Download Understanding By Design Unit Template

Grade 4 UbD Math Unit Planning 2015 to 2016
PS 105
Unit #/Book/Topic
Unit 2 / Books 1 and 3 / Mult. and Div. Concepts and
Approximate Days or Dates
28
Factors and Multiples
Stage 1 - Identify Desired Results
Learning Outcomes
What relevant goals will this unit address?
(must come from curriculum; include specific Common Core standards)
Multiply and Divide within 100
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3.OA.7: Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division.
Use the four operations with whole numbers to solve problems
 4.OA.1: Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5.
Represent verbal statements of multiplicative comparisons as multiplication equations.
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4.OA.2: Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown
number to represent the problem, distinguishing multiplicative comparison from additive comparison.
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4.OA.3: Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which
remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers
using mental computation and estimation strategies including rounding.
Gain familiarity with factors and multiples
 4.OA.4: Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a
given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or
composite.
Use place value understanding and properties of operations to perform multi-digit arithmetic
 4.NBT.5: Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value
and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Enduring Understandings
What understandings about the big ideas are desired? (what you want
students to understand & be able to use several years from now)
What misunderstandings are predictable?
Essential Questions
What is the Go Math Chapter Essential Questions?
Are there any potential cross-curricular connections during this chapter?
Students will understand...
Essential Question:
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The array model of multiplication.
There are many strategies for solving multiplication problems and
good mathematicians are flexible in choosing the best strategy.
The distributive property helps us multiply larger numbers.
Multiplication and division can be used to compare quantities.
(e.g., 36 is 6 times and many as 6)
Related misconceptions…
 Comparisons are only made through subtraction.
Knowledge:
What knowledge will student acquire as a result of this unit? This content
knowledge may come from the chapter’s goals, or might also address
pre-requisite knowledge that students will need for this unit.
Students will know...
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The meaning of the following vocabulary words: compatible
numbers, factor, product, prime, composite, square
How can arrays help us understand multiplication?
Cross-curricular connections…
Skills:
What skills will students acquire as a result of this unit? List the skills and/or
behaviors that students will be able to exhibit as a result of their work in this
unit.
Students will be able to…
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Relate multiplication equations and comparison statements.
Model multiplicative comparison situations by using a bar model.
Fluently answer multiplication and division basic facts.
Multiply tens, hundreds, and thousands through 10.
Use place value and multiplication properties to multiply by 10.
Use mental math and estimate products by using rounding or
compatible numbers.
Represent and solve multi-step word problems using equations.
Find the factor pairs for numbers 1 to 100.
Identify if a number up to 100 is a multiple of a given single digit
number.
Identify numbers 1 to 100 as either prime or composite.
Stage 2 – Assessment Evidence
Evidence
Through what evidence (work samples, observations, quizzes, tests,
journals or other means) will students demonstrate achievement of the
desired results? Formative and summative assessments used throughout
the unit to arrive at the outcomes.
Assessment: Solving 18 x 7 (p. M43)
Student Self-Assessment
How will students reflect upon or self-assess their learning?
Stage 3 – Learning Plan
#
Content Goal
Book 1 Notes:
6
sessions
Book
1/Investigation 1:
Representing
Multiplication with
Arrays
1.1
1.2
Things That Come in
Arrays
Making Arrays
1.3
Making Arrays
Lesson Notes/Planned Differentiation
Although fluency with multiplication facts (through 10 x 10) is a third grade
standard, many children will not yet have achieved it. Even for those students
who have achieved fluency, the model of the array is critical for the coming work
this year on multi-digit multiplication and division. Also, all students need to
extend the grade 3 focus on fact fluency to the grade 4 focus on being able to
identify all the factors of numbers through 100. As such, it is better not to skip
lessons from book 1. However, do plan to differentiate activities based on your
assessment of students’ multiplication (and division) fact fluency.
Don’t forget about the very important Common Core add on lesson 1.6a, which
introduces Multiplicative Comparisons to students for the first time. You may
want to spend an extra day on these or else plan on revisiting the concept in the
future.
Additional Resources or Math
Centers
Multiplication Top-It (from
Everyday Math) or other
multiplication practice centers
could supplement or replace
Factor Pairs.
Do as written.
No Centers Today
Do as written. Be sure to clarify in the mini-lesson that students do NOT need to
make separate arrays for turn-around facts. In other words, if they have made a
2 x 6 array for the number 12, they do NOT need to make a 6 x 2 array. Note
that this lesson is leading to the definition of the words prime, composite, and
square numbers in session 1.3. Also, be sure to read all the margin notes for this
lesson.
Do this lesson as suggested. Be sure to come to a class discussion about the
meaning of prime, composite, and square numbers. In case it comes up, the
number 1 is neither prime nor composite.
Students have just been introduced to the concepts of factors and how they
determine whether a number is prime, composite, and/or square. But they need
more practice with finding all factor pairs of the numbers 1 to 100 (4.OA.4).
