Friday, September 20, 2024

# 2 5 Ell Support Reasoning In Algebra And Geometry

## Algebra & Trigonometry 100 Course Number M312 5 Mathematics Credit Grades 11

MATHIS GEOM Lesson 2-5 Reasoning in Algebra and Geometry

This course is intended for college bound students as an alternative to taking Pre-Calculus . The course will include a review of algebra skills as well as the further development of advanced algebra concepts covered in Algebra 2. Time will be given for preparation for standardized tests such as the ACT and SAT. The fundamentals of trigonometry will also be explored. Students looking for 1 full credit of math are encouraged to also sign up for Probability and Statistics.

Prerequisites: Teacher recommendation and Algebra II

## Using Student Writing In The Classroom

After reading through the students papers, I have discussions with individuals, especially if Im having difficulty understanding their reasoning or if their reasoning was incorrect or incomplete. Sometimes I focus on their writing errors other times I keep the focus just on the mathematics. Making this decision depends on the paper, the student, and the mathematics involved.

Then I have some students share their papers with the class so that students can benefit from one anothers thinking. For example, I asked Mariah to show the class how she arrived at fractions with common denominators. Her method wasnt conventional, but it was mathematically correct and effective. She explained, I kept changing them to equal fractions. I knew when I hit fortieths that it would work for six-eighths.

Its okay for you to use your own method, I responded. Its also important that you learn different ways to think about fractions. Then I wrote two fractions on the board3/4 and 2/3and asked each student to take out a sheet of paper and convert these to common denominators in two ways Mariahs way and Rauls way. I had them compare their work with partners and then had volunteers demonstrate each method so that Mariah and Raul could judge if their methods had been correctly applied.

## Question: Four Students In Mrs Burges Math Class Were Comparing Locker Numbers They Made The Following Observations:

Our four locker numbers are relatively prime to one another.

Exactly two of our locker numbers are prime.

What might the students locker numbers be?

Discuss the terms prime and relatively prime and the distinction between them. Then have students work on answering the question. Post students answers and, for each, have the class verify whether the numbers meet the given criteria. Finally, ask students to write their own definitions of prime and relatively prime. Have them share their ideas, first in pairs and then with the whole class.

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## Getting Your Math Message Out To Parents: Making The Home/school Connection

Were excited about our newest Math Solutions publication, Getting Your Math Message Out to Parents, by Nancy Litton. Nancy is a classroom teacher with almost thirty years of experience as well as a Math Solutions instructor. Shes thought a great deal about how to bridge the gap between home and school and knows that teachers

## Iii: Have Students Respond To The Task

After a brief discussion about the task expectations, students are normally eager to begin their task. Some students might think for a minute or so before beginning to record their ideas, but most begin immediately. Following are examples, including authentic student work, of how students responded to the above task.

Some students used shape templates, while others preferred drawing freehand. Most students began by drawing a shape on their paper and then writing some words above or below it. A few students began by writing an idea or the name of a shape, which they then illustrated.

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## Reasoning With Properties From Algebra Geometry

2. 5 Reasoning with properties from Algebra GEOMETRY

Goal 1: Using Properties from Algebra Properties of Equality In all of the following properties Let a, b, and c be real numbers

Properties of Equality Addition property: If a = b, then a + c = b + c Subtraction property: If a = b, then a – c = b c Multiplication property: If a = b, then ca = cb Division property: If a = b, then for c 0

Addition Property This is the property that allows you to add the same number to both sides of an equation. STATEMENT x=5 REASON given 3+x=8 Addition property of equality

Subtraction Property This is the property that allows you to subtract the same number to both sides of an equation. STATEMENT x=5 REASON given X-2=3 Subtraction property of equality

Multiplication Property This is the property that allows you to multiply the same number to both sides of an equation. STATEMENT x=5 REASON given 3 x = 15 Multiplication property of equality

Division Property This is the property that allows you to divide the same number to both sides of an equation. STATEMENT x=5 REASON given Division property of equality

More Properties of Equality Reflexive Property: a = a. Symmetric Property: If a = b, then b = a. Transitive Property: If a = b, and b = c, then a = c.

Reflexive Property: a = a I know what you are thinking, duh this doesnt seem too difficult to grasp. Just remember this one, when we begin to prove that triangles are congruent. STATEMENT x=x REASON Reflexive property of equality

## Question: For Homework Kim Lee Was Practicing Adding Integers He Looked At One Problem And Said I Know The Sum Will Be Negative Based On Kim Lees Statement What Do You Know About The Problem

This question aims to help students generalize about the relationship between the sign of the sum and the numbers in an integer addition problem. Students may need to make a list of integer addition problems whose sums are negative and look for commonalities among them in order to answer this question.

x 40

x 50

Youll play with a partner. The goal of the game is to be the player closest to three hundred. Its OK to get less or more than three hundred, but the goal is to be the closest.

