# Instructional Materials

To ignite the excitement and love of math in young minds of today for future success tomorrow.

Meaningful mathematics learning is usually image based; therefore, students need opportunities to form necessary images of numbers, mathematical patterns and relationships.  (Wheatly, G.H. and Reynolds, A.M., 1999)

The DMG has developed resources that aid in the development of the essential math concepts as well as concepts found in the Common Core State Standards. Instructional materials have been developed for students Pre-K through Grade 2 to build conceptual knowledge and make sense of the math that they are learning.  Each of our boards, mats, and lines comes with an activity guide, which has been written in a sequence to build understanding.

Our instructional materials:

• Provide conceptual rather than procedural methods

• Emphasize the essential math concepts

• Encourage students to make sense of the math

• Motivate students to be responsible for their own learning

• Enhance mathematical learning

EXAMPLES OF THE DIFFERENT BOARDS, MATS AND LINES

Visual models are an excellent way for students to build an understanding of the math they are learning.  The following are the boards, mats and lines that Developmental Math Group has produced.

99/100 and 120 Boards

The popular 99/100 board was developed to help students see the patterns and relationships within our base-ten numeration system.  It can be used for addition (100 board) and subtraction (99 board) with two-digit numbers to increase mathematical meaning.  The student can use the knowledge of tens and ones to construct their answers.   Students learn about the structure of the written numbers in our base-ten numeration system by using these boards.

We also have a 120 board with blank spaces on the back so grade one students can count beyond 100 and see how the pattern continues.  The open squares on the back provide a place for writing the numerals.

Math Mats

Math Mats were developed so students can act out the addition and subtraction word problems involving different types of situations presented in the Common Core (add to, take from, put together, take apart and comparison).  Students need to develop an understanding of what is being asked and then construct their understanding by acting out the problem.  Later, the students move to the symbolic representation by writing expressions or equations to represent the problems.

Students can use the Math Mats to create their own word problems. This is a wonderful way to enhance each student's math vocabulary as well as his or her mathematical thinking.

We suggest using the Bears and Chairs with these mats.

5 Frame, 10 Frame and Double 10 Frame Boards

Our five, ten and double ten frame boards provide the opportunity for students to form mental images so they can see numbers as a visual collection relating to the important numbers of five and ten.  We want students to understand the importance of 5 and 10. These relationships are important when thinking about various combinations of numbers. The most important model is the ten frame. Before the ten frame, we suggest using the five frame, which builds numbers 6-10 in relationship to the five. The Double 10 Frame has students building the basic facts that have a sum above ten by “making a ten.”

Part/Part/Whole Mats

To conceptually understand that any number can be decomposed into two parts is the most important relationship that needs to be developed about numbers.

This part/part/whole mat can be used to decompose a number into its two parts and to see how many combinations can be made for a particular number.  Each combination can be shown with manipulatives, in order to build a visual as well as a conceptual understanding of the number.  Later, the symbols can be written to show the equations. For example, a five can be decomposed into 0 and 5, 1 and 4, 2 and 3, 3 and 2, 4 and 1 and 5 and 0.  To begin, the children show the whole and then create different combinations on the mat.  Then, students can write 5= with the different combinations.  This mat can also be used when students are solving word situation problems involving put together or take apart listed in the Common Core.

Place and Value Mats (Ones and Tens; Hundreds, Tens and Ones)

Place and value mats were developed to provide students a visual connection to the quantity using base-ten materials. One side has two ten frames to help students organize the count and see how many more to make a ten.  The mat provides students a connection to the written symbol. For first grade, use the tens and ones mat and grade two and up, use the hundreds, tens and ones.

Number Lines

Research supports the importance of the number line as a tool for:

•  Helping students actively construct mathematical meaning

•  Building number sense

•  Understanding number relationships

•  Performing operations.

The number line is an easy model to understand and has great advantages in helping students understand the relative magnitude, position of numbers, comparing and ordering as well as visualizing operations.

Number lines are difficult for children below grade two and do not appear in the Common Core until second grade.

Number paths are used instead because they are a count model and support students counting, cardinality, comparing and numeral recognition.  We recommend using number paths with ages three through grade one.

The following is a list of the various number paths and lines that we have created for students:

1-10 Number Path

Number lines are very difficult for young children to understand.  To get them started understanding and using a number line, we begin with a 1-10 number path.  The side with one-inch squares with numerals can be used as a counting board to count a collection, for word and numeral identification and sequence, for one-to-one correspondence, cardinality, magnitude, ordering, comparing numbers, one more, one less, number after, number before, and count on one or two more.  Activities that involve moving along a number path are important for strengthening foundations and building conventional number knowledge.

Side two of this board has ten blank squares.  The same activities can be done with students but will be more difficult because numerals are not there to support the concepts.  Students can write the numerals in the squares for practice or to answer questions about concepts.

1-20 Number Path

The idea of the number path continues with this 1-20 path.  This two-sided dry erase board has one inch squares for use as a counting mat with numerals on one side and an empty number path on the other.  We recommend this path for kindergarten and grade one students.

1-20 Number Path and Open Number Line

This two-sided board has a smaller 1-20 path on one side to support students who are moving to an understanding of the number line and an open number line (a line with no markings) on the other that can be used to bridge the move for students from counting with manipulatives to symbolic representations.  The open number line side provides students an opportunity to place any numbers they want and learn about spacing, sequencing, counting on from a given number, magnitude, ordering, comparing and perform operations.  It can also be used as an “empty number line” (no markings). The empty number line enables students to create a mental image of the strategies they discover for operations, supporting them to make the leap more easily towards mental calculations without paper. Use of the open number line also increases students” confidence in their ability to use numbers flexibly which leads to further development in their understanding of number sense.

This side can also be used for line plots.

Interactive Number Line

This interactive number line can be used as both a number path and a number line.  With young students, it replicates the number path with the small rectangles almost touching.  These can be removed and the line is then visible and can be used as a number line model.  The board comes with eleven dry-erase pieces and links that can be used to show quantities of 1-5.  This number line is an excellent model to use with the students and then as a tool for students to interact with.

Time Lines and Analog and Digital Clocks

It is common to think of time passing in a straight line.  Yet, clocks show time passing in a circle.

Using a number line to determine elapsed time (time that has passed since a certain point) makes the process easier because it restores our linear way of thinking about time.

First grade teachers should order one time line for demonstrating, along with a classroom set of analog and digital clocks. Second and third grade teachers should order a classroom set of time lines and clocks.

Conceptual activities are provided allowing the student to understand how the circular clock can be stretched into a number line and how time is told in one and five minute intervals.

The day line has two colors, one for am and one for pm.   The line is divided into 12-hour segments as well as 5-minute intervals. There are markings for 15, 30 and 45 minutes and hour markings with numbers 1-12.  Activities are provided for elapsed time (time that has passed since a certain point).

Fraction Number Line/Multi-purpose Number Line

In the third grade, students begin to explore and develop an understanding of fractions.  This fraction number line/multi-use number line has two sides. On one side there are two lines, one that has been marked off into halves, fourths and eighths and the other marked off into halves, thirds and sixths. Students can:

•understand a fraction as a unit,

•partition into equal parts,

•determine size,

•determine equivalence of fractions,

•express whole numbers as fractions,

•locate a/a and 1 at the same time,

•compare two fractions,

•perform operations, and

•justify decompositions.

The other side is a “multi-use” two-line diagram, which can be used as two parallel number lines, double number line diagrams and as a tape diagram.  The students can use these as a “thinking tool” when solving word problems and transform the words into an appropriate numerical model.