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Teaching Long Division
Need to have it make sense!
Have them divide by each at a time.
If they want to divide 865 by 5 for ex.
First have them figure out 800/5 on own
Then show them how they can short-cut that with long division.
Then show them 60/5
Then 5/5.
Then we put it all together with long division and show what each step means!
Have them divide by each at a time.
If they want to divide 865 by 5 for ex.
First have them figure out 800/5 on own
Then show them how they can short-cut that with long division.
Then show them 60/5
Then 5/5.
Then we put it all together with long division and show what each step means!
Process Focused Assessment
Ask students to write the hardest problem they can think of and then have them try to solve it! Or write out a method that could maybe solve it. This will give you an idea of how skewed their perspective on math could be.
Estimation
This is hard to teach
Students do not see the point always.
There are times to over estimate and under-estimate
(COULD BE USED IN THE GROCERY STORE THING TOO!)
Show students some images of a bunch of little blocks AND THEN have them see the actual amounts and then estimate once we show them other clusters of blocks.
Useful for capacity too. This should happen with ALMOST EVERY new math skill.
Students should learn estimation so that they can check their answers!
Learn when to underestimate and over-estimate
Over = gas for a trip, food for quests, money needed
Under = cooking things, books on a bookshelf, weight on something, carrying things in a basic
ALWAYS MAKE THEM EXPLAIN OUT THEIR REASONING!
Students do not see the point always.
There are times to over estimate and under-estimate
(COULD BE USED IN THE GROCERY STORE THING TOO!)
Show students some images of a bunch of little blocks AND THEN have them see the actual amounts and then estimate once we show them other clusters of blocks.
Useful for capacity too. This should happen with ALMOST EVERY new math skill.
Students should learn estimation so that they can check their answers!
Learn when to underestimate and over-estimate
Over = gas for a trip, food for quests, money needed
Under = cooking things, books on a bookshelf, weight on something, carrying things in a basic
ALWAYS MAKE THEM EXPLAIN OUT THEIR REASONING!
Problem Solving in Different Ways
Ask students to solve problems in 4 different ways. Use pictures!
Teach students to anchor to 5 and/or 10.
8+6? 8+2 = 10+4=14
Set up tasks to get students to learn and invent their own strategies to solving problems.
What patterns do you see?
1+1 = 2 1+2 = 3
2+2 = 4 2+3 = 5
3+3 = 6 3+4 = 7
4+4 = 8 4+5 = 9
5+5 = 10 5+6 = 11
Teach students to anchor to 5 and/or 10.
8+6? 8+2 = 10+4=14
Set up tasks to get students to learn and invent their own strategies to solving problems.
What patterns do you see?
1+1 = 2 1+2 = 3
2+2 = 4 2+3 = 5
3+3 = 6 3+4 = 7
4+4 = 8 4+5 = 9
5+5 = 10 5+6 = 11
Mastery of Facts = 3 seconds to respond
Performance Assessment
Grocery Shopping!
Set up boxes all over room and create problems with them.
Students have to solve real life things!
Given Money and have add up to get EXACT amount or do some simple subtraction to see what they end up with
Servings (need to serve ___ people, each eat 1 serving, how many can this or that feed?)
Multiples (they sell ____ hotdog buns and ____ hotdogs in a packet. How many should I get to get same amount of both?)
Factors...
Capacity and Measurements (They find gallon and go around and see what things can be filled with gallons. This would be good for seeing conservation of things.) (can also have them fill up things and figure out how many of things add up to others. Like an experiment)
Set up boxes all over room and create problems with them.
Students have to solve real life things!
Given Money and have add up to get EXACT amount or do some simple subtraction to see what they end up with
Servings (need to serve ___ people, each eat 1 serving, how many can this or that feed?)
Multiples (they sell ____ hotdog buns and ____ hotdogs in a packet. How many should I get to get same amount of both?)
Factors...
Capacity and Measurements (They find gallon and go around and see what things can be filled with gallons. This would be good for seeing conservation of things.) (can also have them fill up things and figure out how many of things add up to others. Like an experiment)
Writing Math Problems.
