Author Archive

Summer Activities for Future Third Graders!

Written by ValerieW. Posted in 2nd Grade

Dear Parents,

School is almost out, and summer vacation will be here very soon!  Here are some suggestions for activities to do with your future third grader.

  • read every day:  read to your child, and have your child read to you, his or her siblings, or to read to himself or herself.  In English and any other language!
  • write every day:  have your child write a summer journal of what they have experienced daily, write to each other in an interactive journal, inspire your child to write POETRY!
  • read the poetry of Amy Ludwig Van Derwater, Zoë Ryder White, Kristine O’Connell George and others.  Inspire your child to look at the world with poet’s eyes.
  • for more math practice, find the Common Core codes, and google loads of activities
  • above all, treasure the time you spend together!

Here are some online resources:

a.  Online games in math and reading:

b.  Work with reading  and reading fluency: http://www.readnaturally.com/  search for summer activities, and you will find, among other activities:

c.  Work with math fluency

d. Work with math word problems (rich tasks, when they genuinely inspire children to think and stretch their capabilities)

e.  Ideas for writing:

f.   Work with reading comprehension:

g.  Summer reading books/ suggested books

h.  free online books

The future third graders have worked very, very hard this year and it has been an honor working with them.

We thank all of you for all the support you have given your child, and also the support you have given us. Take care and have a wonderful summer!

The Second Grade Team

Eureka Math Module 5: Addition and Subtraction to 1,000 with Word Problems to 100

Written by ValerieW. Posted in 2nd Grade

Dear Parents,

We are just starting Eureka Math Module 5, which is addition and subtraction within 1000, with word problems to 100.

In Module 5, students will continue to develop their conceptual understanding  of addition and subtraction  to numbers within 1,000, always  modeling with materials or drawings.  A huge focus is  strengthening and deepening the students’ conceptual understanding and fluency.

Students will:

  • relate 100 more and 100 less to addition and subtraction of 100
  • add and subtract multiples of 100
  • use simplifying strategies for addition and subtraction
  • use compensation to subtract from three-digit numbers
  • use written method which include number bonds, HTO charts,, arrow notation, open number lines, and tape diagrams.

Here are some great LearnZillion videos you can watch at home with your child:

Add or subtract a number within 1000 in parts:

https://learnzillion.com/lesson_plans/263

Add within 1000 using expanded form

https://learnzillion.com/lesson_plans/261

Subtract within 1000 using expanded form

https://learnzillion.com/lesson_plans/265

 

 Adding by grouping tens into hundreds

https://learnzillion.com/lesson_plans/259

 

Please don’t hesitate to ask us if you have any questions!

The Second Grade Team

Writing Workshop: sharing our opinions about books

Written by ValerieW. Posted in 2nd Grade

Dear Parents,

Second grade students are starting  a new unit in writing workshop:  opinion writing.  The great thing  about this unit is that it calls upon our students’ love of books and reading, as the opinions they will be writing  are opinions about their favorite books and the characters that inhabit those  books.

Image result for charlotte's web

Fern Arable from Charlotte’s Web

To begin with, the children will be writing letters about the books they are reading, in class and at home, to other children who may be interested in reading the same book.  They’ll be writing about the characters they meet in their books, discussing their opinions and ideas about the characters, while also including reasons for these ideas and opinions, providing details and examples from the books with which to support their opinions. 

Later, children will deepen their thinking as we encourage them to think up even more ideas, details and evidence from their reading  to support their opinions clearly, in a way that makes perfect sense to the reader. We’ll take a look at the conventions they notice in published books, and incorporate them into our own writing. 

Lastly, we will ask the children to write persuasive essays, in which they try to convince readers that their favorite books should win an award!

The second graders are passionate about books and reading, and will no doubt have much to tell us all about their favorite characters.  We hope they will inspire their classmates to read even more!

