Lesson 1.4 · Decimal Smart Calculation

Use Letters for Repeated Groups

In this lesson, a letter is not a mystery to solve. It is a short, consistent name for a group of numbers that appears again and again.

Spot it. Name it. Use it. Cancel it.

Grade 5 enrichment 40–50 minutes Book Example 6 Your work saves on this device

Spot

Find a group of numbers that repeats inside a long expression.

Name

Give the same repeated group the same letter everywhere.

Simplify

Expand small letter expressions and cancel equal opposite terms.

Warm-up · Look before calculating

Spot the groups that keep returning

The book warns that direct multiplication would create many messy decimal products. First, look for familiar chunks that appear more than once.

(1 + 0.12 + 0.23) × (0.12 + 0.23 + 0.34)
− (1 + 0.12 + 0.23 + 0.34) × (0.12 + 0.23)

purple: the shorter group green: the longer group

A · Count the short group

How many times does the consecutive group 0.12 + 0.23 appear, including inside the longer group?

B · Count the long group

How many times does 0.12 + 0.23 + 0.34 appear as a complete group?

C · Choose the useful habit

Why might a letter help here?

Complete all three observations to finish Mission 1.

Core idea · A letter can name a known group

Letters are name tags, not mysteries

Here, the values are already known. We use letters only to shorten the writing and expose the structure.

Name tag M

0.12 + 0.23

M always means the whole group 0.12 + 0.23 in this problem.

Name tag N

0.12 + 0.23 + 0.34

The longer group is the M-group plus one extra piece: N = M + 0.34.

The consistency promise

Once we choose M = 0.12 + 0.23, every M in this problem must mean exactly that same group.

Visual model · See the extra piece

Build N from M

The most useful fact is not the full value of M or N. It is the small difference between them.

M
0.12 + 0.23

0.34
extra piece

N
M + 0.34

What is the quickest thought?

Find the difference

N − M = ?

Difference shortcut

If N = M + d, then N − M = d. The shared M-part disappears, leaving only the extra piece.

Book Example 6

Compress the long expression

Replace each complete repeated sum with its chosen letter. Keep every extra number, bracket, multiplication, and subtraction in place. In this notation, (1 + M)N means (1 + M) × N.

(1 + 0.12 + 0.23) × (0.12 + 0.23 + 0.34)
− (1 + 0.12 + 0.23 + 0.34) × (0.12 + 0.23)

1 + 0.12 + 0.23

→

1 + M

0.12 + 0.23 + 0.34

→

1 + 0.12 + 0.23 + 0.34

→

1 + N

0.12 + 0.23

→

Choose the correctly compressed expression

(1 + M)N − (1 + N)M

The long decimal expression now uses the labels M and N. Each letter still represents its original sum, so the value is unchanged.

Distributive property · One factor reaches every term

Expand the letter expression safely

The letters shorten the groups, but the multiplication rules stay exactly the same.

Expand the first product

(1 + M)N

Expand the second product

(1 + N)M

What does MN mean?

Multiplication order

MN = M × N = N × M = NM. Switching the order does not change a product.

Book Example 6 · Let equal opposites disappear

Cancel the matching product terms

After expansion, select one positive term and one negative term whose sum is zero. Here M = 0.12 + 0.23 and N = M + 0.34, so M and N are different.

Subtract the whole second group

We expanded the two products separately. Now put the subtraction back:

(N + MN) − (M + NM)

Subtracting the group M + NM means removing both M and NM. That gives N + MN − M − NM.

Check with numbers: what is 10 − (3 + 2)?

Remove 5 altogether, or remove 3 and then 2: 10 − 5 = 10 − 3 − 2 = 5. Writing 10 − 3 + 2 would put 2 back, so it is different.

N + MN − M − NM

Select the two signed terms that add to zero: their products must be equal and their signs opposite.

N + MN − M − NM

= N − M

= 0.34

Numerical value of the whole original expression:

Book Classroom Practice 3(2)

Use the same bridge with new decimals

This is a new problem, with new definitions of M and N shown below. Use these definitions throughout this problem; do not reuse their earlier decimal values.

(2 + 1.23 + 2.34) × (1.23 + 2.34 + 3.45)
− (1.23 + 2.34) × (2 + 1.23 + 2.34 + 3.45)

M = 1.23 + 2.34 N = 1.23 + 2.34 + 3.45

Choose the compact form

(2+M)N − M(2+N)→ 2N + MN − 2M − MN→ 2(N−M)

Because N = M + 3.45, the difference N − M is the extra 3.45.

N − M =

Value of the whole expression, 2(N − M) =

Pattern extension · Inspired by the book’s structure

Letter Lab

Each challenge states its own letter definitions. Use those definitions to select an equivalent shorter expression, then enter the numerical value of the whole original expression.

Pattern discovered from Example 6 and its classroom practice

(c + M)N − M(c + N) = c(N − M)

Here c is the same added number in both brackets; it may be a whole number or a decimal. For example, cN means c × N. Expanding gives cN + MN − cM − MN = c(N − M): the two MN products cancel.

Which shortcut is correct?

Numerical value of the whole expression:

Independent work · Keep names consistent

Repair mistakes, then practice

For each error card, select the explanation that identifies the mistake. Then solve at least six of the eight practice cards. Each card is a separate problem: use any letter definitions stated on that card.

Error A · The extra 1 vanished

M = 0.12 + 0.23, so 1 + 0.12 + 0.23 = M

Error B · N did not reach 1

(1 + M)N = 1 + MN

Error C · Different terms were canceled

N + MN − M − NM = MN − NM

Repair all three errors, then continue to the practice grid.

Solved: 0 of 8. Reach 6 to complete this mission.

Mastery check · Work without lesson hints

Exit Ticket

Answer all five questions, including all three parts of Question 5. Questions 2 and 3 use M = 1.4 + 0.6 and N = M + 0.5. Question 4 states a new relationship; use it for that question only.

1. Choose a useful name

For a repeated group 1.4 + 0.6, which definition is consistent?

2. Compress an expression

Let M = 1.4 + 0.6 and N = M + 0.5.

(3 + 1.4 + 0.6)(1.4 + 0.6 + 0.5) − (1.4 + 0.6)(3 + 1.4 + 0.6 + 0.5)

3. Expand safely

(3 + M)N = ?

4. Use the extra piece

For this question, use N = M + 1.25. Find the numerical value of the whole expression:

(4 + M)N − M(4 + N)

5. Check the letter strategy

Complete all three parts. Use the three steps to explain what the letters mean and why the shortcut works.

Part A. In this lesson, what does a letter such as M mean?

Part B. Why do +MN and −NM cancel?

Part C. What remains after simplifying N + MN − M − NM?

Optional: Explain the idea in your own words.

This writing is for reflection and is not automatically graded.

Certificate of Mathematical Structure

Algebra Bridge Builder

This certifies that

can spot repeated number groups, name them consistently with letters, expand safely, and cancel matching opposite terms.

Lesson 1.4 · Grade 5 Math Explorer

The book defines M = 0.12 + 0.23 and N = 0.12 + 0.23 + 0.34, rewrites the original as (1+M)N − (1+N)M, expands it to N+MN−M−NM, and simplifies to N−M=0.34.It describes letters as a bridge that makes a complicated calculation concise.The visual name-tag models, pattern extensions, feedback, games, and exit-ticket items are original teaching adaptations.