Grades K-2 Math Professional Learning 9-17-16
Transitional Learning Center
Presented by Dr. Michele Douglass
Fluency is key for first grade, as students must know addition and subtraction facts
to 10, and then in second grade students are to become fluent in facts to 20.
MD book reference, “You are a juror, and students have to prove themselves guilty
of understanding,” from Understanding by Design (McTighe).
Rekenreks
Group discussed how to make Rekenreks at minimal expense with pipe cleaners,
cardboard, beads and duct tape:
MD presented an overview of Rekenrek use, beginning in TK:
1. Counting 1-10 on the top row
2. "Can you move 5 beads without counting? 7 beads? 3 beads?" How to
instantly recognize amounts is called “subatizing.” Students who play games
at home (with dice) often have a better sense of numbers. Do students "see"
groups of numbers without having to count? MD suggests to be on watch for
this ability, and to give numerous opportunities for students to visualize
amounts.
3. Show 6 beads on Rekenrek. "How many more to make 10?" According to
MD, children often do not visualize how 6 and 4 make 10 on number bonds
right away, so practice on the Rekenrek is key.
Additional Rekenrek strategies:
1. Begin by using Rekenrek to show one more, one less, and then write the
number sentence.
2. Counting on – start at 6, then count 4 more.
3. Doubles strategy – in counting and initial addition problems, students can see
the groups of 5 first.
4. Doubles plus one – once students see a variety of different doubles, try
problems such as 7 + 8 so that students can see the common value (7), and add
the doubles plus one.
Ten Frames
After using Rekenrek for counting and number relationships to 10, MD
recommends using a white board or putting a 10 frame in a sleeve. Begin by asking
students to show various numbers (e.g. 6), and look for different representation
patterns.
After practice with ten frames, use them to show representations of numbers larger
than 10, first by allowing students to “show 12” or various numbers without
guidance:
Ten frame templates are on Educational Services web site in various forms –
teachers can make copies, or use in sleeves to write on/wipe off.
Mini Ten Frames
To build larger numbers and extend understanding of place value, mini ten frames
are available to copy in groups (also on Educational Services web site). MD
recommends that students start with different colors for ten groups versus another
color for partial groups.
MD example problem:
1. Make 36 from 3 tens, 4 ones, 2 ones. "What does 3 represent?" 3 tens = 30
2. When taking this further: "Take one ten off, and replace it with a value that
makes 10." This helps students with decomposing 10s. Example - 16 ones
and 2 tens.
3. Use mini ten frames to visually support “counting up,” then move to larger
100s charts.
As students are able to use the mini ten frames from K-2, they should go back to
them to model addition and subtraction of larger numbers. In particular, this will
help students see partial sums of two digit numbers. Partial sums are helpful as
they make the cognitive load easier when initially solving these problems. The
textbook also uses “numbers below,” which may make it easier for students to see
the number as it is moved to a greater place value.
Discussion topics
MD review - Students should be used to modeling and drawing a problem. This
will shorten the time on operations if students are able to deconstruct numbers (via
ten frames and other strategies) and model. The recommended path is from place
value models, to mini ten frame, to quick numbers, to algorithm.
Place value blocks are also useful, as they fit together to form 10s.
Some of this relates to “Math Their Way” strategies. In addition, EngageNY
utilizes “arrow math” for counting up:
123 + 34 = 123 + 30 = 153 + 4 = 157
Grade two needs to spend more time on tools and place value at the beginning of
the year before moving to symbols. Keep this in mind in planning pacing.
Skip counting is important for all grades, both forward and backward. Even in
upper grades students can “count by 10s from 368.” Be careful with numbers as
they move from 90s to next group. Students should also count up by fractions.
Compensation – from Rekenrek to Number Line
MD demonstrated the use of compensation - a strategy in which students will make
sense of problems by assigning values to other numbers. In addition, students may
take from one to create a double. Or, they may take from one number to make a
multiple of ten or another “friendly number” to add. This should begin with
Rekenreks:
Larger number examples include 37 + 98. Subtract 2 from 37, while adding 2 to
98. 35 + 100 = 135. Also with 24 + 56, subtract 4 from 24 while adding 4 to 56.
