Protect
Rewrite an expression without changing its value.
Lesson 1.3 · Decimal Smart Calculation
Transform Before Calculating
A difficult expression can become easy when you protect its value, change its form, and reveal a friendly number.
Protect the value. Change the form. Reveal the shortcut.
Transform
Scale, regroup, split, or rename parts to reveal structure.
Choose
Select a useful route before beginning heavy arithmetic.
1
Warm-up · Do not calculate yet
Stop and scan the expression
A smart calculator pauses first. The question is not “What do I multiply now?” but “What can I safely change so the arithmetic becomes friendly?”
PProtect the value
AAnalyze the pattern
UUse a safe rewrite
SSpot a friendly target
EEvaluate at the end
A · A difficult divisor
0.27 ÷ 0.25
Which move keeps the quotient unchanged and makes the divisor 0.25 become 1?
B · Three factors
2.5 × 1.25 × 3.2
Which move lets you regroup the multiplication into one pair with product 1 and another with product 10?
C · A lonely term
3.74 × 2.85 + 8.15 × 3.74 − 3.74
How can you write the final 3.74 as a product containing the factor 3.74? Keep the subtraction sign before this term.
Complete all three scans to finish Mission 1.
2
Toolbox · Preserve equality
Two safe decimal transformations
The book opens with two rules. Products balance in opposite directions; quotients scale together in the same direction.
0.8 × 1.25
⇄
8 × 0.125
If one factor becomes 10 times as large, the other becomes one tenth as large (divide by 10). The product stays unchanged.
Which displayed equality preserves the value of the original expression?
0.16 ÷ 0.04
⇄
16 ÷ 4
The dividend is the number being divided; the divisor is the number you divide by. Multiply both by the same nonzero number to preserve the quotient. The divisor must not be zero.
Which displayed equality preserves the value of the original expression?
Memory rule
Product: multiply one factor by a nonzero number and divide the other factor by that same number. Quotient: multiply the dividend and divisor by the same nonzero number, or divide both by the same nonzero number.
3
Book Example 3
Transform the divisor into 1
The original division is possible, but the divisor 0.25 is awkward. A better form is waiting.
0.27 ÷ 0.25
By which number should you multiply both 0.27 and 0.25 so that the divisor becomes 1?
Why target 1?
Dividing by 1 leaves a number unchanged:
1.08 ÷ 1 = 1.08
The transformation removes the difficult part before calculation.
0.27
0.25
× ? →
?
?
0.27 ÷ 0.25=(0.27 × 4) ÷ (0.25 × 4)
=1.08 ÷ 1
4
Book classroom practice
Regroup consecutive division
With consecutive division, first divide by the first divisor, then divide that result by the second divisor. This equals dividing by the product of those two divisors. Both divisors must be nonzero.
320 ÷ 1.25 ÷ 8
Which equivalent rewrite combines the two divisors, 1.25 and 8, into a single product in brackets?
Friendly divisor pair
1.25 × 8 = 10
So two divisions collapse into one easy division by 10.
320 ÷ 1.25 ÷ 8=320 ÷ (1.25 × 8)
=320 ÷ 10
Why it works: imagine 320 metres of ribbon cut into 1.25-metre pieces. Then bundle eight pieces together. Each bundle uses 1.25 × 8 = 10 metres. Counting bundles is therefore 320 ÷ 10 = 32. Dividing twice gives the same result as dividing by the product of the two divisors. The divisors must be nonzero.
5
Book Exercise 2
Split one factor to create two friendly pairs
The factor 3.2 can be replaced by a product in more than one useful way.
2.5×
1.25×
3.2
Choose either product split of 3.2 that lets you form one pair making 1 and another making 10. There are two accepted choices; you only need to select one.
1.25 × 0.8 = 1
and
2.5 × 4 = 10
(1.25 × 0.8) × (2.5 × 4) = 1 × 10
2.5 × 0.4 = 1
and
1.25 × 8 = 10
(2.5 × 0.4) × (1.25 × 8) = 1 × 10
6
Book Exercises 3 and 5
Give every term the same costume
A lone number may secretly be one copy—or one tenth of a copy—of the factor you want.
Challenge A · Add the hidden ×1
3.74 × 2.85 + 8.15 × 3.74 − 3.74
Rewrite the last term so every term contains 3.74.
3.74 × (2.85 + 8.15 − 1) = 3.74 × 10
Challenge B · Add ×1 and ×0.1
2.4 × 7.6 + 7.6 × 6.5 + 7.6 + 0.76
Which pair of rewrites makes every term a multiple of 7.6?
2.4 × 7.6leaves 2.4
7.6 × 6.5leaves 6.5
7.6 × 1leaves 1
7.6 × 0.1leaves 0.1
7.6 × (2.4 + 6.5 + 1 + 0.1) = 7.6 × 10
Solve both challenges to complete Mission 6.
7
Book Exercise 4
Decompose one number to win twice
The hint in the book tells us to split 43.9. The right split creates a shared 31.4 and a friendly product with 12.5.
3.6 × 31.4 + 43.9 × 6.4
Which equality splits 43.9 into a sum containing 31.4, so that the remaining part can be multiplied by 6.4?
