Teaching Multiplication for Understanding

One thing I love about the new math standards is teaching for depth of understanding.  It makes so much sense to teach fewer things in depth, than a lot of things with very shallow coverage.  In third grade we spend a great deal of time on multiplication, and students now develop a real understanding of what multiplication is and how it works, rather than just memorizing facts.  Below are some of the strategies that I teach students to use when looking for the product of an unknown fact.

1.  Draw equal groups.  Students model equal groups by drawing a circle of each one, then putting the correct number of objects in each group.

2.  Skip count.  To find the product of 3 X 4, count three fours: 4, 8, 12.  Teach students that these numbers are multiples of 4.

3.  Draw equal jumps on a number line.

4.  Relate multiplication and addition.  3 X 4 = 4 + 4 + 4 + 4

5.  Make an array.

6.  Use the Commutative Property of Multiplication.  If you do not know the product of 3 X 4, perhaps you do know the product of 4 X 3.

7.  Use doubles.  To find the product of 4 X 4, think of the product of 2 X 4 and double it.  To find the product of  6 X 4, think of the product of 3 X 4 and double it, and so on.

8.  Use the Distributive Property of Multiplication to break apart larger factors into smaller factors.
5 X 8 can be thought of as (2 X 8) + (3 X 8).

9.   Make a bar model.

I have developed several items to help my students master both the concept of multiplication as well as multiplication facts.  The products shown below, as well as others, are available in my Teachers Pay Teachers store.

Multiplication: Adding Equal Groups Power Point

Mutliplication: Adding Equal Groups Printables

 

 Big Multiplication Flash Cards

Distributive Property of Multiplication

Multiplication Facts Practice: X8

Ideas for Teaching Rounding to Young Learners

1.  Use a number line.  If you are rounding 44, make a number line that begins with 40 and ends with 50.  Put a red line on the number that is halfway between 40 and 50.  Circle 44.  Which side of the line is on?  Is it closer to 40 or 50?  Circle the answer.  Have students practice and practice until they can do the steps mentally.

 

2.  Make a number line on the floor using masking tape and have students use their bodies.  Have them stand on the number line where their number fits.  Then decide which ten it is closer to.

3.  Make “rounding mountains”.  Draw number lines that are shaped like mountains with 5 at the peak, 0 at the bottom on the left, and 10 on the bottom at the right.  Students think of a train trying to cross the mountain.  If it makes it to the top, it will coast down the other side.  If it doesn’t make it to the top, it will slip back down to where it began.

4.  Use place value blocks.   Give students tens strips and one cubes.  Have them show the number 44 using 4 ten strips and 4 ones cubes.  Then tell them that ones are no longer allowed and they need to show the number using only ten strips.  Would they be closer to the actual number if they used 4 ten strips or 5 ten strips.  Do this repeatedly and help them to see that when they have 4 or fewer ones, they stick with the number of ten strips they are already displaying.  When they have five or more ones, they need to trade them in for another ten strip.

5.  Use the picture book Coyotes All Around by Stuart J. Murphy to help students understand why rounding and estimating is helpful.

6.  One thing that has really worked for me is using the “Strong Man” of rounding and the “Wimpy Guy”.  I created these characters to help my students decide whether the digit in the place being rounded to stays the same or gets pushed up to the next number.  After they have spent some time with number lines and are pretty good at using them to round, it is time to switch over to doing the steps in your head.  I have students underline the digit in the tens place if they are rounding to the nearest ten.  They then look at the digit behind it to see if it is a “Strong Man” number-5,6,7,8,9 or a “Wimpy Guy” number-0,1,2,3,4.  A strong man number will push the digit up to the next number.  A wimpy guy number isn’t strong enough to do that and the number will stay the same.

I have created two power points and set of printables that provide lots of practice with the “Strong Man” and the “Wimpy Guy”.  They are available here:

Rounding is a skill that takes a lot of practice to develop proficiency.  I have found that my kiddos master rounding to the nearest ten and are doing great, then when I introduce rounding to the nearest hundred, I have mass confusion!  Lots of practice is the cure for this malady.  Once I have finished the rounding unit, I do not leave rounding behind.  I have my students round numbers to the nearest ten and hundred almost every day.  They soon become masters of rounding!

Happy Teaching!