Spend a math session today working on that skill. Differentiate by the size of the
numbers you ask students to use.
No Centers Today
No Centers Today
1.3B
Extra Lesson on
Factors
1.4
SKIP
1.5
Which Combinations
Do I Know?
Using Arrays to
Multiply
SKIP this lesson. Students that need it can play factor pairs during lesson 1.5.
1.6A
Multiplicative
Comparison
Book
1/Investigation 2:
Multiplication
Do this lesson as written.
 Array Picture Problems
 Factor Pairs (or Division Dash)
 Assessment: Representing 8 x
6
No Centers Today
In this investigation, skip the use of “Multiplication Cards” (unless you like
them), but do continue working to ensure students are fluent in multiplication
facts. Students already fluent should work on division facts. A nice, challenging
Consider using activities for OA.4
from: http://www.k5mathteachingresources.com/4th-
3
sessions
Do this lesson as needed. Students who know all of their multiplication
combinations should substitute Division Dash or some similar activity to work on
division facts.
No Centers Today, but optionally,
you could have students play the
Factor Game on the computer
from this link:
illuminations.nctm.org/Activity.as
px?id=4134
Combinations
2.3
Multiple Turn Over
2.4
&
2.5
Multiplication/Multiples
3
sessions
3.1
3.2
3.3
5
sessions
3.1
Book 1-Investigation 3:
Finding Factors
Factors of 100
Factors of the
Multiples of 100
Factors of Related
Numbers
Book 3 Notes:
Book 3-Investigation 1:
Breaking Apart
Multiplication
Problems
Solving Multiplication
Problems
3.2
3.3
Making Big Arrays
Small Array/Big Array
3.4
Small Array/Big Array
3.5
Assessment: 18 x 7
6
sessions
Book 3-Investigation 2:
Division
2.1
Looking at Division
game is Division Dash (from Everyday Math). Students pick 3 digit cards and
form a two-digit number and divide it by the remaining digit. They earn points
equal to the quotient (ignoring the remainder).
Skip 2.1 and 2.2, starting this investigation with 2.3. Do as written or if you
prefer, have half the class play Multiple Turn Over while the other half plays
either Factor Pairs or Division Dash. Don’t skip “Multiple Turn Over” even though
it is a bit difficult to teach. There should be decks of multiple cards in your card
kits to use for this game.
Do the mini-lesson as suggested and then continue having students play Multiple
Turn Over and whatever center is appropriate for them for working on
multiplication or division fact fluency. Continue for two days using the time to
assess students for multiplication and division fact fluency.
During this investigation, continue to provide fluency practice for students as
needed and/or practice with activities from K-5 Math Teaching Resources site.
grade-number-activities.html
Do this lesson as written.
Do this lesson as written. Note the Ongoing Assessment advice on TG p. 100
and the Differentiation advice on TG pp. 101-102.
Do this lesson as written. It is about the very important concept that factors of
one number are common to factors of multiples of that number.
The standard algorithm for multiplication (and division) is not in the Common
Core until grade 5, so this unit teaches students to use the distributive property
(aka array model or partial products) to multiply larger numbers.
“Small Array/Big Array” is a bit complicated for students at first, but it is worth
the effort. They need to practice combining arrays and attending carefully to the
dimensions. Also, be sure to use the Ten-Minute Math activity, “Quick Images:
Seeing Numbers” during this unit.
No centers today.
No centers today.
Be sure to do Quick Images: Seeing Numbers today. Then do the lesson as
written, being sure to show students how to break apart multiplication problems
as illustrated on TG p. 31.
Do this lesson as written.
See TG pp. 156-157 for a detailed explanation of this game. Even though it is
suggested as a whole class activity, it would be easier to do it with half the class
at a time. Choose a familiar center activity for the other half to do.
Do this centers lesson as suggested. No mini-lesson other than to explain the
centers. Save the time for a discussion at the end of the session as suggested
on TG pp. 49-50. Note that the centers continue the next day.
The same centers continue today, but first start with the assessment page.
No centers today.
There are two types of division problems (unknown number of groups and
unknown number in each group) and students must become accustomed to both
types. The other huge challenge for students here is reading the question
carefully in order to interpret the remainder. Expect significant struggle with
interpreting remainders.
Do this lesson as written. If students complete the SAB pages too quickly they
 Multiple Turn Over
 Factor Pairs (or Division Dash)
 Multiple Turn Over
 Factor Pairs (or Division Dash)
No centers today.
Note: See expanded
differentiation advice for this
investigation (following page).
No centers today.
 Small Array/Big Array
 Teacher’s Choice
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Small Array/Big Array
Breaking Up Arrays
Solving Multiplication Problems
Small Array/Big Array
Breaking Up Arrays
Solving Multiplication Problems
No centers today.
can work on one of the centers from the previous day.