   Annie and Jack from The Magic Tree House

Image result for mercy watson image    Mercy Watson

The Second Grade Team

More addition and subtraction strategies: tape diagrams and number bonds

Written by ValerieW. Posted in 2nd Grade

Dear Parents,

Here are two more addition and subtraction strategies we are teaching the second graders:  tape diagrams and number bonds.  You are already familiar with number bonds, but we thought we’d explain it again to clear up any doubts your child may have.

Tape Diagrams

A tape diagrams is really just a model or a drawing of a word problem!  As we know, a picture is worth a thousand words.  It is incredibly helpful for children to actually see a diagram of a word problem before they try to do any calculations.  It helps them see the “big picture” or the whole idea, and gives them clarity as to just what it is they need to figure out.  As we will see in the examples below, having an illustration of the relationships between the numbers involved in the word problem can help children take a look at how reasonable their answers are.

Tape diagrams are also known as strip diagrams, bar models, fraction strips or length models.  We are calling them tape diagrams at our school, because they look like segments of tape:

Up to this point, the children have, of course,  been making visual representations of problems, drawing apples, trees, or base ten blocks to represent the numbers and give them a better sense of what they need to do to solve problems.  Now the drawings become more abstract, showing quantities as opposed to actually showing groups of objects.

If a word problem calls for adding 13 apples to 37 apples,  when using tape diagrams, children are drawing something more abstract, not one group of 13 apples and another of 37, but rather two rectangles (tape segments)  that bear a relationship to one another, the “13 apple tape” being shorter than the “37 apple” one. 

Part-Whole Model for Addition and Subtraction

In the problems below,  we need to think about what the WHOLE is, and what the PARTS are. 

Sara brought 4 apples to school. After Mark brings some more apples to school, there are 9 apples altogether. How many apples did Mark bring?

9 apples = the WHOLE, shown here with with parenthesis at the top of the problem. 

4 apples brought by Sara = just a PART. We still need to figure out what PART Mark is bringing. 

This is a very simple example!  But when children see that the top part of the drawing is going to be 9, they can see that Mark bringing 12 apples is a very unreasonable answer, because it’s way too big!  (12 is bigger than the WHOLE of 9 apples.)

Comparison Model for Addition and Subtraction

1. Mateo has 5 toy cars. Josiah has 2 more than  Mateo. How many cars do Matteo and Josiah have altogether?

In this problem, we are comparing the PARTS.  We are comparing the amount of Mateo’s cars to Josiah’s cars.   We know the smaller quantity. To find the larger quantity, that is, the amount of cars Josiah has, we add 5 + 2.  Later, to get the WHOLE, we then add up the PARTS.

Here it is very useful to have a diagram that shows us the relationship between the amount of cars that Mateo and Josiah  have.   Since we are making a comparison between how many they each have, it is helpful to have the PARTS of the problem one above the other.  In Mark and Sara’s apple problem above, in which there was no comparisons of the PARTS,  the tape diagram was set up differently.  The PARTS were side by side, with the WHOLE stretching above them.

2. Sam has 45 stamps.  He has 20 more stamps than Joe.  How many stamps does each boy have?  How many do they have in all?

Again, it is very useful to have a diagram to show us the relationships between the quantities of stamps that Joe and Sam have.   Since we are making a comparison between how many they each have, we have have the PARTS of the problem one above the other, and thus we see how much bigger one part is than the other.

We finish by adding up the parts together to get the whole.

This is just a start!  Our students will need time and practice to understand how they need to set up the diagrams, depending upon whether the word problems call for a comparison of the parts, or for taking a part from the whole, or adding up the parts. 

The second graders will be using tape diagrams  in third, fourth and fifth grade, and beyond.  Tape diagrams can be set up differently for multiplication and division.  They help to give “a thousand words” more to word problems, and thereby aid children’s understanding. 

Number Bonds

We worked on number bonds at the beginning of the school yearTo review, a number bond is a  visual way to see a number and the parts that can make up a number. In this way, number bonds help children visualize  addition as putting things together and subtraction as taking things apart.

