Find
Spot a factor that already appears in every term.
Lesson 1.2 · Decimal Smart Calculation
Finding and Creating a Common Factor
Turn several difficult products into one easy multiplication by making every term share the same multiplying piece.
Find it. Make it. Pull it out.
Create
Rewrite products without changing their value so a common factor appears.
Factor
Use the distributive property backward and finish with an easy bracket.
1
Warm-up · Turn on the factor lens
What is a common factor here?
In this lesson, a common factor means the same multiplying piece in every term. It does not have to be the greatest common factor.
A term is a part of an expression separated from the next part by + or −. In the expression below, the two terms are 6 × 4 and 6 × 3:
6 × 4 + 6 × 3
Which factor is explicitly written in both products?
Tap the number that both products use as a factor.
Distributive property — used backward
a × b + a × c = a × (b + c)
The shared factor a is pulled outside the bracket.
2
Visual proof
Pack two rectangles into one
Both rectangles have the same height, 6. That shared height is the common factor.
height 66 × 4
6 × 3
Two areas: 24 + 18
What changes — and what stays the same?
- The common height stays 6.
- The widths join: 4 + 3.
- The total area stays the same.
6 × 4 + 6 × 3 = 6 × (4 + 3) = 42
Teacher thought: factoring does not change the amount. It only repacks the calculation.
3
Book Example 1 · Create the same factor
The 2014 mission
The three terms do not look alike at first. We will use balanced decimal changes so that each one contains the factor 2014.
2014 × 18
−
201.4 × 90
+
20,140 × 0.1
Balanced-change rule
If one factor becomes 10 times as large, the other must become one tenth as large (divide it by 10). Then the product stays unchanged.
201.4 → 2014× 10
×
90 → 9÷ 10
Choose the useful equivalent products
A. 201.4 × 90 = ?
B. 20,140 × 0.1 = ?
To keep a product unchanged, balance a ×10 change in one factor with ÷10 in the other. You may also do the reverse: ÷10 in one factor and ×10 in the other.
Now every term speaks “2014”
2014 × 18 − 2014 × 9 + 2014 × 1
= 2014 × (18 − 9 + 1)
4
Book Example 2 · Build a multi-part common factor
Create the factor 3 × 25
Sometimes the useful common factor is not a single visible number. We can build it from smaller factors.
75 × 4.7 + 15.9 × 25
Useful decomposition for 75
Which true decomposition explicitly shows both factors in 3 × 25?
Useful decomposition for 15.9
Which true decomposition explicitly shows a factor of 3?
Choose the decompositions that will make both terms contain 3 × 25.
3 × 25 × 4.7 + 3 × 5.3 × 25
= 3 × 25 × (4.7 + 5.3)
Friendly-number check: What is 4.7 + 5.3?
3 × 25 × 10 = 750
The important discovery was not the multiplication. It was creating the same factor in both terms.
5
Book Example 4 · Strategic flexibility
One expression, two smart routes
Strong problem solvers do not search for one “official” path. They notice relationships and choose a useful route.
7.816 × 1.45 + 3.14 × 2.184 + 1.69 × 7.816
Find the two friendly relationships
Select both true relationships. They are the map for the solution.
Route A: factor twice
- Combine the first and third terms because both contain 7.816:
7.816 × (1.45 + 1.69) = 7.816 × 3.14. - Now the new product and the middle term both contain 3.14.
- Factor again: 3.14 × (7.816 + 2.184) = 3.14 × 10 = 31.4.
Answer: 31.4
Route B: create a zero bracket
Because 7.816 + 2.184 = 10, we may write:
2.184 = 10 − 7.816
7.816 × 1.45 + 3.14 × (10 − 7.816) + 1.69 × 7.816
= 7.816 × 1.45 + 3.14 × 10 − 3.14 × 7.816 + 1.69 × 7.816
= 3.14 × 10 + 7.816 × (1.45 − 3.14 + 1.69)
= 31.4 + 7.816 × 0 = 31.4
First expand 3.14 × (10 − 7.816). Then collect the three terms containing 7.816 and factor it out. The remaining bracket is 1.45 − 3.14 + 1.69 = 0, so only 3.14 × 10 remains.
7
Game · Meet a bracket-total target
Factor Forge
For each expression, choose a listed common factor that makes the bracket add to the stated target. Then calculate the whole expression. Other factors may give equivalent rewrites, but this game checks the stated bracket target.
Challenge 1 of 5
Score: 0
Which listed factor gives the stated bracket total?
Choose a factor that can be made to appear in every term.
8
Reasoning · Protect the value
Error Detective
A rewrite is useful only when it is equivalent to the original expression.
Case A: unbalanced decimal move
201.4 × 90 = 2014 × 90
What went wrong?
Case B: wrong operation inside the bracket
8 × 3 + 8 × 5 = 8 × (3 × 5)
Which correction is equivalent?
9
Independent practice
Find it, make it, factor it
Complete at least six of the eight problems. A hint should help you notice structure, not replace your thinking.
Solved: 0 of 8. Reach 6 to complete this mission.
10
Mastery check
Exit Ticket
Try all five questions without opening hints, including all three parts of Question 5. Then select “Submit exit ticket” to check your responses.
1. Name the factor already written in both products
36 × 2.5 + 36 × 7.5
2. Calculate the expression in Question 1
36 × (2.5 + 7.5)
3. Create a common factor, then enter the value of the whole expression
4.2 × 38 + 42 × 6.2
4. Protect the product
To keep a product unchanged, if one factor is multiplied by 10, what must happen to the other factor?
5. Create a common factor step by step
Complete all three parts about the book-based expression 22.8 × 98 + 45.6. Together they check how you create, factor, and use a common factor.
This writing is for reflection and is not automatically graded.
Question 5 is graded from the three choices above. The optional reflection is not scored.
Mastery result
CERTIFICATE OF MATHEMATICAL THINKING
Common-Factor Architect
This certifies that
can find, create, and factor a shared multiplying piece while preserving the value of an expression.
Lesson 1.2 · Grade 5 Math Explorer
The visual models, question sequencing, games, feedback, and additional practice are original teaching adaptations for this web module.