English-Math

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Di~!~Arithmetic
and Connecting Math Concepts K
Although Connecting Math Concepts A is a kindergarten program, many kindergartners need
some extra practice in counting and writing numbers. Distar Arithmetic and CMC K prepare
children for CMC A. The main difference between the programs is that Distar Arithmetic takes
almost a whole year to complete, while CMC K takes less than 2 months .
Students work in small-groups, so they will have as much teacher time as possible. Usually, they
spend most of the lesson (about 30 minutes) in chairs facing the teacher, while she teaches and
they practice what they've learned. When they are finished, they complete a worksheet called a
take home. (A take home is c1asswork , not homework . It's called a take home because they take
it home to show you.) The take home gives them a chance to practice writing numbers and other
skills.
Here are some of the skills taught in Distar Arithmetic:
Skill
What the skill means
Matching lines with numerals
5 means the same as III II (five lines).
Counting
Children practice counting . They learn how to start with 1 (1,2,
3, etc.) and from other numbers (for example 6,7,8, etc .).
Counting objects
Children practice counting pictures of objects on a page.
Using symbols
Children recognize the numerals 1 through 10, along with
and 0 . (The box means an unknown number.)
Drawing lines for numerals
Children can look at 9 and draw nine lines.
The equality rule
Both sides of the equals sign must be the same.
Addition
Children can solve simple addition problems , like 5 + 2 =D
(They figure out what number to use for the box, usually by
drawing lines and counting them.)
Algebra addition
Children can solve addition problems where the box has been
moved , like 5 +0 =7.
Counting backward
Children count backward from a number. They use this skill
when they learn how to subtract.
Dictation
Children can write numerals or math problems when the teacher
says them out loud.
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+, -, =
.
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Facts
tp Information for
Parents
Children memorize addition and subtraction facts.
Solving story problems
Children solve simple story problems.
More than/Less than
Children understand these ideas and recognize the symbols>
and <.
Ordinal counting
Children can count like this: first, second, third ... all the way
up to tenth. They can use this skill to pick out the sixth picture,
for example.
If you look at your child's work, you may notice a strange figure: 10. It means 10. Children are
taught to write it that way because it prevents confusion when they learn about two-digit
numbers. If they write 14, it's easy for them to mix it up with 1 and 4. If they write 1 04, they
remember that the number is 10 and 4, not 1 and 4.
What children learn in CMC K:
CMC K is like Distar Arithmetic, except that it moves much faster. It is for children who already
have some number skills from pre-kindergarten or home.
Children practice counting - counting from 1 (1,2,3,4), from another number (6, 7,8, 9), and
to a certain number (1,2,3,4,5, stop). They practice counting lines and other objects.
They learn to identify numerals - that the numeral 4 is the same as the wordfour, and both are
the same as 4 lines on a page. They practice writing numbers and making lines for numbers.
Finally, students learn to answer questions about whether a number is more than or less than
another number.
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Connecting Math Concepts A
( C:nr~ \L') .
Children begin with practice in counting, the number skill they know best. They learn to count
forwards, backwards, and by 10. This skill becomes the basis for harder math skills such as
addition and subtraction. When teachers present new information - how to add, for example they explain it so that students can see how addition connects with counting.
Along with counting, children develop problem-solving skills. For example, they learn to choose
between addition or subtraction to solve a story problem. They also learn to measure, estimate
and count change. Each lesson is taught so that students can see the connection between new
skills and the facts they already know. These connections make new and difficult math skills
easier for children to learn.
In most lessons, the teacher uses a number line and a "bug" that jumps from number to number.
The number line is used to illustrate all kinds of topics - addition, subtraction, more than, less
than, and different kinds of counting. Here is a number line with the bug:
rtf
•••••• •••••
1
2
3
4
5
6
7
8
9
10
This shows: 8 - 1 = 7
In this example, the teacher says, "The bug starts at 8 and minuses 1. Where does the bug end
up?" Later, students will learn that this question is the same as 8 - 1, and they will not need the
number line. You can see how the number line helps make a connection between counting and
addition and subtraction .
Below is a list of some of the skills covered in Level A.
Counting
Children learn several different kinds of counting . From the beginning of Level A, they count the
usual way (l, 2, 3 , 4, etc .). They practice counting pictures and objects . They also learn to draw
lines to show how many . By lesson 40 they can count to 35. By lesson 100 they can count to
150.