Students should put problems into their own words. This would help them.
Get students to write word problems to use. This would help them with a new way of thinking.
Get students to write word problems to use. This would help them with a new way of thinking.
Spacing for Word Problems
Students should have plenty of space to do work.
Students NEED to draw what is in the problem.
Students need HALF a page to show the problem and work it out.
SUGGESTION have a piece of paper with dark lines that students can place UNDER their word problem worksheets to write out their explanations when and where they need it.
Students NEED to draw what is in the problem.
Students need HALF a page to show the problem and work it out.
SUGGESTION have a piece of paper with dark lines that students can place UNDER their word problem worksheets to write out their explanations when and where they need it.
Objective
For every task you want students to be able to tell you: what they did to get the answer, why they did it that way, why they think the solution is correct or reasonable.
Can use this idea to write a new style of objectives
SWBAT tell me what they did to solve ______________ problems.
Can use this idea to write a new style of objectives
SWBAT tell me what they did to solve ______________ problems.
**Find the patterns in math every time!**
What do you do with Mistakes in Math?
Use mistakes as an opportunity to learn!
Examine errors in reasoning and raise everyones level of analysis.
Don't cover them up!
Examine errors in reasoning and raise everyones level of analysis.
Don't cover them up!
Conceptual vs. Abstract Thinking
Students should learn Abstract first and THEN Procedural
Why?
Because students learn tricks but do not know why they work and therefore cannot remember them forever, accurately, or re-teach themselves them if they forget.
Students do long division but they do not know why it works!
Why?
Because students learn tricks but do not know why they work and therefore cannot remember them forever, accurately, or re-teach themselves them if they forget.
Students do long division but they do not know why it works!
Learning Facts
Practice 5-10 minutes a day. NO MORE!
Mix it up.
Asking basics first and then get harder every day!
Around the world?
Do 111 * 2
Then 112 *2. Have students experience how things change with each change in numbers, only 1 up or down at a time.
Mix it up.
Asking basics first and then get harder every day!
Around the world?
Do 111 * 2
Then 112 *2. Have students experience how things change with each change in numbers, only 1 up or down at a time.
Word Problems
Students are made to think that word problems are the hardest because they always come last.
A problem should be something you don't know!
A problem should be something you don't know!
Math Centers
Go to MCPS website and look up flip charts that have them.
Cooperative Math Problem Solving
Steps of Cooperative Problem Solving
Even if your students record the answers on a worksheet, the answers are not a true assessment of their skills. You still need to assign independent math problems on a regular basis. Doing so holds students accountable, not only for completing the work, but for learning the skill.
- Teacher Presents the Problem - Display a math problem on the board, hand out a worksheet, or ask students to turn to a problem in the math book. Read the problem aloud or ask them to read it silently. You’ll find free Daily Math Puzzler worksheets on my Problem Solving page that would work well for this activity.
- Students Work Alone - Ask students to work the problem alone, preferably on dry-erase boards so they can easily erase their work and try different strategies. They turn their boards face down when they have a preliminary answer or you tell them that time is up.
- Students Work Together - Students compare and discuss answers with a partner or with a team. I generally prefer partner work in math, but if the problem is really challenging, I allow the entire team to talk it over and work it out together. If students realize that their answer was wrong, they may change it, but they must show the work to go with their new solution. They don't all have to agree, but each person should be prepared to explain his or her answer.
- Class Discusses Solutions - Reveal the answer to the class and call on students to share how they solved the problem. Instead of focusing on a single "right" way, challenge your class to come up with as many ways to solve it as possible. Allow different students to hold up their dry erase boards or place them under a document camera as they explain their solutions. If students are required to record an answer in a journal or on a worksheet, allow time to do this now, without talking to anyone.
Even if your students record the answers on a worksheet, the answers are not a true assessment of their skills. You still need to assign independent math problems on a regular basis. Doing so holds students accountable, not only for completing the work, but for learning the skill.