Allie asked, Does that mean that one person could have two hundred eighty, and be twenty off, and the other person could have three hundred ten, and three hundred ten would win because its closer to three hundred?

I asked the class, What do you think is the answer to Allies question? Hands went up.

Rachel said, You said it was OK to go over three hundred. So the player with three hundred ten wins.

Youre right, I said and continued, Youll each take six turns. When its your turn, roll a die. Then decide if youll multiply the number you rolled by ten, twenty, thirty, forty, or fifty. Remember, you want to get closest to three hundred, and you must take all six turns.

Oh, this will be fun! Steve said. Do we have to write?

I responded, Yes. To show you how to do the recording part, Id like to have a partner play the game with me. Hands immediately shot into the air. I called on Ben because I knew he had a good grasp of multiplying.

I rolled a six! I said.

Ben won.

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## Exploring Ones Tens And Hundreds With Base Ten Blocks

In this lesson, excerpted from Maryann Wickett and Marilyn Burnss new book, Teaching Arithmetic: Lessons for Extending Place Value, Grade 3 , children use base ten blocks to cement their understanding of how ones, tens, and hundreds relate to our number system. Before class, I gathered the base ten blocks and enough

## Twenty Questions: A Lesson Using The Hundreds Chart 3

Geometry 2.5: Reasoning Using Properties from Algebra part 1

Overview of Lesson This lesson is a math variation of the popular 20 questions game. The teacher chooses a secret number on the 1100 chart. Students ask 20 questions to try to ascertain the secret number. Students mark their 1100 charts to keep a visual record of information they have gathered and to see the

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## The 512 Ants On Sullivan Street

A Lesson for Third Graders by Maryann Wickett and Marilyn Burns This lesson is excerpted from Maryann Wickett and Marilyn Burnss new book, Teaching Arithmetic: Lessons for Extending Place Value, Grade 3 . Childrens understanding of place value is key to their arithmetic success with larger numbers, and this book is important

## Question: The Students In Mr Milas Class Want To Know How Old He Is Mr Mila Told Them My Age Can Be Written As The Sum Of Consecutive Odd Numbers Starting From One How Old Might Mr Mila Be

Adding consecutive odd numbers produces the sums of 4, 9, 16, 25, 36, 49, 64, 81, 100, and so on. Of these, only some are reasonable predictions for Mr. Milas age25, 36, 49, and 64. All of these sums, however, are square numbers. Using different-colored square tiles or by coloring on squared paper, represent square numbers as squares to help students see that they can be represented as the sum of odd numbers. Start with one tile or square colored in. Then, in a different color, add three squares around it to create a 2-by-2 square, then five squares to create a 3-by-3 square, and so on.

Talk with students about extending this pattern.

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## Directions For Playing The Game

• Player 1 writes down a number greater than one and less than 100.
• Player 2 writes down a factor of the rst number underneath it.
• Player 1 writes down a factor of this new number.
• Each player, taking turns, writes down a factor of the preceding number.
• If a player writes down a prime number , the next player adds seven to it and writes down the resulting sum as his or her turn.
• The player who can no longer contribute a new number loses the game.
• ## Shifting The Conversation To Include Remainders

I then wrote on the board: Remainder of 1.

I asked, Can you think of any division problems that have a remainder of one in the answer? Turn and talk to your neighbor and see if you can think of any. The room got noisy as the students conferred. After a moment, I called them to attention and had them report. They came up with a long list that included problems like the following:

10 ÷ 3 = 3 R1

15 ÷ 7 = 2 R1

5 ÷ 2 = 2 R1

I continued until I had listed about a dozen or so problems. As I recorded, I noticed that the first numbers were all odd. I shared this with the class. What I notice about these, I said, is that the first numbers in the problems, the dividends, are all odd. I wonder if its possible to get a quotient of two remainder one if the dividend is even, say ten? I wrote on the board:

10 ÷ ? = 2 R1

I hadnt thought about this before, which is always a risk when posing a problem to a class. But after some students made a few unsuccessful tries, Alexis came up with the answer of dividing by 4 12 . I wrote her idea on the board:

10 ÷ 4½ = 2 R1

Alexis explained, If you want a remainder of one, you have to find a number that you can multiply by two and get nine.

Alexis replied, Because if you have a problem where you divide nine by a number and get two, then if you divide ten by the same number, youll have a remainder of one.

Then I asked what would happen if I changed the problem. I wrote on the board:

100 ÷ ? = 2 R1

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## Question: How Is Each Number Below Different From Each Of The Others

81 81 36 14

List students reasons as they offer them, encouraging them to incorporate math vocabulary into their reasons, such as divisor, factor, and divisible by. Discuss the meanings of the math terms they use and the relationships among them. For example, suppose one student says, The number fourteen is the only number that doesnt have nine as a factor, and another student says, The number fourteen doesnt belong because its the only number thats not divisible by nine. Use these two statements to discuss the relationship between the terms factor and divisible by.