As we do addition and subtraction problems in second grade, we create number bonds by decomposing or “breaking up” larger numbers into smaller parts, so that we can add and subtract more easily.

Addition with number bonds

When doing addition, we often break up numbers into tens and ones, then add tens to tens, ones to ones, and then everything together.

48 + 23 =  _______

a. We can break up the numbers into tens and ones:

b. Then we add the tens together, the ones together, and then both parts together:

Subtraction with number bonds

61 – 27 =  _______

Subtraction will be a bit trickier.  When we look at the tens and ones in this subtraction problem, we see that we need to take away more ones (7) than there are in the first number ( only 1 in 61).  When the second graders use base ten, they know that in order to be able to take away 7 ones, they’ll need more ones to do so – – and so they exchange a ten.

a. Here, they will need to break 61 up in such a way that there will be enough ones to subtract from.  They will need to be careful about the number they place in the red circle:  Here is where most of the second graders make their mistakes!

b. With number bonds, we can be creative when breaking apart or decomposing numbers, and we don’t NEED to stick to tens and ones.  Here is an example:

c. You can break up 61 this way (50 and 11) and you could just as easily do it as 51 and 10.  It’s up to you.  You just need to make sure there is a number larger than 7 (in this case) in the red circle.

Continuing on,  we will subtract 50 – 20, and 11 – 7.  

The tricky part to remember is that we then ADD UP the two partial answers, because the 30 and the 4 represent the PARTS of the initial equation of 61 – 27.

d. The way we have just done 61 -27 with number bonds connects with the way in which the children have done subtraction with the place value (HTO) chart.  Here, we have decomposed 61 into 50 and 11.

e.  We continue to solve the problem:  11 – 7 = 4 (in the ones column) and 50 – 20 = 30 (in the tens column).  We end up with 30 + 4, or 34.

Our goal is for our students to have a large repertoire of strategies at their fingertips;  we want them to have lots of tools in their tool box to choose from!  Later on, with lots of practice and equipped with strong mental representations of these strategies they’ll have loads of flexibility and be much better able to do the math mentally.

Here are two videos you can watch, to see how the second graders are using number bonds.

https://learnzillion.com/lessons/3142-find-unknowns-by-creating-number-bonds

 https://www.engageny.org/resource/grades-pk-2-math-supporting-coherence-across-grades-number-towers-number-paths-and-number

Please let us know if you have any questions!  Thanks for all the help you are giving our students!

The Second Grade Team

How to do addition and subtraction on the place value chart (HTO chart)

Written by ValerieW. Posted in 2nd Grade

Second Grade Math

How to do addition and subtraction on the place value chart (HTO chart)

ADDITION:

48 + 24 = ________

  1. DRAW the numbers in base 10 (meaning, lines for the tens, and circles or dots for the ones) on the place value chart.  Ask your child to organize the numbers; tens like tally marks, ones like dots on a ten frame, as shown below).  The two numbers are placed in middle part of mat, the first number above the second one. (The answer will be placed below the line.) We start out by adding up the ones.  Ooops!  There are too many ones for the ones column (there are 12).

 

 

  1. Take a red pen, and circle a group of 10 ones (you are composing them into a group of ten).

 

 

  1. With the red pen, draw an arrow over to the tens column, where you will be putting your group of ten.  Remember to cross off the circle of ones, so you don’t count it twice.

 

 

  1. Now add them all up, and DRAW the answer (the sum) at the bottom of the table in base ten (lines and circles).  Fill in the number sentence at the bottom.  You are finished!

SUBTRACTION

WHAT’S THE DIFFERENCE? 

When we subtract, we are actually looking at two quantities to see the difference between them, that is, we are trying to figure out how much more one group has than the other.

Before human beings had numerals with which to subtract symbolically, they could figure out which of two groups was bigger by taking away one from one group, one from the other, and so on, till one group disappeared and the larger group remained. 

In the example below, we have butterflies and flowers. We have taken away one from the group of flowers, and one from the group of butterflies.  