Math Anxiety
Cognitive scientist Sian Beilock of the University of Chicago, for example, has theorized that math anxiety affects students’ performance in the subject by using up mental resources, such as working memory, that could otherwise be deployed in solving math problems. One way to relieve this burden on working memory, Beilock and her colleagues have found, is to spend ten minutes writing about one’s thoughts and feelings about a math exam just before taking it. Students effectively offload their worries onto the page, enabling them to tackle the test with a mind free of rumination and distraction.
Other approaches that have proven successful at reducing math anxiety and improving performance include having students reaffirm their self-worth by listing important values like relationships with friends and family, and having students think about why they might do well (“I am a student at a high-level university”) rather than poorly (“I am a girl taking a difficult math test”). These interventions are simple but effective: By deliberately shifting their frame of mind, students can make that creepy-crawly feeling of anxiety go away.
Other approaches that have proven successful at reducing math anxiety and improving performance include having students reaffirm their self-worth by listing important values like relationships with friends and family, and having students think about why they might do well (“I am a student at a high-level university”) rather than poorly (“I am a girl taking a difficult math test”). These interventions are simple but effective: By deliberately shifting their frame of mind, students can make that creepy-crawly feeling of anxiety go away.
Geometry
Created by me in Adobe Illustrator
My 5th graders had trouble with the different types of polygons long after the tests were taken and we had moved on!
There is something about trapezoids that is hard to remember!
Here is a GIANT Venn Diagram about all the polygons (except types of triangles and regular shapes)
There is something about trapezoids that is hard to remember!
Here is a GIANT Venn Diagram about all the polygons (except types of triangles and regular shapes)
The Learner and Learning
- The student group that I worked with was the "Extensions Grous"
- This group of students was on grade level.
- It was my mentor and my job to teach this group on grade level math and to create little extension lessons
- After each unti students are pre-assessed and based on their knowledge on the coming unit they are grouped accordingly. So the groups are always changing and so the curriculum is avaivable to all students who are capable.
- Students had opportunities to work individually and with their peers.
- Students work with peers first and then individually to work on worksheets
Content
- Prior to this lesson, students worked with Circumference but they used 3 instead of Pi.
- Students had to, for my lesson, use 3.14. This was just as simple as using 3, I thought, and they had been using 3 for almost a week and solving problems with no obstacles. I they could do the math the first time but they do not know math. I saw this with this lesson because they could not see the similarities and use Pi with ease.
- Following the classes first Circumference lesson, I tried to create a similar one but on a larger scale. Students measured random objects and saw that it was about Dx3=C. So I got a trashcan and I did the same thing. I measured the C and D out to 2 decimal places and I altered them a little so that the students could see that it equals about 3.14. I explained that mathematicians did this with many circles until they saw the similarities. So I showed them how mathematicians did it so many years ago.
- They also saw a video about it during a different lesson so we were reinforcing their knowledge multiple times.
- Students saw where Pi came from
- Students create formulas from this that were similar to ones that they had already been using for about a week.
- Students did problems similar to previous ones using simpler formula from beginning of week.
- Clas worked together to solve some ont he front board. I got to check for understanding.
- Students worked with peers.
- Students tried some challenges, unlike previous problems they used with simpler formulas. THIS is where problems arose.
Instructional Practice
Assessments
- Students had to fill out a classroom worksheet as their assessment.
- I could not give them homework for it because they had to have the same homework as the other groups that are on grade level.
- We went over some of the problems as a class and I got to informally observe their understanding
- I added challenge questions to see how much extension they could handle. We modeled them after MSA questions.
- CANNOT FIGURE OUT HOW TO DO THE: DEMONSTRATE A DEVELOPING CAPACITY TO USE A VARIETY OF INSTRUCTIONAL PURPOSES AND LEARNING STRUCTURES TO ENCOURAGE LEARNERS TO DEVELOP DEEP UNDERSTANDING OF MATHEMATICS CONCEPTS AND THEIR CONNECTIONS, AND TO BUILD SKILLS TO ACCESS AND APPROPRIATELY APPLY THESE CONCEPTS. "MAYBE SNAP SHOT OF PLAN TO HAVE THEM EXPLAIN, IN THEIR OWN WORDS WHERE TEH FORMULA COMES FROM.