Starting in lesson 18, students practice counting backwards from a number (for example, 20,19,
18, etc.) In lesson 11, they begin learning ordinal numbers (first, second, third, and so on up to
tenth).
Writing Numbers
Children practice writing their numerals correctly. Once they can form the numerals through 9
correctly, they begin writing 2-digit numbers . Starting in lesson 14, children work on writing 2-
°
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digit numbers in a column, being very careful to keep everything in the right place. They learn a
new way to draw lines for 2-digit numbers, which you might recognize if you see some of their
work. It involves writing a T (for 10), plus regular lines (for the ones). Here is how students
would draw lines for the number 13: T I I I
More ThanlLess Than
By lesson 30, children know the meaning of more than and less than. A few lessons later, they
use these ideas to add or subtract 1 or 2. For example, the teacher might ask, "One more than 24
is how many?" Starting in lesson 91, children use the symbols> and < (greater than and less
than).
Addition and Subtraction
Using number lines and their knowledge of more than and less than, students learn to add 0, 1,2
or 3 to any number. They learn to subtract 0, 1 or 2 from any number. Starting in lesson 42,
students learn that you can "turn around" any addition problem - 8 + 3 is the same as 3 + 8, for
example.
As you can see, this means that children learn some, but not all, of their addition and subtraction
facts. Don't be surprised if your child knows 31 + 2, but not 8 + 9, at the end of Level A.
Problem Solving
Children solve simple story problems. Starting in lesson 45, they solve problems like this one: 4
+
= 7. By lesson 50, they work on challenging problems that look like this: 5 C=:J = 7. This
kind of problem, called algebra addition, is actually a kind of subtraction. It reinforces children's
understanding of the connection between addition and subtraction.
0
Counting Change
Students learn to count simple combinations of change (a quarter and five pennies, for example).
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Connecting Math Concepts B
CGYode I ')
Students continue to work with basic skills like counting, adding ~nd subtraction . For every skill,
teachers show the relationship with other skills students have learned. Below is a list of some of
the skills taught in CMC B.
Special Kinds of Counting
Students already know how to count by ones (1,2,3 ,A, etc.). In this level, they learn to count by
twos (2,4, 6, 8, etc.) and other numbers . The count bys" will help them learn addition,
subtraction, multiplication, how to tell time, and other important skills.
II
If you know what lesson your child is working in, you can ask him or her to count out loud for
you . Use the chart below to see what numbers your child can count by.
By the time your child is in this lesson
He or she can count by ...
Lesson 29
Twos (2, 4,6,8)
Lesson 49
Fives (5 , 10,15,20)
Lesson 59
Tens (10,20,30,40)
Lesson 77
Twenty-fives (25,50,75)
Lesson 84
Hundreds (100,200,300)
Lesson 91
Nines (9, 18 , 27,36)
Lesson 102
Fours (4, 8, 12, 16)
Addition and Subtraction Facts and Number Families
Students learn to start with any number and add or subtract 0, 1, 2,5 , 9 or 10. The number they
start with. can be as small (like 2) or large (like 204).
Students solve many problems using number famili es . A number family looks like this:
_ _ _ _9-1•• 10
Children use this diagram to solve all kinds of addition and subtraction problems , especially
word problems. It refers to 4 different facts : 1 + 9 = 10
9 + 1 = 10
10-1=9
10-9=1
Obviously , you could use any two numbers on the arrow - 1 and 9 are just examples. By
teaching arithmetic facts as "families," we can make sure that students understand the
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relationship between addition and subtraction. If you want to know more about how a number
family works, ask your child. It is included here so that you will recognize it on your child's
work.
More than, less than and equals
Students are introduced to the idea that one number is more than, less than or equal to another
number. Beginning in lesson 9, they practice using the symbols >, < and =.
Measurement
Children learn to measure lines, starting in lesson 8. They also learn to use a ruler to make a line
longer or shorter. This relates to what they know about counting, addition and subtraction.
Telling Time
By lesson 65 , students can tell time on a clock face to the nearest 5 minutes. They do this by
counting by fives, which they learned earlier.
Column addition and subtraction
Students learn how to add and subtract numbers in a column, including carrying (lesson 74) and
borrowing (lesson 96) . For example: 64
+ 39
103
Mental Arithmetic
Children practice solving problems like 98 + 1 without using paper or pencil.
Money
Using their "counting by" skills, students add and subtract amounts of money using coins and
dollar bills.