Crossing off one from the bottom, one from the top, we continue until the smaller group (butterflies) is entirely crossed off.  We see that the group of flowers still has 14 flowers. It is larger by 14.  The difference between the two groups is 14. 

 

When we subtract, we use the above rationale. 

 

  1. DRAW 35 and 19 in base ten on the place value chart.

 

  1. We know that 35 is the larger of the two numbers, but we don’t know by how much.  We can see that there are only 5 ones in the top part of the table – – not enough to be able to successfully cross off all those 9 ones in the bottom number!   We need to break 35 apart (decompose 35) in such a way that there will be enough ones in the top group to be able to successfully cross off all those 9 ones in the bottom group.  So, with a red pen we’ll take a group of ten, cross off that ten (so we don’t accidentally count it later), draw an arrow to the ones column, and draw 10 ones in red in the ones column.

 

 

  1. Now in red we will cross off the 9 ones (from the 19), and also cross off 9 ones (from the 35).  (Remember the procedure from “what’s the difference?“)

 

  1. Now we will cross off 1 ten (from 19), and also 1 ten (from 35). (Remember “what’s the difference?“) Finally we draw what is left in base ten:  (1 line and 6 circles), and we write that number in the number sentence: 35 – 19 = 16

 

 

If we teach our students the shortcut (the traditional algorithm of “carrying” and “borrowing”) for doing addition and subtraction without previously giving them the rationale for their actions, learning this shortcut will be meaningless to them.

But by understanding the rationale behind addition and subtraction, these operations become meaningful, and easy to visualize and understand.  The traditional algorithm will come later.

If you have any questions please don’t hesitate to ask us!

The Second Grade Team

Eureka Math Module 4: Addition and Subtraction Within 200 with Word Problems to 100

Written by ValerieW. Posted in 2nd Grade

Dear Parents,

We are about to start yet another math module!
Grade 2 Module 4: 
Addition and Subtraction Within 200 with Word Problems to 100

In Module 4, students will learn place value strategies to  add and subtract within 100; they’ll do one- and two-step word problems of varying types within 100; and they will develop conceptual understanding of addition and subtraction of multi-digit numbers within 200.  Starting from concrete, then moving  to pictorial and lastly to a more abstract approach, students will use manipulatives and math drawings to help deepen their  understanding of the composition and decomposition of units.

They will not be taught the traditional algorithm (carrying and borrowing to add and subtract), but rather other strategies that help them develop their conceptual understanding of math.

Many of us adults were taught the PROCEDURE of adding and subtracting with “carrying and borrowing.”  We were not taught the CONCEPT, that is, the why of adding and subtracting, only how to do it using a shortcut.

When we saw that there were too many units in the ones column,  we were taught how to “carry a ten” – – pass it over to the tens column.  We were also taught how to “borrow a ten” – – get numbers from the tens column to put in the ones column, in case there were not enough units in the ones column for use to carry out a subtraction problem.  It was if the number 35 could start having a conversation with itself, with one part asking the other to give it something!

Giving children this algorithm is like telling them how to get somewhere by taking a shortcut:  go left, take the second right, go three streets, turn left again, etc. etc.  They will need to memorize all the ins and outs in order to get to their destination.

Giving children addition and subtraction strategies based on a strong conceptual knowledge of place value is giving them the understanding of how to use a map to arrive at a destination:  they are given the big picture and guided to discover and use multiple pathways to get to their destination.  Memorization is not required. 

In second grade we are striving to teach the students the CONCEPT of addition and subtraction.   What is really happening is that we are REGROUPING numbers by composing and decomposing them, that is, showing numbers in different ways in order to make addition and subtraction easier to do. The children are encouraged to be flexible thinkers and to learn that 100 can also be shown as 10 tens, 234 could be shown as 23 tens and 4 ones, or 22 tens and 14 ones, for that matter!