- OWN WAY: Students first did the yarn test to see the relationship and then I solidified it with the Trashcan.
- Use effective questioning: the clip where I ask why did divide by three? Going over with Raeann. Why did divide? Wont that make it smaller? Ask them in this way so that they try to think about how operations affect things, they make them bigger or smaller, which do you want?
Professional Responsibility
- Mentor teacher suggested to not do the hands on for Area.
- I chose a good video.
- I related the video back to each problem we did.
- half the class wanted to see it in that way and I listened because it would not hurt the others so I made that choice.
- I wanted to test doing the activity with a student.
I kept realizeing that students were just choosing a formula and going with it.
They thought that squaring something was the same and adding it twice.
I kept writing up the formula again (there is a clip for this).
Realizing when students are not getting it. What are they confused about, why?
Videos In Math...?
Watched a video about LCM, it went fast and used big words all over the place...
I say NO
MAYBE if explaining ONE vocab./concept and is explaining in pictures
I say NO
MAYBE if explaining ONE vocab./concept and is explaining in pictures
Taking Time
Take the time to do it right, or take the time to do it twice.
Calculator Use
Students need to learn about the technology that they will be using for the rest of their life!
Clear the Mat
Create a 'mat' for your team.
Each team starts with 3 flats.
Roll one die or two die.. (if come up with number that is too large to take away than loose term!) Once start rolling die, stuck with it!
Form a number with the digits.
Model the subtracting of the amount rolled, other team checks it.
Record subtractions, other team checks.
Take turns.
First team to get to 0 wins.
Each team starts with 3 flats.
Roll one die or two die.. (if come up with number that is too large to take away than loose term!) Once start rolling die, stuck with it!
Form a number with the digits.
Model the subtracting of the amount rolled, other team checks it.
Record subtractions, other team checks.
Take turns.
First team to get to 0 wins.
Pico, Fermi, Bagels
A secret 3 digit number number.
All have to be different numbers.
Bagels means none of the digits are correct.
Fermi means a digit it right but not in the correct place.
Could use words: place, digit, none.
How Many Different Ways Can You Solve This - Warm Up
Give students a math problem. ___ + ___ or ____ - _______
Have them give you as many ways as possible to solve for it.
Like a challenge.
Try to get them to not use calculators and not worry about actually figuring it out!
This way they can just imagine breaking it down first.
Have them give you as many ways as possible to solve for it.
Like a challenge.
Try to get them to not use calculators and not worry about actually figuring it out!
This way they can just imagine breaking it down first.
GET OUT OF YOUR SEAT learn multiples/divisibility rules
Have students count from 1 and up
Pick some multiple rules for them to identify.
Raise right hand if you get a number that is a multiple of ___
Raise your left hand if you get a number this is a multiple of ___
Don't mention raising both hands, let them figure it out.
This is great to teach them their facts. You can try to add more multiples and they have to kick a leg or something. This will get them to memorize hopefully
Pick some multiple rules for them to identify.
Raise right hand if you get a number that is a multiple of ___
Raise your left hand if you get a number this is a multiple of ___
Don't mention raising both hands, let them figure it out.
This is great to teach them their facts. You can try to add more multiples and they have to kick a leg or something. This will get them to memorize hopefully
Fractions
- Use manipulative and drawing firsts!
- Then have them create their own fractions and explore how they look and how they are simplified and the equivalent ones.
Maybe have them find as many fractions as they can that are in simplest form. 1/x are all simplest form.... 2/odd #'s.... 4/odd
- Use specific vocab. DO NOT SAY "CANCEL OUT", OR "REDUCE" say simplify. Reduce implies that fractions are getting smaller!!
