# 1st Grade Math – Operations and Algebraic Thinking

## Developing Algebraic Thinking for 1st Graders

Common Core math expects that students will understand math concepts rather than learn processes to apply.  Because of this, often the practices of early grade math cause frustrations for parents as they ‘see’ the way to do the problems without going through the steps that are required by the practice assignments.  The problem lies when the steps that are asked of the students at these early grades are building a foundation for understanding, but seem to be more complication that necessary from the parents perspectives. 1st grade math standards in building the Algebraic foundations have several of these types of standards.  They are explained below: #### Represent and solve problems involving addition and subtraction.

A-1.  Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
In this standard, students would be expected to use cubes or other manipulative objects or drawings to represent problems.  So, given the problem “If Sally has 7 cookies and gets 4 more from John”  the student would be able to explain the problem by drawing (or using objects – cubes, etc.) to show the 7 cookies and then adding 4 more and then, count the total to get the solution.  Additionally students should be able to represent the problem with an equation: 7 + 4 = 11.
A – 2. Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
This standards expects the same as the previous, but extends the expectation with the addition of a third number in the problem.

#### Understand and apply properties of operations and the relationship between addition and subtraction.

B-3  Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)
This standard expects that students will begin to make some associations with general understandings about addition and subtraction.  By using those understandings, the students should begin to apply those understandings to mental math to become more fluent in their math skills.  So, if a student ‘knows’ that 5 + 7 = 12 then they will quickly recognize that 7 + 5 = 12.  Also, they should begin to see number relationships (for example, addends of 10).
B-4  Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.

#### Add and subtract within 20.

C – 5  Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
This standard expects that students can use a number-line (for example) to ‘start on a number’ and count forward to represent addition and, for subtraction, count backwards. So, for example, “If Sally has 4 cookies and John gives her 5 more” the student could explain that starting ‘from 4 count 5 more’ to get to 9, or, “If Sally had 8 cookies and gave 3 to John” the student would start at 8 and count back 3 to get to 5.
C – 6  Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
This standards often gives parents difficulty because the practice problems often requires students to ‘break apart’ numbers in order to ‘make ten’, and then use that information to arrive at the sum.  Parents often feel that this step is unnecessary and causes them frustration.  It does provide practice in a skill that will later help with math fluency when the numbers get larger.  For example:  When students are asked to add 54 and 78, the students will be able to quickly recognize that ‘if they take 6 from the 78 to make 60 , they will have 60 +72, which become an easier mental math problem.  But, as 1st graders this seems like an unnecessary step.

#### Work with addition and subtraction equations.

D – 7  Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
This standard expects that students will understand an ‘equal sign’ and will recognize when ‘both sides are equal’.  This is a fundamental concept for algebra.
D – 8  Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.
For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ – 3, 6 + 6 = _.  Again, another fundamental concept for algebra.

# 5th Grade : Measurement and Data Standards #### The standards described below are those specific to the geometry and measurement.  These standards, generally, are pretty straightforward and similar to the more traditional standards that parents may be familiar with. #### A.  Convert like measurement units within a given measurement system.

1. Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

In this standard, the students may be provided a problem such as:

Johnny purchased 16 quarts of milk at the local grocery.  How many gallon containers of milk would he have to purchase to get the same amount?

#### B. Represent and interpret data.

2.  Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots.

A line plot is a number line with the number of occurrences of each unit in a data set.  Example : From the data set, the student will solve problems related to the data.

• If the line plot above represents the height of bean plants after 3 days, what is the difference between the shortest and tallest bean plants height?
• What is the total of the the heights of the bean plants?
• Students will also be expected to collect data and represent that data on a line plot.

#### C. Geometric measurement: understand concepts of volume.

3.  Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

A.  This part of this standard requires that students understand a ‘unit cube’.  Students should recognize that volume is represented by cubic units and that a unit cube is a three dimensional measurement unit that is one unit of height, width and depth.