Multiplication
Starting in lesson 71, children learn to do some multiplication problems by counting. For
example, they can solve the problem 5 x 4 simply by counting by 5 four times: 5, 10, 15 , 20.
Geometry
Geometry work begins in lesson 97. Students learn to recognize circles, triangles and rectangles .
They learn to find the distance around a rectangle or triangle. They also use their multiplication
skills to find the area of a rectangle.
Tables
Starting in lesson 38 , students learn to use information tables and tables of number facts.
Calendars are introduced in lesson 49 .
You can see that counting is a very important skill throughout Level B. Most of the skills taught
are closely related to each other through counting.
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Connecting Math Concepts C
(Grode J)
In Level C, students practice the skills they know and learn many challenging new skills. As in
the other levels of CMC, teachers are constantly reviewing and adding to old material so that
students will not forget anything. Some of the topics reviewed in Level C include more than/less
than, place value, addition, subtraction and how to use a table. Below is a description of the new
material taught in Level C.
Facts: Addition, Subtraction, Multiplication and Division
Students finish learning addition and subtraction facts up to 10. (Most, but not all, were covered
in Level B.) They also work to build their multiplication skills. In Level B, students learned to
multiply by counting. For example, they would solve the problem 5 x 4 by simply counting by 5
four times: 5, 10,15, 20. Students continue to multiply by counting in Level C.
After lesson 80, they begin to solve division problems, including problems with remainders. For
many multiplication and division problems, students use number families.
A number family looks like this:
10
~
90
It refers to 4 different facts:
10 x 9 = 90
90 7 10 = 9
9 x 10 = 90
9079= 10
Of course, you could use any 2 numbers next to the arrow shaft - 9 and 10 are just examples. If
you want to know more about how a number family works, ask your child to show you.
The number families help students make a connection between mUltiplication and division.
(Addition and subtraction "families" are also taught, starting in Level B. They look almost the
same as the example above.)
At the end of Level C, students have learned part of the multiplication table. The following table
shows when your child will be able to multiply and divide by certain numbers. For example, by
lesson 20, students can multiply any number by 1. By lesson 95, they can divide any number
by 1.
Your child can use this number
1
In multiplication problems
by this lesson
Lesson 20
In division problems
by this lesson
Lesson 95
10
Lesson 45
Lesson 95
2
Lesson 55
Lesson 95
4
Lesson 80
Lesson 100
5
Lesson 105
Lesson 110
9
Lesson 90
Lesson 100
3
(not in Level C)
Lesson 125
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At the end of Level C, children know some parts of the "times table" by heart - 7 x 5, for
example, or 24 + 4 - because they know how to multiply by 5 and divide by 4. To figure out 6 x
7, they may have to count, because they don't know their 6 or 7 times table yet.
Mental Arithmetic
Students learn to solve problems like 64 + 7 without using paper or pencil.
Column Multiplication
Students learn to mUltiply 2-digit numbers by I-digit numbers, for example 47 x 5.
Estimation
Students learn to estimate by rounding. For example , they might have a problem in which two
items cost $2.98 and $1.09. The problem asks, "About how much will you spend in all?" The
answer is $4, not $4.07.
Another type of estimation problem involves rectangles. For example, students
might be shown the rectangles at right. They are told to pick the rectangle that is
8 units tall and 1 unit wide. The problem might seem easy, but for children it can
be difficult. It requires them to make. a connection between numbers and shapes.
D
I
Fractions
Students learn to identify fractions as parts of a whole. They learn to add and subtract fractions
1 4
with the same denominator. For example, they can do this problem: 3
-+-=-
5
5
5
Students also learn the relationship between fractions and division. They learn to recognize that
the following statement is true:
20
-=5}20=4
5
Geometry
Students learn to plot points on a graph using X and Y coordinates .
Word Problems and Calculator Skills
Students practice solving different kinds of word problems. They also learn the basics of using a
calculator.
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Connecting Math Concepts D ~ro..d-c
'6 ')
In Level D of CMC, students practice addition, subtraction, multiplication, division, and other
skills they learned earlier. As you will see, some types of problems (like long division) require
them to use all their math facts and pay careful attention to many steps. For other problems, the
arithmetic is very easy, but the ideas are powerful, paving the way for later work in algebra. For
example, students graph simple functions, like y = x + 5.
Below is a description of each topic covered in Level D.
