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Eureka Math Modules 7 and 8

Written by ValerieW. Posted in 2nd Grade

Dear Parents,

As we finish up the year, we are working on Eureka Modules 7 and 8.  Here is what the modules entail.

Module 7: Problem Solving with Length, Money,
and Data

In Module 7, students represent and interpret categorical data by drawing both picture and bar graphs (2.MD.10). They have been practicing some of these skills already,  as we have been conducting class-wide surveys, then organizing and interpreting the data.  New in second grade is the use of line plots, which are graphs that display data as points above a number line:

In the line plot above, the Xs recorded above the numbers show us the number of occurrences a response appears in the data set.  Here, four dogs each weigh 15 kg, one dog weighs 18 kg, two dogs weigh 20 kg each,  and so on. The Xs represent the frequency of a particular response.  Students will continue using line plots throughout elementary and beyond.

Module 8: Time, Shapes, and Fractions as Equal
Parts of Shapes

In the final Eureka Math module, students will consolidate  their understanding of part–whole relationships using geometry. Just as the second graders have composed and decomposed numbers, now they will  compose and decompose shapes, and begin to develop an understanding of unit fractions as equal parts of a whole.

In Topic A, students  recognize and draw  polygons with specific attributes: the number of sides, corners, and angles (2.G.1). They study the cube while focusing on its attributes, counting the number of edges, faces, and corners (2.G.1).

Student use tangrams in Topic B.  A tangram is a set of seven shapes that can be put together to form a square and many other shapes.

Image result for tangram

Farther along in this topic, the students build and partition shapes by putting together two or more smaller shapes, while relating the parts to the whole. (2.G.3).

In Topic C, students start by decomposing circles and rectangles into equal parts and describing them as halves (a half of), thirds (a third of), and fourths (a fourth of) or quarters (2.G.3). Later, students will decompose a rectangle into four parts that have equal areas but different shapes (2.G.3).

In Topic D,  students will apply their knowledge of partitioning the whole into halves and fourths to tell time (using both analog and digital clocks)  to the nearest five minutes (2.G.3, 2.MD.7).

Module 6: foundations for multiplication and division

Written by ValerieW. Posted in 2nd Grade

Module 6 helps give children the  conceptual foundation for multiplication and division that they will use  in Grade 3.  In Topic A the students began by making equal groups using concrete materials (counters and plastic tiles), creating  equal group of objects.

In Topic B, students began to organize equal groups  into arrays, that is, an arrangement of a set of numbers or objects in rows and columns.

An array could be represented by a repeated addition sentence, and tape diagram.  When looking at COLUMNS, our repeated addition equation is  shown in the first example below.  When looking at ROWS, we use the second example below.


5 + 5 + 5 = 15



3 + 3 + 3 + 3 + 3 = 15


In Topic C the children start using same-size squares to tile a rectangle with overlaps or gaps, then they count  on to find the total number of squares.  After composing rectangles, students will  partition, or decompose, rectangles.  Students will later strive to  develop spatial structuring skills by copying and creating drawings on grid paper.

Lastly, we will review the characteristics of even and odd numbers:

1. A number that occurs as we skip-count by twos is even: 2, 4, 6, 8 . . .

2. When objects are paired up with none left unpaired, the number is even.

3. A number that is twice a whole number (doubles) is even.

4. A number whose last digit is 0, 2, 4, 6, or 8 is even.

5. when a whole number is not even, it is odd.

As always, our daily lesson have a fluency component.  Through repeated work with increasingly more complex problems, our aim is for the children to become more and more fluent in math.  Bit by bit, they will be easily able to recall and apply knowledge rapidly and accurately.

Lately, some of our fluency activities have been centered on the age-old game, “20 questions,” with money.  Before playing, the teachers  choose a random quantity of coins (we use fake coins!) and tell the children that they can ask ONLY 20 questions to determine how much we’ve got in our pocket. Our only answers can be “yes” or “no.”  Little by little they are honing their questioning skills, asking smart questions, listening to others’ questions (so as not to waste those 20 precious questions!).