This question can be asked for any set of four numbers. As an extension, ask students to choose four numbers for others to consider. Then have them list all the ways the numbers differ from one another. Use their sets of numbers for subsequent class discussions. Finally, have the student who suggested the numbers describe any differences that the class didnt find.

## How To Support Ell Students In Math

When supporting English language learners in math class, we often dont know if the problem stems from a language need or a math need. Yet it is important to think constantly in terms of assets, not deficits. A class that collectively speaks multiple languages is a cultural gold mine!

With Spanish language components, multilingual family letters, and ELL activity guides, HMH mathematics solutions are designed with the multilingual learner in mind.

• Education Research Director, Core Literacy & Early Learning

Dr. Vytas LaitusisEducation Research Director, Supplemental & Intervention Math

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## Ap Calculus Honors 300 Course Number M122 1 Mathematics Credit Grades 11

This course is the culmination of the honors program in mathematics. Students are expected to enter the course with knowledge of limits and basic derivatives. The curriculum for the course will include techniques and applications of derivatives, integration techniques and applications, the calculus of transcendental functions, the calculus of parametric and polar equations, and infinite series. A CAS TI-Nspire graphing calculator is highly recommended for all students in this course. Students are required to take the Advanced Placement test.

## Relationships In The Metric System

Algebraic Reasoning, BJU Press Geometry 4th Ed, Lesson 2.5–CCCS Flipped Geometry #14

The metric system is particularly easy to work with since its units relate to each other in the same way that units in place value relate to each other: powers of ten. This activity helps make that connection for students. Here students compare centimeter cubes, decimeter rods, and meter sticks and find all the ways

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## Use Tools Visual Models And Manipulatives

It is important to remember that the barrier is language, not necessarily knowledge. These strategies aren’t just good practice for multilingual learners. All learners who have working eyesight benefit from visual learning, and the same applies to physical learning with manipulatives. Math is unique in how often and how flexibly it can make use of visual and physical models. Yet when the learner is multilingual, models can go from best practice to critical mathematical scaffolds.

When learning is digital, the line between visual and physical models becomes blurred. For many students, in fact, digital models are especially helpful. Some learners feel more comfortable using tools when they can do so independently, and potentially anonymously.

When it comes to arithmetic in particular, students may draw calculations using algorithms and models that work but are new to you. Arithmetic falls into this gray area between procedural math and conceptual math, and as such is prone to regional differences. Just look at how division “looks” to a Venezuelan, U.S., and French student:

In general, have students show you what they know. And lean into the differences! There is rich discourse to be had when comparing different ways to divide numbers, along with other possible cultural differences such as how to notate numbers or draw graphs.

## Using Protractors In Middle School

From her past experiences teaching middle school students about angles, Cathy Humphreys knew that students often have difculty learning how to use protractors. Often they dont see the need for the tool, so Cathy does not introduce protractors until after the students have had concrete experiences measuring angles several ways.

When Cathy distributed protractors to her class in San Jose, California, she told the students, The protractor is a useful tool for both measuring angles and drawing angles of specic sizes. She asked the students to work in pairs and explore the protractors.

It may be helpful to use a right angle as a reference, she suggested, since you already know a right angle is ninety degrees.

After a while, Cathy called the class to attention and asked the students to share what they had noticed. Then Cathy gave them the challenge of guring out how to use the protractors and writing directions that someone else could follow. She said, Your directions should tell how to measure angles and also how to draw angles of different sizes. You can include drawings if they will help make your directions clear.

Before the students began, Cathy wrote protractor and angle on the overhead for their reference and asked them what other words about angles they might use. She listed all the words the students suggested: acute, right, obtuse, straight, degrees.

Figure 1.

In Figures 2 and 3, students explained in different ways how to use a protractor.

Figure 2.

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## The Importance Of Increasing Student Language Production In The Content Area

As I’ve worked with content area teachers in my district to develop Sheltered Instruction lessons and activities to enhance ELL learning, I’ve told them, “If a student doesn’t say it in your class, they’re never going to say it.” This is a bit dramatic, but it’s true to some extent. When students learn new vocabulary, the opportunity to use it must be presented in class, because students are unlikely to try it out on their own especially academic words like “parallelogram” or “function”!

Here are some tips to increase student-to-student interaction with academic language in the math classroom:

## A Suggestion For Your Classroom

You might try sharing one of my students methods with your class. Reproduce one of them, give copies to students in pairs, and have them see if they can figure out why it makes sense. Have them explain the method in their own words. Then give them practice applying it to other fractions. In this way, you can use the work of some of my students to help develop your own students skills and understanding. But, if possible, share the ideas from your own class. Honoring your students thinking is a way to involve them as important contributors to their learning and to the learning of their classmates.

From Printed Newsletter Issue Number 24, Fall/Winter 19981999

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