- Give them visual representations of fractions. Like pizza! There is a whole on the bottom. And then the top represents the parts. And then this way when you add they see that they do not just add the top numbers and the bottom numbers! Can also see how 2/1 is 2!
COMBINING MATHS INTO ONE PROBLEM AREA
- A market place idea, there a bunch of problems that they have to explore prices, serving sizes etc.
- Using pizza! The area of it, fractions of it. Expressions. Cost!
What do you Notice? What do you wonder? (Use when introduce word problem)
Why do this?
See if there are words students do not know, can then go over
Get their brains to start thinking.
It takes the pressure off of the students. If there is a lot of text then they way get over-whelmed and shut down.
They can think as a group and see things that someone else might not have seen.
Scaffolds.
Get them to actually notice what information then they and maybe do not need.
A GREAT WAY TO STRETCH A PROBLEM OVER A WEAK. GOOD FOR YOUNGER STUDENTS TO BE INTRODUCED TO PROBLEM SOLVING. THIS COULD BE DAY ONE
See if there are words students do not know, can then go over
Get their brains to start thinking.
It takes the pressure off of the students. If there is a lot of text then they way get over-whelmed and shut down.
They can think as a group and see things that someone else might not have seen.
Scaffolds.
Get them to actually notice what information then they and maybe do not need.
A GREAT WAY TO STRETCH A PROBLEM OVER A WEAK. GOOD FOR YOUNGER STUDENTS TO BE INTRODUCED TO PROBLEM SOLVING. THIS COULD BE DAY ONE
Special Needs in Math
Go up to the Board and show us!
Better to have more than one up at a time because the rest of the class will be bored!7YY7Y6
What is a Problem?
Start every lesson with word problems like this! Students should not be seeing problems at the END of their text book because the end implies that they are harder!
- they have to not KNOW right away how to solve it!
- (IF THEY DO KNOW ONE WAY THEN ASK THEM TO FIND 2 OR 3 MORE AND THEN IT BECOMES A PROBLEMS)
- a task or activity
- there are no perceptions about how to solve it or an answer
- not ONE way to solve it
- not set of rules that a students knows or thinks they have to use while solving it, no memorized formula's etc.
- at an approp. level for the students
- engaging
- MUST COME WITH A JUSTIFICATION UPON SOLVING
Head Spinners
Give students numbers and then use a spinner or a slot machine if you want to use their class numbers to call on them
Or use it for anything where a student needs to do something.
Or use it for anything where a student needs to do something.
New Manipulatives
* When introduce new manipulatives give students a chance to play with them for a little bit, THEY WILL ANYWAY!
Guess this number!
- Give vocab. and then learn their vocab.
- Place value, digits, product, divisible, sum, multiple, prime
- abstract thinking, deductive thinking, narrowing something down by using examples and non-examples, concept attainment
- model how to ask questions
- Then when they know, they will ask questions that they know the answer is yes, and then maybe do ones that they know it is no
- Start them out simply, no noise!
- Let them think and start to make lists etc.
- Let them partner off and share their thought paths
- Give more clues when needed
- When students think they have the right answer they need to then ask questions that they know the answer is yes
- Give more clues if needed
- Go to larger groups if needed
- Eventually, make sure all or most know the right answer and get them to answer together
Multiples with Names
Sarah writes her name over and over, what would the 53rd letter be? How many A's would she write if she wrote her name 30 times?
The letter ____ is 1/5 of the letters in my name.
If put all the letters of my name in a bag and randomly pulled them out what is the most likely?
LCM, give them a bunch of flash cards, right their name out, a letter a card, get with a partner and have to see how many need to spread out to make same length.
The letter ____ is 1/5 of the letters in my name.
If put all the letters of my name in a bag and randomly pulled them out what is the most likely?
LCM, give them a bunch of flash cards, right their name out, a letter a card, get with a partner and have to see how many need to spread out to make same length.
Find your Group
- Group by math (same representations of fractions)
- Pieces of a pie, have to fit all into one pie, they will be one color for each fraction type.