B. Volume represents the number of ‘cubic units’ that will ‘fill’ the space without gaps or overlaps.

4.   Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

With this standard, students should be able to measure the volume when given a visual representation and counting the unit cubes.  The students should be able to ‘count’ even those cubes that they can not see in the representation. 5.  Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

A)

Find the volume of a right rectangular prism by packing it with unit cubes, and then show that the volume is the same as would be found by multiplying the edge lengths, OR  by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.

In this part of the standard, using only rectangular prisms – boxes, the students are expected to show that volume is equal to the area of the base of the structure times the height and is equal to multiplying the edge lengths (height, length and width).  The students will be expected to prove that length x height x width will result in the number of cubes it takes to ‘fill the structure’ and represent volume.  Additionally, the students should be able to explain why the order of the multiplication of the edges (length, width, height) does not matter, the volume will be the same (associative property).

B)

Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.

With the understandings of part A of the standard, the students should be able to use the formula l x w x h to find the volume of rectangular prisms.

For example:  Johnny is given a box of cereal and asked to determine the volume of the box.  He measures the height of the box and finds that it is 10 inches tall.  He measures the width of the box and finds that it is 6 inches wide. If the box measures 2 inches deep, what is the volume of the box? C)

Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

With this part of the standard, the students should be able to understand that some solid figures are made of multiple parts each with a different volume that may be added together to find the total volume.  For example, the volume of the figure to the right could be calculated by adding two parts together.  There are multiple ways to add them together, but the volume will be the same regardless. (2 x 2 x 4) added to (3 x 4 x 1)  OR (5 x 4 x 1) added to (2 x 1 x 4)

This standard includes a lot of skills and the students should have amply opportunity to practice this standard.  The good news is, it is relatively straight-forward and most parents will understand the tasks given in this standard.

# Grocery Store Math – 5th/6th Grade Unit Pricing

### Unit Pricing

Students in 5th grade and early middle school must begin to develop an understanding of ratios and proportions. An easy application of the concept is related to unit pricing.  If  a package of six onions costs \$2.84, what is the price per item?  This is a fundamental skill that all skilled shopper must possess.  By practicing ‘unit pricing’, student will develop an ‘beginning understanding’ of ratio and proportion which is a critical skill at the 6th grade level.  Additionally, they will understand how to make better consumer decisions.

## Ideas to Consider

As you are moving through the grocery store with your child, ask them to approximate the cost ‘per onion’ of the bag of onions when given the price of the bag.  (As an example, of course, there are examples to practice all over the grocery store.) .

Make the activity fun.  We don’t want to make the trip to the grocery store like ‘going to school’.  The point is to make it engaging and fun so that the children get the practice of the concept while in a real-world setting.

Another example would be to have your child try to determine the ‘best buy’ when the grocery store provides more than one packaging option for a particular item. Learning should not only happen in classrooms, learning should be in experiences!  Learning is not in books!

# 3rd Grade: Operations and Algebraic Common Core Standards Explained (for parents)

## Common Core Math – 3rd Grade

Common Core gets a lot a press these days and most of it results from the fact that it is not ‘what we are used to’.  When most adults went to school, math was a series of ‘rules’ and ‘processes’ that we executed in order to ‘solve’ a problem.  The premise behind Common Core is that students should develop and understanding of math concepts so that, when faced with problems of a complex nature, they have the mathematical understanding to ‘figure it out’ rather than relying on a series of steps that may not necessarily apply.  The ‘rub’ lies in that it makes what seems to be ‘simple math’ into a complex task that makes no sense to the parents at home trying to help with homework.  Regardless of your feelings, here’s some information that may help parents understand the standards and provide some insight as to how to ‘help my kid with his homework’.

## The standards: Operations and Algebraic Thinking Strand

OA.A

### Represent and solve problems using multiplication and division.  So…

• 3 x 5 = 15…..not because we memorize that fact, but because we need to understand that this represents 3 groups, each containing 5 objects.
• 15 ÷ 5 represents 15 objects shared amongst 5 groups equally…how many are in each group?
• As third graders, the students are expected to be able to ‘do’ the above for any problem involving a total up to 100.  This includes using drawings to represents the groupings, arrays and equations where a number is missing (for example:  5 x ___ = 15)

This standard is why you see your child drawing arrays (columns and rows of objects) to represent the groups of objects.  Why not just memorize the facts?  Because rote memorization does not provide understanding.