This is a game which you can also do at home with your child.  Give him or her a piece of paper to do calculations.  You can divide a piece of paper in two, writing the “no” answers on one half and the “yes” answers on the other half.  In this way your child can keep track of the information he or she has been asking about, which helps in figuring out what else to ask.  Have fun!

Writing Workshop: sharing our opinions about books!

Written by ValerieW. Posted in 2nd Grade

Dear Parents,

Second grade students have started  a new unit in writing workshop:  opinion writing.  What’s best 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 to the rescue  Mercy Watson


The Second Grade Team

Addition and subtraction strategies: making a problem easier to solve

Written by ValerieW. Posted in 2nd Grade

Dear Parents,

This week we will be working on one more strategy for doing 2-digit addition and subtraction, which entails looking at an equation, and changing it to an easier equation which gives the same answer.  

In teaching this and all the other strategies to the second graders, our hope is that they will develop their number sense and increase their confidence in math as they’ll be able to approach a problem in a variety of ways.  Ultimately they will be able to do these calculations mentally.

Thus we are not teaching them the traditional algorithm, which robs them of flexibility, creativity and autonomy.  Once they use the traditional algorithm, it’s the only strategy they’ll ever use, and you really can’t do it mentally.


We’ll use a tape diagram to show how to simplify subtraction equation, making the equation a little easier to get the same answer.

It’s easy to illustrate with unifix cubes.  In the first picture, we have a tape diagram (made of unifix cubes) which represents the equation 8 – 5.  We want to find the different between 8 and 5, which is the little gap above the red cubes.   But if we add another cube to the top part and the bottom part, we get a new equation, 9 – 6.    We can see that the difference stays the same.  Each part is getting longer, but the gap or difference stays the same.

These are both pretty simple equations that second graders would not even need to simplify;  they just illustrate the concept.  But if we have a problem like 34 – 28

we could simplify the problem by adding 2 on to each part while maintaining the same difference:

It is easier to do 36 – 30 than it is to do 34 – 28, as it’s easier to subtract a multiple of 10 (a friendly number).

Here’s another example with 50 – 29 being simplified to 51 – 30:

In order to make a subtraction problem easier, children need to:

– make a tape diagram to represent the equation.

– look at the subtrahend (the number that is to be subtracted).

– think:  what do I need to add to the subtrahend to make it a “friendly number” (a multiple of ten)?

– add the same amount to each part of the same diagram.

– subtract (since there will be a zero in the ones column, we only have to worry about the tens column)



For addition problems, we will simplify them with the help of number bonds. The idea here is to:

– look at the leading number

– think what we need to add to it to make it a friendly number (multiple of 10)

– decompose the addend in order to add what we need to leading number

In this example, we start with 9.  By decomposing the 4 into 1 and 3, we can add that 1 to the 9, and we get 10 + 3, and it’s easy to add anything to a multiple of 10.

Once again 9 + 4 only illustrates our process and the second graders wouldn’t need to simplify that equation.   But in the example below, 28 + 43, we know that

– if we add 2 to 28, we’ll get 30, a multiple of 10.

– it’ll be easier to add any number to a multiple of 10.


Now my problem will be 30 + 41 (41 is the other part of the number bond that I still have to add on to 30).  So my answer will be 71.  This way of doing addition with number bonds will be different for the children as they are used to making a number bond for each number (decomposing each number). The second graders can add with number bonds the way they are used to doing, that is, by decomposing each number and adding up tens and ones.  This method of only decomposing the second number is only another strategy we are offering them.

The tricky part could be having kids remember when to use the tape diagram (for subtraction) and when to use number bonds (at this point, just for addition).

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

The Second Grade Team


Two 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 her some more apples, she has 9 apples altogether. How many apples did Mark bring her?

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

In this problem,  We are comparing Matteo’s cars to Josiah’s cars.   We know the smaller quantity. To find the bigger quantity we add 5 + 2..  To get the WHOLE, we then add up the PARTS.