4th Grade - Math in Focus
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August 2013
We have new curriculum maps that have more information, Common Core Standards on the document, as well as coding to match the PARCC Content Frameworks. The links to the right are the word documents. We will continue to build these maps. |
General Recommendations
4th grade – students were taking the quick check or finishing an assessment from the day before. In helping one student, she was struggling with doing the multi-step problem, getting confused over what the numbers represented. Was it the amount of pens or the amount of money for the pens. Students do not want to show multiple representations of the math. The expectation is that they will show multiple representation.
What to do . . .
- Provide more opportunities for students to interact with the problems by presenting as open-ended problems (not the teacher demonstrating right away how to solve it) – put up the problem and let students begin to think about how to solve it. Try not to show everything first. Students need to begin developing a sense of learning mathematics by trying different strategies, not just learning steps to get to a right answer.
- Fluency is a problem for students. Teachers can spend some time – about 15 minutes only- building fluency with higher level games, number talks, fluency practice. The focus should be on the strategies. Students need to be able to explain the strategy, show how it works with concrete materials, before applying to “naked” math. Send home some of the fluency practice sheets or the transition guide resources for homework.
- Try to keep the book closed during the teach/learn. Have students work on brainstorming different methods to do the mathematics. Elicit “lots” of student telling and describing.
- Use visuals – concrete, pictorial, abstract – to build understanding of the concepts. Students have to be able to manipulate the concrete, then draw it on their own, before going to abstract. Do not pull the old naked math worksheets. Students need more time with the concrete and describing the processes.
- IEP students or the struggling students – much more time developing the concept of the mathematics (apply fractions to real-life problems), estimation, placing numbers on the number line.
4th grade – students were taking the quick check or finishing an assessment from the day before. In helping one student, she was struggling with doing the multi-step problem, getting confused over what the numbers represented. Was it the amount of pens or the amount of money for the pens. Students do not want to show multiple representations of the math. The expectation is that they will show multiple representation.
What to do . . .
- In Chapter 5, Data and Probability, do lesson 5.1 on averages. This will help students practice with division concepts. Stick with estimation first, and partial quotients. Use the chip models if needed. Remember you can bring those up online.
- Skip lessons 5.2 and 5.3.
- Do Lesson 5.4 on outcomes to provide some practical application of using fraction concepts, which is the next unit.
- Give the assessment, and do not have students do the problems related to lesson 5.2 and 5.3. These are not common core for 4th grade.
- Continue to work out the multiple step problems. Students should build understanding of the general strategies to solve the problem before getting to the equation. For example, the problem had students look at a table with a list of pens and pencils sold each day of the week. They had to figure out the amount of money collected if pens were sold for $2 and pencils for $1. Have them generalize the process first. We need to figure out the totals of how many pens/pencils were sold for the week. How would we do that? Then what happens with those totals? What will be the labels for the numbers?
- To prepare for the fraction unit I have put in the 5th grade notes for students who are behind. This could be done for a couple days before beginning the next chapter lessons. Do the Chapter Opener and pre-assessment.
- Spend much more time with concrete models, especially having the students make their own fraction kit may help. If time is an issue, have them just cut apart a fraction kit. I have posted the ideas on by website. This document provides the basis for building understanding of fractions. It includes a copy of the fraction bars, how to make them, and games that can help build understanding. It also lists the concepts that should be developed first.
- The article posted here, also describes how to develop concrete, pictorial, and abstract representations of fractions. It includes a nice comparison game of fractions. Remember to keep using the number line as a strategy besides the fraction notation and the visual fraction strips.
2012-13 School Year
February 2013
We discussed the basics of fractions that students need to know and understand BEFORE getting to procedures.
What is a fraction and unit fraction? A fraction is a number that represents a quantity as a part of a whole – a number less than one. Fractions represent an amount or quantity, we can do all operations with fractions, fractions have a location on the number line that represents how big or small it is compared to other numbers on the number line.
Identify fractions as an area and a set. They need to understand what the numerator and denominator stands for.
They are comfortable with area models, but not the set model. From analyzing the test, they are confusing what the whole unit is compared to the parts of the unit.
Equivalency – we need to constantly reinforce they are renaming fractions. The equivalent fraction is another presentation of the same number. We need to not say “reduce” as this give an unreal expectation that the fraction is smaller. It names the same number. We can use rename or simplify. There is some discussion with the common core list serv about using simplify, but we will continue to use this.