OA.B

### Understand the properties of multiplication and division and the relationship between multiplication and division

• 4 x 6 = 24 and 6 X 4 = 24.  So, this explains why your child is asked in his homework to ‘draw the arrays’ which show that this is true.  Students are asked to ‘prove’ that 4 x6 = 24, not just ‘know it’.
• 3 X 5 X 2 can be worked out by adding together 3 x 5 = 15 and then multiplying the 15 x 2 to equal 30.
• 3 X 5 = 15 and 3 X 3 = 9, so 3 X 8 = 24.  Understanding that this ‘works’ explains why your child is asked to draw those arrays and combining them to see that when two arrays come together they can ‘prove’ that 3 x 8 = 24.
• When posed with 15 ÷ 5 = ____, we don’t ‘know the answer’ because we memorized it, but because we know that the problem represents 5 x ____ = 15    The process helps students understand that division problems are just multiplication problems with a missing number (relationship between multiplication and division)

OA.C

### Math Fluency

This is the standard that expects that all 3rd graders can multiply all one-digit numbers, from memory,  with fluency, and, since they understand the concepts of multiplication, they will be able to fluently divide, as well.

OA.D

### Solve problems using the 4 operations, and identify and explain patterns

• Solve two-step problems and be able to represent those problems using equations.  Students should be able to begin using letters to represent the unknown quantities.  So, given the problem:

Sally is 4 years older than Robert.  If Robert is 12, how old is Sally?

The student should be able to write that S = R + 4. So, if Robert is 12…

S = 12 + 4, and so, Sally is 16.

• This standard also expects that 3rd graders can begin seeing patterns in numbers and can explain ‘why’ it is true.

# 3rd Grade – Grocery Store Math – Arrays

### 3rd Grade – Arrays:  The foundation for multiplication A walk through the grocery store provides lots of opportunity for parents to help growing mathematicians understand multiplication.  With the common core math standards, the use of arrays (grids of columns and rows) to represent multiplication is fundamental.  Gone are the days where we memorized our math facts, Common Core expects students to understand why 3 x 5 = 15, not just remember it! Consider practicing basic math facts as you approach the canned food section and have your child count the rows and columns of the arrangement of cans.  Use this opportunity to explain that the array created can provide information as to the total of cans (without counting each one) by multiplying the number of rows and number of columns.  Your child has probably been exposed to ‘arrays’ as they are learning the concepts of multiplication.  So, don’t fear, they will know what you are talking about when you say ‘array’, even if you would have never used the term.

As you continue down the aisle, have your child count the number of items in the ‘rows’ and ‘columns’ and predict the number in the section.  Start small!  What is important is that the child sees that ‘math’ is something that they will use outside the classroom.  It is not just an exercise for school.  Make it real!  Learning is not in books, Learning is in experiences!

# Counting Experiences for Kindergarteners

Counting Standards that Kindergartners should master by end of Kindergarten:

• Counting to 100 (including counting forward beginning on a number other than one)
• Writing numbers 1 – 20
• Counting the number of objects
• Identifying numbers are more or less (greater or smaller)

These standards can be practiced in the grocery store environment.  Examples:

• Ask your child to count the number of eggs in a carton
• Ask your child to count out 10 oranges to put into the cart.
• Compare two different containers with different number of items and ask which has more / less.

It is important that children see the practical application of math concepts and the grocery store is the perfect setting. # Math Blog

The grocery store is a great place to begin exposing your child to math concepts. Experiences can be developed around counting, measurement, unit ratios, algebra, etc. in a very practical way.

Check out the resources available at:

www.teacherspayteachers.com/Store/suitcasestudies-educationalresources

As an example: Pre-Schoolers can practice the skills related to equal, more and less. (Common Core Standard: K.CC.C.6 – Identify whether the number of objects in one group is greater than, less than or equal to the number of objects in another group.)