3.6 × 31.4 + (31.4 + 12.5) × 6.4=31.4 × (3.6 + 6.4) + 12.5 × 6.4
=31.4 × 10 + 80
For the second product, use 6.4 = 8 × 0.8: 12.5 × 6.4 = (12.5 × 8) × 0.8 = 100 × 0.8 = 80.
8
Book-based game · Select the transformation first
Transformation Lab
For each expression, select one equivalent rewrite that meets the stated target. Then enter the value of the whole expression, not just one bracket or intermediate product.
9
Independent work · Explain before expanding
Practice the transformation habit
For each error card, select the explanation that identifies the error. Then calculate the value of each practice expression. Correctly answer at least six of the eight practice problems and both error cards to complete this mission.
Error A · Only the dividend changed
0.27 ÷ 0.25 = (0.27 × 4) ÷ 0.25
Error B · The hidden 1 disappeared
3.74 × 2.85 + 8.15 × 3.74 − 3.74 = 3.74 × (2.85 + 8.15)
Repair both errors, then continue to the practice grid.
Solved: 0 of 8. Reach 6 to complete this mission.
10
Book Exercises 10, 12, and 13
Three more ways to transform
Use the same habit in three new situations. Try each answer, then open the worked solution if you need another step. You can pause here; your work will be saved.
Exercise 10 · Work backward to find a missing number
9.4 × [□ − (1.54 − 0.31)] = 0.47
First simplify the small parentheses. Then undo the outer multiplication before undoing the subtraction.
First hint
1.54 − 0.31 = 1.23. So 9.4 times the bracket is 0.47. What is 0.47 ÷ 9.4?
Worked solution — shows the answer
9.4 × 0.05 = 0.47, so the bracket must be 0.05.
□ − 1.23 = 0.05, so □ = 1.23 + 0.05 = 1.28.
Check in the original: 9.4 × [1.28 − (1.54 − 0.31)] = 9.4 × 0.05 = 0.47.
Exercise 12 · Compare without long multiplication
A = 9.8732 × 7.2345
B = 9.8733 × 7.2344
From A to B, the first factor increases by 0.0001 and the second decreases by 0.0001. Adding and subtracting the same amount is different from multiplying and dividing by the same scale factor: it does not generally preserve the product. Rewrite A and B to reveal a product they share, then compare the remaining parts.
First hint
Write 7.2345 as 7.2344 + 0.0001, and 9.8733 as 9.8732 + 0.0001. Distribute each product.
Worked solution — shows the answer
A = 9.8732 × 7.2344 + 9.8732 × 0.0001.
B = 9.8732 × 7.2344 + 7.2344 × 0.0001.
The first product is the same in both. Compare just the extra pieces: 9.8732 × 0.0001 is greater than 7.2344 × 0.0001. Therefore A > B. We never needed the large shared product!
Exercise 13 · Look carefully at a patterned sum
1.1 + 3.3 + 5.5 + 7.7 + 9.9 + 11.11 + 13.13 + 15.15 + 17.17 + 19.19
Watch the place values: the first five numbers use tenths; the last five use hundredths. Do not assume the whole list increases by 2.2.
First hint
Use 1.1 as the common factor in the first five terms, and 1.01 in the last five. For example, 3.3 = 1.1 × 3, while 13.13 = 1.01 × 13.
Worked solution — shows the answer
First group: 1.1 × (1 + 3 + 5 + 7 + 9) = 1.1 × 25 = 27.5.
Last group: 1.01 × (11 + 13 + 15 + 17 + 19) = 1.01 × 75 = 75.75.
To check each bracket, pair its outside numbers: 1 + 9 = 3 + 7 = 10; and 11 + 19 = 13 + 17 = 30.
Total: 27.5 + 75.75 = 103.25.
Solved: 0 of 3. Complete all three to finish this mission.
11
Mastery check · Work without lesson hints
Exit Ticket
Complete all five questions, including the three parts of Question 5. For Questions 2–4, enter the value of the entire displayed expression. Question 5 asks about the rewrite and the bracket value; its optional written explanation is not graded.
1. Product balance
Which expression is equal to 0.6 × 1.5?
2. Quotient scale
0.42 ÷ 0.35
3. Hidden ×1
4.25 × 3.6 + 6.4 × 4.25 − 4.25
4. Regroup division
48 ÷ 1.25 ÷ 8
5. Check one transformation strategy
Complete all three parts about 4.25 × 3.6 + 6.4 × 4.25 − 4.25. Use the three steps to explain why your transformation works.
This writing is for reflection and is not automatically graded.
Mastery result
Certificate of Mathematical Strategy
Expression Transformer
This certifies that
can preserve an expression’s value, transform its form, and reveal a simpler calculation.
Lesson 1.3 · Grade 5 Math Explorer
This module uses the introductory product and quotient transformations, Example 3 (0.27 ÷ 0.25), the classroom practices 320 ÷ 1.25 ÷ 8 and 41.2 × 8.1 + 11 × 1.25 + 53.7 × 1.9, and selected exercises involving 2.5, 3.74, 7.6, 20.05, 0.375, and related decimal structures.The visual models, sequencing, interaction design, feedback, and additional exit-ticket items are original teaching adaptations.