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.
Task 1: 6 x 3
For the first task, 3 x 6, students modeled 6x3=6x2+6x1 using the array model. Other students found Multiple Strategies for 6x3, including crossing off the 3, making it a 4 and solving 6 x 4, then subtracting a six later on.
Task 3: 12 x 3
During the next task, students made an 11x3 array and then added on a 1x3 array to get 12x3. I loved hearing the Multiple Strategies for 12x3.
Task 3: 106 x 3
To begin, I reviewed yesterday's lesson, Day 7: Capacity Conjectures & Number Line. I wanted to help students connect prior learning to the new learning that would happen today! I also wanted students to truly understand that measurements are quantities that can be represented on a number line, similar to numbers. I asked each pair of students to get their Number Line back out from yesterday and to be ready to add new information! I used my own Capacity Number Line to help model and review concepts.
Taking Jumps of One Cup
First, we went over cups. I asked students to take jumps of one cup with me on the number line. As I modeled these jumps on the board, students took jumps on their own number lines with their partners at their desks. As we took each jump, we said, "One cup, two cups, three cups..." I asked the following guiding questions. Each question was followed by rich conversation:
1. How many cups are in a pint? Students responded, "Two cups!" How do you know? One student said, "Because it took two jumps of one cup to get to one pint."
2. How many cups are in a quart? Students responded, "Four cups!" Can you prove this to me? Students justified this idea, "See... It took four jumps of one cup (one cup... two cups... three cups... four cups...) to get to one quart."
3. How many cups are in a gallon? Students responded, "Sixteen cups!" How did the number line help you decide this? Students explained, "When counting by cups, it takes 16 jumps to get to one gallon."
4. How many cups are in a half gallon? Students responded, "Sixteen eight!" Can anyone explain why? Students reasoned, "If it takes 16 jumps of one cup to get to one gallon, then it would take half of 16 to get to 1/2 a gallon, which would be 8 cups."
I knew each of these questions would help students further understand the relative sizes of these measurement units in relation with each other.
Taking Jumps of One Pint
After that, we counted the same process (counting & questioning) with quarts and then gallons. Students excitedly began to see patterns and making conjectures! One student said, "If you 1/2 the number of pints, you get the number of quarts because 2 pints = 1 quart." Each time one student shared a conjecture, this inspired others to create their own Conjectures. (Supporting Math Practice 3: Construct viable arguments and critique the reasoning of others.)
Making More Conjectures
This willingness to take risks also provided opportunities to challenge a Student Misconception 1 oz = a half cup. In fact, a student provided a beautiful explanation to help Correct the Misconception. Then, this conversation inspired another student to figure out the number of ounces per gallon. He anxiously came to the board to share: How many Ounces are in a Gallon?.
All during this time, students also made changes to their own time lines and developed a deeper understanding of customary capacity units. Here, a student uses her number line to make sense of capacity conversions: Student Number Line.
During the Guided Practice time, some students sat up close to the board while others preferred to remain seated at their desks.
It was at this point that I introduced the students to today's Goal: I can show equivalent measurements in a 2-column chart. I explained: Today, we are going to create t-charts to convert cups to pints, quarts, gallons, and ounces, but first, I'd like you to meet Gallon Guy! Let's make Gallon Guy together!
Prior to today's lesson, I constructed posters using these Capacity Conversion Charts. I purposefully related all units to cups (cups to pints, cups to quarts, cups to gallons, and cups to ounces) to create a sense of connectedness. I also made sure that initial conversions began with multiplication. For example, to find the number of cups in one pint, students will multiply one x 2 to get 2 cups. Once they understand the rule, I gradually increase the complexity of the conversion tasks.
At this point, I asked students to create each conversion table in their journals as we completed them together as a class. This way, I could ask students to try completing each conversion task on their own before we discussed the solution as a class.
Pints vs. Cups
We started with the Pints vs. Cups chart. I provided students with the first three number of pints: 1, 2, and 3 and asked students to complete the chart on their own. When finished, I encouraged students to turn and talk about their solutions. Next, we came back together as a class and to share student thinking. I asked one student in particular to share: 1 Pint, 2 Pints, 3 Pints.
Then, I gave the next two number of pints: 10 and the variable x. Again, after giving students some time to think on their own, one student explained how she converted 10 Pints into 20 cups and another student showed how she solved for x Pints. Finally, I provided students the number of cups instead of the number of pints and asked students to convert 16 cups, 24 cups, and 1 cup to pints. Here, a students explains her reasoning: 16 c., 24 c., 1 c.
Other Conversion Charts
1. Students completed the task independently.
2. Students turned & talked about solutions.
3. We discussed student thinking as a class.
Great Class Discussions
As one student shared how he converted Quarts to Cups, there was a PERFECT opportunity to show the relationship between division and fractions. I pointed out that 1 divided by 2 is the same as 1/2.
As this student shared her understanding of Gallons to Cups, I loved how she so fluently spoke about fractions. It is clear that integrating fractions throughout this measurement unit has truly helped students develop a deeper understanding of both fractions and measurement.
Finally, this student uses his understanding of converting Cups to Ounces to figure out ounces to cups. This final problem provided students with the opportunity to connect conversations from earlier on today about the number of ounces in a gallon to our conversion charts! How great!