Operations with fractions – students get confused when “operating with fractions”. They have grown up operating with whole numbers and have conceptions of when I add numbers the sum is bigger, when I subtract numbers the difference is smaller, when I multiply the numbers the product is bigger, when I divide numbers the quotient is smaller. Now, operating with fractions has changed some of these thoughts. When the 4th graders had to multiply a whole number by a fraction, they subtracted. They knew the number should be smaller, so they subtracted, trying to change the fraction to fractions with like denominators. So, 2/5 of 20 they changed the 2/5 to 20ths and the 20 they tried to change to 20ths. Working with multiplication of fraction by a whole number, reinforce with the model drawing what represents the 1/5 or unit fraction. Then they will see that it is multiplication of the number.
January 2013
Focus on the partial products and partial quotients. We worked out several problems. Show the distributive method for the students by writing it next to the second factor. Reinforce any of the problems by doing 2 digit x 1 digit, then doing the problem 2 digit x 2 digit with the same numbers from the previous problem. For example, 26 x 8, then 26 x 18.
We worked out several examples of teaching fractions. They have pulled some of the activities from 3rd grade. We can begin with having students build fractions and equivalent fractions, then let them work on the 3rd grade problems from the transition guide. Really reinforce the vocabulary of equivalent, simplify as a process.Simplify fractions by writing out the factors and then factor out the “ones”.
Students either have the fact fluency or they are struggling. We need to focus on the thinking strategies to learn the facts.
The program comes with a fact fluency book. Different pages provide the strategies for learning different sets, as knowing my 2 facts. There are timed drills so students can practice.
We discussed the need for the number talks to develop fluency in students. The students have not been taught the strategies for learning their facts and computation.
50 Fact Times Drill - Students do not worry about how many, but practice using the strategies to complete this timed
drill. There are one minute and three minute sections on this drill. One minute must follow down the columns equentially with no skipping. Then students switch to a different color pen or pencil. They now have 3 minutes to hop around and use their strategies to aswers problems.
Times Table - is a Excel workbook with different sheets to help students learn the strategies for multiplication. One
is to fill in the multiplication table. This is to be used with the Strategies sheet. Students record the strategies and how many facts that relates to. This activity is to show students that there are not 169 facts to memorize, but much
less when we learn out strategies. One sheet has different sets of times tables that students have to fill in using their strategies.
Multiplication strings will help build fluency and use strategies of doubles and making tens to help build fluency. The visuals of arrays, area model, and dots build understanding.
For our after school training, this is the PPT that I used with reference to Chapter 6 - lessons with fractions.
November 2012
With multiplication and division, arrays are good models to use to show the concepts. Division with a zero can be tricky. Here is a problem 2,117/7 using partial quotients. Using this strategy can help students see the values instead of focusing on digits when they do not have understanding and make careless errors.
We discussed using a simpler problem to have students work with the unit bars for the model drawing strategy. If Grandma's age is twice that of her grandson, and their total ages is 120, what is Grandma's age. Click here for a model drawing explanation.
We discussed the basics of fractions that students need to know and understand BEFORE getting to procedures.
What is a fraction and unit fraction? A fraction is a number that represents a quantity as a part of a whole – a number less than one. Fractions represent an amount or quantity, we can do all operations with fractions, fractions have a location on the number line that represents how big or small it is compared to other numbers on the number line.
Identify fractions as an area and a set. They need to understand what the numerator and denominator stands for.
They are comfortable with area models, but not the set model. From analyzing the test, they are confusing what the whole unit is compared to the parts of the unit.
Equivalency – we need to constantly reinforce they are renaming fractions. The equivalent fraction is another presentation of the same number. We need to not say “reduce” as this give an unreal expectation that the fraction is smaller. It names the same number. We can use rename or simplify. There is some discussion with the common core list serv about using simplify, but we will continue to use this.
Operations with fractions – students get confused when “operating with fractions”. They have grown up operating with whole numbers and have conceptions of when I add numbers the sum is bigger, when I subtract numbers the difference is smaller, when I multiply the numbers the product is bigger, when I divide numbers the quotient is smaller. Now, operating with fractions has changed some of these thoughts. When the 4th graders had to multiply a whole number by a fraction, they subtracted. They knew the number should be smaller, so they subtracted, trying to change the fraction to fractions with like denominators. So, 2/5 of 20 they changed the 2/5 to 20ths and the 20 they tried to change to 20ths. Working with multiplication of fraction by a whole number, reinforce with the model drawing what represents the 1/5 or unit fraction. Then they will see that it is multiplication of the number.