9
Multiplication and Division
Students finish learning their times tables, which they began in Level C. The program
uses number maps to help students see how numbers are related. Once they see certain
relationships, it's easier for them to remember facts. For example, there is a trick to
remembering the 9 times tables. At right is the number map for 9, which is arranged so
that students will see the trick and remember it. You can see how the numbers in the ones
column decrease as you go down, while the numbers in the tens column increase. The other
numbers have similar tricks and number maps that students can use to help them remember
and understand multiples.
18
27
36
45
54
63
Students also learn to solve harder multiplication problems , for example 49 x 51. They
begin long division, learning to solve problems like 610 -;- 3. Also, for the first time,
students add and subtract numbers larger than 1,000. All of these activities require them
to practice and improve their skills.
72
81
90
Mental Arithmetic
Students solve problems like 64 + 7 without using paper or pencil.
Calculator Work
For some lesson parts , students use calculators to check their work or change fractions to
decimals.
Equality and Other Number Relationships
One of the most important ideas in Level 0 is the idea that math problems can be written more
than one way, and still mean the same thing. For example , 8 divided by 4 can be written any of
the following ways: 8
4'8
4
Students learn that 4 x 2
;0
4xD=8
= 8 can also be written as 8 =4 x 2.
They solve problems in which they are gi ven 3 numbers: 10 , 5 and 2, for example. From the
numbers , they are asked to write an equation that makes sense, such as 5 x 2 = 10. This reminds
students of the relationships between different numbers , and between multiplication, division and
fractions .
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Fractions
Students review fractions and learn how to multiply them. They learn to recognize equivalent
fractions (112 = 2/4) and to compare fractions (5/8 is less than 11116). They also use fractions to
solve problems involving ratio, proportion or probability. For example, students can use their
knowledge of fractions to answer this question: If there are 5 girls for every 4 boys, how many
girls are there in a class of 27 children?
Decimals
Students learn to use decimals and convert them to fractions.
Word Problems
Students apply their skills to different kinds of word problems.
Functions and Coordinate Systems
In Level C, students learned how to plot points on a graph, using coordinates (X = 4, Y = 7, for
example). In Level D, they continue to practice plotting points. They also graph linearfunctions.
(An example of a linear function would be the equation y = x + 5.)
You can see that this activity goes beyond what children usually learn in elementary school, even
though the addition and subtraction required are fairly easy .
Geometry and Measurement
Students practice finding the perimeter and the area of rectangles and other shapes. They learn to
recognize lines that intersect or are parallel. They also learn about units of measurement - for
example, that 12 inches = 1 foot.
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Connecting Math Concepts E
(GrOGk L/ )
In Level E , students become masters of calculation, working with whole numbers, fractions, and
decimals. At the end of this program, they will be ready for pre-algebra (Level F). In addition to
a review of multiplication , division, and calculator skills, students learn about the topics listed
below.
Number Relationships
Students practice converting numbers . If the teacher gives them a number in any form - whole
number, fraction, decimal or percent - they can change it into any other form. For example, they
can change the number 2 into 6/3 , or 2.00, or 200%.
Students practice writing equivalent fractions, mixed numbers and improper fractions.
Students also review inverse relationships. In other words, they learn that to "undo" addition, you
subtract , and to "undo" multiplication, you divide.
Number Families
This is what a number family looks like:
31
268. 299
From this number family, you could write four different addition and subtraction facts. Your
child can write these facts and explain how a number family works .
Students use number families starting in Level B. By the end of Level D, they know how to use
them to add , subtract, multiply, divide and solve word problems . In Level E, they learn to use
them to add and subtract fractions. They also use number family tables, in which they add or
subtract in two directions.
The number family helps students see the relationship between three numbers , instead of
learning the facts separately . It is very helpful when they have to organize information from a
word problem.
Decimals
Students learn to add, subtract, mUltiply and divide decimal numbers.
Place Value and Rounding
Students learn to work with numbers greater than 1 million. They also learn to round to a
specified place. For example , if the teacher says , "Round 78,483 to the nearest hundred," they
write 78 ,500 .
Coordinate SJstems
Using an X-axis and a Y-axis, students plot points on a graph , starting at lesson 49 . They also
learn about functions. (A function is an equation with x and y. For example , y = x + 5 .) They
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learn to make a table of x and y values. From that, they can plot points and draw a line
representing the function. (If you haven't studied algebra recently, this description may not make
much sense. You can ask your child to show you some work and explain it.)