The superintendent of our district visited my classroom the other day.  He is a very positive person who has made a significant difference in our district over the last eighteen or so years.   He quietly watched the lesson, asked a couple of questions, paid a compliment and then moved on to visit another classroom.  He has, for the past few years, visited the classrooms of teachers in their first three years of teaching.  This year he decided to spend an entire day at each school.  So, after visiting with beginning teachers, he rounded out his day by scheduling visits with veteran teachers.  That's how I got on the schedule.  I enjoyed his visit, though observations by anyone always make me nervous, even after 37 years.  At the end of the day, we had a staff meeting with him.  One thing he complimented the entire staff on, was being dressed professionally.  He said he recently visited with the education department at the state university in our community and asked them to stress the importance of dressing appropriately with college students preparing to become teachers.  He said he was so frustrated, during interviews with potential teachers, with the impression they projected, that he finally took action and requested that the university address the issue.  My personal opinion is that this is a reflection of prevalent attitudes in our society.  Perhaps we have become a little too casual overall.  Perhaps the way we present ourselves adds to the lack of respect for our profession.  I think that, as professionals, who represent the district that employees us, we need to give this some thought.  We don't need to spend a lot of money for clothes, and we can wear jeans occasionally and still present a professional image.  It has as much to do with grooming and attitude, I think, as it does with the way we are dressed. When we respect ourselves and our profession, it will show.

Why I Give the Comprehension Test First, Instead of Last

My district uses Pearson Reading Street for Tier 1 reading instruction.  I really like most of the passages in Reading Street, as they are tied to either science or social studies.  I don't like the fact that weekly comprehension tests are not tied to the passage we work with  all week.  So I wrote my own comprehension activity and test to go with every passage in the book. I also wrote my own comprehensive vocabulary tests.  At the end of every week, I test comprehension, vocabulary, and spelling. The comprehension tests are open book, and each one has only fourteen questions.  Even though we have read the story two or three times by Friday, I insist that my students use their books.  This  goes back to the Common Core standard of having students refer specifically to evidence in the text when answering questions.

These weekly comprehension tests have been a great source of frustration for me.  Most of my students would rather guess, or go on memory alone, instead of finding answers in the text.  They do not want to put forth the few minutes of effort that would almost always give them a good test score.  They have been content to get low scores, as long as they don't have to work for a high score.  I have talked, coaxed, pleaded, and explained the reason for referring to the text.  All to no avail.  I have sat next to kids and made them show the answer in the book before marking it on their paper.  I have gone over the test together when handing back the corrected papers, pointing out where the answer is found in the story.  I have given them a certain amount of time that they must spend on the test, and have not allowed to do anything else until that time is up.

This week, I gave the comprehension test first, on Monday morning.  We had not read the story together, I had not introduced the story or the vocabulary words.  I simply told them to open their books to the beginning of this week's story, gave them the test, and told them to use their books.  Guess what?  They used their books.  They had to.  They had no memory of the story to rely on.
Of twenty-three students taking the test, 2 had the same score as on last week's test (which was taken last Friday), 1 had a lower score, and 20 students had higher scores.  My class average last week was 79%, when taking that comprehension test after reading, discussing, and studying the story.  This week, on the first reading, my class average was 93%.

From now on, we will take the comprehension test with the first reading of the story.  This makes sense, because on standardized tests, they answer comprehension questions the first time they read the text, not after a week of discussion and re-reading.  They are in a situation of relying on the words of the text, rather than their own memory.

Family Literacy Night

My school hosts a Family Literacy Night every year during the first week of March.  We invite students and their families to spend the evening celebrating reading and literacy.  The first grade classes usually open the evening with a couple of songs about books and reading.  Then we have break-out sessions.  Each session lasts 20 minutes and there are three different choices during each time slot.  Our evening involves presentations by the students, as much as possible.  They give book talks, present reader's theater, showcase book projects, or share research reports that they've done.  Our kindergarten teachers do a puppet show that is a big hit every year.  We have "Are You Smarter than a . . .(third grader, fourth grader, or fifth grader) where parents compete against their kids in answering test questions similar to those found on end-of-year tests.  The local public library is invited to share resources such as free e-books. One of the big hits of the evening are the free books that are given to each of our students and their pre-school age siblings.  These books are purchased with Title 1 funds.  There is a room assigned to each grade level where student can choose a book of their liking.  This evening also is the culmination of our One School, One Book project.  About a month ago, each family was given a copy of a novel, this year it was "The World According to Humphrey", along with a reading schedule.  Students have been winning prizes all month for correctly answering questions about the book.  At Family Literacy Night, the whole family gets to work together to answer questions about the book.  Prizes are given to the families with most correct answers.  The evening is capped off with ice cream.  As students and their families leave the building, they are all given ice cream sandwiches.  This evening has become an important tradition at our school, and is looked forward to by students and their parents, alike!

