity to prove or disprove their own thinking!
Next, students began Using Number Lines to Prove & Disprove Conjectures on the class poster! With some time, this group disproves a conjecture then Changes 2 qts = 1 gal. to 4 qts = 1 gal on the Conjecture Poster.
I got the biggest kick out of watching this student disprove his OWN conjecture! This first clip shows the Identification of a False Conjecture. Next, the student works at Providing Evidence to Disprove the Conjecture using the gallon balance. Here, the student is Correcting the Conjecture by changing "Four pints are in a gallon." to read "Eight pints are in a gallon." SO great!
Other students worked on Finding Correct Conjectures and Proving a Conjecture is True!. As this student excitedly modeled evidence that supported the conjecture, another student asked for her to explain how she knew 1 quart equaled 2 pints. I absolutely loved watching her Provide Evidence to Support Thinking which helped the other student develop a deeper understanding of equal units.
To bring closure to this lesson, I asked students to clean up to return to their desks. Next, I asked students to explain to a partner: Explain at least one conjecture that you either prove or disproved today! Student responses varied:
"I found that there really are four quarts in a gallon!"
"I found that there are 8 pints, not 4 pints, in a gallon!"