January 2013
Focus on the partial products and partial quotients. We worked out several problems. Show the distributive method for the students by writing it next to the second factor. Reinforce any of the problems by doing 2 digit x 1 digit, then doing the problem 2 digit x 2 digit with the same numbers from the previous problem. For example, 26 x 8, then 26 x 18.
We worked out several examples of teaching fractions. They have pulled some of the activities from 3rd grade. We can begin with having students build fractions and equivalent fractions, then let them work on the 3rd grade problems from the transition guide. Really reinforce the vocabulary of equivalent, simplify as a process.Simplify fractions by writing out the factors and then factor out the “ones”.
Students either have the fact fluency or they are struggling. We need to focus on the thinking strategies to learn the facts.
The program comes with a fact fluency book. Different pages provide the strategies for learning different sets, as knowing my 2 facts. There are timed drills so students can practice.
We discussed the need for the number talks to develop fluency in students. The students have not been taught the strategies for learning their facts and computation.
50 Fact Times Drill - Students do not worry about how many, but practice using the strategies to complete this timed
drill. There are one minute and three minute sections on this drill. One minute must follow down the columns equentially with no skipping. Then students switch to a different color pen or pencil. They now have 3 minutes to hop around and use their strategies to aswers problems.
Times Table - is a Excel workbook with different sheets to help students learn the strategies for multiplication. One
is to fill in the multiplication table. This is to be used with the Strategies sheet. Students record the strategies and how many facts that relates to. This activity is to show students that there are not 169 facts to memorize, but much
less when we learn out strategies. One sheet has different sets of times tables that students have to fill in using their strategies.
Multiplication strings will help build fluency and use strategies of doubles and making tens to help build fluency. The visuals of arrays, area model, and dots build understanding.
For our after school training, this is the PPT that I used with reference to Chapter 6 - lessons with fractions.
November 2012
With multiplication and division, arrays are good models to use to show the concepts. Division with a zero can be tricky. Here is a problem 2,117/7 using partial quotients. Using this strategy can help students see the values instead of focusing on digits when they do not have understanding and make careless errors.
We discussed using a simpler problem to have students work with the unit bars for the model drawing strategy. If Grandma's age is twice that of her grandson, and their total ages is 120, what is Grandma's age. Click here for a model drawing explanation.
October 2012
Double division is a strategy to find the lowest common factor. Is does not always have to begin with the greatest factor. Students begin with what they know, but the result will be the same.
Fluency is inhibiting students' ability to do the problems and understand the concepts. We should continue to do the number talks daily for 5 minutes. Remember to focus on the strategies for the number talks.
Estimation strategies are using rounding and front end estimation. Students should be fluent with either, and understand how the approximation differs with each strategy. When would the appoximations be the same? When are the approximations different?
Fluency is inhibiting students' ability to do the problems and understand the concepts. We should continue to do the number talks daily for 5 minutes. Remember to focus on the strategies for the number talks.
Estimation strategies are using rounding and front end estimation. Students should be fluent with either, and understand how the approximation differs with each strategy. When would the appoximations be the same? When are the approximations different?
September 2012
Chapter 1 begins with place value of whole numbers to 100,000. The strategies to compare numbers are using a number line, using a place value chart, lining up numbers vertically as a place value chart, and looking for patterns in the digits. Students should be able to record the number in expanded notation as well.
Use a quick write strategy to build understanding. Post a number as 23,568. Students can use the words place and
place value to describe this number. For example, 23,568 is a number with the largest place as ten-thousand. Twenty-three thousand five hundred sixty-eight can be written using expanded notation as 20,000 + 3,000 + 500 + 60 + 8. This helps see that the value of the 2 is twenty thousand. The value of the digit 5 is five hundred and it is in the hundred's place.
Use a quick write strategy to build understanding. Post a number as 23,568. Students can use the words place and
place value to describe this number. For example, 23,568 is a number with the largest place as ten-thousand. Twenty-three thousand five hundred sixty-eight can be written using expanded notation as 20,000 + 3,000 + 500 + 60 + 8. This helps see that the value of the 2 is twenty thousand. The value of the digit 5 is five hundred and it is in the hundred's place.