20 + 60 = 80.
Compensation can be applied to subtraction, but it must be handled differently.
Using a number line, students will “shift” both numbers in the same direction to
find the difference (or distance) between them. This will differ with regards to the
mental math, so must be demonstrated using number lines and other strategies to
ensure that students understand the difference. An example is 56 – 24, in which
you subtract four values from both numbers (shift them 4 to the left on the number
line) before subtracting. So 52 – 20 = 32.
Mini Ten Frames for Subtraction
MD cautions to be sure to eliminate the word “borrow” from your vocabulary and
that of your students – use words like “trade,” “exchange,” “substitute,” “replace,”
or “regroup.”
Students use their ability to deconstruct numbers on the ten frames in order to do
subtraction. In the example, MD demonstrates on the mini ten frames, bond and
algorithm in order to show relationships.
Place value on white boards for subtraction
MD demonstrated using a place value mat for addition or subtraction. In this
example, different shapes are used. You can also use different color chips for each
place value. Demonstrate that by “trading” ten 10s for one 100, that the value
remains the same before you subtract. MD recommends to show the algorithm at
the same time in order to demonstrate how regrouping works.
MD recommends that students check each place value (ten of one group to make
100), and to show care in what is already in a given place value space. When all
areas are regrouped, students should add up all items to ensure that they match the
original number.
Addition and subtraction on number lines
After modeling with ten frames, students should work on counting by groups on
number lines, both forward and backward. This supports “counting up” and
“counting down” mental math.
When presenting addition and subtraction on number lines, students should look at
multiple ways of grouping numbers as they count. In an example of 67 + 58,
students can count up from 67 by 5 tens to 117, plus 3 to 120, then plus 5 to 125. In
a different method, students may be able to add 67 + 40 to 107, and then add 18
more to 125.
When teaching on a number line, students have to take care with their record
keeping. With subtraction, students will set up their number lines with distances
between, and use strategies of counting up or down. Initially students should be
taught to relate the subtraction problem to the inverse addition problem. For
example, 9 – 7 = ___ is the same as asking 7 + ___ = 9.
In a larger example of 63 – 7, MD sets up a number line between 7 and 63. When
counting up, 7 + 3 = 10, 10 + 50 = 60, and 60 + 3 = 63. By keeping track of those
numbers, 3 + 50 + 3 = 56, and 63 – 7 = 56.
Another example of counting up when subtracting is with the problem 2807 – 1476.
1476 + 4
= 1480
1480 + 20
= 1500
1500 + 500
= 2000
2000 + 807
= 2807
1331 is the sum of numbers when counting up
Problem solving
MD recommends using current ELA stories from which to make up word problems
for your students.
Have students address problem solving first by addition and subtraction actions,
then later by relationships. Initially set up equations as start, addition or subtraction
change, and then end result.
Start
+
Change
=
Result
Start
-
Change
=
Result
Students need to be able to write and associate action problems by equations and
number bonds, and to be able to go back and forth between them.
Later, students will learn that relationships are about equality. Tape diagrams can
be introduced to support these problems to help students see the relationships:
“Ronald has 87 baseball cards. Molly has 24 less than Ronald. How many baseball
cards does Molly have?” First of all, students have to have a good sense of who has
more of the items when doing comparisons. The tape diagram will help.
This should be started with smaller numbers, as well as looking at part-part-whole
relationships (addition).
Number Talks
MD introduced Number Talks, 10-minute routines for composing and decomposing
numbers. Students will be exposed to visual images and patterns first, in small
groups or rows at a time. The book works in a scaffolded approach, and students
move to a separate area without pencil or paper. As students explain their thinking,
the teacher records symbolically what they are saying.
The day concluded with sample videos from the Number Talks, as well as a review
of the documents that introduce the strategy. Emphasis was placed on Talk Moves
during Number Talks, and principles of computational fluency.