Reading Comprehension: Fix-up Strategies

Good readers automatically monitor their own comprehension.  They know when they don't understand the text, and they use various fix-up strategies to make sense of what they are reading.  Poor readers, however, often do not recognize that they don't understand.  They muddle through, reading in the same way and at the same rate, even though the text is meaningless to them.  Sometimes poor readers don't expect the words they read to make sense.  They are so used to failing at reading comprehension, that they don't know any different.  For these kids, explicit teaching of fix-up strategies is essential.  Listed below are strategies that can help poor readers improve comprehension.  They can be used separately, or in combination:

1.  Stop and re-read.  This is the most important fix-up strategy.  Don't just keep going if you don't  understand.  When you reread, you might find a word that you skipped or misread.  Sometimes a single word can make a huge difference. 

2.  Slow down.  Read slowly and carefully when things don't make sense.  Vary your reading rate, depending on your understanding.

3.  Question yourself as you read.  Think about whether or not the sentence or paragraph makes sense to you.  Could you retell it to someone in your own words?

4.  Read ahead a few sentences.  Maybe getting a little more information will make things clear and help with the confusion.

5.  Visualize.  Make your own movie in your head and "see" what is happening or what is being described.

4.  Figure out words you don't know.  Look for prefixes, suffixes, and base words.  Use a dictionary if you need to.

5.  Make predictions.  What do you think will happen next?  Check to see if your predictions were right.  Make new predictions as you read further and get more information.  Predicting can keep you focused on what is happening in the text.

6.  Look at pictures, diagrams, charts, and maps.  Read captions and sidebars.  Especially in expository text, a lot of understanding comes from sources other than the paragraphs. 

Struggling readers can benefit greatly from the explicit teaching of these strategies.

 

Academic Vocabulary

 

One of the key shifts in Common Core standards is an emphasis on academic vocabulary.  These are words that are used in the classroom, in textbooks, and on assessments.  Understanding academic language is essential to student success.  Academic vocabulary needs to be taught explicitly.  It is not likely to be acquired through students’ independent reading or incidental classroom exposure.

Vocabulary words can be divided into three tiers.  Tier 1 words appear frequently in spoken language.  They are commonly used in various contexts, not just in classroom or educational settings.  They rarely require explicit instruction. These would be words like clock, house, or walking.

Tier 2 words are used less commonly, and are likely to appear more often in written text than in spoken language.  These words are generalized, and used in many different contexts.  They are often more precise ways to say rather simple things. They are essential to understanding, and need to be taught explicitly.  Examples of these words are: conclude, define, and dialogue.

Tier 3 words are highly specialized, and domain-specific.  They are not generalized, or used widely. These words must be explicitly taught. Tier three word examples are: isotope, tectonic plates, phoneme

Here is a suggested procedure for teaching academic vocabulary:

1.  Introduce the word by providing a description, explanation, or example in student-friendly terms.  Do not use dictionary definitions when introducing the word.  Write the word for students to see.  Have students repeat the word, and spell it orally.  Provide several opportunities for students to read and say the word.

2.  Have students explain the term in their own words and let them share examples, synonyms, or uses of the word.

3.  Have students represent the word in graphic or picture form.

4.  Working with partners or in small groups, have students discuss their ideas and understanding of the word.

5.  Provide opportunities for the word to be used in context.  Teach a word when it will be incorporated into the lesson activities.  For example, teach the word develop when students will be working on writing assignments and can be encouraged to develop their ideas and add more information.

6.  Be sure to teach other forms of the word.  For example, when students learn conclusion (a noun), they should also learn conclude (a verb).

7. Review and revisit the words after they have been taught.  Use the terms frequently during instruction in all subjects.

 

I have  new academic vocabulary packet in my Teachers Pay Teachers store.  It has page-size posters for 48 academic vocabulary terms that are taken directly from Common Core standards.  I purposes chose words that were not tied to one domain, but are very general and are useful in any subject area.  You can find it here:

Academic Vocabulary from Common Core Standards

 

 

 

Best Fraction Strips EVER!
I bought a class set of these fraction strips from EAI for an amazingly low price because they were made for use with overhead projectors and were on a close-out sale.  They are great for kids to manipulate.  Each denominator is a separate strip, so any two strips can be moved to compare side-by-side. They are sturdy plastic and are much more kid-friendly than the fraction circles that I had been using. 
The circles were really hard for the kids to put all the pieces together to make an actual circle.  These are so easy to use!