North Carolina 5th and 6th Grade Math – (Wake County)

As my son heads into 5th grade math I thought I would look at the curriculum goals for the year.  Although I knew the curriculum was weak, I was still surprised at the lack of expectations in math.

This is ALL they expect 5th graders to accomplish in a full year of math:

  1. Adding and subtracting rational numbers (decimals, fractions, mixed numbers).
  2. Basic rates of change
  3. Solving and working with simple equations and inequalities
  4. Basic properties of 2-D shapes
  5. Bar graphs and stem & leaf plots

I could teach these 5 topics in 12 weeks.  Four weeks for goal 1 and 2 weeks for the rest of the goals.

Let’s add in what in the 6th grade curriculum:

  1. Negative numbers
  2. Percents
  3. Transformations in the coordinate plane
  4. Basic Probability
  5. More with equations and inequalities
  6. Multiplication and division with decimals and fractions

Okay – so if we add these next 6 concepts, if we are generous, we can do 3 weeks with negative numbers, 3-4 weeks for percents if you want to do some really fun, real world activities with the unit, 2 weeks for transformations (although could be done even quicker), 2 weeks for probability, 3-4 weeks if we really extend equations and inequalities, and finally 4-6 weeks for multiplication and division of decimals and fractions.  That adds another 19 weeks.

Maybe after we finish the 12 weeks for 5th grade and the 19 weeks for 6th grade math, we spend another 6 weeks integrating all the concepts taught together and do more applications as a review.  This is a total of 26 weeks.  This would still leave approximately 8 weeks more in case any of the topics take longer or need more practice with 2 additional weeks for days when we don’t do math because we have field trips or parties and such.

The fact that we really move that slowly through the curriculum is pathetic.  We also don’t take the time to link all the concepts together.  We will teach each unit in isolation and then we move on to the next topic instead of finding activities and applications that will allow the children to practice many of these goals at the same time.

Let’s say that we are teaching equations.  We can teach the following concepts integrated within our goal of equations:

  • Arithmetic with decimals
  • Arithmetic with fractions
  • Concept of inverse

This meets 6 out of the 12 goals in both the 5th and 6th grade Standard Course of Study.  One-half of our goals all at the same time!

When teaching the unit on percents, we can meet these goals all at the same time:

  • Multiplication and division of rational numbers
  • Equations and inequalities

In fact things like Rates of Change, Percent, and Probability can all be applications for practice of Arithmetic of rational numbers and Equations/Inequalities.

The biggest GOAL needs to be in providing a better curriculum and teacher education so that our teachers can do a better job in instruction.  They need to do more teaching as opposed to asking students to do rote memorization.

If we put the bar really low (as we have) and provide curriculum materials that ask students to do the same problems on worksheets over and over while throwing in these higher level thought questions that aren’t even related to what the student is working on, we will continue to be way behind mathematically as a nation.  We will continue to hear parents say (in front of their children), “well, I was never good at math either.”  It seems that being BAD in math is a badge that you can wear on your shirt and be proud of.

Come on North Carolina – and the USA – is it really that hard to get it right?!

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Easy step by step, concept building, Math Curriculum and Workbook

What do we look for in a mathematics curriculum?

First – let’s think for a moment about the traditional math curriculum.  Is it working?

Not really – why are so many kids still struggling with math?  Why do so many families have to get their children outside help for their children?  Why do we have to repeat the same concepts over and over and over again each year?

What is in the traditional curriculum?  Children learn from many different goals.  Although I understand the desire to cover all of mathematics and give children exposure to multiple areas of mathematics – I think we should focus on the basics and overall number sense before we worry about teaching children all the vocabulary needed for Geometry.  Why does a 4th grader need to know the name “Obtuse” angle when they still haven’t grasped multiplication and divisibility?  Most of the Geometry concepts we teach can really be taught later while we use our time more effectively to build number sense among children. Each year, children need to re-learn the Geometry vocabulary we throw at them because we don’t actually “use” Geometry, we just insist that they need to know the names of angles, triangles, and which angles are congruent when 2 parallel lines are cut by a transversal.  We should teach all of this Geometry when students have learned enough mathematics that they can use and see the importance of Geometry.