This is a very important lesson that is difficult for many students. Do the lesson
as written and be sure to read the Math Note in the margin on TG p. 66. It is
very important that students think of a division problem like 64 ÷ 4 as “How
many 4’s are in 64?”
Do these two lesson as written. On the second day, there is no mini-lesson other
than introducing Missing Factors. But be sure to save 20 minutes for the
“Developing Strategies for Division” discussion on TG pp. 81-83.
Do this lesson as written. See the Ongoing Assessment advice on TG p. 86.
2.2
Division with
Remainders
2.3
and
2.4
2.5
Division Stories and
Strategies
2.6
Assessment: Writing
and Solving a Division
Problem
Book 3-Investigation 3:
Multiplying 10s
Building Multiple
Towers
Multiplying Groups of
10
Do this lesson as written.
3.3
Multiplying 2-Digit
Numbers
Do NOT discuss any algorithms for the opening problem 15 x 13. This discussion
should be related to towers of 13 or towers of 15.
0
sessions
Book 4-Investigation 4:
Strategies for
Multiplication
Unit Assessment
Skip this investigation
3
sessions
3.1
3.2
Related Multiplication
and Division Problems
Do the first three sessions of this investigation. Note that this is a conceptual
introduction to multiplying larger numbers. Students should NOT use any
algorithms yet. There will be time for that during the next unit.
Read this lesson carefully and implement it as exactly as written. The purpose of
the lesson is for students to examine what ten groups of a number looks like.
Do this lesson as written. See the Ongoing Assessment and Differentiation
advice on TG pp. 108-109.
No centers today.
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Small Array/Big Array
More Division Stories
Missing Factors (second day)
Missing Factors
Related Multiplication and
Division Problems
No centers today.
 Multiplying by Multiples of 10:
What Happens?
 Multiple Towers
 Multiples of 10: Related
Problems
 Story Problems About 10s
 Multiplying by Multiples of 10:
What Happens?
Expanded Differentiation Advice for Book 3/Investigation 1
When doing your lesson plans with the Investigations curriculum, it is really important to plan for a whole investigation together, rather than going day by
day. Here are my thoughts for how to plan for the first investigation of Book 3:
One problem to anticipate is that some students will already know the standard algorithm for multiplying two-digit numbers. Since the purpose of this
investigation, however, is to learn to use the Distributive Property (break-apart method), the standard algorithm should not be allowed. The Distributive
Property has major importance in mathematics. Here are just a couple of examples:
1) Mental Math: Can you solve the problem 29 x 40 mentally? I would use the distributive property and calculate 30 x 40 = 1200 and then
subtract one group of 40 to arrive at 1160. (29 x 40 = 30 x 40 – 1 x 40)
2) Algebra: The distributive property is critical for multiplying algebraic expressions. It explains why “FOIL” works for (x+1)(x+2) .
3) State Test: The state test requires students to understand partial products and partial quotients, essentially the area model for multiplying
and dividing numbers.
Consequently, you must do everything in your power to convince those students who want to use the standard algorithm that the point of this
investigation is not merely to get the answer, but to learn to use the distributive property, so they must work to learn to do these problems without the
standard algorithm.
Once this has been established, this investigation becomes valuable for the full range of students in the classroom, and differentiation will not be needed
for the CORE new materials here. However, there are some important opportunities for differentiation that can be worked into this investigation and
throughout the rest of this unit. At this point, students may need more practice in any of the following (listed in order):
1) Multiplication Fact Fluency: Students should be fluent in the product of pairs of numbers from 1 to 10. If not, practice in this area should be
prioritized.
2) Division Fact Fluency: After students become fluent with Multiplication Facts, they should work on developing fluency with Division Facts.
3) Factors: Once students become fluent with Division Facts, they can practice identifying the factors of numbers up to 100.
These skills can be practiced through a differentiated center based on what the students need.
Multiplication Fact Fluency: Multiplication Top-It, Product Game http://illuminations.nctm.org/Activity.aspx?id=4213, Array Card Games, Studying
Multiplication Cards, Games listed under 3.OA.7 from http://www.k-5mathteachingresources.com/3rd-grade-number-activities.html
Division Fact Fluency: Division Dash, Games listed under 3.OA.7 from http://www.k-5mathteachingresources.com/3rd-grade-numberactivities.html
Factors: Factor Game http://illuminations.nctm.org/Activity.aspx?id=4134
Post-Unit Reflection
Considerations
Comments
Required Areas of Study:
Was there alignment between outcomes, performance
assessment and learning experiences?
Adaptive Dimension:
For struggling students:
Did I make purposeful adjustments to the curriculum
content (not outcomes), instructional practices, and/or
the learning environment to meet the learning needs and
diversities of all my students?
For students who need a challenge:
Suggested Changes:
How would I do the unit differently next time?