In class, we also us the term exchange to emphasize the fact that 10 ones is the same as one ten, 10 tens is equal to 100.  We can exchange ten 10 € bills for one 100€ bill.  We study how an exchange can be made for units of equal value. Here are two excellent videos to watch to help understand how to explain this process to second graders:

http://learnzillion.com/lessons/3118-explain-addition-using-place-value

http://learnzillion.com/lessons/3119-explain-subtraction-using-place-value

Here is an excellent article titled, “Nix the Tricks:  A Guide to Avoiding Shortcuts that Cut out Math Development” by Tina Cardone et al.

NixTheTricks

Below are some simple drawings which we hope you will find useful to see the WHY and HOW of what we are doing in class.  When your child does the homework, he or she will need a pencil and a pen or marker in a contrasting color.  Below, we have used red to show contrast, but any color is fine.  We are DRAWING the numbers in base ten, in a way that simplifies the shape of base ten blocks:  

b10

For example, 48 will be 4 tens (4 lines) and 8 ones (8 circles or dots).

 

ADDITION:

48 + 24 = ________

DRAW the numbers in base 10 (meaning, lines for the tens, and circles or dots for the ones) on the place value chart.  Ask your child to organize the numbers;  tens like tally marks, ones like dots on a ten frame, as shown below).  The two numbers are placed  in middle part of mat, the first number above the second one. (The answer will be placed below the line.) We start out by adding up the ones.  Ooops!  Too many ones for the ones column (there are 12).  

addition 1

  1.  Take a red pen, and circle a group of 10 ones (you are  composing them into a group of ten).   addition 2
  2. With the red pen, draw an arrow over to the tens column, where you will be putting your group of ten.  Remember to cross off the circle of ones, so you don’t count it twice!  addition 3

 3Now add them all up, and DRAW the answer (the sum) at the bottom of the table in base ten (lines and circles).  Fill in the number sentence at the bottom.  You are finished!  addition 4

SUBTRACTION

WHAT’S THE DIFFERENCE?

 

When we subtract, we are actually looking at two quantities to see the difference between them, that is, we are trying to figure out how much more one group has than the other.

  1.   We have two groups here.

 2. Before human beings had numerals with which to subtract  symbolically, they could figure out which  of two groups was bigger by taking away one from one group, one from the other, and so on, till one group disappeared and the larger group remained.  Here, we have taken away one from the group of  flowers , and one from the group of butterflies.  

difference 2

3.  We continue until the smaller group (butterflies) is entirely crossed off.  We see that the  group of  flowers still has 14 flowers. It is larger by 14.  The difference between the two groups is 14. 

difference 3

 

  1. When we subtract, we use the above rationale.  DRAW 35 and 19 in  base ten on the place value chart. We know that 35 is the larger of the two numbers, but we don’t know by how much.  We can see that there are only 5 ones in the top part of the table – – not enough to be able to successfully cross off all those 9 ones in the bottom number !   (See HTO chart below).
  2. We need to break 35 apart (decompose 35) in such a way that there will be enough ones in the top group to be able to successfully cross off all those 9 ones in the bottom group.  So,  with a red pen we’ll take a group of ten, cross it off (so we don’t accidentally count it later), draw an arrow to the ones column, and draw 10 ones in red in the ones column.  subtraction 2
  3. Now in red  we will cross off the 9 ones (from the 19), and also cross off 9 ones (from the 35).  (Remember the procedure from “what’s the difference?“)  subtraction 3
  4. Now we will cross off the 1 ten (from 19), and also the 1 ten (from 35). (Remember “what’s the difference?“) subtraction 4
  5. Finally we draw what is left in base ten:  (1 line and 6 circles), and we write that number in the number sentence: 35 – 19 = 16  subtraction 5

 

 If we teach our students the shortcut (the traditional algorithm of “carrying” and “borrowing”) for doing addition and subtraction without previously giving them the rationale for their actions, learning this shortcut will be meaningless to them.

But by understanding the rationale behind addition and subtraction, these operations become meaningful, and easy to visualize and understand.  The traditional algorithm will come later.

If you have any questions please don’t hesitate to ask us!

The Second Grade Team