Concepts like perimeter and area can be taught and used as an application to addition and multiplication but so much of the rest of the Geometry we teach to elementary and middle school children is just done to justify that we are teaching “all” topics in math instead of teaching what is needed at that particular time.

The only Geometry topics that I think the elementary child needs to know are:  what different degree measures mean (what does it mean to turn 90 degrees), perimeter, area, and basic shapes.

What about the number sense?  We teach basic facts but we do not try to build number sense with children.  Even if you ask a child this second grade goal questions, “Write 3 different ways to represent 250.”  Does the student even understand the question?  If they respond 250 ones or 25 tens – do they really get what that means?  No, because we don’t teach things in the right order.  If you want a child to understand that 250 is the same as 25 tens – you should have introduced multiplication by then.  Yet, multiplication isn’t taught until 3rd grade and our students are supposed to answer this question in 1st and 2nd grade.  They learn to repeat what they are told but don’t really understand at a deeper level – how can they?  They haven’t learned all the needed pieces to full grasp it.

The curriculum needs to stop thinking, “Multiplication is taught after addition and subtraction.”  Multiplication links into a lot of concepts we teach younger children.  When it links in, we should introduce it.  If we do that, multiplication will have a lot more meaning.

Progressive Math is a curriculum that is designed to make the links as they come up.  It has no rules of that say “A must be taught before B.”  The curriculum guides itself.  If a so called “later topic” (such as multiplication) presents itself conceptually in a unit taught to a kindergartener (think  4+4 means 2 sets of 4 means 2X4) then the concept is taught.  Math makes more sense since topics are linked to other topics.  It is a chain of links together rather than our current curriculum which is a bunch of disjoint sets.  We teach kids to count by 5′s – what concepts link to that?  Telling time, multiplication, counting coins, etc.  All these things link to counting by 5′s and therefore we connect them.  This is something that is missing in current curricula.

Another problem when traditional math is taught, is that a concept is taught and then forgotten about.  Maybe the student will remember, maybe not.  Progressive Math constantly reviews old topics so students are constantly getting practice of the older material so they don’t forget as well as moving forward.  How many times has a child taken a unit test and once it was over, forgot everything.  If you gave that student the same unit test 2 months later, how would they do?  If they really understand, link concepts, and have regular practice, they should do fine.

Another issue we hope to resolve is the steps involved from getting from point A to point D. Many traditional curriculum teach all the steps A to B, B to C and C to D together in one “step,” this is fine for some students who can make that leap but what about those that can’t and get lost or even if they get it but find it “hard.”  What if, instead, we gave mini-lessons that taught students to just do step A to B.  When they had some practice, we teach B to C, and then finally from C to D.  Now we can do the A to D jump and all students are right there with us because we took the time to break down the intermediate steps.  Here is an example:

Goal:  Teach time to the 5 minute interval.

Traditional Lesson: Students have been taught time to the hour, half-hour, and now (a year later) must go to 5 minute interval.  They must see a clock and be able to say the time is 2:35.

Progressive Lesson:

1.  Student learns to count by 5′s.

2.  Student learns how to associate the numbers on the clock that the big hand points to as values when one counts by 5′s. (They only focus on the big hand and the concept of counting by 5′s, there is no telling time here – it is just a basic lesson on counting by 5′s but using the clock numbers as our counters.)

3.  Students learn how to tell what time the small hard is referencing.

4.  Student takes step #3 and combines it with step #2 and can tell the time to the 5 minute interval.

The progressive lesson is a more natural approach.  It links things the student already knows (counting by 5′s) to telling time.  The traditional approach has students memorize what a clock looks like when it is on the hour or half hour.   Next the student has to make a large jump from the half hour time to the 5 minute interval with out any bridge.

If we linked all our mathematics together, if we encouraged overall number sense, and we provided intermediate steps our children would learn faster and better.

