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The Standards…Stairsteps to Success

As we finish our second wave of state testing, I am reminded how critically important it is that teachers know and teacher their standards for each math concept. Whether your standards are the CCSS or the Texas TEKS, they are vertically aligned to ensure that concepts are introduced in a way that allows students to build upon a solid foundation. Let’s look, for example, at fractions. I am paraphrasing the Texas TEKS, but you’ll find that the CCSS are very similar.

1st Grade

At the most basic level, students learn to partition (divide) shapes into two or four equal parts and describe the parts using words. They also learn to identify examples and non-examples of halves and fourths.

2nd Grade

Students continue to partition objects, but now they add eights to their repertoire. In addition, they understand that the more parts a whole is divided into, the smaller the parts. That is a huge concept! Up until this point, 8 has always been bigger than 4. But now, we tell kiddos that 1/8 is smaller than 1/4. Students needs lots of concrete experiences to truly understand that concept. Finally, students understand how many fractional parts it takes to make one whole and count fractional parts beyond one whole. In other words, it takes 4 fourths to make a whole, so if I have 6 fourths, that’s more than a whole. In 2nd Grade, students do not use formal fraction notation at all. Everything is done using words only–one fourth, two eighths, etc.

3rd Grade

In 3rd grade students are introduced to formal fraction notation–the numerator and denominator. They learn about unit fractions–the idea that 1/b is one part of a whole divided into b parts. So 1/4 is one part of a whole divided into 4 parts. Their understanding of fractions goes beyond partitioning a whole and extends to identifying fractions on a number line and with sets of objects. Equivalent fractions are introduced with concrete materials, pictures, and number lines. Students compare fractions with either the same numerator or denominator, further reinforcing that the larger the denominator, the smaller the pieces.

4th Grade

Students learn to compose and decompose fractions with objects and pictures. Essentially, they are learning the concept of adding and subtracting fractions with like denominators. This concrete foundation helps them to avoid the common misconception that 1/4 + 1/4 = 2/8. If I put together 1/4 and 1/4, now I have 2/4. Equivalency and comparison are now accomplished using a variety of methods, which should be built on fraction sense. Understanding, for example, that 7/8 is very close to a whole, while 2/6 is much closer to 0. Or that 4/6 is greater than 1/2, while 3/8 is less than 1/2. The relationship between fractions and decimals to the tenths and hundredths is introduced.

5th Grade

At this point students are asked to add and subtract fractions with unlike denominators–a skill totally dependent on an understanding of equivalency. They are multiplying a whole number and a fraction (1/4 x 3 or 3 x 1/4). There is no way this will make sense to them without an understanding of unit fractions and addition of fractions with like denominators (3 x 1/4 is the same thing as 1/4 + 1/4 + 1/4). Students divide a unit fraction by a whole number (1/4 ÷ 2) or a whole number by a unit fraction (2 ÷ 1/4). The standards stress that this should be taught with objects and pictorial models.

Each and every step in this process is crucial. Personally, I like seeing the vertical progression of each math concept from Kindergarten up to 5th Grade. As a classroom teacher, of course you must be most familiar with your grade level standards. Keep in mind that often includes not only what you will teach, but how you will teach it (concrete, pictorial, abstract). But I would suggest you take it a step further and familiarize yourself with the standards of the grade level before and the grade level after. Knowing the standards the students should come to you with will help you do a quick preassessment prior to teaching and it will also help you pinpoint and fill in the gaps students might have. Understanding what is to come will give you a better understanding of the standards you are supporting.

The standards might not always be easy to understand, but unpacking them must be a part of planning for instruction. Make sure your step up that staircase is solid!


· Teacher choice of STAAR like practice – Choose topics that your group of students need extra practice on

· Repeat Coordinate Tic-Tac-Toe from Mini-lessons for Math Practice pg 30


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Diagnostic Window:

January 18, 2016–February 19, 2016


· Unit 8 Session 2.4B: Multiplication Algorithm (pg 111 from the TEKS book)

· Unit 8 Session 3.5A: Dividing 4 Digit Numbers (pg 117 from the TEKS book)


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Concept Sorts--Deep Thinking about Fractions

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One thing that I really like to work on with my students is organizing their learning in different ways. One method I have found to be very successful is to ask students to take work samples or problem types and sort them into categories. People who use Words Their Way do this all the time with students as they look for sound or letter patterns that help them understand how words are put together.

I find concept sorts particularly useful in math and the content areas as well--and I really believe it helps students make connections among ideas and help them to see how things are connected. In fact, two of Marzano's effective teaching strategies--cooperative learning and identifying similarities and differences--are addressed with this activity. If research has shown solid evidence that categorizing and finding similarities and differences works—I’m all for it!

Here is how I tend to use these sorts—although please know that there is no RIGHT way! (And another way will be blogged about soon) Get creative and let the students guide you as you go. Here is one example of a sort I did with my students. I started by splitting my class into groups of 3. (I like trios for lots of reasons…because it allows for better dialogue, it helps strugglers, and also allows for one extra person in case a third person gets pulled out of the group or has to leave for any reason!) For this sort, I gave each group a small piece of bulletin board paper for them to do their sort. They grabbed their sort cards, cut, and started to organize them!

As my students sorted, I simply walked around and eavesdropped! It was a great time to listen for math language, to listen for any misconceptions, and to see who was feeling confident and who was not.

Students begin to question each other, ask questions of each other, and help one another come to higher levels of understanding. My role is simply to be an observer—I really don’t get involved at this point…even if I see errors. Trust me on this!

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This sort was an example/counterexample sort…I wanted the students to decide if the

fraction card I gave them was an “example” of 1/2 or a “counterexample” of 1/2.

This group was getting their T-chart all set up.

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Now…it would be plenty easy to have the students stop at this point, but I am working hard with mine on to get them to use their writing skills in math and to work on that “critiquing reasoning” standard. So I had my students actually write on their posters to explain why all their “counterexamples” were NOT 1/2!

I gave the students a total of 15 minutes to do this job—and there were a few groups that did not get every card sorted. That’s okay. If I gave some groups 45 minutes, they may not have finished! I like to keep things moving—I even had a visual timer up on my Smartboard so they know how much time they had left.

After the time was up, I continued the activity with a gallery walk. Although actually DOING the sort is a valuable activity, gallery walks can add a whole new level of critique to the lesson! Each trio took a post it note and cut it into three “tags”. They were allowed to “tag” up to three spots on other groups’ papers where they felt an error was made. Here’s what it looked like!

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Finished? Not yet! Each group then went back to their “home base” and looked at the results. They then discussed any “flagged” items and we went through some of the most challenging ones as a class. The entire activity took us 25 minutes from start to finish! Are you ready to give it a try? I have included FIVE different fraction sorts for you—including the one pictured above.

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· Multiplication/Division Fact Fluency-each student is working on activities to support their individual goal

· Motivational Math Unit 33- TE 359 TIME INTERVALS

· Mini-Lessons for Math Practice pg 35: Digit Place


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