I'm Back!I apologize for not updating the blog the last couple weeks! I took a few days off for a Professional Development and to go to Kansas for my sister's wedding.
The students have been working on multiplying and dividing rational numbers. We talked about how to multiply and divide integers and how those rules can be applied to any rational numbers. They completed their SWYK over this today. Our next unit will be over proportional and linear relationships.
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The past few days we've been solving problems involving addition and subtraction of rational numbers. We've done sample problems on a worksheet, Tenmarks and a challenging activity called Twists and Turns. Here is a synopsis of the problem:
Together as a class, we tangled our ropes in this form: Twist, twist, turn, twist, twist, twist, turn, twist, twist, twist, turn. This is the sequence of numbers it produced: 0, 1, 2, -1/2, 1/2, 3/2, 5/2, -2/5, 3/5, 8/5, 13/5, -5/13... The challenge for the students is to create a set of twists and turns that will get them back to zero. So far, only 1 group has been able to solve the problem using ropes and mathematically. Due to the heat, we have postponed taking the activity outside to explore more, but after it cools down a bit, we will come back to this problem! We started today off with a 24 problem: Level 1: -2, 2, -8, 4 Level 2: -8, 4, 2, 9 Level 3: 8, -6, -6, 5 Next, I had the students brainstorm in their groups the rules they know or have learned about how to add and subtract integers and fractions. This should be review from last year, but now, we are going to combine these types of problems in to multi step addition and subtractions of all rational number problems. We discussed the rules as a class and this is what they came up with: Students spent the rest of the class working on the following worksheet.
Today, students worked on a problem involving looking for patterns with consecutive integers. I was not in class during this class period, but here are some of the answers that were turned in. Today we started off with a number talk: 19 x 21. I am always amazed at the ways students manipulate numbers in order to mentally solve problems such as this. There was one student who described this method:
I was fascinated that this worked and wanted the students to be able to explore if it always worked or was just special in this case. What they came up with was pretty cool! They came up with a few rules about how it can be useful in other cases, but we were quickly taking up too much class time trying to determine why it worked. I told them we'd take a class period to explore this problem in more depth another day. Next, we played some games that involved integers. Different students played different games depending on their current understanding of adding and subtracting integers. Some students played the Balloon Race game. They wrote down the equations that were created through out their games and looked for patterns or what they noticed about how the answers were found. They completed the Analyzing the Balloon Game worksheet in their notebooks. Others played the game Connect 3. In this game, students use their understanding of how to add and subtract integers to attempt to manipulate numbers rolled on a dice to get 3 in a row. Today we started off with an activity called 24 which we will continue to do on the block days. 24 is very similar to the Four 4's activity. They are given 4 numbers and they may add, subtract, multiply and divide them to get 24. During this activity we focus strongly on creating one equation that can be solved to equal 24 as opposed to breaking it up in to steps. This activity is a great way for students to be able to problem solve as well as practice order of operations and writing equations. Eventually we will include different forms of 24, such as using integers and fractions. Today's 24 problems were: Level 1: 1, 12, 4, 5 Level 2: 12, 3, 9, 3 Level 3: 1, 4, 24, 12 We are starting our unit on Operations with rational numbers. In order to better understand how to manipulate all forms of numbers (fractions, integers, etc...) we had a discussion about the 4 operations (addition, subtraction, multiplication and division) and what we were actually doing with numbers in each of these cases. Students were asked to brainstorm everything they can think of about each of the operations in groups and then created posters summarizing their ideas. We ended in a gallery walk where they took their notebooks around with them and added any ideas that were not discussed in their groups. Here are the posters the groups came up with. We will extend each operation to manipulate different types of numbers. We started off today with a visual pattern in which they had to come up with a rule to describe the pattern. Next, we explored repeating decimals. We used this worksheet which goes on page 11 of the students' notebooks. They did a great job finding the pattern of what makes repeating decimals. Many took on the challenge of trying to apply the rule to 0.4888.... I don't think anyone found the answer yet, so we'll explore that more on Monday. The last question posed the best conversation. Students were amazed to realize that 0.999... actually equaled 1! Finally, I had the students log on to their Tenmarks accounts. This will be where most of their homework will be assigned.
We started today with a number talk which resulted in LOTS of different answers. The problem was 3.76 - 1.99. Most of the students attempted to mentally line the numbers up and subtracted the traditional method. I'm always impressed when students can work with this many numbers in their head accurately! However, many made mistakes while using this method. Another student shared the "add to both sides" method: 3.76 - 1.99 = ____ They added 0.01 to 1.99 on the left side, to get 3.76-2, which is 1.76. To balance out the equation, they had to then add one to the answer, which gave them 1.77. One student shared this method: "I added 0.01 to the 1.99, so I took 0.01 away from the 3.76, but I didn't get the same answer, I got 1.75." I love these questions! It led to a great discussion about how the rules for addition are different from the rules of subtraction. We looked at a simpler example of 5-2 and saw that it wouldn't work to change the problem to 4 - 3. Essentially what she was doing was taking 1 away from what she was subtracting from and also adding to what was being subtracted. She would have had to have added back 0.02 to get the answer. Next we talked about rational numbers. I started out just giving them the definition of what rational number are: numbers that can be written as a ratio of 2 numbers (in other words, a fraction). I then gave them a list of numbers to try and sort between rational and irrational just based on the definition. There were some great discussions and disagreements in the groups! We came back together to share our thinking. Here were the end results: We then made generalizations about which kinds of numbers are rational and irrational and added them to our notes.
First, students finished their pre assessment that was started on Monday.
To start off our unit on operations, I gave the students a problem using fractions: Fair Share Problem |
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June 2016
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