The traditional approach also relies a lot on memorization.  Students are told to memorize their addition facts.  How many kids are still counting on their fingers in second, third, fourth, and later for addition?  I have high school students I see who still use their fingers.  That is the only strategy they have!  They don’t have any number sense because no one ever taught them number sense.  Instead of memorizing 6+7=13, a student with number sense who has learned to link concepts can use a variety of approaches to solve the problem in their head without counting on their fingers.  For some reason most kids are able to learn their doubles easily – so why not rely on the doubles.  If a child knows that 6 + 6 = 12 but they don’t know 6 + 7 they can apply number sense and know that 6 + 7 must be just one more than 6 + 6, therefore it is 13.  Our curriculum doesn’t teach strategies, they don’t teach or build number sense so students have to memorize a whole bunch of unrelated pieces of information rather than using what they do remember and pulling from that to get what they don’t have memorized.

This concept of number sense goes far beyond just addition or multiplication facts.  It serves as backbone of how to think logically and do what mathematicians do all the time – use prior knowledge and link that together to find answers to things they don’t know.  This is a skill that helps students through all their years of math and in real life.  Our schools, our curricula, just don’t take advantage of that.  Instead we teach third graders the name “Obtuse angle.”

We apply this idea of Progressive teaching to all of our mathematics curriculum.  Visit us at Apex-Math to see what we have to offer.

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How to Teach Division to Children

Teaching Division to children
Teaching Division

In order to learn division, the student must first have a good understanding of multiplication. The child doesn’t need to be perfect but should know the majority of the facts or have a reasonably quick strategy to figure out the answer.

When teaching division, we will be going in steps.

Step 1: Have the student understand the concept of division and be able to solve division problems with manipulatives.

For example: Given 12 pennies. Have the student evenly divide the 12 pennies among 2 people. From that practice writing the division statement 12 / 2 = 6 and have student explain the meaning: “12 pennies divided into 2 groups gives 6 in each group.” Then have the student divide the 12 into 3 groups and repeat the math sentence and the explanation. Next, divide 12 into 4 groups, then 6 groups. Repeat with other numbers such as 16 (divide into 2, 4, and 8 groups) until the student shows mastery of the concept and writing the corresponding math sentence.

Step 2: Have the student understand the concept of a remainder. You will continue with manipulatives in this exercise. Give the child 5 pennies. They have to share the pennies among 2 people. Let them try, if they split the pennies into 2 and 3 then discuss how that isn’t fair. If they divide it 2 and 2 with 1 left over, explain that happens sometimes with division: we don’t have an even amount to divide and therefore get a remainder. Have the student write the problem as: 5 / 2 = 2 with Remainder 1. Continue this process with other numbers: 9 divided by 2. 11 divided by 3, etc.

Step 3: Have the student understand the link between multiplication and division. Go back to your manipulatives and have them show you 3 x 4 = 12 with manipulatives. Remember that multiplication should have been taught as the x means “groups of” so 3 x 4 means 3 groups of 4. Put your 3 groups of 4 equals 12 to the side. Now have them take 12 and divide it 3 groups as you did in step 1. 12 / 3 = 4. Show them that their result is the same as their multiplication piles they made. In the end they have 3 groups of 4 items. Have the student write the fact family: 12 / 3 = 4; 12 / 4 = 3; 3 x 4 = 12; 4 x 3 = 12. Show them how if you read a division problem “backwards” you have a multiplication problem. Also show them that they can think of 12 / 3 as “what times 3 equals 12?” Continue with practice – 1) writing fact families and 2) finding missing factors: 4 X ___ = 16 (After they get the answer, convert to a division problem – 16 / 4 = 4).

Step 4: The next step is to teach the long division process. The key here is that we are focusing on the process, not on learning division – although doing the practice will help reinforce the concepts of division. Print out and cut the division cards, you will need to use these are you teach the process.

We are going to start with 215 / 5. We actually start with a 3 digit number because it shows the repetitive process. Get out the X5 division card. On a white board, write the problem. The steps to the division process are:

1. Look at the first number, 2, does 5 go into 2? No, 5 is too big.

2. Look at the first 2 numbers together: 21. Looking at the division card, find the number closest to 21 without going over. You see that the number is 20.

3. Write the number to the left (blue number) above the 1 on top. Write the number to the right (red number) below the 21. So, a 4 goes on top and the 20 goes below.

