Transcript Wheeler Lower School Mathematics Program

Wheeler Lower School
Mathematics Program
Grades 4-5
1.
2.
Goals:
For all students to become
mathematically proficient
To prepare students for success
in middle and upper school
What do children need to become
mathematically proficient?
 Strong Understanding
of Mathematics
Understanding what
concepts mean and
how they interconnect
with one another
 Computational Fluency
Ability to add, subtract, multiply and
divide efficiently and accurately
 Problem Solving Skills
Ability to use knowledge to
solve problems in a variety
of contexts and real world
situations
 Mathematical Reasoning
Ability to use logic
and deductive thinking
 Meta-Cognition
Ability to reflect on
one’s own thinking
and process
 Communication Skills
Ability to use
mathematics language to
share thinking and justify
solutions
 A Set of Beliefs, Dispositions & Attitudes
A willingness to engage with mathematics,
seeing it as sensible, useful, and doable
Mathematics in Fourth Grade – Content & Skills
Number and Operations: Whole Numbers
Work focuses on extending knowledge of the base-ten number system to 10,000.
Multiplication and division are the major focus of students’ work in number and
operations in Grade 4. Students use models, representations, and story contexts to
help them understand and solve multiplication and division problems. In addition
and subtraction, students refine and compare strategies for solving problems with
3-4 digits. By the end of the year, students are expected to solve addition and
subtraction problems efficiently; know their multiplication combinations to 12 x 12
and use the related division facts, and to solve 2 x 2 digit multiplication problems
and division problems with 1-2 digit divisors.
Number and Operations: Fractions and Decimals
The major focus of work is on building students’ understanding of the meaning,
order, and equivalencies of fractions and decimals. They work with fractions in the
context of area, as a group, and on a number line. Students are introduced to
decimal fractions as an extension of the place value system. They reason about
fraction comparisons, order fractions on a number line, and use representations
and reasoning to add fractions and decimals.
Geometry and Measurement
Students expand their understanding of how the attributes of two-dimensional (2-D) and threedimensional (3-D) shapes determine their classification. Students consider attributes of 2-D
shapes, such as number of sides, the length of sides, parallel sides, and the size of angles.
Students also describe attributes and properties of geometric solids (3-D shapes). Measurement
work includes linear measurement (with both U.S standard and metric units), area, angle
measurement, and volume. Students work on understanding volume by structuring and
determining the volume of a rectangular prism.
Patterns and Functions
Students create tables and graphs for situations with a constant rate of change and use them to
compare related situations. By analyzing tables and graphs, students consider how the starting
amount and the rate of change define the relationship between the two quantities and develop
rules that govern that relationship.
Data Analysis and Probability
Students collect, represent, describe, and interpret numerical data. Their work focuses on
describing and summarizing data for comparing two groups. They develop conclusions and make
arguments, based on the evidence they collect. In their study of probability, students work on
describing and predicting what events are impossible, unlikely, likely, or certain. Students reason
about how the theoretical chance (or theoretical probability) of, for example, rolling 1 on a
number cube compares to what actually happens when a number cube is rolled repeatedly.
Mathematics in Fifth Grade – Content & Skills
Number and Operations: Whole Numbers
Students practice and refine the strategies they know for addition, subtraction,
multiplication, and division of whole numbers as they improve computational fluency
and apply these strategies to solving problems with larger numbers. They expand
their knowledge of the structure of place value and the base-ten number system as
they work with numbers in the hundred thousands and beyond. By the end of the
year, students are expected to know their division facts and to efficiently solve
computation problems involving whole numbers for all operations.
Number and Operations: Fractions, Decimals, and Percents
The major focus of the work with rational numbers is on understanding relationships
among fractions, decimals, and percents. Students make comparisons and identify
equivalent fractions, decimals and percents. They order fractions and decimals, and
develop strategies for adding fractions and decimals to the thousandths.
Geometry and Measurement
Students develop their understanding of the attributes of 2D shapes, examine the
characteristics of polygons, including a variety of triangles, quadrilaterals, and regular
polygons. They also find the measure of angles of polygons. In measurement, students
use standard units of measure to
study area and perimeter and to determine the volume of prisms and other polyhedra.
Patterns and Functions
Students examine, represent, and describe situations in which the rate of change is
constant. They create tables and graphs to represent the relationship between two
variables in a variety of contexts and articulate general rules using symbolic notation
for each situation. Students create
graphs for situations in which the rate of change is not constant and consider why the
shape of the graph is not a straight line.
Data Analysis and Probability.
Work focuses on comparing two sets of data collected from experiments developed by
the students. They represent, describe, and interpret this data. In their work with
probability, students describe and predict the likelihood of events and compare
theoretical probabilities with actual outcomes of many trials. They use fractions to
express the probabilities of the possible
outcomes.
Multiplication Strategies Used by
Wheeler Students
Building Arrays
This strategy provides a concrete visual model of
‘how many rows of how many squares.’
Example: 4 x 8 = 32
Arrays model multiplication in terms of
the area of a rectangle.
4 rows of 8 squares is 32 squares or the
product of the dimensions of the
rectangle.
Example 12 x 8 =
(10 x 8)
/
80 + 16 = 96
+ (2 x 8)
Children begin to
develop a sense of
double digit by
single digit
multiplication by
seeing larger
arrays as the sum
of smaller arrays
they already know.
Example 12 x 8 =
Children develop
deeper
understanding of
multiplication
when they find
more than one
way to solve a
problem.
4x8
+
4x8
+
4x8 =
96
Box Method Multiplication
Example: 12 X 6
6
60
12
10
60 + 12 = 72
+ 2
This box is an
abstraction of
the gridded
array. The
smaller boxes
show the
partial products
of the
multiplication
problem.
Double Digit by Double Digit Example: 12 x 13
10
+
3
10
+
2
100
30
20
6
100 + 30 + 20 + 6 = 156
Expanded Notation Multiplication
Example:
X
24 x 5
20 + 4
5
20
100
120
Example: 12 x 23
x
10 + 2
20 + 3
6
30
40
200
276
Partial Products Multiplication
27
x
6
42
120
162
323
x
7
21
140
2100
2261
No need for x-ing out decimal places. No need for funny little
numbers scribbled on the top of the problem.
Students make less place value errors when they write down
the entire partial product.
Division Strategies Used At Wheeler
Subtracting Out Convenient Groups
168 ÷ 14 = 12
645 ÷ 6 = 107 3/6
12
14
168
- 140
28
- 28
0
107
6
10
2
645
- 600
45
- 36
9
6
3
R3
100
6
1
Expanded Notation
945 ÷ 9 = 105
100 + 5
9 900 + 45
777 ÷ 24 = 105
20 + 10 + 2 R 9/24
24 480 + 240 + 48 + 9
How Can Parents Help Children Become
Mathematically Proficient?
 Help them develop automaticity of
basic facts with addition, subtraction,
multiplication and division.
 Encourage children to independently
work through nightly homework before
you step in and help. Instead of showing
them the way you learned in school, ask
about the different strategies they are
learning.
 Ask them to share their thinking with you and
talk about the steps they took to solve a problem
and how they know their solutions are correct.
 Keep pointing out the different ways
that you use mathematics in your life so
your children understand how
meaningful it is.
 Make sure they understand that being
good at mathematics doesn’t come from
a special gene, but is something that is
learned and developed over time.