Probability
Students learn how to determine the probability of a random event. For example, if a bag
contains 5 XiS and 30's, and you pull one object out, what are the chances that you'll pick an X?
The answer is 5 to 3, they learn, and they also are taught to write that as a fraction, 5/3.
Geometry
Students learn to find the perimeter and area of triangles, rectangles and circles. They study the
vocabulary of circles: diameter, radius, circumference and pi (Jr, or 3.14). Parallel and
intersecting lines are also reviewed. Students learn to calculate the volume and surface area of a
box or pyramid . They also learn all about angles, so that they are able to recognize right angles,
corresponding and opposite angles. Students can also add and subtract angles.
Projects
At the end of Level E, students work in teams to complete projects that review and reinforce
their work. The projects can be challenging. In lessons 116 and 117, they construct a scale model
of the solar system. In order to get the planets in the right places, they have to use their
knowledge about circles and proportion, as well as multiplication, division, and, rounding. In
their model, Mercury is 4 inches from the Sun, while Pluto is almost 400 inches away.
In lesson 125, students complete a project about probability. They calculate how often a flipped
coin should come out heads-up, then they flip a coin 30 times to see how close the result matches
their prediction. They use their results to construct a frequency-distribution table.
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Connecting Math Concepts
FeGroclst
5 j(p ')
In Level F, the highest level of CMC , students penect their arithmetic abilities and learn
important pre-algebra skills. At the end of the program many students will be ready for algebra.
Students continue to review skills they learned in earlier levels: division , fractions, decimals,
place value, rounding , ratios, the coordinate system, and geometry . (For an explanation of how
these skills are taught, please pick up the information about CMC E.) The new skills they learn
are described here ;
Long Division
Students learn to do long di vision problems, for example 4086 + 23 . They also learn to sol ve
division problems where each part contains a decimal, like 534.87 + 40.9.
Fractions
In many lessons, students practice working with fractions. Starting in lesson 44, students learn
about reciprocal fractions. For example, the reciprocal of!t2 is 2. If your child has reached that
point in the program, see if she can tell you the reciprocal of 3/4.
Positive and Negative Numbers
Students learn the difference between positive and negative numbers (like 5 and -5). For the first
time, they are told that you can subtract a big number from a small one; the answer will be less
than zero. They use a number line that goes in both directions:
-6 -5
I I
-4 -3 -2 -1
I I
0
2
I 4I
3
5
6
Once they know the number line , students can solve problems like 3 - 20 . They can also mix
signs to write longer equations like -3 + 5 - 1 - 9 = -8. Students learn about absolute value ,
starting in lesson 66.
Exponents
Starting in lesson 74, students learn to recognize exponents . They learn to solve exponent
problems by mUltiplying , for example, 63 = 6 x 6 x 6 .
Simple Machines
Students learn the math used with simple machines such as levers, pulleys and wheels. Starting
in lesson 88, they use formulas to figure out how long a lever needs to be , or how hard someone
has to push to turn a wheel.
Averages
Some students may already know how to average a set of numbers . Others will not. Level F
covers two different ways to find an average.
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Working with Time, Measurements and Other Units
Students learn to add and subtract amounts of time, the same way they would add and subtract
other numbers. For example , if someone leaves at 4 :15 and their trip takes an hour and a half, a
student could write the problem 4: 15 + 1:30 =5:45.
They also learn to make fractions out of units such as days , feet or years. For example, they can
write 7 days, 5 hours as 7 5/24 days.
Geometry
Students learn to recognize triangles , quadrilaterals, pentagons, hexagons, and octagons. In
lesson 47, they learn that a circle has 360 degrees, a semicircle or a triangle has 180 degrees, and
a right angle has 90 degrees .'They also learn to use a circle graph (pie graph) and calculate the
angles used for the graph.
Projects
From lesson 91 until the end of the program, students work in teams on projects that review and
reinforce their work. Some of the projects are very challenging. In lesson 93, each team makes a
probability chart for a pair of dice, showing how often someone would roll 2, 3,4 and so on .
Then the team uses their chart to make a circle graph. Having made a prediction based on
probability , the teams conduct an experiment in which they roll two dice 72 times. They compare
their results to their prediction . The project gives students practice in the scientific method
(making a prediction and testing it) , along with a chance to use their math skills on a real
problem.
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