4. Now, subtract 21-20. You get 1. Using an arrow (make sure they use the arrow) bring down the 5 so it is next to the 1, making 15.

5. Repeat process: find a number as close to 15 without going over. We find 15 in the table. The red number (3) goes on top above the 5. The blue number (15) goes below the 15, now subtract. We get 0. Note, there are no more numbers to bring down and since we ended with 0, we have no remainder.

Example: 3426 / 5

1. 5 doesn’t go into 3.

2. Find closest to 34 on chart without going over: 30 (5X6=30).

3. Put 6 on top and 30 under the 34.

4. Subtract, get 4.

5. With arrow, bring down the 2, to get 42.

6. Find the closest to 42 without going over: 40 (5X8=40)

7. Put the 8 on top and the 40 under the 42.

8. Subtract, get 2.

9. With arrow, bring down the 6, to get 26.

10. Find the closest to 26 without going over: 25 (5X5 =25)

11. Put the 5 on top and the 25 under the 26.

12. Subtract, get 1.

13. Since that was our last number, 1 is the remainder.

14. The answer is 685 Remainder 1.

Keep practicing with different divisors until the student can do it independently with the division cards.

Step 5: The student now knows the concept of division and the process of division. Now they need to practice division without the help of the division strips. First give them problems with one of the easy divisors such as 2 or 5 and a 2 digit dividend. Even if it goes in evenly such as 5 into 25, make sure they write out the 25 underneath and show the remainder goes to 0. Give them remainders that go to zeros and ones that don’t so they get practice taking the problem to the end.

Step 6: Once the student is successful with 2 digit dividends with divisors of 2 and 5, have them work with 3 and 4 digit dividends but still use the divisors of 2 and 5.

Step 7: Expand with new divisors one at a time. Do 2 digit dividends first and then expand to 2or 4 digit dividends before moving onto a new divisor.

To view division strips visit:

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Apex-Math Products

Why Apex Math Products Work

  • Apex-Math Products are progressive in nature.  Each lesson is linked into previous knowledge.  We show the student how to build on their current knowledge to learn new information.
  • Apex-Math teaches concepts with small steps that gradually get more difficult so students build mathematical self-confidence and see the work as easy with each step only adding one small additional piece that is easily mastered.
  • Apex-Math provides continual review.  In traditional math programs, math topics are taught in a random order and students forget old topics when new topics are introduced.  Our program provides continual review so that the student gets practice and stays sharp from previous material.
  • Apex-Math is skill based, not grade-level based.  In other words, students begin in Level One and proceed through each level.  Upon completion of all levels they are ready to begin our Pre-Algebra Course.  Even students who are further ahead but have gaps in their knowledge can move through our earlier levels to help solidify their mastery and learn new connections and strategies.
  • Apex-math provides lessons that either include the curriculum to guide a parent or teacher or can be used as a self-paced program for more advanced students.  We are both a teacher text-book and a student practice book.
  • Our scope and sequence may seem a little different.  The reason is because if a student learns something like counting by fives, we want to link that skill to all areas where one would count by five.  Students will learn to count by fives, count tallies, count nickels, and learn how to tell the time on the clock in 5 minute intervals – because of all these items are relative and use the same skill – so we teach it together.
  • Students in the program will move through it faster than a traditional program and be ready for higher level mathematics classes sooner because our program teaches the topics correct to begin with so constant review year after year is not needed.  We also use the old topics within the new topics so that the student is always reviewing and moving forward in all our lessons.

We are a new company with a novel and better approach to teaching mathematics!  We will be adding materials that are available for free and other will be available at a fee.

Our ultimate goal is to provide educators, families, and students, math materials that work, make sense, and increase a student’s math self-esteem.

Our products come with a 7-day Guarantee.  If you purchase a program and decide it does not meet your expectations, you can get a full-refund.  For print materials, the purchase price will be refunded for books that have not been used, shipping costs are not refundable.

Our current projects included:  Pre-Algebra, Algebra Review, and Elementary Math Curricula.

Take a look around the website, and if you have any questions at all, get in touch with us.

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